I njinite dimensional
Lie algebras


Third edition


VICTOR G. KAC


Professor of Mathematics
Massachusetts Institute of Technology


CAMBRIDGE
UNIVERSITY PRESS




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DetlictJ'ed 'D m, fe.cAer,
Ernest Bori80vich Vinberg
wi,1a gra'i,.tle anti admira'ion

La Nature est un temple ou de vivants piliers
Laissent parfois sortir de confuses paroles;
L thomme y passe a travers des forets de symboles
Qui I'observent avec des regards familiers.
Charles Baudelaire, Le FlelJrs du Mal
Ii . BItIXO)ICY 83 npOCTpaHCTB&
B 3&OYl1\eHRl»dI cap' BenH'IHB
Osip Mandelstam
Introduction
50.1. The creators of the Lie theory viewed a Lie group as a group
of symmetries or an algebraic or a geometric object; the corresponding
Lie algebra, from their point or view J was the set of infinitesimal trans-
formations. Since the group of symmetries of the object is not necessarily
finite-dimensional, S. Lie considered not only the problem of classifica-
tion of subgroups of GL n , but also the problem of classification of infinite-
dimensional groups of transformations.
The problem of classification of simple finite-dimensional Lie algebras
over the field of complex numbers was solved by the end of the 19th century
by W. Killing and E. Cartan. (A vivid description of the history of this
discovery, one of the most remarkable in all of mathematics, can be found
in Hawkins (1982J.) And just over a decade later, Cartan classified simple
infinite-dimensional Lie algebras of vector fields on a finite-dimensional
space.
Starting with the works or Lie, Killing, and Cartan, the theory of finite...
dimensional Lie groups and Lie algebras haa developed systematically in
depth and scope. On the other hand, Cartan's works on simple infinite-
dimensional Lie algebras had been virtually forgotten until the mid-sixties.
A resurgence of interest in this area began with the work of GuiJlemin-
Sternberg [1964] and Singer-Sternberg [1965], which developed an adequate
algebraic language and the machinery of filtered and graded Lie algebras.
They were, however, unable to find an algebraic proof of Cartan's clas-
sification theorem (see Guillemin-Quillen-Sternberg [1966] for an analytic
proof). This was done by Weisfeiler [l968}, 'Who reduced the problem to the
.
IX

x
Introduction
classification of simple Z-graded Lie algebras of finite CCdepth n 11 = EB 8J t
j-d
where dim gj < 00 and the go-module g-1 is irreducible.
jO.2. At the present time there is no general theory of infinite-dimension-
al Lie groups and algebras and their representations. There are, however t
four classes of infinite-dimensional Lie groups and algebras that underwent
a more or less intensive study. These are, first or all, the above-mentioned
Lie algebras of vector fields and the corresponding groups of diffeomor-
phisms of a manifold. Starting with the works of Gelfand-Fuchs [1969J,
[1970A,B], there emerged an important direction having many geometric
applications, which is the cohomology theory of infinite-dimensional Lie
algebra.s of vector fields on a finite-dimensional manifold. There is also a
rather large number or works which study and classify various classes of
representations of the groups of diffeomorphismB of a manifold. One should
probably include in the first class the groups of biregular automorphisms
of an algebraic variety (see Shafarevich [1981]).
The second class consists of Lie groups (resp. Lie algebras) of
Imooth mappings of a given manifold into a finite-dimensional Lie group
(resp. Lie algebra). In other words, tbis is a group (resp. Lie algebra) of
matrices over some function algebra but viewed over the base field. (The
physicists refer to certain central extensions of these Lie algebras 88 current
algebras.) The main subject of study in this case has been certain special
families of represent.tioDS.
The third class consists of the classical Lie groups and algebras of op-
erator. in a HiJbert or Banach apace. There is a rather large Dumber of
scattered results in this area, which study the structure of these Lie group.
and aJgebras and their representations. A representation which plays an
important role in quantum field theory is the Segal-Sha1Weil (or meta-
plectic) representation of an infinite-dimensionalaymplectic group.
I shall not discuss in this book the three classes of infinite.dimensional
Lie algebras listed above. with the exception of those closely related to
the Lie algebras of the fourth class, which we consider below_ The reader
interested in these three classes should consult the literature cited at the
end of tbe book.
Finally, the fourth class of infinite-dimensional Lie algebras is the class
of the so-called Kac-Moody algebras. the subject of the present book.
50.3. Let us briefly discuse the main concepts of the structural theory of
Kac-Moody algebru. Let A = (aiJ )=1 be a generalized C."'can mca',;z,
i.e., U1 integral n x n matrix such that au = 2, a'i S 0 for i :I: j, and Oij = 0

I.trod.clio"
.
Xl
implies Oji = O. The associated Kac-Mood, IIlgell,.. ,'(A) i8 a complex Lie
algebra on 3n generators ei, Ii. hi (i = 1,..., n) and the following defining
relations (i,j, = 1,..., n):
(hi,hj] = 0, [ei, Ii] = hi, [ei,/j] = 0 if i  j,
(0.3.1) (hi,ej] = Gijej. [hit/i] = -Oij!j,
( ade i)l-..iej = 0, {ad Ji)I- Cl i; Ii = 0 if i  j.
(The definition given in the main text of the book (see Chapter 1) is dif..
ferent from the above; it is more convenient for a number of reaeon8. The
proof of the fact that the derived algebra of the Lie algebra g(A) defined in
Chapter 1 coincides with the Lie algebra 8'(A) defined by relation. (0.3.1)
bas been obtained by Gabber-Kac [1981J under a "symmet,rizability" ....
8umptionj this proof appears in Chapter D.)
I came to consider these Lie algebras while trying to understand and
generalize the works of Guillemin-Quillen-Singer-Sternberg-Weu.reiler OD
Cartan)s classification. The key idea was to consider arbitrary simple Z-
graded Lie algebras 9 =., 9j j but since there are too many such Lie
algebras. the point was to require the dimension of Ij to grow no Caster
than some polynomial in j. (One can show tbat Lie algebras of finite
depth do satisfy this condition, and that this condition is independent of
the gradation.) Such Lie algebras were classified under some technical
hypotheses (see Kac [1968 B]). It turned out that in addition to Cartan's
four series or Lie algebras of polynomial vector fields, there is another cl..
or infinite-dimensional Lie algebras or polynomial growth, which are called
affine Lie algebras (more preciseJy, they are the quotients of affine Lie
algebras by the I-dimensional center). At the same time, Moody [1968]
independently undertook the study of the Lie algebras g'(A).
The class of Kac-Moody algebras breaks up into three 8ubcl8l8e8. To
describe tliem, it is convenient to assume that the matrix A is i.tleeom-
pOlalJle (i.e., there is no partition of the set {I,..., n} into two honempty
subsets 80 that OJ} = 0 whenever i belongs to the first subset, while j be-
longs to the second; this is done without los8 of generality since the direct
8um of matrices corresponds to the direct 8um or Kac-Moody algebru).
Then there are the following three mutually exclusive possibilities:
a) There is a vector' of positive integers such that all the coordinates or
the vector AD are positive. In such case all the principal minors of the
matrix A are positive and the Lie algebra ,'(A) is finite--dimeD8ional.
b) There is a vector 8 or positive integers 8uch that AcS = O. In luch cue all
the principal minora of the matrix A are nonnegative and det A = OJ the al-
gebra g'(A) is infinite..dimensional, but is of polynomial growth (moreover,

..
XII
Introduction
it admits a Z-gradation by 8ubspaces of uniformly bounded dimension).
The Lie algebras of this 8ubclass are called (jffint Lie tJIge6ra..
c) There is a vector Q of positive integers such that all the coordinates
of the vector Ao are negative. In such case the Lie algebra S'(A) is or
exponential growth.
The main achievement of the Killing-Cartan theory may be formulated
as follows: a simple finite-dimensional complex Lie algebra is isomorphic
to one of the Lie algebras of the lubclass a). (Note that the classification
or matrices of type a) and b) is a rather simple problem.) The existence
of the generators satisfying relations (0.3.1) was pointed out by Chevalley
[1948] and Barish-Chandra [1951]. (Much later Serre [1966] and Kac [1968
B) showed that these are defining relations.)
It turned out that most of the classical concepti of the Killing-Cartan-
Weyl theory can be carried over to the entire class of Kac-Moody algebras,
such 88 the Cartan 8ubalgebra, the root system. the Weyl group, etc. In
doing 80 one discovers a series of new phenomena, which the book treats in
detail (see Chapters 1-6). I shall only point out here that g'(A) does not
always posseSl a nonzero invariant bilinear form. This is the case if and
only if the matrix A is .ymmetriza6/e. i.e., the matrix DA is symmetric for
some invertible diagonal matrix D (see Chapter 2).
10.4. It is an important property of affine Lie algebras that they pO&-
1e88 a simple realization (see Chapters 7 and 8). Here I shall explain this
realization for the example of the Kac-Moody algebra associated to the ex-
tended Cartan matrix A or . simple finite-dimensional compJex Lie algebra
8. (All such matrices are "affine" generalized Cartan matrices; the corre-
sponding algebra g'(A) is called a nontwisted affine Lie algebra.) Namely,
the affine Lie algebra g'(A) is a central extension by the I-dimensioDpJ
center of the Lie algebra of polynomial maps of tbe circle into the simple
finite-dimensional complex Lie algebra 9 (so tbat it is the simplest example
of a Lie algebra or the second cl888 mentioned in 10.2).
More precisely, let us consider the Lie algebra 8 in 80me faitbful finite-
dimensional representation. Then the Lie algebra g'(A) is isomorphic to
the Lie algebra on the complex space (C[t,t- 1 ]0c g)EBCc with the bracket
[(t m  a) $ c, (t n 06) $ pc} = (t m + n 0 [a, 6]) $ mcS m .- n (tr (6)c,
10 that Cc is the (l.dimensional) tenter. This reaJization allows us to study
affine Lie algebras from another point of view. In particular, the algebra
or vector fields on the circle (the simplest algebra of the firat class) playa
an important role in the theory of affine Lie algebras.

In t",duc 'ion
...
XIII
Note also that the Lie aJgebras of the fourth class are closely related to
the affine Lie algebras of infinite rank, considered in Chapters 7 and 14.
U nrortunately, no simple realization has been found up to now for any
nonaffine infinite-dimensional Kac-Moody algebra. This question appears
to be one of the most important open problems of the theory.
SO.5. An important concept missing from the first works in Kae-Moody
algebras was the concept of an integrable highest-weight representation
(introduced in Kac [1974J). Given a sequence of nonnegative integers A =
(1'... ,An), the integrable AigAe6t-weiglat repre$eratat;on of a Kac-Moody
algebra g'(A) is an irreducible representation 'itA of g'(A) on a complex
vector space L(A), whicb is determined by the property that there is a
nonzero vector VA e L(A) 8uch that
1I'A(ei)VA = 0 and 1rA(h.)VA = it1A (i = 1,..., n).
(This terminology is explained by the fact that A is called the highest-
weight, and the conditions on A are necessary and sufficient for being able
to integrate 1fA and obtain a representation of the group.)
Car tan t 8 theorem on the highest-weight asserts that all the represen-
tations A of a complex simple finite-dimensional Lie algebra are finite-
dimensionaJ, and that every finite-dimensional irreducibJe representation is
equivaJent to one of the "A.
That the representations 'irA are finite-dimensional (the most nontrivial
part or Cartan t s theorem) was proved by Cartan by examining the cases,
.one by one. A purely algebraic proof was Cound much later by C. Chevalley
(1948) and Harish-Chandra (1951) (a Utranscendental" proof had been found
earlier by H. Weyl). This brief note by Chevalley. appears in retrospect
as the precursor of the algebraization of the representation theory of Lie
groups. This note also contains, in an embryonic form, many of the basic
concepts of the theory of Kac-Moody algebras.
The algebraization or the representation theory of Lie groups. which has
undergone such an explosive development during the last decade, started
with the work Bernstein-Gelfand-Gelfand (1971) on Verma modules (the
first nontrivial results about these modules were obtained by Verma [1968]).
In particular, using the Verma modules, Bernstein-Gelfand-Gelfand gave
a transparent algebraic prooC of Wey}'s Cormula Cor the characters of finite-
dimensional irreducible representations or finite-dimensional simple Lie
algebras..
At about the same time MacdonaJd [1972] obtained his remarkable jden..
tities. In this work he undertook to generalize the WeyJ denominator

xiv
1 fa 'rotl.cCio n
identity to the cue of affine root Iy.terns. He remarked that a straight-
forward generalization i. actually (alae. To salvage the situation he had to
add lame "mylterious" factors, which he waa able to determine aa a result
of lengthy calculations. The simplest example of Macdonald t. identities is
the famous Jacobi triple product identity:
II (1 - unvn)(t - u n - 1 v n )(1 - unv n - 1 )
nl
= E (_I)m u !m(m+1)v!m(m-l),
mez
The "mysterious" factorl which do not correspond to affine roots are the
factors (1 - unv").
After the appearance or the two works mentioned above very little re-
mained to be done: one had to place them on the desk next to one another
to understand that Macdonald's result is only the tip of the iceberg-the
representation theory ot Kac-Moody algebras. NamelYt it turned out that
a 8implified version or Beroatein-Gelrand-GeICand t. proof may be applied
to the proof of a formula generalizing Weyl'. formula, for the formal char-
acter of the representation "A of an arbitrary Kac-Moody algebra g'(A)
corresponding to a .ymmetrizable matrix A. In the case of the simplest
l.dimensional representation "'0 I this formula becomes the generalization
of Weyl', denominator identity. In the case of an affine Lie alsebra, the
generalized Weyl denominator identity turDS out to be equivalent to the
Macdonald identities. In the proceea, the "mysterious" factor. receive a
simple interpretation: they correspond to the so-called imaginary roots
(i.e., roots that one should add to the affine roots to obtain all the rootl of
the affine Lie algebra). Note that the simplest example of the Jacobi triple
product identity turns out to be jU8t the generalized denominator identity
for the affine Lie algebra corresponding to the matrix (_22 -,2).
The exposition of these results (obtained by Kac [1974J) may be found
in Chapter 10. Chapters 9-14 are devoted to the general theory of highest-
weight representations and their applications.'
The main tool or the theory or representations with highest-weight is the
generalized Casimir operator (see Chapter 2). Unfortunately, the construc-
tion of thi. operator depends on whether the matrix A is Iymmetrilable.
The question whether one can lift the hypothesis of .ymmetrizability of the
matrix A remains open.
Once the integrable highest-weight representations had been introduced,
the theory of Kac-Moody algebras lot off the ground and haa been devel-
oping since at an accelerating speed. In the past decade this theory has

I,.frotl8ction
xv
emerged as a field that hu close connections with many areas of mathe-
matics and mathematical physics. luch as invariant theory, combinatorics,
topology, the theory of modular Corlnl and theta functions, the theory of
singularities, finite simple groups, Hamiltonian mechanica, soliton equa-
tions, and quantum field theory.
10.6. This book contains a detailed exp08jtion of the foundations of
the theory of Kac-Moody algebras and their integrable representations.
Besides the application to the Macdonald identities mentioned above (Chap-
ter 12), the book di8cu88e8 the application to the classification of finite-
order automorphisD18 or simple finite-dimensional Lie algebras (Chapter
8), and the connection with the theory or modular forms and theta func-
tions (Chapter 13). The last chapter (Chapter 14) discusses the remarkable
connection between the representation theory of affine Lie algebras and the
Korteweg-de Vries-type equations, discovered by the Kyoto schoo).
A theory of Lie algebras is u8ually interesting, insofar as it is related to
group theory, and Kac-Moody algebras are DO exception. Recently there
appeared a series of deep results on groups associated with Kac-Moody
algebras. A discussion of these resuJts would require writing another book.
I chose to make only a few comments regarding this subject at the end of
some chapters.
SO.7. Throughout the book the base field is the field of complex num-
bers. However, all the ..eslt8 of the book, except, or course. for the ones
concerning Hermitian forms and convergence problems, can be extended
without difficulty to the case of an arbitrary field of characteristic zero.
SO.8. Motivations are provided at the beginning of each chapter, which
ends with related bibliographical comments. The main text of each chapter
is followed by exercises (whose total number exceeds 250). Some of them
are elementary, others constitute a brieC expOIition of originaJ works. I
hope that these expositions are sufficiently detailed for the diligent reader
to reconstruct all the proofs. The square brackets at the end of some
exercises contain hints for their IOlution.
The exposition in the book is practically self..contained. Although I had
in mind a reader familiar with the theory of finite-dimensional semisimple
Lie aJgebras. what wouJd suffice for the most part is a knowledge of the
elements of Lie algebras. their enveloping algebras and representations. For
example, the book of Humphreys [1972] or Varadarajan (1984] is more than
sufficient.

.
XVI
In'rod.clion
One finds a rather extensive bibliography at the end of the book. I hope
that the collection o( references to mathematical works in the theory of
Kac-Moody algebras is at least everywhere dense. This is not at aU eo
in the case of the works in physics. The choice or references in this eue
was rather arbitrary and often depended on whether I had a copy or 'he
paper or discussed it with the author. The same should be said u regards
the references to the works on the other classes of infinite-dimensional Lie
algebras.
SO.9. This book is bued on lecturel given at MIT in 1978, 1980. and
1982, and at the College de France in 1981. I would like to thank thOle
who attended for helpful comments and corrections of the notes, in partic-
ular F. Arnold, R. Coley, R. GrOls, Z. Haddad. M. Haiman, G. Heckman,
F. Levstein, A. Rocha, and T. Vongiouklis. I am grateful to M. Dufto,
G. Heckman, B. Kupershmidt t and B. WeisCeiler (or reading some parts of
the manuscript and pointing out errors. I apologize Cor those errors that re-
main. My thanks go to F. Rose, B. Katz, and M. Katz without whOle help
and support this book would never have come out. I also owe thanks to
K. Manning and C. Macpherson for help with the language. The book W8I
prepared using D. Knuth'. TEX. Finally, I would like on this occasion to
express my deep gratitude to D. Peterson, whose collaboration had a sr eat
influence not only on this book, but also on mOlt of my mathematical work
in the past few years.
The author was supported in part by & Sloan foundation grant and by
grants from the National Science Foundation.
July 1983, L'lsJe Adam, France.
Preface to the Second Edition
The most important additions reftect recent developments in the theory
of infinite-dimensional groups (some key facta, like Proposition 3.8 and
Exercise 5.19 are among them) and in the soliton theory (like Exercieee
14.37-14.40 which uncover the role o(the Viruoro algebra). The moet im-
portant correction concerns the proof or the complete reducibility Propo-
sition 9.10. The previous proof used Lemma 9.10 b) of tbe firlt edition
which is false, as Exerciee 9.15 abows. A correct version of Lemma 9.10 b)

Introduc'ion
xvii
is the new Proposition 10.4 which gives a characterization of integrable
highest-weight modules.
In addition to correcting misprints and errors and adding a few dOlen
of new exercises, I have brought. to date tbe list of references and related
bibliographical comments. I want to thank thOle who have pointed out er-
rors and suggested improvements, in particular: J. Dorfmeister, T. Enright,
D. Freed, E. Getzler, E. Gutkin, P. de la Harpe, S. Kumar, B. Kupershmidt,
S.-R. Lu, D. Peterson, L.-J. Santharoubane, G. Schwarz, V S. Varadara-
jan, M. Wakimoto, Z..-X. Wan, X.-D. Wang, Y.-X. Wang, B. Weisfeiler,
C.-F. Xie, Y...C. Yt>u, H.-C. Zhang.
April 1985, Cambridge. Massachusetts.
Preface to the Third Edition
This edition differs considerably from the previous ones. Particularly, more
emphasis is made on onnection8 to mathematical physics, especially to
conformal field theory. ,.
Below is a list of the most important improvements and additioDs:
Chapter 3.. A simplest example of a quantized Kac-Moody algebra,
U,(6/ 2 ), is given, along with its repre8entations (Exercises 3.23 and 3.34).
Chapter 5. The hyperbolic Weyl group theory is applied to the study of
the unimodular Lorentzian lattices or rank S 10 <55.10).
Chapter 6. An explicit construction of all finite type root and coroot
lattices is given, along with the associated Weyl group, root systems, etc.
(16.7).
. Chapter 7. The field theoretic approach to affine algebras is briefly
outlined (S7.7). An explicit construction of all simple finite-dimensional Lie .
algebras is given in terms of the root lattice anc;l an "asymmetry function"
on it (557.8-7.10).
Chapter 8. A simple and self-contained proof is given of the basic fact
about twisted affine algebras: the equivariant loop algebra .c(9, fT, m) de-
pends only on the connected component of Aut 8 containing tr (!8.5).
Chapter 9. Elements of the representation theory of the Virasoro alge-
bras are discussed (19.14). A free field construction of representations of
the Virasoro algebra and the affine algebra of type Ail} is given (Exercises
9.17-9.20).
Chapter 11. Unitarizability of representations of the Virasoro algebra ill

..,
XVIII
Introtluc'iora
iscuued (511.12). A theory of generalized Kac-Moody algebras is outlined
<511.13).
Chapter 12. The Sug&wara construction and the coset construction,
which are the basic construction. or conformal field theory, are explained,
The general branching functions and vacuum pairs are introduced in this
context (5112.8-12.13).
Chapter 13. General branching functions are studied along with 8tring
functions. The matrix 5 of the modular transformations of characters is
studied (this wu impJicit in earlier editions). Explicit estimates of the
orders of all poles and or levels of branching functions are given. Asymp-
toties of characters and branching functions at high temperature limit is
studied, along with the related positivity conjecture. The interplay be-
tween the modular and conformal invariance constraints is demonstrated
(5S13.8-13.14). This is used to study unitarizable representations of the
Virasoro algebra, and to calculate the fusion rules (Exercises 13.18-13.26,
and 13.34-13.36).
Chapter 14. The homogeneous vertex operator construction is derived
via the vertex operator calculus (114.8). The infinite wedge representation
is constructed (514.9). By making use of the boson-fermion correspondence
<514.10) tbe whole KP hierarchy is studied (SS14.11 and 14.12). By mak-
in use of the principal and homogeneous vertex operator constructions of
All) I the whole K dV and N LS hierarchies are described U 14.13). The
BK P hierarchy is constructed (Exercises 14.13-14.15). A theory of the
infinite Grassmannian and ftag manifold is sketched and their connection
to the K P ,and M K P hierarchies is explained (Exercises 14.32. 14.33).
A pseudodifrerential operator approach to the K P and K dV hierarchies is
outlined (Exercises 14.44-14.51). A basis free theory of the Lie algebra and
group or type Aoo is discussed (Exercises 14.55-14.58), and lome c)assicaJ
theorems of the theory or algebraic curves are derived from this discussion
(Exercises 14.59-14.63).
In addition to correcting misprints and errors and adding some hundred
new exercises, I have brought up to date the list of references and related
bibliographical comments. The explosion or activity in the field between the
second and the third editions, due to a great extent to physicists working
in ,tring theory and conformal field theory, made it an impossible task
to compile a reasonably compJete bibliography. I hope, however t that the
collection of references compiled (or this edition at least reftects all the
major directions of research in the field. Needless to say that every sentence
or my bibUographical comments could be prefixed by an "It is my opinion
that .. . ."

In'"'tl.ciDn
.
XIX
I want to thank thOle who have pointed out errors and suggested im-
provements, in particular:
R. Borcherds, J. Dixmier, D.Z. Djokovic, E. Frenkel, S. Friedberg, M.-J.
Imben., R. Iyer, R. C. KinS. M.F.R. KrueJJe, J. van de Leur, S.-R. Lu, P.
Magyar, G. Rousseau, G. Seligman, M. Wakimoto, Z.-X. WaD, Y.-C. You.
September 1989, Newton, MUBachusettl.

Z
z+
N
Q
R
R+
C
SX
Re z and 1m z
logz
zQ
U Ef) V or E9 U 0
a
EU a
Q
nU a
Q
ItS
uv
U.
k"
Iv or In or 1
(. . .)
lul 2 = (ulu)
ISI
P mod Q
Notational Conventions
the set or integers
the set of non-negative integers
tbe set of positive integers
the set of rational numbers
the set of real numbers
the set of non-negative real numbers
the set of complex numbers
the set of invertible elements of a rins S
real and imaginary parts of z e C
for z E CX : etos' = z and
-1r  Imlogz < 
= e o1ol ' for Q e C, z E CX
direct sum of vector spaces
sum of 8ubspaces of a vector space
direct product of vector spaces
the linear k-span of S (k = Z,Z+.Q.R.
or C)
t.ensor product of vector "spacel over
I: (k = Q,R, or C)
the dual of a vector k-space over
, (k = Q, A, or C)
direct sum or n copies of the vector space
k (n E Z+ U {oo})
the identity operator on the n-dimensional
vector space V
pairing between a vector space and its dual
square length of a vector u
cardinality of a set S
a set of representatives of coset. of an
abelian group P with respect to a
subgroup Q
xx

NottJ'ioftGI ConVen'iOR$
.
XXI
Uo(g) = U(g)g
g( V) or 9 · v
means that (I - 6 E C.
universal enveloping algebra of a Lie
algebra "
the augmentation ideal of U(g)
action of an element 9 of a Lie algebra or
a group on an element 11 of a module; all
mod ulea are assumed to be left mod u lea
unless otherwise specified
= {g. vii E G} the orbit of v under the
action of a group G
union of orbits of elements from a set V
linear span of the set {u · vlu e U J v EM},
where U is a 8ubspace of an algebra and M
is a subset of a module over this algebra.
CI =" mod C
U(g)
G · v or G( 11 )
G · V or G(V)
U.M

Contents
Introduction ix
Notational Conventions xx
Chapter 1. Baaic: Definitions 1
Chapter 2. The Invariant Bilinear Form and the Generalized
Casimir Operator 16
Chapter 3. Integrable RA!presentations of Kac-Moody Algebras
and the Weyl Group 30
Chapter 4. A CI888ification of Generalized Cartan Matrices 47
Chapter 5. Rea) and Imaginary Roots 59
Chapter 6. Affine Algebras: the Normalized Invariant
Form, the Root System, and the Weyl Group 79
Chapter 7. Affine Algebras as Central Extn8ion8 of
Loop Algebras 96
Chapter 8. Twisted Affine Algebras and Finite Order Automorphiama 125
Chapter 9. Highest-Weight Modules over Kac-Moody Algebras 145
Chapter 10. Integrable Highest-Weight Modules:
the Character Formula 171
Chapter 11. Integrable Highest-Weight Modules:
the Weight System and the Unitarizability 190
Chapter 12. Integrable Highest-Weight Modules over Affine
Algebras. Application to q-Function Identities.
Sugawara Operators and Branching Functions 216
Chapter 13. Affine AJsebras, Theta Functions, and Modular Forms 248
Chapter 14. The Principal and Homogeneous Vertex Operator
Constructions of t.he Basic Representation. Boson-Fermion
Correspondence. Application to Soliton Equations 292
Index of Notations and Definitions 353
References 367
Conference Proceedings and Collections of Papers 399
..
VII

Chapter 1. Basic Definitions
51.0. The central object of our study is a certain class of infinite-
dimensional Lie algebras alternatively known as contragredient Lie alge-
bras. generalized Cartan matrix Lie algebras or Kac-Moody algebras. Their
definition is a rather straightforward "infinite-dimensional" generalization
of the definition of semisimple Lie algebras via the Cartan matrix and
Chevalley generators. The slight technical difficulty that occurs in the case
det A = 0 is handled by introducing the "realization" (, n. n Y ) of the ma-
trix A. The Lie algebra SeA) is then a quotient of the Lie algebra g(A)
with generators e.. Ii and , and defining relations (1.2.1), by the maximal
ideal intersecting  trivially. Some of the advantages of this definition as
compared to the one given in the introduction, as we will see. are as follows:
the definition of roots and weights is natural; the Weyl group acts on a nice
convex cone; the characters have a nice region of convergence.
51.1. We start with a complex n x n matrix A = (a.j)j=l of rank l
and we will associate with it a complex Lie algebra g(A).
The matrix A is called a genralized Cartan mGtriz if it satisfies the
following conditions:
(Cl)
(C2)
(C3) Gij = 0 implies aji = O.
Although a deep theory can be deveJoped only for the Lie algebra asso-
ciated to a generalized Cartan matrix At it is natural (and convenient) to
begin with an arbitrary matrix A.
A realization of A is a triple (. 0, nV), where  is a complex vector
space, n = {Ol,'.' ,an} C . and n v = {or,... ,a:} C  are indexed
subsets in . and . respectively, satisfying the (ollowing three conditions
(1.1.1) both sets nand n Y are linearly independent;
(1.1.2) (Qr,Oj) = Oij (i,j = 1,... ,n);
(1.1.3) n - t = dim - n.
Two realizations (,D,nV) and (l,nJ,nr) are" called ilomorplaic if
there exists a vector space isomorphism  :  -. 1 such that (nY) = Dr
and ;.(DI) = n.
aii = 2 for i = 1,.. . , n i
aij are nonpositive integers for j  j;
1

2
Baic Definitions
Ch.l
PROPOSITION 1.1. There exists a unique up to isomorphism realization for
every n x n-matrix A. Realizations of matrices A and B are isomorphic jf
and only jf B can be obtained (rom A by a permutation of the index set.
Proof Reordering the indices, if necessary, we can assume that
A = (:)
where At is an ex n submatrix of rank t. Consider the following n x (2n -l)
matrix:
c - ( AI 0 )
- A 2 In-t ·
Taking  = C 2n -l.. elements Ql t . . . ,On to be the first n linear coordinate
functions and elements or,..., Q to be the rows of the matrix C, we
obtain a realization of the matrix A.
Conversely, given a realization (, n, n V ) we complete n to a basis by
adding elements 0'"+1, · · . ,Q2n- E . so that we have for some t x (n - t)
matrix B and invertible (n - l) x (n - i)-matrix D:
«(ar,aj» = ( ).
Adding to Qn+ 1, · · · suitable linear combinations of ai, . . . , Ql, we can make
B = o. Replacing Qn+l,..' ,Q2n-t by their linear combinations, we can
make D = 1. This proves the uniqueness. The second part of the proposi-
tion is now obvious.
o
It is clear that if (, 0, n V ) is a realization or a matrix A, then (., n v , 0)
is a realization of the transposed matrix' A.
Given two matrices Al and A 2 and their realizations (l,nl.nr) and
(2, n 2 , Un, we obtain a realization of the direct Bum ( I :2) of the two
matrices:
(1 $ 2, n 1 x {OJ U to} x O 2 , nT x to} U to} x n),
which is called the direct 6um of the realizations.
A matrix A (and the corresponding realization) is called decompola6/e
if, after reordering the indices (i.e. a permutation of its rows and the same
permutation of the columns), A decomposes into 8 nontrivial direct .um.

Ch. 1
Balic Definitionl
3
It is clear that after reordering the indices, one can decompose A into a
direct sum of indecomposable matrices, and the corresponding realization
into a direct sum of the c.orresponding indecomposable realizations.
In analogy with the finite-dimensional theory, we use the following ter-
minology. n is called the root 6a.s;, n v the coroot 66.9;", elements from n
(resp. n V ) are called &imple roo's (reap. limple coroOt8.). We also set
n
Q = LZa s ,
i=l
"
Q+ = 2: Z+ai.
i=l
The lattice Q is called the root lattice.
For Q = L kiQi E Q the number ht Q := E ki is called the height of Q.
i i
Introduce a partial ordering  on . by setting  > JJ if  - IJ E Q+.
Sl.2. Let A = (aij) be an n x n-matrix over C, and Jet (, n, n V ) be a
realization of A. First we introduce an auxiliary Lie algebra g(A) with the
generators f., I.(i = I, . . . . n) and , and the following defining relations:
(1.2.1)
[eil/j] = 6iji
[h,h'] =0
[h, t;] = (a., h)e;,
[h,!;] = -(ai, h)/i
(i,j = l,...,n),
(h, h' e ),
(i = 1, . . . f ni h E ).
By the uniqueness of the realization of A it is clear that g(A) depends only
on A.
Denote by "+ (resp. n_) the subaJgebra of g( A) generated by el , . . . f en
(resp. /1'...' In). OUf first fundamental result is
THEOREM 1.2. a) g(A) = n_ $  ED n+ (direct sum of vector spaces).
b) n+ (rup. n_) i, fret" gtntnJttd 6, el, . . . ,en (rtsp. It,..., In)'
c) The map ei ..... - Ii, Ii t-+ -tfi( i = 1".., n), h t-+ -h(h e ), can be
uniquely ezttnded to an involution w of tlae Lie alge6rtJ g(A).
d) With respect 10  one htJ$ 'he root $pace tlcompo$;t;on:
(1.2.2)
g(A) = ( Ea g-o) $  $ ( E9 go),
aEQ+ aEQ+
oo aO
where go = {x E g(A)l[h,x] - a(h)x for all h E }. Furthermore,
dim go < 00, and go c n:t: for :!:a E Q +, Q :f; O.

4
BGic Definition,
Ch.l
e) Among the ideals of g(A) intersecting  trivially, there exists a unique
maximal ideal t. Furthermore}
(1.2.3)
r=(rnn_)$(tnn+>
(dirtct ,urn of .deal).
Proof. Let V be the n-dimensional complex vector space with & basis
V1 t . . · t Vn and let  be a linear function on . We define an action of
the generators of g(A) on the tensor algebra T(V) over V by
Q) f.(a) = v. 0 (J for a e T(V);
13) h(l) = ('\, h) 1, and inductively on a,
h( Vj @a) = - (Qj, h)tJj 0 a + Vj 0 h( a) for a e T' -1( V), j = 1, . . . ,n;
1) e.(l) = 0, and inductively on "
ei(Vja) = 6ijQi(a) + Vj  l!i(a) for a E T,-l(V)t j = 1,... f n.
This defines a representation of the Lie algebra g(A) on the space T(V).
To see that. we have to check all of the relations (1.2.1).
The second relation is obvious since  operates diagonally. For the first
relation we have
(ei/j - fje,)(a) = ti(Vj @ a) - tJj  ei(a)
= 6ijQ;'(a) + Vj tg) e.(a) - Vj @ ei(a) = 6ijai(a),
by Q) and 1).
For the fourth relation. we have
(h/ j -/j h)(a) = h( Vj 0 a) - Vj 0 h(a)
= -(Qj, h)vj  a + Vj 0 h(a) - Vj  h(a)
= - ( 0 j 1 h) /j ( t1 )
by 0) and {3).
Finally, the third relation is proved by induction on 6. For 8 = 0 it
evidently holds. For 8 > 0 take 0= Vt001. where 01 E T'-l(V). We have
(hej - ejh)(v}  01) = h(6j}Qj (al») + h(v, 0 ej(al»
- ej(-(Q.t,h)(v @ al) + Vet  h(oJ»
= 6 i I:Q'1(h(Ol» - (Qi,h)Vi @ fj(01)
+ Vj: @ hei(at)
+ (0 It , h)6 j i Q j (a1)
+ (aA:,h)vA:  ej(al)
- 6jQ;h(al) - v @ejh(ol)
= (Qj, h)6 j A:oj (al) + v @ (hej - ej h)(Ol).

Ch.l
Ba,ic DefiniCion,
6
To complete the proof. we apply the inductive 888umption to the second
summand.
Now we can deduce all the statements of the theorem. Using relations
(1.2.1), it is easy to show by inuctjon on . that a product of 8 elements
from tbe set {e., 1.(; = 1,..., n); } lies in n_ +  + n+. Let now u =
n_ + h + n+ = 0, where n:l: e n:f: and h E . Then in the representation
T(V) we have u(l) = R_(l) + ('\, h) = O. It follows that (, h) = 0 for
every  E . and hence h = O.
FUrthermore, sing the map Ii .... Vi, we see that the tensor algebra
T(V) is an associative enveloping algebra of the Lie algebra n_. Since
T(V) is a free associative algebra, we conclude that T(V) is automatically
the universal enveloping algebra U(ii_) of ii_, the map n_ t-+ n_(l) being
the canonical embedding ii_ -+ U(ii_). Hence n_ = 0 and a) is proven.
Moreover, by the Poincare-Birkhoff-Witt theorem, ii_ is freely generated
by /1,.. ., In. The statement c) is obvious. Now applying w we deduce
that n+ is freely generated by e 1 , . . . , en, proving b).
Using the last two relations of (1.2.1), we have
n: = E9 9*11"
aEQ+
QO
We also have the following obvious estimate:
( 1.2.4)
dim Sa  nlhtal.
These together with a) prove d).
To prove e) note that for any ideal i of g(A) one has (by Proposition 1.5
below): .
i = E!1<ia n i).
a
Hence the Bum of ideals which intersect  trivially, itself intersects 
trivia)ly, and the Bum of aU ideals with this property is the unique maximal
ideal t which interseds  trivially. In particular, we obtain tbat (1.2.3) is
a direct sum of vector spaces. But, clearly, II., t n n+J c ii+. Hence
. [g(A), t n "+J C t n "+; similarly, [g(A), t n ft-J C tn R_. This shows that
(1.2.3) is a direct 8um of ideals.
o

6
BtJ,ic Definition,
Ch.l
51.3. Given a complex n x n-matrix A, we can now define the main
object of our study: the Lie algebra g(A). Let (.n,IIV) be a realization
of A and let g(A) be the Lie algebra on generators fi, Ii (i = 1,..., n) and
, and the defining relations (1.2.1). By Theorem 1.2 a) the natural map
 -+ g(A) is an imbedding. Let t be the maximal ideal in g(A), which
intersects  trivially (see Theorem 1.2 e». We set:
g(A) = g(A)/r.
The matrix A is called the Carl4n matriz of the Lie algebra g(A), and
n is called the nJRk of g(A). The quadruple (g(A), t n, n V ) is called the
quadruple associated to the m4tri% A. Two quadruples (g(A),, n, n V )
and (g(A 1 ), 1, Ut. Dr> are called isomorphic if there exists a Lie algebra
isomorphism tP : g(A) -+ g(A 1 ) such that tP() = Ql' (nV) = nr and
.(nl) = n.
The Lie algebra g(A) whose Cartan matrix is a generalized Cartan ma-
trix is called a K(Jc-Moody alge6ro.
We keep the same notation for the images of ej J ,. ,  in g( A). The
subalgebra  of g(A) is called the Gartan ,u6alge6rtJ. The elements ei, Ii
(i = 1,..., n) are called the Claevo.Ile1l generators. In facts they generate
the derived subalgebra g'(A) = [g(A), g(A)]. Furthermore,
g(A) = g'(A) + 
(g(A) = g'(A) if and only if det A  0).
We set ' = El CQr. Then g'(A) n  = '; g'(A) n 9a = go if Q  O.
It follows from (1.2.2) that we have the following root !pace decompoi-
lion with respect to :
( 1.3.1)
g(A) = Ef1 go.
oEQ
Here, ga = {z E g(A)llh, z] = Q(h)z for all h E } is the root space
attached to o. Note that 90 = . The number mult Q := dim go is ca)led
the multiplicitg of a. Note that
( 1.3.2)
muJtQnlhtQI
by (1.2.4).
An element Q E Q is called a root if a  0 and mult Q  O. A root Q > 0
(resp. Q < 0) is called p05iiive (resp. ntgolive). It follows from (1.2.2) that

Ch. 1
Ba,ic Dfini'ion'
7
every root is either positive or negative. Denote by &. A+ and i1_ the sets
of all roots, positive and negative roots respectively. Then
A = A+ U A_ (a disjoint union).
Sometimes we will write A(A), Q(A), . .. in order to emphasize the depen-
dence on A.
Let n+ (resp. n_) denote the 8ubalgebra of g(A) generated by el, . . . ,en
(resp. 11,... ,In). By Theorem 1.2 a), e) we have the triangular decompo-
$ition
g( A) = n _ EB  E9"+ (direct sum of vector spaces).
Note that 90 C n+ if Q > 0 and 90 C n_ if 0 < O. In other words, for
Q > 0 (resp. Q < 0), So is the linear span of the elements of the Corm
[. · · ([eip ei;l)' ei,} .. · ei.] (resp. [... ([/;., At),!i,} .. . Ii.)), such that 0i, +
· · · + Qi. = Q (resp. = -Q). It follows immediately that
(1.3.3)
90 i = Cei, g-ai = C/.; g,oi = 0 if 1st > 1.
Since every root is either positive or negative, (1.3.3) implies the follow-
ing important fact:
LEMMA 1.3. IffJE+\{Qi},then(P+lai)ndCA+.
It follows from Theorem 1.2 e) that the ideal t C g(A) is w-invariant (see
Theorem 1.2 c». Hence w induces an invoJutive automorphism eM of the
Lie algebra g(A), called tbe ChlJtJllt, involu'ion of g(A). It is determined
by
( 1.3.4)
wee,) = -Ii. w(/.) = -eSt .w(h) = -h if h E .
As w(go) =: 8-0' we deduce that
(1.3.5)
mult Q = mult( -Q).
In particular,
- = -+.

8
Ba,ic Dfini'ion
Ch.l
51.4. The following simple statement is useful.
PROPOSITION 1.4. a) Let 9 be a Lie algebra,  C 9 a commutative subal..
gebra, el..... en, 111... tIn elements o£g, and Jet n v = {Qj t'." o} C ,
n = {Qlt..., Qn} C . be linearly independent sets such that:
( 1.4.1 ) lei, Ii] = 6 ij Qj E  ( i. j = 1,.. . , n),
(1.4.2) [h, fi] = (ai, h)ei, [h,/il = -(ai, h)/i. (h E ; i = 1...., n).
Suppose that ei, Ii (i = 1,..., n) and  generate 9 as a Lie algebra, and
that 9 has no nonzero ideals which intersect  trivially. Finally, set A =
((Qr,Qj»j=l' and suppose that dim = 2n-rankA. Then (g..n,nV)
is the quadruple associated to the matrix A,
b) Given two n x n-matrices A and A' J there exists an isomorphism of
the associated quadruples jf and only jf A' can be obtained (rom A by
reordering the index set.
Proof follows from Proposition 1.1 and Theorem 1.2 e).
o
COROLLARY 1.4.. The quadruple associated to a direct sum of matrices
Ai, is isomorphic to a djrect sum of the quadruples associated to the A..
The root system of g(A) is a union of the root systems of the g(A.).
Proof. follows from Proposition 1.4 a).
o
51.5. Now we need a short digression on gradations. Given an abeJian
group M. a decomposition V = e Va or the vector space V into a direct
oEM
sum of its 8ubspaces is called an M.gradation of V. A 8ubspace U C V
is called grnded if U = e (U n Yo). Elements from Va are called homo.
oeM
gneou or degree Q. The following fact is widely used in representation
theory.
PROPOSITION 1.5. Let  be a commutative Lie algebra, V a diagonalizable
-module, i.e.
(1.5.1) V = ED V,\, where V,\ = {v E Vlh(v) = '\(h)v (or all h E }.
E.
Then any submodule U o(V is graded with respect to the gradation (1.5.1).

Ch. 1
BGic Defini'ion,
9
Proof Any tI e V can be written in the form tJ = EI:I 1J j, where Vj E VJ'
and there exists h E  such that Aj(h) (j = 1,. . . , m) are distinct. We have
for v E U:
m
hl:(v) = L >.j(h)l: vj E U (i = Ot 1... .. m - 1).
j=l
This is a system of linear equations with a nondegenerate matrix. Hence
aU Vj lie in U.
o
One introduces the so-called formal 'opolog, on a graded vector space
V = ED V o 88 follow8. Given a finite subset F C M, we put V F =
oEM
ED Vcr, and declare all the subsets VF of V to be the fundamentalsys-
aEM\F
tern of neighbourhoods of zero. The completion of V in the formal topology
is, clearly, n Va. Given a subset C of this (complete) topological vector
oEM
space, its closure in tbe formal topology is called the fonnal completion of
c.
An M-gratltJtion of a Lie algebra 9 is its gradation as a vector space,
such that [go, g] C go+. For example, (1.3.1) is a Q-gradation of the Lie
algebra g(A).
In order to introduce an M -gradation in a Lie algebra 9 one chooses a
system of generators of g, saYt (11,..., On, and elements At,... ,'\n E M
and prescribes degrees to each Os : deg (Ii = i. This defines & ( unique) M-
gradation of g with deg Oi = i if and only if the ideal of relations between
the (Ji is M-graded. For if (Jl,... ,a" is a free system of generators or g,
such a gradation does exist.
Let now. = (81, . . ., &n) be an n-tuple or integers. Setting
degei=-deg/;='i (i=l....,n), deg=O
defines a I-gradation
,(A) = ED 'j(8)t
lEI
called the gratlation 01 '"e .. Explicitly:
9J(.) = E9 1o,
CI
where the 8um is taken over Q = E k;Q; E Q such that E i;8. = j. It is
i i
clear that if I; > 0 for all i, then 80(1) =  and dim 'j(.) < 00 (j E Z).

10
BtJlic Definition,
Ch. 1
A particularly important gradation is the principal gradation. This is
the gradation of type 1 = (1,. .. t 1). Explicitly:
8j(l) = $ go.
o:ht o=j
Note that
go(l) =, g-1(1) = E C/i, g1(1) = E Ceil
.
i
so that ":!: = EI:) 9%;(1).
.:> 1
J_
The following simple lemma is useful for computations in g(A).
LEMMA 1.5. Let a E "+ be such that [a, Ii] = 0 for all; = It. .. tn. Then
a = o. Similarly, for a E n_. if[atei] = 0 (or all i = 1,... f n, then a = o.
Proof. Let a E n+ be such that [0,9-1(1)] = o. Then it is easy to see that
Ei.jO(8d gl(l)i(ad Ya is a 8ubspace of n+ C g(A), which is invariant
with respect to ad 91(1), ad  and ad 9-1(1) (the condition on a is used only
in the last case). Hence if a 1: 0, we obtain a nonzero ideal in g(A) which
intersects  trivially. This contradicts the definition of g(A).
o
Remark 1.5. Sometimes it is useful to consider the Lie algebra g'(A) instead
of g(A). Let us give a more direct construction of g'(A). Denote by ;'(A)
the Lie algebra on generators ei, Ii, Qr (i = 1,..., n) and defining relations
[ei,/j] = 6ijQj,[Qr,QJJ = O,[ojtej] = aijejt(Qr,!j] = -atilj.
Let Q be a free abelian group on generators Ql t . . . ,Qn' Introduce a Q-
gradation g'(A) = E9 g setting
cr
degei=cri=-deg/i, degar=O.
There exists a unique maxima) Q-graded idea) r C g'(A) intersecting go
(= E Car) trivially. Then
i
g'(A) = g'(A)/t.
Note that this definition works for an infinite n as well.

Ch.l
BSI;c Definitionl
11
S 1.6. The folJowing statement is an application of Lemma 1.5.
PROPOSITION 1.6. The center of the Lie algebra g(A) or g'(A) is equal to
c := {h E 1(Qi. h) = 0 for all i = 1,... ,n}.
Furthermore, dim c = n - I.
Proof. Let c lie in the center; write c = L: Ci with respect to the principal
i
gradation. Then [C,9-1(1)] = 0 implies [ci.9-1(1)] = 0 and hence, by
Lemma 1.5, c; = 0 for i > O. Similarly, c; = 0 for i < o. Hence c E  and
[e,e;] = (Q;,c)e; = 0 implies that (oi,e) = 0 (i = 1,...,n). Conversely,
if c E  and the latter condition hoJds, c commutes with all Chevalley
generators and, therefore, lies in the center. FinaHy, c C ' since in the
contrary case, dim, > n - land n would not be a linearly independent
set.
o
Another application of Lemma 1.5 is
LEMMA 1.6. Let 1 1 ,1 2 C {I,..., n} be disjoint subsets such that Oij =
Oji = 0 whenever i e II, j E 1 2 . Let p, = L: k!.)Qi (s = 1,2). Suppose
iE/.
that l1' = /31 + {J2 is a root of the Lie algebra g(A). Then ejther PI or {J2 is
zero.
Proof Let i E 1 1 ,j E [2' Then [Qr,ej] = O,[QJ,e.] = O,[e.,hJ =
O,[ej,I,] = O. Using Lemma 1.5, one checks immediately that [ei,ej) =
OJ(fj,/jJ = O. Denote by g(') the subalgebra of g(A) generated by ei,ji
with i E 16. We have proved that g(l) and g(2) commute. Since 9a lies in
the subaJgebra generated by g(l) and 9(2), we deduce that Sa lies either in
g< 1) or in g(2).
o
SI. 7. We conclude the chapter with a description of the structure of
ideals of g(A).
PROPOSITION 1.7. a) g( A) is simple if and only if det A  0 and (or each
pair of indices i and j the following condition holds
(1.7.1)
there exist indices i I. i2, . . . , I,
such that Oii 1 Q . 1 i2 ...Oi,j  o.

12
BG;C Definition,
Ch.l
b) Provided that (1.7.1) holds, every ideal of g(A) either contains g'(A) or
is contained in the center.
Proof. The conditions in a) are obviously necessary. Now suppose that
the conditions are satisfied and let i C g(A) be a nonzero ideal. Then i
contains a nonzero element h E . Since det A 1: 0, we have c = 0 by
Proposition 1.6 and hence [h, ej) = aej t. 0 for some j. Hence ej E i and
Ql = (ej,/j] E i. Now it follows from (1.7.1) that ej,/j,aj E i for all j.
Since detA :F 0,  is a linear span of o:J's and we obtain that i = g(A).
proving a).
. The proof or b) is similar.
o
Sl.8. Exercises.
1./. Show that if Oij = 0 implies OJ. = 0, then (1.7.1) is equivalent to the
indecomposability of A.
I
1.2. Let A' = (aij)j=l be a submatrix of A of rank f'. One can choose a
subspace I)' of fJ containing il'Y = {0'1', . # . t Q', } of din1ension 2n' - .i' such
that n' = {0'1, · · . , Qn' } I' is a linearly independent set. Then (, n' I D /V )
,
is a realization of A'. Set Q' = E?=l ZQi, and Jet g(A) = EBoEQg(l be the
Lie algebra associated to A. Then
S(A')  ' $ ( $ Sa).
oEQ'\{O}
1.9. Show that if (, n, n V ) satisfy (1.1.1 and 2). then dim   2n - t.
1.. Suppose that A satisfies condition (1.7.1). Then, provided that there
is no root Q such that QI' = 0, the Lie algebra g'(A)/c is simple.
1.5. Show that mult(oj + SOj) < 1 and mutt 2(oj + OJ) < 1 in every g(A).
1.6. Let A = (_23 ;3). Show that mult(20 1 + 3( 2) = 2 in g(A). Show that
for an arbitrary 2 x 2-matrix A, mult(2Ql + 3(2)  2; find the conditions
when it is = 2.
J.7. Define the Lie algebra g'(A) in characteristic p > 0 as in Remark 1.5.
Prove that the Lie algebra g' ((:1 )) of characteristic 3 is a simple Lie
algebra of dimension 10 for any ;\ 1: 0, -1, and that two such algebras
corresponding to  = A1 and ,\ = A2 are nonisomorphic unless '\1 = '\2 or
'\1 = -A2 - I.

Ch. 1
Ba,ic Definition,
13
1.8. A direct sum of vector spaces g-1 E& go ED 91 is caJ)ed a local Lie algebra
if one has bilinear maps: gi x Gj -+ 9.+j for Iii, Iii, Ii + jl  1, such that
anticommutativity and Jacobi identity hold whenever they make sense.
Prove that there exists a unique Z-graded Lie algebra 9 = EB 9i such that
.
g-1 EB go $ 91 is 8 given local Lie algebra and EB "*i are free Lie algebras
i>O
on g::!: 1.
1.9. Let g be a Lie algebra,  c 9 a finite dimensional diagonalizable suba1-
gebra, g EB go the root space decomposition such that
QEQ
go = . Show that der" = (del 9)0 + ad g, where (der 9)0 consists of
endomorphisms preserving the root space decomposition.
[Choose h e  such that (0, h) :F 0 (or all Q E Q, cr  O. Adding to
d E der 9 an inner derivation, one can assume that d(h) E . Deduce that
d() C  and d(9a) C 90].
1.10. Deduce from Exercise 1.9 that all derivations of the Lie algebra
g(A)/c are inner, provided that A has no zero rows.
1.11. Show that ad induces an isomorphism
g( A)/ c ..... der g' (A),
if A is a generaJized Cartan matrix and al,,' 1: 0 for every Q' E A.
[Show that no root is equal to a simple root when restricted to /).
1.1 e. Let 9 be a complex semisimple finite-dimensional Lie algebra with
the Cartan matrix A. Then a choice of a Cartan 8ubalgebra  C g and a
root basis n c . provides 9 with a structure of a quadruple associated to
the matrix A.
1.13. (This is for a Jess advanced reader.) Prove that the Lie algebra
g = Slt+l of traceless (t + 1) x (l + 1) matrices with the usual bracket
[A, B] = AB - BA is a Kac-Moody algebra. In more detail, let Eij (i,i =
1, · · · t l + 1) denote the standard basis of the space of all (l + 1) x (l + 1)-
matrices. Let  be the space of all traceJes8 diagonal matrices. Then
Otr = Ei. - Ei+l,i+l (i = 1,..., I)
form a basis of. Define f:i e . (i = 1,... ,t + 1) by
f:.(diag(al J. .., Ot+l» = 0i.
Then
Qi = li - fi+ 1 (i = 1,. . . , t)

14
BOlie Dtfinition.
Ch.l
form a basis of .. Set
n = {Ql,...t Q t}, n V = {Qi,...,Qi}.
Then (, n, n V ) is a realization of the matrix
2 -1 0 . . . 0 0
A= -1 2 -1 0 0
......................... .
0 0 0 . . . -1 2
The root space decomposition with respect to  is
9 =  $ (EI)CEij).
ij
Set ei = Ei,i+lt Ii = Ei+l,i (i = 1,...,t). Show that 9 = g(A), with the
Chevalley generators ei, fie Show th3t {li - lj (i f; j)} is the set of all
roots, {fa - fj (i < j)} being the set of positive roots, and that n:J: are the
subalgebras of strictly upper and strictly lower triangular matrices. Show
that the Chevalley involution is a t-+ -'a.
1. J 4. Let 9 = e j 9j be a Z-graded Lie algebra such that every graded ideal
is trivial, g-l + go + 91 is finite-dimensional and generates 9, and the So-
modules 9-1 and gl are irreducible and contragredient. Show that 9 is
isomorphic to a Lie algebra g(A).
1.15. Let ' and V be two n-dimensional complex vector spaces with bases
II v = {() r , · · · , Q} and {VI,.'.' tJ n } respectively. Define a map C(J : 1)' 
V. by (cp(Qr),vi) = a,j. Choose a subspace o of V. complementary
to cp('), and let  = o $ '. Define Oi E . by (Oil ho + L: Cjoj) =
(oi,h o ) + EajiCj, and let n {al,...,a n }. Show that (,ntrrV) is a
j
realization of A = (aij).
1.16. Let Aut(A) be the group of permutations t7 of the set {l,...,n}
such that Qij = aa(i).a(j). This group acts by automorphism of g'(A) by
O'(ei) = e C7 (i)1 O'(/i) = fC7(i). O'(an = o(i)' Using Exercise 1.15. show that
this extends (non-canonically) to the whole Kac-Moody algebra g(A).
SI.9. Bibliographical notes and comments.
The study of Kae-Moody algebras was started independenlly by Kae [1967) I
[1968 A, B), and Moody (1967], [1968], {1969].

Ch.1
Basic Definitions
15
The proof of a statement much more general than Proposition 1.1 can
be found in Vinberg [1971].
Theorem 1.2 should be probably attributed to Chevalley [1948]. In this
2-page note (presented by E. Cartan) Cbevalley introduces & general alge-
braic approach to the construction of finite-dimensional simple Lie algebras
and their finite-dimensional representations. A detailed exposition of this
was given by Harish-Chandra [1951] and Jacobson (1962)" The proof in
Kac [1968 B) and Moody (1968J is a simple adaptation of these.
The material of SSl.5 and 1.6 is taken from Kac [1968 B).
Exercises 1.4, 1.8 and 1.14 are taken -Cram Kac [1968 B]. A somewhat
different a.pproach to the construction of graded Lie algebras W88 developed
by Kantor [1968], [1970]. Exercise 1.7 is taken from Weisfeiler-Kac [1971].
Exercise 1.11 is taken from Berman [1976]. Exercises 1.15 and 1.16 are
taken (rom Kac-Peterson (1987].
The rest of the material of Chapter 1 is fairly standard.
The problem of isomorphism of Kat-Moody algebras has been settled
quite recently. Namely, as shown by Peterson-Kat (1983], any wo maximal
ad-diagonaJjzable subaJgebras of a Kac-Moody aJgebra are conjugate, and
hence two Kac-Moody algebras are isomorphic if and only if their Cartan
matrices can be obtained (rom each other by reordering the index set. The
question for arbitrary g(A) remains open.

Chapter 2. The Invariant Bilinear Form
and the Generalized Casimir Operator
S2.0. In this chapter we introduce two important tools of our theory,
the invariant bilinear form and the generalized Casimir operator n. The
operator 0 is a "second order" operator which, in contrast to the finite-
dimensional theory. does not lie in the universal enveloping algebra of g(A)
and is not defined for all representations of g(A). However, {} is defined in
the so-called restricted representations, and commutes with the action of
g(A). Remarkably, one can manage to prove a number of results (including
some classical ones) using only O.
52.1. Note that resealing the ChevaJJey generators: ei ....... ei J ,. f-+ £."
(i = 1,.... n) where fi are nonzero numbers, we get or ....... fiQr, which
extends to an isomorphism  ......  (nonunique, if det A = 0). This extends
to an isomorphism: SeA)  g(DA), where D = diag( fl, . . . ,En).
An n x n matrix A = (a;}) is called ymmetrizCJble if there exists an
invertible diagonal matrix D = diag( iI, . . . , (ra) and a symmetric matrix
B = (b. j ), such that
(2.1.1)
A = DB.
The matrix B is then called a symmttrization of A and g(A) is called a
!ymmetrizable Lie algebra.
Let A .be a symmetrizable matrix with a fixed decomposition (2.1.1) and
let () n, n V ) be a realization of A. Fix a complementary 8ubspace " to
' = E CQ r in , and define a symmetric bilinear C- val ued form (.1.) on 
by the following two equations:
(2.1.2)
(2.1.3)
(oilh) = (Qi,h)l; for h E ,i = 1,.. .,n;
.
(h'Ih") = 0 for hi, h" E ".
Since or I .. · I o are linearly independent and since (by (2.1.1 and 2» we
have
(2.1.4)
( Q  I Q i) = bij (i (j (i. j = 1 J · · · t n) .
there is no ambiguity in the definition of (.J.).
16

Ch. 2
TAe Invariant Bilinelr Form
17
LEMMA 2.1. 8) The kernel of the restriction of the bilinear fOlm (.'.) to '
coincides Mrith c.
b) The bilinear (orm (.1.) is nondegenerate on .
Proof. a) (ollows from Proposition 1.6. If now for all h E  J we have
n n n
o = (E ciarl h ) = (E CiliO;, h), then E Clf;Q; = 0 and hence Ci = 0,
1=1 i:l 1=1
i = 1,.. . . n. proving b).
o
Since the bilinear form (.1.) is nondegenerate, we have an isomorphism
II :  --+ .. .defined by
(v(h). hi) = (hlh 1 ), h, h 1 E ,
and the induced bilinear (orm (.,.) on ..
It is clear from (2.1.2) that
(2.1.5)
lIe or) = (iOi, i = 1,..., n.
Hence from (2.1.4) we deduce:
(2.1.6)
(Qilaj) = hi; = GijIE,. i,j = 1, .. tn.
52.2. OUf next basic re&uJt is the CoUowing theorem.
THEOREM 2.2. Let g(A) be a symmetrizable Lie algebra. Fix a decompc>
sition (2.1.1) of A. Then there exists a nondegenerate symmetric bilinear
C-valued form ( . J .) on SeA) such that:
a) ( .,.) js invarjant, i.e. ([z, yJlz) = (zJ[y, zJ) for a1J z, 11. z E g(A),
b) (.,. )1" is defined by (2.1.2 and 3) and js nondegenerate.
c) (gol9p) = 0 if Q + P  O.
d) (.'. )"G+'-_ is nondegenerate for Q  0, and hence Ga and 8-c. are
non degenerately paired by ( .1.).
e) (z,yJ = (zJy)v- 1 «() for z E 9al 11 E 9-a, Q E.
Proof Consider the principal Z-gradation (see 51.5)
g(A) = E9 gj
IE I

18
The Invariant Bilinear Form
Ch. 2
N
and set g(N) = ED Sj for N :: 0,1,.... Define a bilinear symmetric
J=-N
form ( .1.) on ;(0) =  by (2.1.2 and 3) and extend it to 9(1) by
(2.2.1)
( e; I'i) = 6 ij l i (i t j = 1, · · · t n ) ;
(gotg:l:l) = 0; (gllg:!:l) = o.
Then the form ( .1. ) on g(1) satisfies condition a) as long as both [z, y]
and [y, z] lie in g(I). Indeed, it is sufficient to check that
([ e i , fj ] 1 h) = (e i I[/j , h]) for h E ,
or, equivalently,
6ij(Qr Ih) = 6ijEi(Oj, h)
which is true, due to (2.1.2).
Now we extend ( .,. ) to a bilinear form on the space g(N) by induction
on N  1 80 that (g.lgj) = 0 if lilt IiI 5 Nand i + j '# 0, and &)80
condition a) is satisfied as long as both [z, y) and [11, z] He in g(N). Suppose
that this is already defined on g(N - 1); then we have on)y to define (zly)
for z e 9:J:Nt'll E 9 T H. We can write y = E[Ui,Vt], where !.Ii and v.
i
are homogeneous elements of nonzero degree which lie in g( N - 1). Then
(x, Ui) E g(N - 1) and we set
(2.2.2)
(zly) = 2:([ZI Ui]lV.).
i
In order to show that this is well defined, we prove that if i,j, 8, t E Z
are such that Ii + it = Js + tl = N, i + j + , + t = 0, IiI, liJ, IsJ, It) < Nand
Zi E 9j, Zj E 9j, z, e 9" z, e g't then we have (on g(N - 1)
(2.2.3)
([[Zi, Zj], %,]Iz,) = (zil[zj, (z" za]]).
Indeed, using the invariance or ( .1 .) on g( N - 1) and the Lie algebra
axioms, we have
([[Zi, Z j], z,] Iz,) = ([[Zi, z,]. Zj ]Iz,) - ([[z j, z,], z.]lz,)
= ([ Z i , z,] I[  j , z,]) + (z i 1[[% j , z,]. z,])
= (z i J [ z, , [z j , z,]] + ([ Z j , z,], z, J )
= (z i I[ Z j , [z, , z c]]) .

Ch. 2
The Invariant BiJinelJr Fonn
19
If now z = E[u, v;], then by the definition (2.2.2) and by (2.2.3) we
.
have
(zly) = E([z, ui]lv,) = E(um v :, y]).
. i
Hence this is independent of the choice or the expressions for z and 11.
It is clear from the definition that a) holds on g( N) whenever (z, 1/] and
[y, z] lie in g(N). Hence we have constructed a bilinear form ( .1.) on 9
such that a) and b) hold. Its restriction to  is nondegenerate by Lemma
2.1 b). The form ( .(.) satisfies c) since for h E  J z e go and y E 81l we
have, by the invariance property:
o = ([ h, z] I y) + (z I [h. y]) = «( (), h) + (13, h) ) ( z I y) .
The verification of e) is standard. For z EGo, y E 9-0" where Q E t and
h E t we have
([z, y] - (zly)v-1(Q)lh) = (zl[y, h) - (zlu)(o, h) = O.
Now e) follows from b).
It follows from b), c) and e) that. the biJinear form ( . , .) is symmetric.
If d) fails, then, by c), the form ( .1.) is degenerate. Let i = Ker( .1.).
It is an ideal and by b), we have i n  = O. which contradicts the definition
of g(A).
o
52.3. Suppose that A = (Gij) is a symmetrizable generalized Cartan
matrix. Fix a decomposition
(2.3.1)
A = d iag( £11 · · · , £") (6;j )j = 1
where (i are positive rational numbers and (h ij ) is a symmetric rational
matrix. Such a decomposition always exists. Indeed, (2.3.1) is equivalent
to a system of homogeneous linear equations and inequalities over Q with
unknowns fi 1 and 6 ij :
l;1 I 0; diag(l1 1 ,... ,l;I)A = (b ij ); ",j = bji.
By definition, it has a solution over C. Hence, it has a so1ution over Q. We
can assume that A is indecomposable. Then for any 1 < j < n tllere exists

20
The [nvGritJnt Bilintdr-'Foma
Ch, 2
a 8equene 1 = i1 < i2 <' · .. <. ii- i <! i,' == j' such .that'. ai, ,i+, <' 'Oi We
have: . \
(2.3.2)
ai. .i.+t ii.+t = ai.+1. i . f (s = i",-,': . ,k'- 1)
. .. .  ... ..... '..  : .... . ' III
Hece £j El > 0 ,for ,I j t .omleing th pJ;o.of. ': :  ' " .....
From (2.3.2) we also deduce .': ": .". : . .:., ,. . '.;  I .
Remark !.3. ](:A is indecomposable, t.hen-the matrii diag.;(€l.....tt,,) is'
uniquely determined by (2.3.1) 'up to a,-constatnt. factor.
. .
Fix a nondegenerate bilinear symmetric tor'm (.',.) asscjated to the
decomposition (2.3.1) as defined in 52.1 From ,(2.1.6)  deduce:
(2.3.3)
(2.3.4)
(Q.'Oi) > 0 (or,.i-=.!.. .:-:f,n. '.
(2.3.5)
(QiIQj)  0 for i  j;
v' 2 -'1 ( . )
Qi = ( r ) 1/ Qi.
Qi Qi
I .' -
Hence we obtain the usual expression for :th generalized Car.tan matrix:
A = ( 2(Qilj» ) n ;
(Qilai) ij==1
. I
We extend the bilinear form (.1.) from  to an invariant symmetric
bilinear form ( .1.) on the entire Kac-Moody algebra g(A). By Theorem
2.2 such a form exists, and satisfies all the pfQperties state4 tbe.re" It is 88Y:
 .
to show that such a form is also unique (see Exercise .2.2). The bilinear
form ( .1. ) on the Kac-Moody algebra g(A) provided by Theorem 2.2 and
satisfying (2.3.3) is called a tandtJrd invariant.. form.
52.4. Let A be a symmetrizable matrix, let g(A) be the associated Lie
algebra and let (.1.) be a bilinear form on' g(A) .provided by Theorem
2.2. Given a root a, by Theorem 2.2 d) we can choose dual bases {e)}
d { (i) } f d · h b h ( (;) I (j» c ( ..
an e_ a 0 go an 9-0, I.e., sue ases t at ea e_ o = Vi; IfJ =
1,. .., mult Q). Then for % E go and y E 9-a we have
(2.4.1)
(zly) = E(zle)(Yle».
i

Ch. 2
TAe InvarianC Bil.MtJr Form
21
The following lemm. is crucial for "many computations:
. . . I .
LEMMA 2.4. If Q, PEA and. z e 9"-a, then we have'in- SeA)  g()
(2.4.2) ,
. Eel [zielJ = E(eJ, zJ  e.'>.
. ,
. "
Proof. We define the bil.inear fom. (.1.) on ,0(.4)  (A)by:
(i: . ylxl  ifl) '= (xlzl)(yJy). Pick'.r E: ga' ana.'f' E gp. it suffices
to check that pairing both sides of (2.4.2) ,with e 0 i, g'iv the same tesJt.
, ,r. .
We have; . . . . . i ' ' ,. "
L(e  [z, e!:)lI e  J) == L(ele)([zte)IIf)


= L(e (.t le)(e)II!, z]) = (ell!, z])
.
by Theorem 2.2 a) and (2.4.1). Similarly,
, ,
. .
E([eJ, z]  e.)te @ f) = 2:<e't[zl e])(e.)If) = ([z, eJl!).
"
,
Applying again Theorem 2.2 a) gives the result.
o
COROLLARY 2.4. In the notatjon of Lemma 2.4, we .ave
(2.4.3)
)e (' ;Izle)n = - )[%,'e],e')] in g(A),
.
,
(2.4.4)
Ee (' [zle)J = - }:)zl e<'; Je) in U(g(A».
, ,
Proof. Apply to (2.4.2) the linear maps from g(A)  g{A) to g(A) and to
U(g(A», defined by z 0 II  [z, y] and z @ II t-+ ZJI, respectively.
o

22
The Invariant Bilinear Foma
Ch, 2
S2.5. Let g(A) be a Lie algebra associated to a matrix A,  the Cartan
subalgebra, 9 = e Gcr the root space decomposition with respect to . A
a
g(A)-module (reep. g'(A)-module) V is called re,tricted if for every tJ E V.
we have 9a(v) = 0 for all but a finite number of positive roots a.
It is clear that every 8ubmodule or quotient of a restricted module is
restricted. and that the direct sum or tensor product of a finite number of
restricted moduJes is also restricted. Examples of restricted modules will
be constructed later (see Exercise 2.9 and Chapter 9).
Assume now that A is symmetrizable and that ( .1.) is a bilinear form
provided by Theorem 2.2.
Given a restricted g(A)-moduJe V, we introduce a linear operator n on
the vector space V, called the (generalized) Ca,imir operator, as follows.
First, introduce a linear function p E . by equations
(P, on = aii (i = 1,..., n).
If det A = 0, this does not define p uniquely, and we pick any solution. It
follows from (2.1.5 and 6) that
(2.5.1) (pIOi) = (QiIQi) (i = 1,... ,n).
Further, fo! each positive root Q we choose a basis {e)} of the space gOt
and let {e} be the dual basis of g-o. We define an operator 0 0 on V by
00 = 2 E Eee).
oe+ i
One could easily check that this is independent of the choice of bases
(see Exercise 2.7). Since for each v e V, only a finite number of sum-
mands ee)(v) are nonzero, 0 0 is well defined on V. Let "'11 U2,." and
u 1 , u 2 , . .. be dual bases of . The generalized Casimir operator is defined
by
o = 2v- 1 (p) + E t';Ui + no.
i
We record the following simple formula:
(2.5.2)
L (, ui)(p, Ui) = (..\Ip),
i
which is clear from
(2.5.3)
..\ = E (..\, u i )v( Ui) = :L (, Ui)V( u i ).
.
i

Ch. 2
The InvGriant Bilinear Fonn
23
We make one more simple computation. For % E go one has
[t.iU',Z] = (IU')ZU. + ui(IUi)Z
. . .
= (I ui)(1 Ui)Z + Z ( ui(1 Ui) + U'(I u i »).
Hence, we have
(2.5.4)
[U'Uil z] = z«ol) + 2v- 1 (o» for Z Ego'
,
52.6.
to :
Consider the root space decomposition of U(g(A» with respect
U(g(A» = EI1 UfJl where
IlEQ
U" = {z E U(g(A»I[h,] = ({J, h)z for all h E }.
Put Up = U(g'(A» n Ufj, 80 that U(g'(A» = e Up' Now we prove the
IJ
following important theorem:
THEOREM 2.6. Let g(A) be a symmetrizable Lie algebra.
a) If V is a restricted g'(A)-module and u E U, then:
,
(2.6.1)
[00, u] = -u (2(pI0) + (Qla) + 2&1- 1 (Q») .
b) IE V is a restricted g(A)-moduJe, then {1 commutes with the action of
g(A) on V.
Proof. b) follows immediately from a) and (2.5.4). If a) holds for u E U
and Ul E Up, then it holds for UUl E U+IJ' Indeed
[OOt UU1] = (0 0 , u]Ut + u[OOt Ul]
= -u(2(pIQ) + (Qlo) + 2&1- 1 (Q»Ul
- uUl(2(pIP) + (PIP) + 211-1(,8»
= -uul(2(plo) + (alo) + 2&1- 1 (0)
+ 2(QlfJ) + 2(pIP) + (PIP) + 211- 1 (,8»
= - uu l(2(pla + (J) + (0 + Pier + {3) + 2v- 1 (0 + (J».

4'
TIa lnw.ri.,.,t B"a.ear Form
Ch. 2
Hence, sinc eo., ,e_ a (> (i = J, . · .f, n), .geneate {t'(A)1 it suffioes tQ' check,
(2.6.1) for t£ = e ai or e_' a .. Applying (2.4.4) to z :::: ea. and using Lemma
1.3, we have: . , .,..t,.
-
: [Oo,eQJ = 2 L L([,eQJe) +e[e),eQJ)
, " oEA+ ..;" : ',.' ,
= 2{e_QjleQJe. + 2 L (E{e,eQ.Je!:).
oE+\{ai) , ' .
. "0 +  e+QJeQ"eQi"]) . . .
= -211- 1 (cr.)e Oi = -2(QiI Q i)e Oi '- 2e oi v- 1 (ai).
Thanks to.(2.5.1)J.this is,,(,.6..1) for u = a; ilaly.,.(nOte_Q.l.:;:,
2e-Oi[eatt e_ a ,] = 2e_'1I1'(Qi)' which. by (2.5.1),' 18 (2.6.1) fOf'
u = e_ ai .
, '
o
"0' :' :
COROLLARY:2.6. It, under: the, hypothe$ of TheQrem 2.6 b), there exists
v E V such that ei(v) = 0 for all i = 1,... tn, and h(v) = (A, h)v for some
A. E . an,1 all .h E . , then. " '.. ,.
(2.6.2)
O(v) = (A + 2pIA)v:;
If, furthermore, U(g(A»v = V':then .
",
(2.6.3)
o  (A + 2pIA)Iv.
. ., .
Proof. Frmula (2.62) follows from the definit,ion o( n and formula (2.5.2).
Formula (2.6.3) Collows Crom (26.2) and Theorem" 2.6.
o
. .
\, ".
Remark 1.6. One can define the Q-grada.tion.U(g'(A» =EB Up and the
peQ
map 11-1 : Q -. ' without using the Lie algebra g(A). (Here, 88 in
Remark 1.5, the symbol Q denotes the free abelian group on generators
Ql, ..., Qn.) Indeed, the Q-gradation of U(g'(A» is induced by the Q-
gradation of g'(A) defined in Remark 1.5. If A is 8ymmetrizable, we fix
a decompatitjon (2.1.1) and define ,,-1 81 a homomorphism of abe]ian
groups such that 1.I-1(ai) = f;lQ (i = 1,..., n). These definitions work
for infinite n as well.

Ch. 2
Tht Invariant Bitintor Form
.25
S2. 7. Let: A b'e an n x n-matrix Oyer R. 'Let.. (. . n t n ) be .& realization
of the matrix A over R" i.e., . is a vector pace of dirnnsiolJ  -,t. 9V
. c , , . . . . . . .,.) .
R, so that ( := C. ..t Dt n V ) is th'e'realization of A oy'er c. "
\Ve <icfine the compact fom. -t(A) of (A) fJS follows.. Denote by Wo the
 .. . " .
antilinear autolnorphism of g(A) determined by:
wO(ei) = -/.t
, Wo ( I) == - ei (i =. 1 , '.' .. , n) r
Wo ( h) = - h for h E ..
We call "'0 the compact involution of g(A). The existence of Wo is proved by
the same argument as that of w in 51.3. Then t(A) is defined as the fixed
pO,int set of o; tbis:j a real Lie algebra whose complexification is g(A).
Note "that thi definition of the compa form conide8 wth. the usual one
in the finite-dimensional case.
, .
. .
Now Jet A be a symmetrizable matrix over R and let ( . , . ) be 'a standard
form on g(A). Define a errQltian fqrm on g(A) by:
(x1y)o := -(wo(z)ly).
Theorm 2..2 implies the foJlowing propert of tis Hermiti fOl"m. The
restriction of ( · I. )0 to go is nondegeeate for all cr E AU {O}; (go (gp)o = 0
if Q  {3; the operators ad u and - ad:wo(u) for U E 'g(A) are adjoint to-each
other, i.e., ([u t z]ly)o = -(zl[wo(u), 11))0 for all z, II E g(A); in particular,
the restriction of ( . J .)0 to erA), is a.'. nondegenerate invariant R-bilinear
form. We will return to the study of the 'Hermitian form (.'.)0 after
developing some representation theory.
, ,
52.8. The following variation of the above results is very useful for
applications.
PROPOSITION 2.8. Let 9 be a Lie algebra with an invariant non-degenerate
bilinear form (.,.), let {z,} ,and {Y.} be dual bases (i.e. (ziIYj) = 6 ij ),
and let V be a g-module'such that fot'every pair of elements U,v E V,
i( u) = 0 or Yi( v) = 0 (or all but a finite number of i. Then the operator
O 2 := E i @ 1!i
( .
is defined on V @ V and commutes with the action of g.
Proof. We have to show that for every % E 9 one has:
(2.8.1) 2)[z, i]  IIi + i 0 [z, IIi]) = o.
i

26
The Invariant Bilinear. Form
Ch. 2
Write: [Z,Zi] = r:aijj, [Z,Yi] = EfJjillj; taking inner product of the first
j j
equation with Yj and the second with Zj' we obtain:
Qij = ([z,z.))Yj), Pji = ([z,YiJlzj).
Using invariance of ( . f .), we deduce:
Oij = (z I [ZI, Yj]), /ljj = (z I [y" Zj)).
It follows that Ctij = - {3ij, proving (2.8.1).
o
S2.9. Here we consider tbe rrlost degenerate example, the Lie algebra
;(0) associated to the n x n zero matrix (including n = (0). In this case
[e i , e j J = 0, IIi, Ii] = O. [e i, Ij] = 6ij Q r, (i, j = 1. . . . , n), so th at
9(0) =  ED E Ce, ED EC/,.
i i
n
The center of g(O) is c = E CQr. Furthermore, dim  = 2n and one can
i=1
choose elements d 1 , . . . , d n e  such that
n
f) = c + E Cd, I
i=l
and
[di t e j) = 6ij e j 1 [d., 'j] = - 6 ij 'i (i ,j = I, . . . , n ) .
One defines a nondegenerat,e symmetric invariant bilinear form on the basis
of 9(0) by:
(eil/i) = 1, (arld i ) = I, all the others = O.
Note that p = 0 and the Casimir operator is
{} = 2 Lord, + 2 E/,ei.
i 4
Set Cl = E C(Q( - oj) C c. Then the Lie algebra 9 / (0)/C1 is a Heien.
6erg Lie algebra of order n, i.e., it has a basis ei, /. (i = 1,... ,n), z, such
that (e;, Ij) = 6iJ Z (I, j = 1,. . . , n), and all the other brackets are zero.

Ch. 2
The Invan"tJnt Bilinear Form
27
S2.10. Exercises.
f.l. The matrix A = (dij) is symmetrizable if and only if
Qij = 0 implies OJ. = 0 and
aiai2a.i3 ... at.ia = 0i 2 i. ai,i 2 ... ai.i, for all i l ,..., i,.
2.2. Show that the bilinear form ( .1. ) is uniquely defined by properties a)
and b) of Theorem 2.2.
e.s. Let (.1.) be a nondegenerate symmetric invariant bilinear form on
g(A). Show that the matrix A is symmetrizabJe, that ( .,. )1,. is defined by
(2.1.2 and 3) for some choice of ", and that ( . I. ) satisfies all the properties
of Theorem 2.2.
[Set l i = (e ill. )] ·
H.4. Let 9 == ED g. be a Z-graded Lie algebra, which is generated by 9-1 +
i
go + gl. Show that an invariant symmetric bilinear form on the subspace
g-1 + 90 + {II such that (giJgj) = 0 whenever i + j # 0, can be (uniquely)
extended to such a form on the whole g. (Here the property of invariance
is understood to hold whenever it makes sense.)
£.5. Let ( .1.)1 and ( .1.)2 be two nondegenerate synunetric invariant bi-
linear forms on g( A) and assume that A is indecomposable. Show that
there exists an automorphism t/J of g(A) which leaves g'(A) pointwise fixed
and preserves , such that the bilinear forms (xIY)t and (4)(z)I(Y))2 are
proportional. Show that any two invariant bilinear forrns on g'(A) are
proportional.
£.6. Show that the adjoint representation of g(A) is restricted if and only
if dim g(A) < 00.
t.7. Let {Zi} and {Yi} be bases of n+ and "_, dual with respect to an
invariant bilinear form on g. Show that no = 2 E !li%i is independent of
i
the choice of these bases. Prove an analogous statement for 02.
e.B. Let 9 = g(A) be a simple finite-dimensional Lie algebra. Choose a
basis V1, · · . t Vd of g and the dual basis vi, . . . , vtl with respect to the form
( .1.). Set {} = E ViV i . Show that {} coincides with the Casimir operator
i
defined in 52.5. Show that p is the half-sum of positive roots of g.
e.g. Show that the g(A)-module T(V) constructed in the proof of Theorem
1.2 has a unique maximal submodule J(V). Show that T(V)/ J(V) is an
irreducible restricted g(A)-module.

28
Tht, InvGnGnt Bilinear Form
Ch. 2
D.10. Let go be a finite-dimensional Lie algebra with the bracket {, Jo and a
- .
nondegenerate invariant symmetric bilinear form ( .1.). Let d be a derIva-
tion or the Lie algebra go such that (d(z)ly) = -(zJd(y» for z, JJ E go. Set
g = go EB Cc ED Cd, where c and d are some symbols, and define a bracket
[, ] on 9 by:
[z, II] = (z, 11]0 + (d(z)ly)c for Zt II Ego,
[c, z] = 0 for z E g; [d, zJ = d(z).
Check that this is a Lie algebra operation. Extend the bilinear form ( .1. )
from go to g by setting
(did) = (cJc) = 0, (cJd) = 1, (cJ9o) = 0, (d)go) = O.
Show that this is a nondegenerate symmetric invariant bilinear form on g.
1.11. Let g be a solvable n-dimensional Lie algebra with 8 nondegenerate
invariant symmetric bilinear form. Show that either g is an orthogonl'J
direct sum of (n -I)-dimensional and I-dimensional Lie algebras, or 9 can
be constructed 88 in Exercise 2.10 from an (n - 2)-dimensional Lie algebra
go.
[Let i be a I-dimensional ideal in g. Show that i lies in the center (using the
fact that coadjoint orbits are even-dimensional). If i is isotropic, consider
il. and the Lie algebra go = iJ. Ii.]
S2.11. BibJiographical Dotes and comments.
Theorem 2.2 is due to Kac (1968 BJ and Moody (1968]. For finite-dimension..
al semi-simple Lie algebras this result is known due to the existence of the
Killing-Cartan form, which cannot be defined in the infinite-dimensional
case .
The generalized Casimir operator {} was introduced by Kac [1974]. The
idea of its definition is borrowe from physics. We take the usual definition
of the Casimir operator (see Exercise 2.8):
(}= EE (e e)+e) e) + LUiU'
0>0 i i
we rewrite it by using commutation relations:
{} = E 11-1 (a) + 2 L E e e) + L u,u i l
0>0 >o i i

Ch. 2
'The IntJtJritJrat Bi/intar Fonn
29
and then replace the first lummand. which makes no sense, by a "finite"
quantity 2v- 1 (p).
The proof of Theorem 2.6 is along the linea of Kac-Peterson [1984 B].
Some of the exercises, such as Exercises 2.10 and 2.11, seem to be new.
(The simplification of these exercises in the present edition was pointed out
to me by G. Favre and L.J. Santharoubane.) The rest of the material of
Chapter 2 is fairly standard.
A complete system of "higher order" Caaimi operators was constructed
by Kac [1984], using ideas of Feigin-Fuchs [1983A].

Chapter 3. Integrable Representations
of Kac-Moody Algebras and the Weyl Group
53.0. In this chapter we begin a systematic study of the Kac-Moody
algebras. Recall that this is the Lie algebra g(A) 8S8ociated to a generalized
Cartan matrix A. The main object of the chapter is the Weyl group W
of a Kac-Moody algebra, which is a generalization of the clU8ical We)'1
group in the finite-dimensional theory. However, in contrast to the finite-
dimensional case, W is infinite and the union of the W-translate8 of the
fundamental chamber is a convex cone, which does not coincide with the
whole "real" Cartan 8ubalgebra ..
S3.1. Let us first make some remarks on the duality of Kac-Moody
algebras. Let A be a generalized Cartan matrix; then its transpose C A
is again a generalized Cartan matrix. Let (, n, n V ) be a realization or
Ai then (.. n v t n) is clearly a realization of fA. So, if (g(A), , n, n V )
is the quadruple associated to At then (g(' A), . I n v , D) is the quadruple
associated to cA. The Kac-Moody algebras g(A) and g('A) are called dUGI
to each other.
Note that the dutJI root l(Jttic of g(A):
n
QV := LZar
_=1
is the root lattice of g('A). Furthermore, denote by v C QV
(c ' c ) the root system (' A) of g(' A). This is called the dual root '11 1 -
'em of g(A). In contrast to the finite-dimensionaJ case, there is no naturaJ
bijection between A and A v .
53.2.. RecaJJ Borne weU-known results about representations of the Lie
algebra Sl2(C). Let
e = ( ) I h = ( 1) I J = ( )
be the standard basis of Sl2(C). Then
[e,J] = h, [h,e] = 2e. [h,/l = -2/.
30

Ch.3
In'egrn61e Repreentation! tJnd 'he Weyl Group
31
By an easy induction on It we deduce the following relations in the universal
enveJoping algebra of st 2 (C):
(3.2.1)
(3.2.2)
[h, IJ = -2Icj", [h, el:] = 2k e l:,
[e,Ji]::: -k(k - 1)/'-1 + k/i-th.
LEMMA 3.2. a) Let V be an st 2 (C)-module and let v E V be such that
h(v) =.Av (or some  E C.
Set Vj = (j!)-l/ i (tJ). Then:
(3.2.3)
h(Vj) = (,\ - 2j)vj.
If, in addition, e( v) = 0, then:
(3.2.4)
e(Vj) = (,,\ - j + l)vj-t.
b) For each integer k > 0 there exists a unique, up to isomorphism,
irreducible (k + I)-dimensional st 2 (C)-module. In some C-basis {Vj H=o of
the space of this module the action of st 2 (C) looks as follows:
h(Vj) = (k - 2j)vi; f(vi) = (j + l)vi+1; e(vj) = (k + 1- j)Vj-l'
Here j = 0,..., k and we assume that Vi+l = 0 = V-I.
Proof Formulas (3.2.3 and 4) follow from (3.2.1 and 2). Let now V be an
irreducible (k+ I)-dimensional sl2(C)-module. Let u E V be an eigenvector
of h with eigenvalue p. It follows from (3.2.1) that if e'(u) 'F OJ then it
is again an eigenvector for h with eigenvalue IJ + 2s. Since V is finite
dimensional, there exists v = e'(u) 'I- 0, such that f(V) = 0 and h(v) = ..\v.
We set Vj = (j!)-l ji(v). As V is finite dimensional, we deduce from (3.2.4)
that  is a nonnegative integer, say m, that {Vj} are linearly independent
for j = 0, · · · . m and that V rn +! = O. Hence m = k and b) follows.
o

32
Inlegra61e Repre.tn'4'ion. Gnd the Wey' Group
Ch.3
53.3. Let g(A) be a Kac-Moody algebra and let ;, Ii (i = 1,. .., n) be
its Chevalley generators. Set 9(i) = Cei +Cor +C/.; then 9(i) is isomorphic
to .t 2 (C), with standard basis {ej, Qr ,I.}.
We can deduce now the following re)atioDs between the Chevalley gen-
erators:
(3.3.1)
(ad e.)l-tl iJ ej = 0; (ad Ji)l- Cl i; Ii = 0, if i 1: j.
We prove the second relation; the first one (oilowl by making use of the
Cheval1ey involution w (see (1.3.4»).
Denote v = 1;1 8 i ; = ( ad /;)1- 0 i j fj. Consider g(A) as a 9(ir..module by
restricting the adjoint representation. We have:
or(v) = -OijV; e;(v) = 0 if i;: j.
Hence Lemma 3.2 a) together with the properties (Cl) and (C2) of A imply
[ei,8 ij ] = (1- Gij)(-aij - (1- Oij) + l)(ad/i)- Cl i j /j = 0 if i  j.
It is also clear that e, commutes with 9 ij if k  i, k ;: j (by relations
(1.2.1», and al80 if i = j but aij :F O. Finally, if k = j and Gij = 0 we
have:
[ej, (lij} = [ej, [lit /jJJ = o,ji/i = 0 (by (C3».
Sot [ei, 8;;] = 0 for all i and we apply Lemma 1.5.
o
S3.4. Now we need a general fact about a module V over a Lie algebra
g. One say. that an element % Egis local', nilpo'enf on V if for any tI E V
there exists a positive integer N 8uch that zN (f) = O.
LEMMA 3.4. a) Let III J Y2, · .. be a system of generators of a Lie algebra
9 and lee z e 9 be such Chat (ad z)N iYi = 0 (or some positive integers Ni,
i = 1,2, · . .. Then ad  i8 locally nilpotent on 8.
b) Let VI, 1J21 · .. be a system of generators of a g-module V, and let  E a
be such that ad z is locally nilpotent on 9 and ZHi (Vi) = 0 for BOrne positive
integers Ni, i = 1.2,.... Then z isloca11y nilpotent on V.
Proof. Since ad z is a derivation of " one baa tbe Leibnitz formula:
i
(ad z)i[y, z] = E (  ) [(ad z)'y, (ad z)'-i z]t
· 0 I
.=

Ch.3
Integrable ReprelenltJtioft' anti 'he Weyl Group
33
proving a) by induction on the length of commutators in the Yi. b) follows
from the (ollowing formula (for ,\ = p = 0):
(3.4.1) {X-.x-JL)k a = t (  ) {(adX-A)ja)(X-'J.L)k-", k > 0, A,J.L E C,
-,=0
which holds in any associative algebra. In order to prove (3.4.1), note
that ad z = L - R, where L and R are the operators of left and right
multiplication by z, and that L and R commute (by associativity). Now
we apply the binomial formula to L1f -  - IJ = (ad z - ) + (R - p).
# 0
Applying the binomial formula to ad z = L - , we obtain another
useful formula (in any associative algebra):
(3.4.2) (adz)t a = (-l)'()zi-'az"
S3.5. LEMMA 3.5. ad ei and ad Ii are locally nilpotent on g(A).
Proof. By relations (1.2.1) and (3.3.1), we have: (adei},a.jf+lz = 0 =
(ad fi)'G,;f+ 1 z , if z is ej or fj. Also, (ad ei)2h == 0 = (ad fi)2h if h E .
Now we can apply Lemma 3.4a).
o
S3.6. A g(A)-module V is calIed -diagonalizdle if V = eAE. VA,
where VA = {v e Vllh(tJ) = (,h)v for. h E ). As usual. VA is called
a weight space,  E . is called a weight if V :F 0. and dim V is called
the multiplicity of  and is denoted by multv . Similarly. one defines an
'-diagonalizable g'(A)-module, its weights, etc.
An -(resp. '-)diagonalizable module over a Kac-Moody algebra g(A)
(resp. g' (A» is called integrable if all ei and Ii (i = 1,..., n) are locally
nilpotent on V.
Note that the underlying module of the adjoint representation of a Kac-
Moody algebra is an integrable module by Lemma 3.5. The following im-
portant proposition justifies the term "integrable.»

34
Integrd6/e Reprelen'tJtion& Gnd 'ht We,l Group
Ch.3
PROPOSITION 3.6. Let V be an integrable g(A)..module.
a) A, G 9(i)-module, V decompo,e, into G direct .urn of finite tlimen.ional
imducib/e -invoriant module, (Aence the oction of S(i) on V cCln 6e -in-
'egraletl" to the action of tAt ,roup SL 2 (C».
b) Let ,\ E . be 4 weight of V ond let O'i be CI ,imple root of g(A). Dtno'e
by M the 6e' 0/ all t e z 6UCla tlad'  + tai ;6 a weight of V I Gnll let
me = multv( + tQi)' TAen
(i) M i, 'he cloled intervGI of integer, [-p, q], whe p Gnd q Clre 60'A
either nonnegative in'eger, or 00 Ind p- q = (, Qr> when 60tA p Gnti
q are finite; il multv  < 00, then p and q Irt finite,-
(ii) ei : VA+'oi --. V,\+('+l)o. i, an injection/or t E [-Pt -t(A,or); in
particular, the function t t-+ m, incG't' on thi, interval;
(iii) the function t .-. m, i, ,ymmetric witla re,pect to t = -i(A, Ql)i
(iv) i/60ll&;\ ond  + Qi are weigh'" then ei(VA)  O.
Proof. By (3.2.2) we have:
(3.6.1)
e./il&(v) = k(l- k + ('\,Qr)/l-l(v) + Il'e,(v) for II EVA.
We deduce that the subspace
U = L Il'e'i(v)
i.mO
is (get> +  )-invariant. As ei and Ii are locally nilpotent on V, dim U < 00.
By the Weyl complete reducibility theorem applied to the 9(i)-module U,
the latter decomposes into a direct sum of finite dimensional -invariant
irreducible g(i)-modules (cf. Exercise 3.11). So, each v E V lies in a di-
rect sum of finite dimensional -invariant irreducible g(i)-modules, and a)
follows.
For the proof of b) we use a) and Lemma 3.2 b). Set U =
L V A + koi ; this is a (g(i) + )-module, which is a direct sum of finite
keZ
dimensional irreducible (g(i) + )-modules. Let p = - inC M, q = sup M.
Both p and q are nonnegative as 0 EM. Now all the statements oC b)
Collow from Lemma 3.2 b), as (A + tas,an = 0 Cor t = -j(A,an.
D
The following corollary of Proposition 3.6 b) (i) and (iii) is very useful.
COROLLARY 3.6. a) If ,\ is a weight of an integrable g(A)...module V and
A + OJ (resp.  - ai) is not a weight, then (A, on > 0 (resp. (A,an  0).

Ch.3
Integrable Reprelenldtion$ and 'he Weyl Group
35
b) If A is a weight of V. then  - (A, ar)aj is also a weight of the same
multiplici ty.
Remark 9.6. Let V be an integrable g'(A)-module. Then, clearly, Proposi-
tion 3.6 and Corollary 3.6, with  replaced by ' t still hold. Furthermore,
the local nilpotency of e. and Ii on V guarantees that V is
hi-diagonalizable, and hence '-diagonalizable provided that n < 00. This
follows from a general fact which will be proved in S3.8.
S3. 7. Now we introduce the important notion of the Weyl group of a
Kac-Moody algebra g(A). For each i = 1,. . . t n we define the fundamental
reflection r. of the space . by
ri(;\) = ,\ - (J Q)Qi, A E ..
]t is clear that r, is a reflection since its fixed point set is 11 = {A e
.I(,Qr) = O}, and r.(Q.) = -Qi.
The subgroup W of GL(.) generated by all fundamental reflections is
called the Weyl group of g(A). We will write W(A) when necessary to
emphasize the dependence on A.
The action of ri on . induces the dual fundamental reflection rl on 
(for the dual algebra g(' A». Hence the Weyl groups of dual Kac-Moody
algebras are contragredient linear groups; this allows us to identify these
groups.
The following proposition is an immediate consequence of Corollary 3.6
b) and Lemma 3.5.
PROPOSITION 3. 7. a) Let V be an integrable module over a Kac-Moody
algebra g(A). Then multv ;\ = multv w('\) for every  E . and w E W.
In particular, the set of weights of V is W -invariant.
b) The root system  of g(A) is W-invariant, and mu)t Q' = muJt w(a) for
every Q E d,w E W.
The following fact will be needed later.
LEMMA 3.7. If Q E L1+ and ri(Q) < 0, then Q = Q.. In other words,
+ \ {.} is r;-jnvariant.
Proof follows from Lemma 1.3.
o

36
Infegro61e Repre6entatioR$ Gnd the Weyl Group
Ch.3
S3.8. In this section we outline a somewhat different approach to the
Weyl group. Let (I be 8 locally nilpotent operator on a vector space V.
Then we can define tbe exponential
11 2
exp ° := Iv + n0 + 2T o +... I
which has the usual properties, in particular,
exp ka = (exp 0)1: (k E l).
Let b be another operator on V, such that (ad a)N6 = 0 for some N. Then
one knows the following formula:
(3.8.1)
(exp a)b(exp -0) = (exp(ad 0»(6).
This easiJy (olJows by using formula (3.4.2).
LEMMA 3.8. Let 1r be an integrable representation of g(A) on a vector
space V. For i = 1 t . . . ,n see
r, = (exp li)(exp -ei)(exp Ii).
Then
a) r; (V) = i()
b) rr e Aut g(A)
c) rr I  = r i ·
Proof Let v e V. Then h(rf{tJ) = (, h)rf(v) if (ai, h) = O. Hence for
a) and c) we have only to check that ar(rf(v)) = -(;\, cr;')r,(v). This
Collows from
(3.8.2)
(r;)-l Q r r ; = -Qj.
By (3.8.1) it is sufficient to check (3.8.2) onJy in the adjoint representation
of at 2 (C); using (3.8.1) again, one has to check (3.8.2) in the 2-dimensional
natural representation of .t 2 (C). But in this representation we have
exp/i= (: ),eXp(-ei)= ( l),r;= ( l),
and (3.8.2) is clear.
b) follows from (3.8.1) applied to the adjoint representation.
o

Ch.3
Integro6/e Reprelentation and the We,l Group
37
Remark 9.8. Let (V, 11') be an integrable g(A)-module whose kernel lies in
. By Proposition 3.6, the action of the 8ubalgebra g(i) (i = I J . . . , n)
on V can be integrated to a representation i : SL 2 (C) -4 GL(V). The
groups 1r.(SL 2 (C» generate a subgroup G- in GL(V). The group Oft can
be viewed as an "infinite dimensional" group associated to the Lie alge-
bra g(A). The elements ri = ,,"i(  -0 1 ) (i = 1,.. . ,n) generate a subgroup
- -
W. C G1t. The group W. contains an abelian normal subgroup Dr gen-
erated by (r;)2 (i= 1,...,n) such that W(A) W/D1t.
We conclude this subsection by the proof of a general fact a special
case of which was mentioned in 53.6. One says that an element z of a Lie
algebra 9 is locally finite on a g-module V if any v E V lies in a z-invariant
finite-dimensional subspace. Note that locally nilpotent and diagonizable
elements are locally finite and that (3.8.1) still holds if a is locally finite on
V and the linear span of the (ad a)i 6 is fi nite-dimensional.
PROPOSITION 3.8. Let 9 be a Lie algebra and let V be a g-module such
that g is generated by the set Fv of all ad-locally finite elements which act
locally finitely on V. Then
a) 9 is spanned over C by Fv. In particular, if 9 is generated by the set F
of all ad-locally finite elements, then 9 is spanned by F.
b) If dim 9 < 00, then V is g-locally finite (i.e. any element of V lies in a
finite-dimensional g-submodule of V).
Proof. Let gv denote the C-span of Fv and let a E Fv. Using (3.8.1)
we deduce that gv is invariant with respect to automorphisms exp '(ad a).
Since lim«exp t ad a)b - 6)/t = [at 6], it follows that [at 9v] C 9v, proving
'-+0
a). b) follows from a) and the Poincare-Birkhoff-Witt theorem.
o
53.9. Let A be a symmetrizable generalized Cart an matrix and let (.1.)
be a standard invariant bilinear form on g(A).
PROPOSITION 3.9. The restriction of the bilinear form (.1.) to . is W-
invariant.
Proof As I r i(Oi)1 2 = I-OiI2 = IOil 2 ;/; 0, it 8uffices to check that PIOi) = 0
implies (ri(A)la.) = O. But ri('\) = A by (2.3.5).
o
For a converse statement see Exercise 3.3.

38
Integrable ReprelentatioRI and 'he Weyl Group
Ch.3
53.10. We return to the study of tbe Weyl group W of a Kac-Moody
algebra g(A). Let us start with the following technical lemma.
LEMMA 3.10. I(Qi is a simple root and ri1 ." r..(Q.) < 0 then there exist.
. (1  a  t) such that
(3.10.1)
ria ...ri, ...ri. r , =rit...ri,_1ri,+1...ri..
Proof. Set {J = ri.+I'" ri,(Qi) for k < t and P, = Q.. Then, by the
hypothesis, /30 < 0; on the other hand, {3, > O. Hence, for some , we have:
{J,-1 < 0, p, > O. But (J,-1 = ri,(p,); hence by Lemma 3.7, p, = Oi., and
we obtain:
(3.10.2)
Qi, = W(Qi), where w = ri.+l · .. ri,.
But
(3.10.3)
W(Qi) = OJ (w E W) => w(Qr> = oJ.
Indeed, w = wl for some w from the subgroup in Aut g generated by
rr (i = 1,... t n) (see S3.8). Applying w to both sides of the equality
[Sa-"S-a-J = Car, we obtain (by Lemma 3.8 b» Cw(on = Co;' Since
w(ai)(W(Qr) = (Qi,aj) = 2, we get w(Qr) = aj.
Now we can conclude from (3.10.2 and 3) that
-1
ri. = wri w .
Multiplying both sides of this by ri 1 ... ri,_1 on the left and by
ri.+1 .. · ri. ri on the right completes the proof.
o
53.11. The expression w = ri l ... ri, E W is called reduced if , is
minimal possible among all representations of w E W as a product of the
ri. Then, is called the length of w and is denoted by t(w). Note that
det. ri = -1 and hence
(3.11.1)
det,. w = (_l)t(w) for w E W.
The following lemma is an important corollary of Lemma 3.10.

Ch.3
Integrable Repreen'tJt;on' anti ,Ae Weyl Group
39
LEMMA 3.] I. Let w = r'a . . . ri. E W be a reduced expressjon and Jet 0;
be a simple root. Then we have
a) l{wr.) < l(w) if and only ifw(£ti) < O.
b) W(G'i.) < O.
c) (Exchange condition) lfl(wr;) < t(w), then there eJdsts 8, such that
1 < B Stand
r.. ri,+l · · · rat = ri,+, · . , ri, ri.
Proof By Lemma 3.10 (applied to w), weal) < 0 implies that l(wr.) <
i(w). If now W(Qi) > 0, then wr.(Q.) < 0 and hence t(w} =
l(wrr) < l(wr.), proving a). b) follows immediately from a). Finally,
if l( wri) < l( w), then a) implies w( Q.) < 0 and applying Lmma 3.10
to w we deduce the exchange condition from (3.10.1), multiplying it by
(ria ...ri._,)-l on the left and by r. on the right.
o
S3.12. Now we are in a position to study the geometric properties of
the action of the Weyl group. Recall the definition of . C  from S2.7.
Note that . is stable under W since QV C .. The set
C = {h E .f(()iJ h) > 0 for i = 1,. .. J n}
is called the fundamental chamber. The sets w(C), W E W, re called
chambers, and their union
x = u w( C)
weW
is called the Tit cone. We clearly have the corresponding dual notions of
C V and XV in ..
PROPOSITION 3.12. a) For h E C, the group Wit = {w E Wlw(h) = h} is
generated by the fundamental reflections which it contains.
b) The fundamental chamber C is a fundamental domain (or the action oE
W on X, j,e., any orbit W · h of hEX intersects C in exactly one point.
In particular, W operates simply transitively on chambers.
c) X = {h E fJ.I(a-, h) < 0 only for a finite number of 0 E A+}. In
particular, X is a convex cone.
d) C = {h E fJ./ for every wE W, h - w(h) = LCiOr where Ci  OJ.
I

40
1,,'egm61e Repre,enltJ'ion. dnd the Weyl Group
Ch.3
e) The foJlowing conditions are equivalent:
(i) 'WI < 00;
(ii) X = .;
(iii) IAI < 00;
(iv) IAvl < 00.
f) If hEX, then tw" I < 00 jf and only jf h lies in the interior of X.
Proof. Let w E W and let w = ri, . . . ri. be a reduced expression of w.
Take h E C and suppose that hi = w(h) E C. We have (Qi,.h)  0 and
therefore (w(Qi,),h')  O. But by Lemma 3.11 b), W(Qi,) < 0. and hence
(W{cri,), hi)  O. So, (W(Qi,), h') = 0 and (Qi., h) = O. Hence ri.(h) = h
and both a) and b) follow by induction on t( w).
Set X' = {h E .I(Q, h} < 0 onJy for a finite number of Q E +}.
It is clear that C C X' and it follows from 13.7 that X' is r.-invariant.
Hence XI :) X. In order to prove the reverse inclusion, let h E X' and
set Mia = {Q E A+f(Q, h) < OJ. By definition, (Mia I is finite. If M,. #= 0,
then Qi E M" for some i. But then it follows from Lemma 3.7, that
IMr.(h») < IMlal. Induction on IMhl completes the proof of c).
The inclusion ::> of d) is obvious. We prove the reverse inclusion by
induction on , = l(w). For t(w) = 1, d) is the definition of C. If
l(w) > 1, et w = ria ... ri.. We have h - w(h) = (h - ria." r.,_,(h» +
ria ... ri._ 1 (h - ri.(h)) and we apply the inductive assumption to the first
summand and Lemma 3.11b) for /1v to the second summand, completing
the proof.
Now we prove e). (i) => (ii) since an element hi of W · h with maximal
height of (hi - h) lies in C. In order to show (ii) => (iii) take h in the
interior of C. Then (0, -h) < 0 for all Q E + and hence 'I < 00 by c).
(iii)  (i) because of
(3.12.1)
{ w( Q) = Q for w e W and all QEd} => w = 1.
To prove (3.12.1), note that if a reduced expression w = ria' .. ri. is non-
trivial, then Lemma 3.11 b) implies that w(Q;,) < 0, contradiction. The
fact that (iv) is equivaJent to (i) follows by using the dual root system.
Finally, to prove f) we may assume that h e C. Then f) follows from a)
by applying the equivalence of e)i and e)ii to W" operating on /Ch..
o
.
Note that, by Proposition 3.12 b)t tbe Weyl group W operates simply
transitively on chambers t the stabilizer of a point from the interior of a
chamber being trivial.

Ch.3
Integrable ReprelenttJ'ionl dntl 'he We,l Group
41
S3.13. Recall that the group with generators rl,... J r n and defining
relations
(3.13.1 ) r l = 1 (i = 1,. . . , n); (r i r j ) m.j = 1 (i, j = 1, . . . , n)
is called a Co%e'er group. Here mij are positive integers or 00 (we use the
convention ZOO = 1 for any z).
PROPOSITION' 3.13. The group W is a Coxeter group, where the mij are
given by the following table:
aij Oji 0 1 2 3  4
mij 2 3 4 6 00
Proof. First we check relations (3.13.1). The relation rl = 1 is obvious.
Furthermore, the 8ubspace t := Rcri +Rcrj is invariant with respect to both
ri and rj. and the tnatrices Ofri and rj in the basis Oi.Oj of l are ( 1 -,j)
and (-ji 1 ). Hence the matrix of rirj in this basis is
(3.13.2)
( -1 + tJijOji a ij )
-aja -1
and we obtain
(3.13.3)
det t(rirj - >'1) = ,\2 + (2 - aijaji) + 1.
Hence rirj has an infinite order if (lajOj. > 4 and the order is given by the
table in the rest of the cases (f). = t E9 { e .I('\, Qr) = (, oj) = O} if
OijOji 1: 4).
Now we can refer to an abstract fact that relations (3.13.1) and the
exchange condition (see Lemma 3.11 c» imply that the group in question is
a Coxeter group (see Bourbaki [1968], Ch. IV, no. 1.6). A more transparent
geometric approach is outlined in Exercise 3.10.
o
53.14. Exercises.
3.1. Let A be a symmetrizable matrix and let A = DB be a decomposition
of the (orm (2.3.1). Show that C A is also symmetrizable; more explicitly
· A = D V BY , where D V = D-l, BY = DB D. Show that the corresponding
standard forms on  and . (defined in 52.1) induce each other.

42
Integrable Repreen'Gtion6 Gnd 'he Weyl Group
Ch.3
9.e. Let e. /, h be the standard basi. of "2(C). Show that one haa in the
enveloping algebra:
[  l::.. ] = min(m.n) /"-; ( h - m - n - 2 j ) em-i. .
ml' nl  (n-i)! j (m-J)!
Deduce that if I is locaUy nilpotent on a 612(C)-moduJe V. then V e is
h-diagonalizable.
[Prove this for m = 1 by induction on n, then use induction on m].
3.3. There exists a nondegenerate synunetric W -invariant bilinear
form on  if and only if A is symmetrizable_ Any such form can be extended
from  to an invariant nondegenerate symmetric bilinear form on the entire
Lie algebra S(A). This (orm satisfies all the conclusions of Theorem 2.2.
3../. Let V be an integrable g(A)-module. Show that
(r;)2(v) = (-l)(tOr)v for v E V,\.
3.5. The set L\+ is uniquely defined by the following properties:
(i) n C d+ C Q+; 2ai;' + for i= 1,...,n;
(ii) if Q E A+, Q  Qi. then the set {Q + /eQ.; It e z} n + is a "string"
{a - pai,...,O' + qlti}, where p,q E Z+ and p- q = (a,ai);
(iii) if Q e A+ \n then Q - Qi E + for some i.
9.6. Prove that l(w) = I{a E A+lw(Q) < O}I.
3. 1. Let A = (:, -;0) be a generalized Cartan matrix of rank 2, and
BSSume that 06  4. Show that the Weyl group V(A) is an infinite dihedral
gr<?up. Let ab > 4; then (zoJ + YQ21zQl + Y ( 2) = bz 2 - o6zy + 01/2 is a W-
invariant quadratic form on ., and XVU-X v = P E .I(.\I.\) < O}U{O}.
9.8. Let A = ( :2 -;2 !l ) , and let W = W(A) be the associated Weyl
o -1 2
group. Show that the map rl t-+ (l)' r2 ...... (l:), r3 t-+ ()
induces an isomorphism 11" : W  PGL2(Z).
[Use the fact that PSL 2 (1) is generated by elementary matrices
1r(r2)7r(rl) and w(rl)1r(r3)].
3.9. Let A = (aij) be the matrix of Exercise 3.8,  the Cartan subalgebra
of g( A), Q1, Q2, 03 simple roots. Define a standard bilinear form on f) by
(odCtj) = aij. Set 11 = -al - 02, ;2 = !Ol' ,3 == -al - 02 - 0'3- Define

Ch.3
Integrable Repreentationa and the Wey' Group
43
a map p : . -+ S2(C) (symmetric 2 x 2 matrices) by p(a"}'1 + 61'2 + C'Y3) =
(.i2 62), and define the action of PGL 2 (Z) on S2(C) by g(5) = g5('g).
Check that p is a W-equivariant map, and that (oIQ) = -2detp(a) for
Q e .. Using Proposition 3.12 b), (5.10.2) and Exercise 3.8, deduce that
a quadratic form oz2 + 6zy + cy2 such that a, 6, c E Z, 4ac  6 2 and
a  0 can be transformed by G £2(1) to a unique quadratic form such that
a > c > h  O.
3.10. In this exercise we outline a geometric proof of Proposition 3.13. Let
W' be the Coxeter group on generators ri (i = 1,..., n) and relations
(3.13.1), and let 1t : W' -+ W be the canonical homomorphism. We con-
struct a topological space U = W' x c /( ), where W' is equipped with the
discrete topology, the fundamental chamber C with the metric topology,
and  is the following equivalence relation: (WI, z)  (W2t y)  {z = y
and w 1 1 w2 )jes in the subgroup of W' generatd by those r. which fix z}.
Define an action of W' on U by: w( wI, z) = (WWl, z). This is obviously
well defined. Show that there exists a unjque continuous W'-equivariant
map  : U ..... X such that t/J( 1 , z) = z for :r E C (W' operates on X via
1r). Let Y = {zlz is fixed by at least three reflections from W} and set
X' = X\Y, U' = U\-I(y). Show that q,' : V' -+ X' is a covering map.
Deduce that 4>' is a homeomorphism and hence,.. is an isomorphism.
3. J J. Show that for an irreducible -diagona)jzab)e module over
Sl2(C) all weight spaces are I-dimensional. Classify these modules. Clas-
sify the ones which are integrable. Prove that every finite-dimensional
sI2(C)-module is completely reducible.
[Use the Casimir operator and Exercise 3.21.
3.1. Let w = ria ... Ti, be a reduced expression of w E W. Let /3 E A+
be such that w- 1 (,8) < 0; show that the sequence (J, ria (tJ), ri ri 1 (13), . . .
contains a unique simple root, say OJ(fj). Let ,\ E .; show that
 - w() = E (t QJ(fJ»)!3.
tJ:w- 1 (P)<o
[Use the identity  - wlri() = (- Wl('\») + Wl(A - ri('\) and induction
on t.)
9.13. Show that the action of T on g(A) preserves every invariant bilinear
form and commutes with the Chevalley involution.

44
Integrable Reprelentation' and 'he Weyl Group
Ch.3
3.14. Show that 0 -+ ( ..... g'(A) -+ g'(A)/c -+ 0 is the universal central
extension of g'(A)/c, where A is a generalized Cartan matrix.
[Let cp ; 9 --. g'(A)/c be an epimorphism with  finite-dimensional kerne
Then <p- 1 (ei + Ii + (' / c») is isomorphic to 9i $, where 9i ':::: !t 2 (C) and 
is commutative, and is completely reducible on g. Let ei, 5:, Ii E 9i be the
preimages of ei,oi'!i; show that they satisfy relations (1.2.1) and hence
generate a quotient g' of g' (A) by  central ideal. Show that 9 is a direct
sum of g' and a central ideal from  J.
3.15. Show that Interior X = {h e .I(Q, h}  0 only for a finite number
of Q e L\+}.
9.16. An element z of a Lie algebra g is called loeall, finite if ad z is locally
finite on g. Put gfln = linear span of {z e glz is locally finite}. For a
g-module V, put Vftn = {v e VI dim E Cz. (v) < 00 for every locally
i:>O
-
finite element z of g}. Show that "fin is a subalgebra of 9 and Vfln is a
g-submodule of V.
3.17. Let 9 be a Lie algebra such that 8 = Sfln; then g is called integrable.
Let V be a g-module such that V = Vftn; then V is called inttgra6/. Let
a. be a free group on generators S := {z e glz is locally finite }. Given an
integrable representation 11' of 9 on V, define a representation 1( 1r) of O. on
V by 1( 1I")(z) = E 1;11"(%)", Z e S. Let N. be the intersection of kernels
n.
nO
of al11(w), where 1r runs over all integrable g-modules. Put G = G. IN.;
denote by exp z the image of z e S under the canonical homomorphism
a. -+ G. The group G is called the group tJIsociated to the integrable Lie
algebra g. Show that if 9 is a simple finite-dimensional Lie algebra, then
G is the associated connected simply connected Lie group, and % H exp z
is the exponential map. (It is a deep fact established in Peterson-Kac
[1983] that the above definition of an integrable module over a Kac-Moody
algebra coincides with that of S3.6.)
3.18. Show that any Kac-Moody algebra g(A) is integrable. (As shown
in Peterson-Kat [1983], the associated group G(A) constructed in Exercise
3.17 is a central extension of the group constructed in Remark 3.8.)
3.19. Let 9 be an integrable Lie algebra and G the associated group. Then
G acts on the space of every integrable representation V of 9. 80 that exp z
is locally finite on V for every z E 5 (defined in Exercise 3.17). Open
problem; let V be a G-modu]e such that every exp tz is JocaJJy finite on V
and is a differentiable function in t on every finite-dimensional %-invariant
subspace. Show that V can be "differentiated" to an integrable 9-module.

Ch.3
In'tgra61e Representalion, and 'he We,t Gro.p
45
9.£0. Let A be the extended Cartan matrix of a complex connected simply
o
connected algebraic group G. Show that the group G associated to the
o
Kac-Moody algebra g(A) is a central extension of tbe group G(C[t, ,-1]).
n
:J.tl. Let 9 be the Lie algebra of polynomial vector fields E  ,,:, such
i:l
that };: g: = const. Show that 9 is an integrable Lie algebra and that the
.
associated group G is a central extension of a group of biregular automor-
phisms of en. Show tbat g is generated by all locally finite elements of the
Lie algebra of all polynomial vector fields.
3.ee. Show that the group Ow described in Remark 3.8 depends, up to
isomorphism, only on the l-span of the set of weights of 1r. Show that if
.
the g(A)-module V is irreducible then the kernel of the adjoint action of
-
W. on g(A) lies in {:i:l v }.
3.e9. Let q be a complex number =F 0, :1:1. Let U,(,l2) denote an associative
algebra on generators e./, 0 and 0- 1 and the following defining relations:
aa- 1 = a-1a = 1, aea- 1 = q 2 e, ala- 1 == q-2 It
ef - Ie = (a - a-1)/(q _ q-l).
Let Ie E Z+ and let V be the (k + I)-dimensional vector space with a basis
VO t ..., v. For j E N let UJ = ( - q-j)/(q - q-l). Show that formulas (we
assume V-I = 0. V+1 = 0) : e{vi) = :I:[k - i + l]v.-t, I(Vi) = [i + l]Vi+l.
a(vi) = q-2.v. define a representation otthe algebra U f (sl2) on V. Show
that this representation is irreducible if and only if 9 is not an n-th root of
unity, where n  k, and that, under this assumption, these are the only two
irreducible (k + I)-dimensional U f (sI2)-moduJes. Show that as q -+ 1, we
get the (It + l).dimensional irreducible representation of .l2(C) (or rather
of U(st 2 (C») described by Lemma 3.2b).
S.t./. Let for BEl and j EN: [0;8] = (aq. - o-lq-6)/(q - 9- 1 ) and
[4j'1 = [0; 8)[0; 8 - 1] . .. [0;' - j + 1]/U]!.
Show that we have the following identity in U,(,l2) (which is a "q-analogue"
of that of Exercise 3.2):
[ m ' ''' ] min(m.n) / n- j m-.
!- _ _ " [ Clt-m-n+2j ] e J
[m]!' [n]! - f;-: [n - j]! j [m - j]!.
3.!5. (Open problem.) Classify all simple Z-graded (by finite-dimensional
subspaces) integrable Lie algebras.

46
Inttgra61e Repre,enlaliora, and 'he We" Gro.p
Ch.3
53.15. Bibliographical Dote. aDd comments.
The We)'1 group of a Kac-Moody algebra was introduced by Kac [1968 B]
and Moody [1968]. The exposition of S53.6-3.9 follows mainly Kat [1968 B].
The importance of the category of integrable modules was pointed out by
Frenkel-Kac [1980] and Tits (1981).
The Tits cone was introduced by Tits in the "synunetrizable" case. In
the framework of general groups generated by reflections it was introduced
by Vinb'erg [1911] (see also Looijenga [1980]). The exposition orSS3.1o-3.13
follows mainly Kac-Peter80n [1984 A].
Exercise 3.2 is due to Steinberg (1967] and Benoist [1987]. Exercise 3.8
is due to Vinberg (see Piatetsky-Shapiro-Shafarevich [1971]). Exercise 3.9
is taken from Feingold-Frenkel [1983]. Exercise 3.10 is taken from Vinberg
[1971]. Exercise 3.14 was independently found by Tits [1982]. Exercises
3.16-3.20 are based on Peterson-Kac [1983], Kac-Peterson [1983] and Kac
(1985 B). The notion of an integrable Lie algebra seems to be new.
Hopefully, one can develop a general theory of integrable Lie algebras,
associated groups and their integrable representations (cr. Kac [1985 B],
Abe-Takeuchi [1989). As shown in Peterson-Kac [1983] and Kac-Peterson
[1983], (1984 B), [1985 A,B]. [1987] one can go quite far in this direction
in the case of arbitrary Kac-Moody algebras (some important previous
work was done by Kac [1969 B), Moody-Teo (1972), Marcuson [1915), and
Tits [1981], [1982]). Various aspects of the theory of Kat-Moody groups
may be found in Arabia [1986], Kumar [1985], [1987 A], (1988], Kostant-
Kumar [1986], [1987], Mathieu [1986C], [1988], [1989A], Slodowy [1985B],
Tits [1985], (1987]. Bausch-Rousseau [1989], and others.
The algebra U t (Sl2) constructed in Exercise 3.23 is the simplest example,
first considered by Kulish-Reshetikhin [1983J, of a "quantized Kac-Moody
algebra U introduced independently by Drinfeld [1985], [1986] and Jimbo
[1985]. This new field, known under the name of quantum groups, is in
a process of rapid development. Exercise 3.24 is due to Kac (cf. Lusztig
(1988]).

Chapter 4. A Classification of
Generalized Cartan Matrices
S4.0. In order to develop the theory of root systems of Kac-Moody al-
gebras we need to know some properties of generalized Cartan matrices. It
is convenient to work in a slightly more general situation. Unless otherwise
stated, we shall deal with a real n x n matrix A = (Oij) which satisfies the
following three properties:
(ml) A is indecomposable;
(m2) aij S 0 {or i  j;
(m3) ai; == 0 implies Oji = O.
Note that a generalized Cartan matrix satisfies (m2) and (m3) and we can
assume (ml) without loss of generality.
We adopt the following notation: for a real coJumn vector u = t ( U 1 , U2, . . .)
we write u > 0 if all Ui > 0, and u > 0 if all Ui > o.
54.1. Recall the following fundamental fact from the theory of linear
inequali ties:
A $ytem of reGI homogeneoul linear inequalitiel i > 0, i = 1,
. . ., nt, has tJ olution if and only if 'here i$ no nontrivitJllinear
dependence with nonnegative coefficitntl among the i.
We shall use a slightly different form of this statement.
LEMMA 4.1. It A = (Oij) is an arbitrary real m x 8 matrix (or which there
is no u  0, u I 0, such that (C A)u  0, then there exists v > 0 such that
Av < o.
Proof Set; = E OjjZj, where the Zj form the standard basis of the dual
j
of R.. Then the lemma is a consequence of the "fundamental fact" for the
system of inequalities:
{ - i > 0 (i = 1 t . . . , m)
Zi > 0 (i = 1 t . . . ,8).
o
41

48
A C'ltJI,ijication of Generalized Carlon M (J'rice
Ch.4
54.2. We need one more lemma.
LEMMA 4.2. If A satisfies (ml)-(m3), then Au  0, u  0 imply that
ejther tl > 0 or u = o.
Proof Let Au  0, u  0, 1.4 t. o. We reorder the indices 80 that Ui = 0 for
i  , and Ui > 0 for i > I. Then by (m2) and (m3), Au > 0 implies that
Qij = Oji = 0 for i S 8 and j > '. in contradiction with (ml).
o
54.3. Now we can prove the central result of the chapter.
THEOREM 4.3. Let A be a real n x n matrix satisfying (mt), (m2), and
(m3). Then one and only one of the following three possibilities holds for
both A and 'A:
(Fin) detA F OJ there exists u > 0 such that Au > OJ Av  0 implies t1 > 0
or v = 0;
(Aft") corank A = 1; there exists u > 0 such that Au = OJ Av  0 implies
Av = 0;
(Ind) there exists u > 0 such that Au < 0,. Av  0, v  0 imply tJ = O.
Proof. Replacing v by -t! we obtain that in cases (Fin) and (Air) there
is no v  0 such that Av S 0 and Av  o. Therefore each of (Fin) and
(Aff) is not compatible with (Ind). Also (Fin) and (Air) exclude each other
because rank A differs. Now we shall show that each A together with' A
is of type (Fin), (AIf), or (Ind).
Consider the convex cone
KA = {u I Au  OJ.
By Lemma 4.2, the cone K A can cross the boundary of the cone {u I u  O}
only at the origin; hence we have
K A n {u I u 2: O} c {u I u > O} U {OJ.
Therefore, the property
(4.3.1)
KA n {u I u 2 OJ # {OJ
is possible only in the following two cases:
1) K A C {u I u > O} U {O}, or
2) KA = {u I Au = OJ is a I-dimensional subspace.

Ch.4
A ClaslificGtion of Generalized Canan MtJlnce!
49
Now, I) is equivalent to (Fin); indeed, det A -F 0 because K A does not
contain a l-dimensional 8ubspace. Clearly, 2) is equivalent to (AIf). We
also proved that (4.3.1) implies that there is no u  0 such that Au < 0,
Au :F O. By Lemma 4.1 it follows that if (4.3.1) holds, then both A and
, A are of type (Fin) or (AIf). If (4.3.1) does not hold, then both A and' A
are of type (Ind), again by Lemma 4.1.
o
Referring to cases (Fin), (Air), or (Ind), we shall say that A is of finite,
affine, or indefinite type t respectively.
COROLLARY 4.3. Let A be a matrix satisfying (ml)-(m3). Then A is of
finite (resp. affine or indefinite) type jf and only jf there exists Q > 0 such
that Aa > 0 (resp. = 0 or < 0).
54.4. We proceed to investigate the properties of the matrices of finite
and affine types. Recall that a matrix of the form (aij)..iES, where S C
{I, . . . , n}, is called a principal su6matriz of A = (Oij); we shall denote
it by As. The determinant of a principal submatrix is called a principal
minor.
LEMMA 4.4. IE A is of finite or affine type, then any proper principal
submatrix of A decomposes into a direct sum of matrices of finite type.
Proof Let S C {I,... , n} and let As be the corresponding principal sub-
matrix. For a vector u, define Us similarly. Now, if there exists u > 0 such
that Au > 0. then Asus > 0 and = 0 only if aij = 0 for i E S, j  s.
The latter case is impossible since A is indecomposable. Now the lemma
follows from Theorem 4.3.
o
S4.5. LEMMA 4.5. A symmetric matrix A satisfying (ml)-(m3) is of
finite (resp. affine) type jf and only jf A is positive-definite (resp. positive-
semidefinite of rank n - 1).
Proof If A is positive-semidefinite, then it is of finite or affine type, since
otherwise there is u > 0 such that Au < 0 and therefore 'uAu < o. The
cases (Fin) and (Air) are distinguished by the rank.
Now let A be of finite or affine type. Then there exists u > 0 such that
Au > o. Therefore. for ;\ > 0 one has: (A + >'l)u > 0, hence A + ).[ is of
finite type by Theorem 4.3. Hence det(A + >'1)  0 for all ). > 0 and all
the eigenvalues of A are nonnegative.
o

50
A CltJ,ifictJtion of GenetGliZtd Ccarian M atnet,
Ch.4
S4.6. LEMMA 4.6. Let A ::: «(lij) be a matrix of finite or a11ine type such
that Gii = 2 and aijOji = 0 or  1. Then A is symmetrizabJe. Moreover, jf
(4.6.1)
ai&i2aii3 · · · a.,i l  0 tor some distinct ;1 J · . ., i, wjth.  3,
then A is of the (orm
2 -Ul
-1 2
-u 1
o
o
_u- 1
n
o
o 0
-Un 0
2
-1
-U n -l
-U n -l
2
where Ul, .. . I Un are some positive numbers such that Ul . . . Un = 1.
Proof It is clear that the second statement implies the first one (cr. Exer-
cise 2.1). Suppose now that (4.6.1) holds. Taking the smallest possible, in
(4.6.1), we see that there exists & principal 8ubmatrix B of A of the form:
2 -6 1 0 -hi
,
-6} 2 0 0
0 0 2 -6'-1
-6. 0 -6_1 2
By Lemma 4.4, B is o( finite or affine type and therefore by Theorem
4.3 there exists u > 0 such that Bu  o. Replacing B by the matrix
(diag u)-l B(diag u), we may assume that C u = (1,. ..,1). But then Bu  0
implies that the sum of the entries of B is nonnegative:
(4.6.2)
.
28 - 1)6i + 6D > O.
4::1
Since 6,bi  It we have 6i + 6  2; hence, by (4.6.2) we obtain that
hi + bi = 2 and therefore bi = 6 = 1 for all i. As in this case
det B = 0, Lemmas 4.4 and 4.5 imply that B = A.
o

Ch.4
A C/d6$;ficat;on of Genem/;zeJ Carlan Matrice6
51
54.7. We proceed to classify all generalized Cartan matrices of finite
and affine type. For this it is convenient to introduce the so-calJed Dynkin
diagrams. Let A = (a.j)j=l be a generalized Cartan matrix. We asso-
ciate with A a graph S(A), called the Dyniin diagram of A as follows. If
OijOj;  4 and lOii I  'OJ;', the vertices i and j are connected by IO;j' Jines,
and these lines are equipped with an arrow pointing toward i if lo.j I > 1. If
OijtJji > 4, the vertices i and j are connected by a bold-faced line equipped
with an ordered pair of integers Jaii" faj.'.
It is clear that A is indecomposable if and only if S(A) is a connected
graph. Note also that A is determined by the Dynkin diagram S(A) and
an enumeration of its vertices. We say that S(A) is of finite, affine, or
indefinite type if A is of that type.
Now we summarize the results obtained above for generalized Cartan
matrices.
PROPOSITION 4.7. Let A be an indecomposable generalized Cartan ma-
trix.
a) A is oE finite type jf and only j( all its principal minors are positive.
b) A is of afflne type if and only if all its proper principal minors are positive
and det A = O.
c) If A is of finite or affine type, then any proper subdiagram of S(A) is a
union of (connected) Dynkin diagrams of Jinite type.
d) If A ;s of finite type, then S(A) contains no cycles. If A is of affine type
and SeA) contains a cycle, then SeA) is the cycle Al) (rom Table Aft" 1.
e) A is of affine type jf and only if there exists 6 > 0 such that A6 = 0;
such a () is unique up to a constant factor.
Proof To prove a) and b) note that by Lemma 4.6, if A is of finite or
affine type) it is symmetrizable, i.e., there exists a diagonal matrix D with
positive entries on the diagonal such that DA is symmetric (see S2.3). Now
a) and b) follow from Lemma 4.5. c) follows from Lemma 4.4, d) (oHows
from Lemma 4.6, e) follows from Theorem 4.3.
o
S4.8. Now we can list all generalized Cartan matrices of finite and affine
type.
THEOREM 4.8. a) The Dynkin diagrams of all generalized Cartan matrices
of Jjnite type are listed in Table Fin.

52
A CltJ,ific(Jt;on of Generalized Carlan Matrice,
Ch.4
b) The Dynkin diagrams of all generalized Cartan matrices of affine type
are listed in Tables Aft" 1-3 (all of them have l + 1 vertices).
c) The numerical labels in Tables Aff 1-3 are the coordinates of the unique
vector 6 = '(00.01,...,01) such that A6 = 0 and the Oi are positive rela-
tively prime integers.
Proof. First, we prove c). Note that AcS = 0 means that 20j = Lmjaj for
j
all i, where the summation is taken over the j's which are connected with
i, and mj = 1 unless the number of lines connecting i and j is 8 > 1 and
the arrow points toward i; tben mj = I. Now c) is easily checked case by
case. .
It follows from c) and Proposition 4.7 e) that all diagrams in Tables
Atr 1-3 are of affine type. Since all diagrams from Table Fin appear as
subdiagrams of diagrams in Tables Aff 1-3, we deduce, by Proposition
4.7 c). that all diagrams in Table Fin are of finite type.
It rema.ins to show that if A is of finite (resp. affine) type, then S(A)
appears in Table Fin (resp. Air). This is an easy exercise. We ,do it by
induction on n. First, det A  0 immediately gives
(4.8.1) A2' C 2 and G 2 are the only finit type diagrams of rank 2; Ai}) and
A2) are the only affine type diagrams of rank 2.
(4.8.2) A3, 8 3 , C 3 are the only finite type diagrams of rank 3; Al), C1),
Gl), D2) A2), D3) are the only affine type diagrams of rank 3.
Furthermore, by Proposition 4.7 d) we have
(4.8.3) S(A) has a cycle, then SeA) = Al).
By Proposition 4.7 and inductive assumption we have
(4.8.4) Any proper connected 8ubdiagram of S(A) appears in Table Fin.
Now let S(A) be of finite type. Then S(A) does not appear in Tables Air
1-3, has no cycJes by (4.8.3), each of its branch vertices is of type D4 by
(4.8.2 and 4), and it has at most one branch vertex by (4.8.4). By (4.8.4),
for S(A) with a branch vertex, the only possibilities are Dt, Es, E1. Es.
Similarly we show that if SeA) is not simply-laced (i.e., has multiple edges),
then it is Bl' C l , F4t or G2. A simply-laced diagram with no cycles and
branch vertices is Ai.

Ch.4
A CltJ'3ifictJtion of Genrulizd Carlan Matrice&
53
Now let S(A) be of affine type. By (4.8.3) we can assume that S(A) has
no cycles. By (4.8.4), S(A) is obtained from a diagram of Table Fin by
adding one vertex in such a way that any subdiagram is from Table Fin.
Using (4.8.1 and 2) it is not difficult to see that only the diagrams from
Tables Aff 1-3 may.be obtained in this way.
o
We fix once and for all an enumeration of tbe vertices of the diagrams
of generalized Cartan matrices A in tables below as follows. In Table
Fin vertices are enumerated by symbols Ql, . . . ,Ql. Each diagram X?> of
Table Aff 1 is obtained (rom the diagram XI. by adding one vertex, which
we enumerate by the symbol QQ, keeping the enumeration or the rest of
TABLE Fin
At. 0-0-...-0-0 (1+ 1)
a. 0, °1-1 at
81. O-o-..'-oo (2)
0, 0'2 0'-1 at
Ct. o - 0 -. · -- 0 <= 0 (2)
aa 0'2 0'1_1 at
°0'
DI. I (4)
0-0-...- 0 - 0
01 0, 0,_, at_I
lOt
E6 0-0- -0-0 (3)
OJ °2 OJ a. CI,
let T
E7 0-0- -0-0-0 (2)
Q1 Q 0'3 0.. 0'5 06
Es fa. (1)
0-0-0-0-0-0-0
O'. a, 0', a. a. a. 0,
F4 0-0::>0-0 (1)
0'1 °a Qa °t
G 2 QO (1)
OJ Q2

54
A C/a"ijica'ion of Generalized Carlan M4,rice.
Ch.4
TABLE Air 1
Al)
oo
1 1
Al)(t 2= 2)
1
o
0-0- · .. -0-0
1 1 1 1
Bl)(t 2= 3)
01
o-b-o-... - oo
12222
c?)(t  2)
0:>0-'. .-oo
1 2 2 1
Dl)(t 2= 4)
1 1 1 1
0-0-0-.. .-0-0
12221
Gl)
0-0S-0
123
Fl)
0-0-0:::>0-0
1 2 3 4 2
E1)
01
I
02
o-o-b-o-o
12321
E(l)
7
o-o-o-xo-o-o
1 234 3 2 1
E(l)
8
1 3
0-0-0-0-0-0-0-0
1 234 564 2
the vertices as in Table Fin. Vertices of the diagrams of Tables Aft' 2
and Aff 3 are enumerated by the symbols 00...., Qt. The numbers in
parentheses in Table Fin are det A. The numerical labels in Tables Air are
coefficients of a linear dependence between the columns or A.

Ch.4
A Cltl,iJictl'ion of Generalized COMtln Matricel
55
A2)
TABLE Aff 2
2 .k:: 1
oo
00 a,
A(2) ( l > 2 )
2l -
2 2 2 1
0<::0-'..- 0 o
00 01 0'-1 Of
A(2) ( l > 3 )
2l-1 -
1
1 ooy 2 2 1
0- 0 2 -0-...- 0 <:=0
01 02 OJ Ot-J at
D(2) ( l > 2 )
l+ 1 - .
1 1 1 1
oo-...- 0 =>0
00 01 0'_1 0t
E2)
12321
0-0-0<:0-0
00 aa 02 03 o.
TABLE Aff 3
D(3)
4
121
o-Oo
00 01 02
S4.9. We conclude the chapter with the following characterization of
Kac-Moody algebras associated with generalized Cartan matrices of finite
type (cr. Proposition 3.12 e».
PROPOSITION 4.9. Let A be an indecomposable generalized Cartan ma-
trix. Then the following conditions are equivalent:
(i) A is a generalized Caftan matrix of finite type;
(ii) A is symmetrizable and the bilinear (orm ( .1. ). is positive-definite;
(iii) II  ;
(iv) IL\J < 00;
(v) g(A) is a simple finite-dimensional Lie algebra;
(vi) there exists 0 E d+ such that Q + Qi  A {or all i = 1, . .. , n.
Proof By Lemma 4.6 and Proposition 4.7, the bilinear form (.1. ),. is
positive-definite if A  of finite type; thus. (i) => (ii). If (ii) holds, then
by Proposition 3.9. the group W lies in the orthogonal group 0« . , . » and
hence is compact. Since W preserves the lattice Q, it is discrete. Hence
IWI < 00, which proves (ii) => (iii). The implication (iii)  (iv) follows

56
A CltJI,ijictJtion of Generalized CtJrltJn MtJtricel
Ch.4
from Proposition 3.12 e); (iv)  (v) is clear by Proposition 1.7 a), and
for (v) => (vi) we can take Q to be a root of maximal height. Finally. let
Q + Qj tJ  for all i. Corollary 3.6 a) implies that (a, oj)  0 for all i.
But then it follows by Theorem 4.3 that A is of finite or affine type, and
in the latter case (Q.or) = 0 for all i. But in this case, by Lemma 1.5.
0' - Q, e A+ (or some i. Hence, if A is of affine type, Proposition 3.6 b)
implies that Q + ct, E +, a contradiction. This proves the last implication
(vi) => (i).
o
Remark 4.9. A root of a finite root system A which satisfies condition (vi)
of Proposition 4.9 is called a higlae,' root. It is easy to see that there exists
& unique such root (cr. Proposition 6.4 in Chapter 6); it is given by the
formula
I.
9 = LOiQil
i=l
where 0; are the labels of the extended Dynkin diagram from Table Air 1.
S4.10. Exerciaea.
-1.1. An indecomposable generalized Cartan matrix A is said to be of
strictly hyperbolic type (resp. hyperbolic type) if it is of indefinite type
and &Ilf connected proper 8ubdiagram of S(A) is of finite (resp. finite or
affine) type. Show that a matrix (:. -,0), such that 0 and 6 are positive
integers. is of finite (resp. affine or strictly hyperbolic) type if and only if
ab  3 (resp. oh = 4 or 06 > 4). Show that there is only a finite number of
hyperbolic matrices of order  3 and that the order of a strictly hyperbolic
(resp. hyperbolic) matrix is  5 (resp.  10). (Note a discrepancy with
Chapter 5 where we assume a hyperbolic matrix to be symmetrizable.)
.e. Let A be of type T',f. r (p 2= II  r). i.e., let its Dynkin diagram be of
the form
/
,,0
0-0- .. · -0-0-... -0
  
......- .....
" t
Set c =  + : +. Then A is of finite (resp. affine or indefinite) type if and
only if c > 1 (reep. c = I, c < 1). Show that for c < I, the signature of A

Ch.4
A Clal,ijictJ,ion of Generalized Carlan Matrict$
57
is (+ + · · · + -). Show that A is hyperbolic if and only if (p, q, r) = (4,3,3)
or (5,4,2) or (7,3,2). Show that detA =pq+pr+qr-pqr.
[To prove the statement about the signature, delete the branch point.]
.3. Introduce the following diagrams with n vertices (n  5):
En(: Tn- 3 t 3 . 2 )
0-0- ... -o-X-o-o
AEn

0-0- · .. -0-0-0
BEn
y
oo- ... -0-0-0-0
CE n
oo- .. · -o-X-o-o
DEn
Show that Eto, BE IO , GElD, and DEI0 are all hyperbolic matrices of rank
10. Show that a hyperbolic matrix of rank 7, 8, or 9 is one of the following
list: T 4 . 3 ,3, T S ,4,2, AEn, BEn, CE n , DEn (n = 7,8, or 9).
O-X- .,. -o-X-o-o
./.. Classify all (strictly) hyperbolic matrices.
4.5. Show that E 10 is the only symmetric hyperbolic matrix with deter-
minant -1. Show that diag(1/2, 1,..., 1)(BE 10 ) is an integral symmetric
matrix with determinant -1.
l.6. Show that if A is symmetrizable and hyperbolic, then the corre-
sponding "symmetrized" matrix has signature (+ + · · · + - ).
-I- 7. Let A be a strictly hyperbolic matrix. Show that the group Aut X of
aU Jinear automorphisms of the Tits cone X acts with a compact funda-
mental domain.
./.8. Let A be a hyperbolic matrix. Show that if A is symmetrizable, then
X is linearly isomorphic to an upper cone over a unit ball. Show that,
conversely, if A is a hyperbolic matrix of order 3, such that Aut X operates
transitively on the interior of X, then A is symmetrizable.
( 2 -1 -1 )
-1.9. Show that the real Tits cone X for the matrix -2 2 -1
-1 -1 2
provides an example or a cone in R3 such that Aut X has a compact fun-
damental domain, but does not act transitively on the interior of X.

58
A Cla".ficGt.on of Generalized Carlan M tJtrice,
Ch.4
.1.10. Using Exercise 1.4, show that if A is an indecomposable generalized
Cartan matrix and A is not from Table Aft', then the Lie algebra g'(A)/c
is simple.
.11. Let A = (Oij) be a symmetric matrix. Set tt = EGii:Ci, where Ci
i
are nonzero numbers. Check that
ti:ZI 1 ( Zi % ) 2
LaiiZiZ = E - - ECiCiai - - - ·
. L I. C. 2. L Ci C.
',15 15 .,.
Assume that Gij  0 for i  j. Deduce that if there exist positive num-
bers Cl, ..., C n such that ti = 0 (k = l,...,n), then the quadratic form
E Qik(i(1: is positive-semidefinite. Use this to prove Proposition 4.7 e). De-
i I
duce also that this quadratic form with an indecomposable A is positive-
definite if and only if there is no Cl > 0, ..., C n > 0 such that ta S 0
(k= l,...,n).
./.1£. Let 5' and S" be affine Dynkin diagrams. Let A be a generalized
Cartan matrix with the Dynki n diagr am
5(A)= 1 5' 1 - 0 - 1 5" I .
Show that det A = O.
4. J 3. Show that an indecomposable generalized Cartan matrix is affine if
and only if it is degenerate and all its principal minors are nonnegative.
.1.14. Show that the following is a complete list of connected Dynkin dia-
grams of generalized Cartan matrices of infinite order such that any prin-
cipal minor or finite order is positive:
Aoo · · · - 0 - 0 - 0 - · · ·
A+ oo 0-0-0- .. ·
Boo 0$:0-0-0-'"
Coo 0::>0-0-0- ...
D oo o-X-o-o- ...
54.11. Bibliographical Dotes and comment..
Theorem 4.3 is due to Vinberg [1971], 88 well as most of the results or 554.4-
4.7. The list of affine diagrams appean in Kac [1967] and Moody (1967].
The notation appears naturally in the context or the theory of finite order
automorphismB (Kac [1969 A]), 88 we shall lee in Chapter 8.
Exercises 4.8 and 4.9 are taken from Vinberg-Kac [1967]. Exercise 4.11
is taken (rom Bourbaki [1968]. The rest oCthe exercises are fairly standard.

Chapter 5. Real and Imaginary Roots
S5.0. In this chapter we give an explicit description of the root system
A of a Kac-Moody algebra g(A). OUf main instrument is the notion of an
imaginary root, which has no counterpart in the finite-dimensional theory.
As an application, we derive the structure of the automorphism groups of
integral unimodular Lorentzian lattices of dimension  10.
SS.1. A root Q E A is called real if there exists W E W such that w( a)
is a simple root. Denote by ,.e and A+ e the sets of aJ) real and positive
real roots respectively.
Let Q E re; then Q = w(Qj) for some Qi E n, w E W. Define the dual
(real) root a V E d Vre by QV = weer,). This is independent of the choice
of the presentation Q = w(a;). Indeed, we have to show that the equality
U(Qi) = OJ implies u(or) = oj for u E W. But this is (3.10.3). Thus
we have a canonical W-equivariant bijection ,.e -+ A v,... By an easy
induction on ht a, one shows, using Proposition 5.1 e) below, that a > 0 jf
and only if QV > o.
We define a reflection ro with respect to Q E A,re by
ra() =  - (, QV)Q,  E ..
Since (a, QV) = 2, this is a reflection, and since ro = wriw-1 if Q == w(O'j),
it lies in W. Note that r Oi is the fundamental reflection ri.
The following proposition shows that real roots have all the "classical"
properties.
PROPOSITION 5.1. Let Q be a real root of a Kac-Moody algebra g(A).
Then
a) rou It Q = 1.
b) kQ is a root jf and only jf k = :t:I.
c) If {J E  then there exist nonnegative integers p and q related by the
equation
p _ q = (P, Q V) ,
such that fJ + ka E L\ U {OJ if and only if -p 5 I:  q, k E Z.
59

60
Redl dntl Imdgindrg ROOt8
Ch.5
d) Suppose that A is symmetrizable and let (.,.) be a standard invariant
biJjnear form on g(A). Then
(i) (Qler) > 0;
(ii) QV = 2v- 1 (a)/(QIQ).
(Hi) if Q = E kiai, then ki(ailai) e (ala)Z.
i
e) Provided that :!:()  n, there exists i such that
Ihtr.(a)1 < Ihtal.
Proof All the statements a)-d) are clear if Q is a simple root: a) holds by
definition, b) holds due to (1.3.3), c) follows from Proposition 3.6b), and
d) is (2.3.3 and 5). Now a), b), and c) follow from Proposition 3.7b), while
d) (i) and (ii) (oHow (rom Proposition 3.9. Statement d) (Hi) (ol)ows from
the fact that Q v e E laj and the following formula:
(5 1 1) v -  (aiIQi) L v
· · Q -  (010) iO. ·
.
Fina1Jy, suppose the contrary to e); we may assume that Q > O. But
then -() E C V and, by Proposition 3.12 d) for the dual root system,
-Q + w(Q)  0 for any w E W. Taking w such that w(Q) E fit we
arrive at a contradiction.
o
The following lemma will be needed in the sequel.
LEMMA 5.1. Suppose that A is symmetrjzable. Then the set of all Q =
E kiQi e Q such that
i
(5.1.2)
ki(Qilai) E (oIQ)Z (or all i
is W -invariant.
Proof. It suffices to check that ri(a) again satisfies (5.1.2),
(k, - (QI()r»(OiIQi) E (QIQ)Z. This is equivalent to
.
I.e. ,
that
(5.1.3)
2(Ql o i) E (olQ)l,
which follows from (5.1.2):
",2(Q.IQ.)
2(0'10;) = L..J ( .J 1 .) kj(ojloj) = LQjikj(ojloj) E (olo)Z.
j Ct J a J j
o

Cb.5
Real and ImaginG,., Root
61
Let A be a symmetrizable generalized Cartan matrix, and let (.1.) be a
standard invariant bilinear form (see 52.3). Then, given a real root a we
have (ala) = (QiIQi) Cor some simple root lri. We call Q a ,laort (resp.
long) real root if «()I) = mioi(o.loi) (reap. = maxi(Qila.». These are
independent of the choice of a standard form. Note that jf A is symmetric,
then all simple roots and hence all real roots have the same square length.
If A is not symmetric and S(A) is equipped with m arrows pointing in the.
same direction, then there are simple roots of exactly m + 1 different square
lengths since an arrow in SeA) points to a shorter simple root. It follows
that if A is a nODsymmetric matrix from Table Fin, then every root is either
short or long. Furthermore, iC A is a nonsyrmnetric matrix from Table Aff,
and A is not of type A) with I > I, then every real root is either short
or longj for the type A) with I > 1 there are real roots of three different
lengths.
In this chapter, we shall normalize (.1.) such that the (Qi(Qi) are rela-
tive)y prime positive integers for each connected component of S(A). For
example, if A is symmetric, then (oilai) = 1 for all i, and all real roots are
short (= long).
S5.2. A root Q which is not real is called an imaginary root Denote by
Aim and m the sets of imaginary and positive imaginary roots, respec-
tively. By definition,
A = re U im (disjoint union).
It is also clear that Aim = dm U (_m).
The following properties of imaginary roots are usefu).
PROPOSITION 5.2. a) The set m is W-invariant.
b) For Q E dm there exists a unique root {J E -Gv (i.e., (fJ,or)  0 for
all i) which is W-equivalent to Q.
c) If A is symmetrizable and (.1.) is a standard invariant biljnear form, then
a root a is imaginary if and only jf (QIQ) S o.
Proof As m C Ll+ \n and the set d+ \ {Qi} is r;-invariant (by Lemma
1.3), it Collows that m is ri-invariant for all i and hence Winvariant,
proving a). Let Q E Am and let P be an element of minimal height in
W · Q C L\+. Then -{J E CV. Such a P is unique in the orbit W . Q
by Proposition 3.12 b), proving b). Let Q be an imaginary root; we may
assume, by b), that -Q E C V (since (.1.) is W-invariant). Let Q = L:ki Q ;,
i

62
Real antllmagintJrr RooCl
Ch. &
kt  0; then (0)0) = E i.(Qlo,) = E lIQiI2.(Q, or> s 0 by (2.3.3 and 5).
i i
The converse holds by Proposition 5.1 d). 0
55.3. For Q = E iiOi e Q we define the ,uppor1 of Q (written 8UPP 0)
i
to be the 8ubdiagram of SeA) which consists or the vertices i such that
k i  O. and of all the edges joining these vertices. By Lemma 1.6, 8UPP 0
is connected for every root Q. Set:
K = {o E Q+\{O} I (o.or) $ 0 for all i and suppa is connected}.
LEMM.A 5.3. In the above notation, K C Am.
Proof. Let 0 = E iiOi E K. Set
i
0 0 = {" E A+ 11 So}.
The set 00' is finite, and it is nonempty because the simple roots, which
appear in the decomposition of Q, lie in 0 0 . Let (J = E miOi be an element
i
of maximal height in Ga. It follows from Corollary 3.6 a) that
(5.3.1)
supp fJ = 8UPP Q.
First, we prove that Q E +. Suppose the contrary; then Q :F {J. By
definition:
(5.3.2)
fJ + Qi  + if i. > m;.
Let Al be the principal 8ubmatrix of A corresponding to 8UPP Q. If At is
of finite type, then {Q e Q+ I (a,or) s 0 for all i} = to} and there is
nothing to prove. If Al is not or finite type, then, by Proposition 4.9, we
have
(5.3.3)
P:= {j e supp, k j = mj}  e.
Let R be a connected component of the subdiagram (suppa)\P.
From (5.3.2) and Corollary 3.6 a) we deduce that
(5.3.4) (fJ,or)  0 if i E R.
Set /1' = E miQi. Since 8UPPQ is connected, (5.3.1) and (5.3.4) imply
iER
(5.3.5) (p',or>  0 if i e R; (fJ',O}) > 0 for some j E R.
Therefore, by Theorem 4.3, the diagram R is of finite type.

Ch.5
ReGI Gnd/maginG,., Root,
63
On the other hand, set
0' = l)k, - m,)oi ·
fER
Since 8UPP Q' is a connected component of 8UPP(O - fJ), we obtain that
(5.3.6)
(Q',Qr) = (Q-J1,or> for i E R.
But (Q, Qr)  0 since Q e K, hence by (5.3.4 and 6):
(Q', Qr>  0 for i E R.
This contradicts the fact that R is of finite type. Thus, we have proved
that Q E L\+.
But 2a also satisfies aJ) the hypotbeses of the lemma; hence
2a E + and, by Proposition 5.1 b), Q E m.
o
S5.4. Lemma 5.3 yields the following description of the set of imaginary
roots.
THEOREM 5.4. m = U w(K).
wEW
Proof. Lemma 5,3 and Proposition 5.2 a) prove the inclusion :>. The re-
verse inclusion holds by Proposition 5.2 b) and by the fact that 8UPP Q is
connected for every root Q (by Lemma 1.6).
o
f5.5. The fonowing proposition shows that the properties of imaginary
roots differ drastically from those of real roots.
PROPOSITION 5.5. IE Q E dm and r is a nonzero (rational) number such
that rQ E Q, then ro E A. 1m. In particular, no E A. 1m if n E Z\ {O}.
Proof By Proposition 5.2 b) we can assume that Q E -Gv n Q+; since
Q E A, it follows that 8UPP Q is connected and hence Q E K. Hence ra E K
for r > 0 and therefore, by Lemma 5.3, ro E Aim.
o

64
RtJl Gntl ImaginG'" Roo"
Ch.6
S5.6. Now we prove the existence theorem for imaginary roota.
THEOREM 5.6. Let A be an indecomposable generalized Cartan matrix.
a) If A ;s of finite type, then the set Aim is empty.
b) If A is of affine type, then
J.1m = {n6 (n = 1, 2, . . . )},
l
where 6 = E a,crl, and the Gi are the labels of S(A) in Table Air.
t=O
c) If A is of indefinite type, then there exists a positive imaginary root
o = 1: ki O i such that ki > 0 and (0, Qr) < 0 (or all i = 1, .. . , R.
i
Proof. Recall (see Chapter 4) that the set {o E Q+ I (0, or) s 0, i =
1, . . . , n} is zero for A of finite type, is equal to Z6 Cor A of affine type, and
there exists 0 = E kiQi such that ki > 0 and (Q, or) < 0 for all i if A is of
i
indefinite type. The theorem now follows from Theorem 5.4.
o
S5.7. It follows from Proposition 4.7 that if 0 is a null-root, i.e., Q e
(A) is such that ol' = 0, then 8UPP Q is a diagram of affine type which
is a connected component of S(A) and 0 = k6. Ie E Z. We proceed to
describe the isotropic roots.
PROPOSITION 5.7. Lei A 6e ,ymme,riztJ6/e. A roo' Q iI ilo'ropic (i.e.,
(010) = 0) il and only il it il W -equivalent '0 tJn imagintJ,., roo' {J .ucA
tlaat 8UPP {J ;$ CI lu6cliagram of Gffin typ of S(A) (then {J = 1;6).
Proof. Let 0 be an isotropic root and let (.1.) be a standard bilinear form.
We can assume that Q > O. Then Q E dm by Proposition 5.1 d), and
Q is W-equivalent to an imaginary root {J e K such that (P, Qr) s 0 for
all i, by Proposition 5.2 b). Let P = E ki()i and P = 8UPP 13. Then
iEP
(,BI) = E ki(PI()i) = 0, where k i > 0 and (PIQi) = tI Q iI 2 (p,Qr> s 0 Cor
ieP
i E P. Therefore (IJ, or) = 0 for all i e P, and P is & diagram of affine
type. Conversely, if P = k6 is an imaginary root for a diagram or affine
type, then (PIP) = k 2 ( 616) = k 2 E ai(6)Qi) = 0, since (6, Qr> = 0 Cor all i.
i
o

Ch.5
Ral Gratl Imagina,., Root,
65
S5.8. Now we give a description or the Tits cone X in terms of imaginary
roots.
PROPOSITION 5.8. a) It A is o{fjnite type, then X = ..
b) 1£ A is of affine type, then
X = {h E . J (6, h) > O} URv- 1 (6).
c) If A is of indefinite type, then
(5.8.1)
x = {h E . I (a,h)  0 (or all a E m},
where X denotes the closure of X in the metric topology of ..
Proof a) holds by Proposition 3.12 e). If A is of affine type, then ,.e+k6 =
,.t for some k (see Proposition 6.3 d) from Chapter 6). Using PropaJition
3.12 c), we deduce immediately that
{h E .I (6, h) > O} C X and {h E .I (6, h) < OJ nx = 0.
If (cS,h) = 0 and h  RII-l(6)) then (0;, h) < 0 for some i, completing
the proof of b).
In order to prove c), denote by X' the right-hand side of (5.8.1). By
Proposition 5.2 a), X' is W-invariant; also it is obvious that X' ::> C. Hence
X'::>X.
To prove the converse inclusion it is sufficient to show that for h E X'
such that (ai, h) E Z (i = 1, ..., n) there exists only a finite number of
positive real roots r such that (1, h) < 0 (see Proposition 3.12 c». Recall
that by Theorem 5.6 c) there exists fJ e m such that ({J,o,/) < 0 for all
i. If,., E +e t then r.,(fJ) = {J+Si E A, where 8 = -({3, 1 V )  ht 1 v .
As h E X', we have (f3+B7 t h)  o. Hence there is only a finite number of
real roots '1 such that (1) h) S -1, which is the same 88 (1. h) < o.
o
Proposition 5.8 has a nice geometric interpretation. Define the imagi-
nary cone Z as the convex hull in . of the set Am U {OJ. Then the cones
Z and X are dual to each other:
x = {h E . I (0, h)  0 for all Q e Z }.
In particular, Z is a convex cone (c{. Proposition 3.12 c). Note also that
Z C -Xv. Exercises 5.10 e) and 5.12 give another description of the cone
z.

66
ReG' tJntl ImaginG,., Roo"
Ch. &
In the next subsection we will need
LEMMA 5.8. The limit rays (in metric topology) {or the set of rays {R+lr J
Q E +e} lie in z .
Proof. We can assume that the Cartan matrix A is indecompOlable. In the
finite type case there is nothing to prove since IdJ < 00 by Proposition 4.9.
In the affine case the result follows (rom the description o( 4 in Chapter
6. In the indefinite case we choose {J e dm such that (P, or> < 0 for
all i (see Theorem 5.6 c». Then (P,OV) S - ht QV (or Q E 4+' .and
ra({J) = {J + ko E 4m t where k  1, proving the lemma in this case alIo.
o
S.9. A linearly independent set of roots 0' = {Q, Q, . . .} ia called a
root 6tJi of L\ if each root 0 can be written in the form Q = :i: E iiO J
i
where k i E Z+.
PROPOSITION 5.9. Let A be an indecomposable generalized Cartan ma-
trix. Then any root basjs 0 ' of A is W-conjugate to n or to -D.
Proof. Set Q = L Z+oi. By Theorem 5.4, the set of rays through a e
i
6.m is dense in Z, which is convex. It follows that 6.m lies in Q'+ or -Q'+ I
and changing the sign if necessary we can assume that l:1m C Q. It follows
by Lemma 5.8 that the set Ll+ n (-Q) is finite. If this set is nonempty,
it contains a simple root Oi. But then I+ n (-ri(Q)) t < ILl+ n (-Q)I.
After a finite number of such steps we get Ll+ c w(Q) for some w E W
and hence II = w(II').
o
Remark 5.9. n is W-conjugate to -n if and only if A is or finite type.
S5.10. A generalized Cartan matrix A is called a matrix of Ayper60/ic
type if it is indecomposable synunetrizable of indefinite type, and if every
proper connected subdiagram of SeA) is of finite or affine type.
Note that if A is symmetrizable. then a standard invariant bilinear
form (.'.) can be normalized such that (QiIQj) are integers. Hence CJ =
min IQ)2 exists and is a positive number.
oEQ:loI3>O
LEMMA 5.10. Let A be a generalized Cartan matrix of finite, affine or
hyperbolic type, and let Q = E kjo j E Q.
j

Ch.5
Real Gnd Imaginary Roo',
67
a) If 101 2 < a, then either 0' or -0 lies in Q+.
b) 1£ Q satisfies (5.1.2), then either Q or -0 lies in Q+.
Proof. Suppose the contrary to a); then 0 = fJ - 1, where {J, 1 E Q+ \{O},
and the supports PI and P2 of (J and 1 have no conunon vertices. Then
(5.10.1)
a  IQI 2 = 1.81 2 + 111 2 + 2( -PI,,).
There are two possibilities: (i) both PI and P2 are of finite type; (ii) PI is
of finite type, P2 is of affine type, and they are joined by an edge in S(A).
In case (i) we have IPI 2  at 111 2  at and (-PI,)  0, which C"ontradicts
(5.10.1). In case (ii) we have IPI 2 > at 111 2 > 0, and (-PI1) > 0, which
again contradicts (5.10.1). The proof of b) is exactly the same using the
observation that 1131 2 ,111 2 e Zlal 2 , since:
1,01 2 = L klail2 + L k i k j 2(ail a j)
i i<j
= L k i (k i jad 2 ) + L a ij k;(k i laiI 2 ) E Zlal 2 .
i i<j
o
PROPOSITION 5.10. Let A be a generalized Cartan matrix or finite, affine,
or hyperbolic type. Then
a) The set of all short real roots is
{o E Q 11012 = a = m.in IOif2}.
,
b) The set or all real roots is
{a:: LkjO'j E Q 110'1 2 > 0 and kjlO'jl2/1QI2 E Z {or all j}.
j
c) The set of all imaginary roots is
{Q E Q\{O} 11 0 1 2 SO}.
d) If A is affine, then there exist roots of intermediate squared length m
j{ and only j{ A = A), I > 2. The set o{ such roots coincides with
{o E Q f 101 2 = m}.
Proof Let Q e Q be such that IQI 2 = o. Then Iw(o)12 = a for w E Wand
hence, by Lemma 5.10 a), w(Q) E Q+ (or every w E W. Without loss of

68
ReGI and Imagindry Roou
Ch.5
generality we may assume that Q E Q+. Let {J be an element of minimal
height among (W .a)nQ+. As (PIP) = a > 0, we have (11Ioi) > 0 for some
i. If now (J'f. Qi, then ri({J) E Q+ and ht(ri(,B) < ht(,8), a contradiction
with the choice of {1. This shows that Q is a real root, proving a). The
proof of d) is similar.
By Proposition 5.1 b) (i) and (iii), l1 r . lies in the set given by b). The
proof of the reverse inclusion is the same 88 that of a), using Lemmas 5.1
and 5.10 b).
Let now Q E Q\{O} and IQI 2 S o. By Lenuna 5.10 we may assume that
o E Q +. The same argument as above showl that W · Q C Q +. As above.
we choose an element fJ E W. Q of minimal height. Then (J1, Qr> S 0
for all i. Furthermore, 8UPP P is connected, since otherwise (J = 11 + 12,
where supp 11 and 8UPP 12 are disjoint unions of finite type diagrams which
are disjoint, and, moreover, are not connected by an edge in S(A); then
1111 2 = 1111 2 + 1121 2 > 0, a contradiction. So, (J E K and therefore /3 E Aim
by Lemma 5.3; hence Q E Aim. Now c) follows by Proposition 5.2 c).
o
Note that Propositions 5.10 c) and 5.8 c) give the following explicit
description of the closure of the Tits cone in the hyperbolic case:
(5.10.2)
xu -X = {h E . J (hJh)  OJ.
We obtain also the following corollary of Propositions 5.9 and 5.10. Note
that an automorphism (f of the Dynkin diagram S(A), induces an automor-
phism of the root lattice Q by D'(Oi) = Q(i); denote the group of all such
automorphisms by Aut(A). Another subgroup or Aut Q is the Weyl group
W. Note that CTri tT - J = r C1 (i) and that W n Aut(A) = 1 by Lemma 3.11
b). Thus Aut Q J Aut(A) K W.
COR.OLLARY 5.10. a) It A is indecomposable, then the group of all auto-
morphisms o£Q preserving A is:l: Aut(A) K W.
b) If A is of a symmetric matrix of Iinite, a/fine, or hyperbolic type, then
the group of all automorphisms ofQ preserving (.1.) is % Aut(A)  W.
Proof. If iT E Aut Q and f1(l1) C , then 0' := cr(n) is a root basis or .
By Proposition 5.9, %w(D') = n for some w e W, proving a). b) followa
from a) and Proposition 5.10 a) and c).
o

Ch.5
ReGI anti ImGginG,., Roo',
69
Remark 5.10. 8) Suppose that in Corollary 5.10. A is not symmetric' but
the square length of one of the simple roots, say Q1. is 1 and that of all
Qi  Ql is 2. Then
d re = {a E Q I (aJa) = 1 or 2}.

Indeed, by Proposition 5.10 b) the only additional condition. occurs if
(of a) = 2, and in this case i 1 E 2Z; but (Qla) == ifmod2, hence (ola) = 2
implies that k 1 e 2Z. Therefore, by Corollary 5.10 a), the conclusion of
Corollary 5.10 b) still holds.
b) By Remark 5.9, if A is o(finite type, one tan drop:!: from the (onclusion
of CoroUary 5.10 b).
55.11. In this section we apply the results of 55.10 to the study of the
standard Lorentzian lattice An (n > 3) with the bilinear form (.1.) which
in some l-basis 110, Vi,".' V n -l is given by
(5.11.1)
n-l
I  1 2 - 2 2 2
L.J%iVi --%O+%I+...+ Z n-l.
1=0
(This is tbe only odd unimodular integral n-dimensional Lorentzian lattice.)
First, we choose another basis of An:
01 == V1 - V2,..., On-2 = V n -2 - tin-I, 0"-1 = V n -l,
Qra = -vo - VI - "2 - 113 (resp. = -Va - VI - 112)
if n  4 (resp. n = 3).
This shows that. An is the root lattice corresponding to the generalized
Cartan matrix (2(Qil o J)/(QiJO;» with the following Dynkin diagram BE,.:
f'.
0-0-0-...-0  0 if n5,
aa a:a OJ Qra-3 0._ I
0-0::>0<:=0 and ooo.
01 O:a 03 a. a, Q2 Q3
Note that in the basis {Vi}, the fundamental reflection ri (or i  n - 2 is
the transposition of Vi and Vi+l, rn-l is the sign change of V n -l, and r n is
represented by the following matrix:
2 1 1 1
-I 0 -1 -1
R,. = -1 -1 0-1
-1 -1 -1 0
if n  4t and  = ( - - - ) .
-2 -2 -1
In-4

70
Real antllmGginary Roou
Ch.5
o
Denoting by W n the group of all permutations of Vi, . · . ,V,,-1 with simul-
taneous arbitrary sign changes, we have thus obtained that the Weyl group
o
W n of type B En is generated by W nand Rn. The fundamental chamber
C C Rn = R 01 An or the group W n is given by the following inequalities:
k 1  k 2 > .. ·  k n - 1  0,
ko  k 1 + 1- 2 + k3
(k 3 = 0 if n = 3).
n--l
Note that C C L:= {Q = E Zitli I Zo  OJ-z+zl + '.'+Z_1  OJ,
1=0
the upper half of the light cone. 'rhus, the dual Tits cone X lies in L:.
Note that W n cot. := {g E GLn(R) I 9 · L: = L:. 9 · An = An}.
Now, B En is hyperbolic if and only if n  10. It follows from 55.10 that,
for 3 < n < 10 we have:
- -
(5.11.2)
( 5 .11.3 )
Aim = {Q E An I (QIQ)  O}\{O},
d re = {Q E An I (oIQ) = 1 or 2}.
(IC n  4, (5.11.3) follows (rom Remark 5.10. For n = 3, due to Proposi-
tion 5.10 b), aU 0 E A3 with (trJo) = 1 are roots, but the
Q = k 1 Q l + k 2 Q 2 + k 3 Q 3 E A3 with (QJ) = 2 should satisfy the addi-
tional condition k 2 , k3 e 2Z; however, this easily follows from (QIB) =
2k 1 (k 1 - k 2 ) + (k 2 - k3)2 = 2.) Thus:
(5.11.4)
L\ = {a E An I (QIQ)  2}\{O}.
It follows from Coronary 5.10 a) that
(5.11.5 )
W n = 0A" for 3 S n  10.
COROLLARY 5.11. Let 3 $ n  10. Then
a) All integral solutions of the equation
-z + z + ... + z = 1 (resp. = 2)
(orm a Wn-orbit of the vector V1 jf n > 3 and a union of Wn-orbita of VI
and Vo + VI + V2 jf n = 3 (resp. (orm a union of Wn-orbits of VI + "2 and
Va + VI + 112 + Va ifn > 3 and aWn-orbit O{VI + "2 ifn = 3).
b) All integral solutions of &he equation
2 2 2 0
-Zo + Zl + · · · + Zn =

Ch.5
ReGI Grad 1mtlgina,., Roo',
71
form a :l:Wn-orbit of Vo + VI jf n  9 and a union of two :l:Wn-orbjts of
Vo + V1 ancl3vo + VI + tJ2 + · · · + V9 jf n = 10.
o
We conclude this section with the folJowing general construction of by-
.
perbolic lattices. Let Q be the root lattice corresponding to a generalized
Cartan matrix of finite type Xl' Let Ql, . . . ,Qt be simple roots and 9 be the
highest root (see Remark 4.9). We normalize the standard invariant form on
o
Q by the condition (819) = 2. Let H 2 be the 2-djmensiona) hyperbolic Jat-
tice with basis UI,U2 and bilinear form (ullu2) = 1, (ullul) = (u2Iu2) = o.
o
Let Q be the orthogonal direct sum of lattices Q and H 2 . Then Q is the
root lattice corresponding to the Dynkin diagram xl' obtained from XJ1)
.
by adding an addition vertex a_I joined with the vertex 00 by a simple
edge. This follows from the following choice of basis of Q:
Q - 1 = U 1 + U2,
0'0 = -Ul - B,
01 t . · . , 0t.
The lattice Q (resp. diagram xfI> is called the canonical hyptr601ic ezlen-
o
"ion of the lattice Q (resp. diagram X,). Note that
(5.11.6) detXl' == -detXi.
Note that if X = A, B, C, or D. then xl' = X Et+2 (cC. Exercise 4.3), and
that Ef = T 4 . 3 ,3, E? = TS.4.2' Ef = E 10 = T7.3.2 (cf. Exercise 4.2).
The most interesting example is Ef = EIO. The corresponding root
lattice Q is the (only) even integra] unimodular Lorentzian to-dimensional
lattice. Since E JO is of hyperbolic type, we can apply the results of 55.10.
Thus, the Weyl group of E 10 coincides with the group of automorpbisffiS
of the root lattice Q preserving the upper half of the light cone, A =
{a E Q I (oIQ) $ 2}\{O}, etc.
S5.12. In conclusion, let us make one useful observation. Recall that
g(A) = g(A)/r, where r is the maxima) ideaJ intersecting  trivially. How-
ever all the proofs in Chapters 3, 4, and 5 used only the fact that
(5.12.1)
(adei)Nijej :: 0 = (ad/i)Nii Ii for all i  j and some Nij.
In other words, we have the following:
PROPOSITION 5.12. Let 9 be a quotient algebra or the Lie algebra i(A)
by a nontrivial Q-graded ideal such that (5.12.1) holds. Then all the state-
ments or Chapters 3, 4. and 5 (or g(A) hold {or the Lie algebra 9 as well.

72
RttJl anti Imaginary Roo',
Ch.5
We deduce the following:
COROLLARY 5.12. Let 9 be as in Proposition 5.12. Then
a) The root system of g is the same as that of g(A), the multjplicities ot
real roots being equal to 1.
b) If A is of finite type, then 8 = g(A).
Proof. a) follows from the proofs of Proposition 5.1 a) and Theorem 5.4,
while b) follows Crom a).
o
Remark 5.1e. We shall see in Chapter 9 that Corollary 5.12 b) holds for
an arbitrary symmetrizabJe generaJized Caftan matrix.
55.13. Exercises.
5.1. Show that for Q E 4,.e{A) one has:
[go, 8-0] = CoY.
Show that if A is a nonsymmetrizable 3 x 3 matrix and Q = Ql + a2 + Qa,
then Q E im(A) and dim[Sa, "_a] > 1 (cf. Theorem 2.2 e».
5.1. Show that dim[80.9-0] = 1 for all Q E 6.(A) if and only if A is
symmetrizable.
5.3. If dim g(A) = 00, then IArel = 00.
5.4. The set +(A) is uniquely defined by the properties (i) and (ii) of
Exercise 3.5 and the following property
(iii)' if Q E +, then 8UPP Q is connected.
[Let + satisfy (i) t (ii)t and (iii)'. Then d \ {Qi) is r,-invariant, hence
t\+. C 6. If now Q E A \+e, then W(o) C + and fJ of minimal
height from W( Q) lies in K.]
5.5. Let A = «(Jij) be a syrmnetric generalized Cartan matrix. Show that
{Q e Q\{O} I (oIQ) S I} :J A.
Show that the converse inclusion holds if and only if A is or finite, affine,
or hyperbolic type.

Ch.5
Real and ImaginG'" Root,
73
5.6. Let A = ((Jij) be a finite, affine, or hyperbolic matrix, let B = (b ij )
be the corresponding "symmetrized" matrix, and let B(z) = E6 ij z;%j be
the associated quadratic form. Show that all the integral solutions of the
equation B(z) = 0 are of the form Bw(6), where, E Zt w E W(A), and
6 is the indivisible imaginary root of (A')t where A' is a principal affine
8ubmatrix of A. (In particular, any solution is 0 if A is strictly hyperbolic.)
Show that if A is symmetric, then all the integral solutions of the equation
E aijZiZj = 2 are of tbe form w(ai), where w E W, Qi E n.
5. 7. Show that the sublattice of the lattice An consisting of vectors with
even sum of coordinates is isomorphic to the root lattice of type D!!_2.
5.8. Show that the Weyt group W is a subgroup of index 3 in the group of
automorphisms preserving (.J.) of the root lattice of type C 4 .
5.9. Let Q e '+ be such that -Q E CV and (0, or) -F 0 for some i E
SUPPQ. Then the subdjagram {i E SUPPQ I (a, a;') = O} C S(A) is a union
of diagrams of finite type.
[Denote by T a connected component of tbis subdiagram and let Q =
E kiQi; set P = E /riOi. Then (P,o:,/) > 0 Cor all i and> 0 Cor some
i iET
i E T.]
In Exercises 5.10-5.15 we assume A to be indecomposable.
5.10. An imaginary root a is called ,trietly imogintJrg if for every -, E 11'"
either Q+r or 0-1 is a root. Denote by d,im tbe set of strictly imaginary
roots.
a) If Q e 6+ and (0, Qr> < 0 (i = 1,..., n), then Q E d,i",. Deduce that
if Q E dim t r(Q) :F Q for all -y E Are t then 0 E d,im.
b) If Q e 4na and (ct, Qr) :s 0 for i = 1,..., n, then Q + {J e &+ for every
(J e d+.
c) If Q E m, fJ E J1m, then Q + fJ E m.
d) dm is a se migroup.
e) Z = R+A  m.
5.11. Let Ai denote the linear function on QR :== R z Q defined by
(Ar. a:j) = b ij (j = 1,." I n). Then Z = {a: E QR n - X / (w(A), a:) > 0
for all w E Wand those s = 1,... t n for which the principal submatrix
(aij )i.j is of indefinite type}.
5.12. Z is the convex hull of the set oClimit points for R+Ai-'.

74
Real GhtllmGginary Root,
Ch.5
5.13. If A is a matrix of finite or affine type, fJ e d, Q e r., then the
string {P + Ita, Ie E Z} contains at most five roots. Show that if A is of
indefinite type, then the number of roots in a string can be arbitrarily large.
5.14. Given fJ E d and Q e r., the string {t1 + ia} contains at most (our
real roots.
5.15. Let A be symmetrizable of indefinite type. Then the (ollowing con-
ditions are equivalent:
(i) A is of hyperbolic type;
(ii) Xu -X = {h e . I (hlh) SO};
(iii) Z = -Xv;
(iv) L\im == {Q e Q\{O} I (QIQ) SO}.
5.16. Let A be symmetrizable and let (.1.) be a standard form. Let 0,
(J e Am. Then (QIP) S O.
[One can assume that -Q E C V .)
5.17. Under the hypotheses of Exercise 5.16 assume that (alP) < O. Then
a + {J E Am.
[Since the cone Xv is convex, we can assume that -( cr + /1) E C V . But
8UPp Q and 8UPP.a are onnected, and since (alP) < 0, supp(o + (J) iI
connected and we can apply Lemma 5.3.]
5.18. Under the hypotheses of Exercise 5.16, assume that Q+{J e Am and
that () and p are not proportional isotropic roots. Then (QIIJ) < o.
[Use Exercise 5.9.]
5.19. Let a, {J E +e be such that (Q,fJ V )  o. Show that (P,OV)  O.
Furthermore, show that (Z+Q + l+/J) n d = {Q,{J,Q + P} n +'.
[To show that (fJ, Q V)  0, one can 88ssume that Q is a simple root; then
(0, (JV) > 0 (resp. = 0) if and only if r() < 0 (resp. = Q), which is
equivalent to r(QV) < 0 (reap. = QV). Furthermore, if mQ + np E A for
some m, n E Z+, then (mo + n{J,{JV)  2n, hence mo e d and m  Ii
similarly, n < 1; in particular, 2( Q + P) is not a root.]
5.fO. Let a = E k.Q. E K and let fJ = E milt.. E Q+ be such that
i i
o - fJ E Q+. Using the identity
 1 1  m. m.
(a - PIP) = £.Jmj(kj - mj)ki (alaj) + 2 £.J6ij(-f - -f>2kikjl
J 'J · J
show that (Q - PIP) < 0 unless (010) = 0 and {J is proportional to Q.

Ch.5
Real and Imaginary Roo'
75
The rest of the exercises deal with Kac-Moody algebras g(A) of rank 2,
.
I.e. ,
A = (b a), where a, b are positive integers, and a > b.
We associate to g(A) the field F = Q( y'0 6(06 - 4» ; when ab ¥ 4 we
denote by   '\' the unique nontrivial involution of F. Fix the following
symmetrization of A:
( 2 -0 )
B = -a 2 a lb
and the corresponding standard form (.1.) on ., 80 that (allOt) = 2,
(021 0 2) = 2a/6, (QlI Q 2) = -0. Introduce the following numbers:
-ab + y'o b(ah - 4)
'1 = 26 ; { = -6'1 - 1;
(0 = t if a  6 and = 'I if a = b.
5.el. Assume that ab  4. Show that in the basis 01, Q2 of ., one has
C V = {(z,y) 12z  ay, 2y > 6z}.
Show that if ab > 4, then
-v
X = {( z, y) I - fI'lI S z < - '1y } ·
5.2D. Show that land l' are eigenvalues of rl r2, and that
k fk - fo'k fk-l - f,k-l
(TIT2) = J TIT2 - I I, k E Z.
f-f f-f
[Use the fact that for a 2 x 2 matrix a with eigenvalues 1 and 2. one has
a = t - t a _ ,\t2 - '\1 I .]
1- 2 1- 2
5.:J. Assume that ab f:. 4. Let xy' + x'y be the trace form of the field F.
Show that the map
t/J : k 1 0 J + k 2 0 2 1-+ k 1 + k 2 '1
is an isometry of the lattices Q and Z['l] C F, 80 that the fundamental
reflections rl and r2 of Q induce automorphisms of the lattice l[1J]:
rl() = -',
r2() = -l',

16
RetJI tJntllmaginary Roou
Cb.5
and the group from Corollary 5.10 a) maps isomorphically onto the group
generated by multiplication by the unit EO, the involution' and -1. Show
that
f/>{Q+) = {z + lI v'o b(06 - 4) e I[']] J JJ  0 and %  0 if J/ = OJ,
<p(e) = {f"'1,t- n ,-l-"'1',_(n+l, where n  O},
;(im) = {z E Z[F1] J zz'  0, z  O}.
5.24 . Show tllat in the case when a = b = m, m :1= 2, we have 1F =
Q( v m 2 - 4), 11 = fO, E = E; if in addition m 2 - 4 is squ are fre e except
possibly for a factor of 4 when m is even. then Z + 1(1 + v m  - 4)Z is tbe
ring of integers of the field F. Ded uce that under the above hypotheses
on m, the number i(m + vm 2 - 4) is a fundamental unit of the ring of
integers of the field Q( vm 2 - 4), i.e., (0 together with -1 generate the
group of all integers of norm I.
5.5. Show that
+e(A) = {Cj Q l + d j + 1 Q2 and Cj+l 0 1 + d j Q 2 (j E Z+)},
where the sequences Cj and d j are defined by the following recurrent for-
mulas for j > 0:
Cj+2 + Cj = ad j + 1
dj+2 + dj = bCj+l
and Co = do = O. Cl = d 1 = 1.
5.26. Show that all integral solutions (z,y) with relatively prime z and y
of the equation
6z2 - a6zy + ay2 = a or "
are (Cjtdj+l) with j odd or :J:(cj+!,dJ') with j event j  O. Show that
more solutions exist only if the right-hand side is a and a/b = r 2 t where r
is an integer> 1 t and they are (rz, ry). where bz 2 - a6zy + ay2 = 6.
5.27. Show that
t1"/ (:2 2) = {j Q l + (j + 1)Q2 and (j + 1)01 + j 0 2 (j E Z+)}.
t1,+ ( :1 4) = {2j o l + (j + 1)02 and (j + 1)01 + lj 0 2
for even j E Z+i jOt + lU + I)Q2 and 2(j + 1)01 + jCt2 for odd j E Z+}.

Ch.5
Real and ImaginG,., Roo',
77
5.18. Show that
+ (!33) = {2;al+ 2H2a2
and 4J2j+2 a l + tlJ2j Q 2 (j E Z+)}.
where q,j is the jth Fibonacci number:
</Jo = 0,
4Jl = 1,
t/Jj+2 = ;J+l +,pj for j E Z+.
S5.14. Bibliographical notes and comments.
The notion of real and imaginary roots were introduced in Kac [1968 B),
where Propositions 5.2 and 5.5 and Theorem 5.6 were proved. (Moody
[1968] introduced, independently, real roots; he called them Weyl roots.) ..
Lemma 5.3, Theorem 5.4, and Proposition 5.7 are proved in Kac [1980
A]; the exposition of SS5.1-5.7 and S5.10 is taken (rom this paper. The
strengthening of Proposition 5.10 b) as compared to previous editions, is
due to Mark Kruelle.
Proposition 5.10 c) was obtained by Moody [1979], where he initiated
a detailed study of hyperbolic root systems. The material of S5.8 is taken
from Kac-Peterson [1984 A). Proposition 5.9 is proved in Kac [1978 A).
Most of the results of 55.11 are due to Vinberg [1972], [1975], who devel-
oped a general algorithm allowing to compute the subgroup W generated by
reflections in the automorphism group of a Lorentzian lattice. In particular,
he showed that W has finite index in Aut An if and only if n < 20 (Vin-
berg (1972], [1975], Vinberg-Kaplinskaya [1978]). Recall that H2fl)nE s are
the only even unimodular lattices (of dimension 8n + 2) (see Serre [1970]).
For n = 1 and 2, their automorphism groups were computed by Vinberg
[1975]; and for n = 3, by Conway [1983]. See Vinberg [1985] for a survey
of hyperbolic reflection groups.
As shown in Kac [1980 A], given a symmetric generalized Cartan ma-
trix A, the set of positive roots d+ (A) describes the set of dimensions of
indecomposable representations of the graph SeA), equipped with some
orientation. Moreover, the number of absolutely indecomposable repre-
sentations of dimension Q E +(A) over a finite field Ff is given by a
polynomial qN + OtqN-l + ... + ON. where Gi E Z. N = 1 - (ola) (see
Kac [1982 A]). In these papers several conjectures are posed; the most in-
triguing of them suggests that ON = multo. Many of these conjectures
(but not the latter) have been solved by Schofield [1988A-C). Ringel [1989]

78
Real antllmtJgintJ,., Root,
Ch.5
found a connection of this field to quantized Kae-Moody algebras via Hall
algebras.
The nature of the root multiplicities in th indefinite case still remains
mysterious: there is no single case when the answer is known explicitly.
Asymptotic behavior or root multiplicities was studied in Kac-Peter80D
[1984 A); in some cases upper-bounds were found by Frenkel [1985] and
Borcherds (1986].
Exercises 5.10 and 5.12 are taken from Kac [1978 A]; and 5.11 and 5.20,
from Kac [1980 A). Exercise 5.19 is taken from Peterson-Kac (1983); it is
one of the key lemmas in the structure theory of groups attached to Kac-
Moody algebras. In Lepowsky-Moody [1979] one can find a detailed study
of the root systems in the hyperbolic rank 2 case by making U8e of the
map ; Exercise 5.23 is due to them. Exercise 5.28 is taken from Feingold
[1980]. The remaining exercises are either new or standard.

Chapter 6. Affine Algebras:
the Normalized Invariant Form,
the Root System, and the Weyl Group
S6.0. The results of Chapter 4 show that a Kac-Moody algebra g(A)
is finite-dimensional if and only if all principal minors of A are positive.
These Lie algebras are semisimple; moreover, by the classical structure
theory, they exhaust all finite-dimensional semisimple Lie algebras. So, the
classical Killing-Cartan theory or simple Lie algebras is, in our terminology,
the theory of Kac-Moody algebras associated to a matrix of finite type.
In this chapter we consider the next case, when the matrix A is of affine
type. Recall that this is a generalized Cartan matrix A, all of whose proper
principal minors are positive, but det A = 0 (A is then automatically in-
decomposable). A Kac-Moody algebra associated to a generalized Cartan
matrix of affine type is called an Gffine (Kac-Moody) algebra.
We describe in detail the standard bilinear form, the root system, and
the Weyl group of an affine algebra g in terms of the "underlying" simple
finite-dimensional Lie algebra g. In particular, we show that the Weyl
group of g is the so-called affine Weyl group of g; this explains the term
CCaffine,t algebra.
At the end of the chapter we give an explicit construction of the root
lattice, the root system and the Weyl group of all simple finite-dimensional
Lie algebras.
56.1. Let A be a generalized Cartan matrix of affine type of order l + 1
(and rank l), and let S(A) be its Dynkin diagram (rom Table Aff. Let
ao, 01, · . · , at be the numerical labels of S(A) in Table Afr. Then Go = 1
unless A is of type A) J in which case 00 = 2.
We denote by ar (i = 0,. . . , l) the labels of the Dynkin diagram S(' A)
of the dual algebra which is obtained from S(A) by reversing the directions
of all arrows and keeping the same enumeration of vertices. Note that in
all cases
(6.1.1)
aX = 1.
19

80
Affine Algebra.: 'he S'ruc'u TlaeorJ
Ch.6
The numbers
l l
h = Eai and h V = Ear
i=O i=O
are called, respectively, the Cozeter number and the dUGI Cozeter number
of the matrix A. We Jist these important numbers below:
A
h
h V
A
h
h V
AP> I + 1 I + 1
Bl) 2/ 2/- 1
Cp> 21 I + 1
DP> 21 - 2 21 - 2
EI) 12 12
EI) 18 18
EI) 30 30
Fl) 12 9
Gl) 6 4
Another important number is r t the number of the Table Air r containing
A.
Remark 6.1. The dual Coxeter number of the affine matrix X) is inde.-
pendent of r.
The matrix A is symmetrizable by Lenuna 4.6. Moreover, we have
A(2)
2'
A(2)
21-1
D(2)
'+1
E1 2 )
D(3)
4
21 + 1
2/- 1
I + 1
9
4
21 + 1
21
21
12
6
(6.1".2)
A d . ( V-I V-1 V -l )B b B ' 8
= lag 00 Go I 01 01 , · · · , a' (J l , were = .
Indeed, let 6 = '(00,'.. ,at) and 6 v = '(a,... ,01); if A = DB where D
is diagonal invertible and B = C B, tben B = 0 and hence 'B = o. On
the other hand, '6 v A = 0 implies (t6 V )DB = 0, and we use the fact that
dim ker B = 1.
56.2. Let 9 = g(A) be the affine algebra associated to a matrix A
of affine type from Table AfF r, let  be its Cartan subalgebra, n =
{Qo,...,ad C . the set o(simple roots, n v = {a,,,.,Q} C  the
set of simple coroots,  the root system, Q and QV the root and coroot
lattices, etc. It follows from Proposition 1.6 that the center or g(A) is
I-dimensional and is spanned by
l.
K = L:ajQ.
i=O

Ch.6
Affine Alge6rtJ: 'Ia Struct.re Tlaeory
81
The element K is called the canonical central element.
Recall the definition of the element 6 (cf. Theorem 5.6):
1
6 = La;o; E Q.
i=O
Fix an element d E  which satisfies the following conditions:
(Qi, d) = 0 for i = 1,..., t;
(00 ,d) = 1.
(Such an element is defined up to a sununand proportional to K.) The
element d is called the ,cGlin, element. It is clear that the elements
Q6' · · · , eri, d form a basis of. Note that
9 = [g, g] + Cd.
We define a nondegenerate symmetric bilinear C-valued form (.'.) on 
88 follows (cr. (2.1.2, 3, and 4) and (6.1.1»:
(6.2.1)
( VI V ) _ V-I
Qi ,OJ - Qjaj Ojj
( ert Jd) = 0
(QId) = 00;
( i J j = 0, . . . , t);
(i = 1,. . . ,t);
(did) = O.
By Theorem 2.2 this form can be uniquely extended to a bilinear form
(.1.) on the whole Lie algebra 9 such that all the properties described by
this theorem hold. From now on we fix this form on g. This is, clearly, a
standard form. We call it the normalized invariant form.
To describe the induced bilinear form on ., we define an element Ao E
. by
(Ao,Qr> = 60i for i= 0,...,1;
(AOt tf) = o.
Then {ero,..., Ql,Ao} is a basis of. and we have
(6.2.2)
(QiJQj) = ala-;lo;J
(a.IAo) = 0
(aoIAo) = 0 0 1 i
( i, j = 0, . . . , t);
(i = It... t I);
(AofAo) = O.
The map 1/ :  t-+ . defined by (.,.) Jooks 88 follows:
(6.2.3)
ar v(cr) = aiai;
II(K) = 6;
1I(d) = GoAo.

82
Affine Alge6ra,.' '/ae St",clure TAeory
Ch.6
We also record some other simple formulas:
( 6.2.4) (  I Q i) = 0 (i = 0 J . . . ,i);
(6.2.5) (Klaj) = 0 (i = 0, .. ., t);
(616) = 0;
(KIK) = 0;
(6IAo) = 1;
(Kid) = 00.
o 0 v v
Denote by  (resp. .) the linear span over C (resp. R) of Ql ,. .. , Q, . The
o 0
dual notions . and . are defined similarly. Then we have an orthogonal
direct sum of subspaces:
o
 =  $ (CK + Cd);
o
. = . e (C6 + CAo).
o 0
We set . = . + RK + Rd, . = . + RAo + R6.
o 0
Note that the restriction of the bilinear form (.).) to .. and . (resp.
o 0
i + ReS and . + RK) is positive-definite (resp. positive-semidefinite with
kernels ReS and RK) by Proposition 4.7 a) and b).
o
For a subset S of . denote by -S the orthogonal projection of S on ..
(This should not be confused with the sign of closure in metric topology.)
Then we have the following useful formula for  E . such that (K)  0:
(6.2.6)
 - X = (, K)Ao + (2, K)-1(112 - rXI 2 )6.
Indeed,  - X = blAo + 6 2 6. Taking inner product with 6, we obtain, by
(6.2.2, 3, and 4) that 6 1 = (, K). As 1,\1 2 = rXl 2 + 26 1 6 2 , we are done. We
also have another useful Cormula:
(6.2.7)
 = X + ('\, K)Ao + (IAo)6.
Define p E . by (p, or) = 1 (i = 0,. .. ,t) and (p, d) = 0 (cf. 52.5). Then
(6.2.7) gives
(6.2.8)
p = p + h v Ao.
56.3. Denote by g the subalgebra of 9 generated by the ei and Ii with
i = 1,.. . ,l. By the results of Chapter 1 it is clear that g is a Kac-Moody
o
algebra associated to the matrix A obtained from A by deleting the Oth row
and column. The elements ei, I. (i = 1,. . . t l) are the Cheval ley generators
0 0 0 0
of g, and  = 9 n  is its Cartan 8ubalgebra; n = {Ql, . . . ,Qt} is the root
o
basis, and n v = {or t . . . , 0'1} is the coroot basis for g. Futhermore, by

Ch.6
Affine Algt6rtJ!: 'he S'ructure Theory
83
o
Proposition 4.9, 9 = g(A) is a simple finite-dimensional Lie algebra whose
o
Dynkin diagram S(A) is obtained from S(A) by removing the Oth vertex.
The set A == 6nt. is the root system of g; it is finite and consists of real
o 0
roots (by Propc»ition 4.9), the set A+ = A n + being the set of positive
o Q
roots. Denote by , and At the sets of short and long roots, respectively,
o 0 0 0 0
in . Put Q = ZA. Let W be the Weyl group of L1.
Recall that the sets of imaginary and positive imaginary roots of g are
as Collows (Theorem 5.6 b):
dim = {:i:6. :1:26, . . . },
dm = {cS, 26, . . . }.
The following proposition describes the set of real roots ,.. and positive
o
real roots d+ e in terms of d and 6.
o
PROPOSITION 6.3. a) d"e = {a+ n61 Q E tn E Z} ifr = 1.
o 0
b) re = {a + no f Q E '" n E Z} U {a + nr6 f a E t, n E Z} JET = 2
or 3, but A is not of type A) .
o 0
c) l:1 re = {!(Q + (2n - 1)6) I Q Edt, n E Z} U {o + n61 0 E £\"n E Z}
Uta + 2n61 a E At,n e Z} jf A is of type A).
d) re + r6 = ,.e.
o
e) A+e == {a E d re with n > O} U A+.
Proof It is clear that d) and e) follow from a), b), c). The proof of a). b),
and c) is based on Proposition 5.10. Let a and b denote the square lengths
of a short and a long root, respectively, and let :e, le be the sets of
short and long real roots.
First, suppose that A is not of type A). Then 6 = 0'0 + al 0'1 + · · '. If
t 0
now Q == E kio i E :et then a = IQI 2 = IQ-k o 6(2 and hence cr-k o 6 E,
i:::O
by Proposition 5.10 a), which gives the inclusion C in the relation
(6.3.1)
o
6: e :::: {a + n61 a E 6"n e Z} ir A  A).
The reverse inclusion also follows from Proposition 5.10 a).
If r = I, then 00 is a long root and the same argument as above gives,
using Proposition 5.10 b):
(6.3.2)
o
L\l"f! = {a + n61 Q E l, n E Z} if r = 1.

84
Affine Alge6rtJl.. 'he St",cfure Theory
Ch.6
If ,. = 2 or 3. then 00 is a short root, hence, by Proposition 5.10b), Q =
koQo + · .. E l' only if io is divisible by r. Therefore we obtain, by
Proposition S.IOb),
(6.3.3)
Ai' = {a + nr61 a E Al' n E l} jf r = 2,3; A  A).
Formulas (6.3.1,2, and 3) prove a) and b).
Finally, let A be of type A). Then short (reap. long) real roots
have square length 1 (resp. 4), and the roots from a := d,.a\
(A: e UAi e ) have square length 2 (d t- 0 if l > 1). We have to show that
(6.3.4)
(6.3.5)
(6.3.6)
o
A:' = {1(Q + (2n - 1)6) J Q E i, n E l},
o
£\ = {Q+ n61 Q E L\"n E Z},
o
Al' = {Q + 2n6 I Q E dt, n E Z}.
By Proposition 5.10 b), Q = koQo + · .. E die only if ko is divisible by 4.
Now the same argument as above proves (6.3.6). A similar argument gives
(6.3.4 and 5).
o
Note that Proposition 6.3 will also follow from the explicit construction
of affine algebras given in the next chapters. Note also that (6.1.1) implies:
(6.3.7)
o
QV = QV  ZK (orthogonal direct sum).
and that (due to (6.2.1) and (6.2.2) we have the isomorphism of lattices
equipped with bilinear fOfI118
(6.3.8)
QV (A)  Q(' A).
o
Warning. A = {O} in all cases except A), in which case A\{O} is
a
& non-reduced root system, and  is the 8S8ociated reduced root system.
56.4. Introduce the following important element:
t 0
9 == 6 - aoao = E aiai E Q.
i=l

Ch.6
Affine Alge6nJ.: tAe Strwctare TAeo,.,
85
It is easy to deduce from the formulas of 56.2 that
(6.4.1)
181 2 = 2ao.
Hence '01 2 is equal to the square length of a long root jf A is from Table
Aff 1 or is of type A>, and it i8 equal to the square length of a short root
otherwise. One deduces now from Proposition 5.10 a), b) that in all cases
o
8 E A+. Again, from the formulas of 56.2 we deduce:
8 = aOJl(9 V );
19 v 1 2 = 20 0 1 ;
Q = 11-1(6 - 9) = K - ao(Jv.
Furthermore, one baa
PROPOSITION 6.4. a) If A is from Table Aff lor is of type A), then
o 0
8 e (6+)t and 6 is the unique root in A of maximal height (= h - 00).
o
b) If A is from TabJe Aff 2 or 3 and is not of type A), then 6 E (d+).
o
and is the unique root in , of maximal height (= h - 1).
Proof. It is easy to check that all simple roots of the same square length
o 0 0 0 0
in dare W-equivalent, hence both . and At are orbits of W. Also,
(9, or> = -00(00, aj)  0 for all i = 1,.. . I t. Now a) and b) Collow from
Proposition 3.12 b).
o
Unless otherwise stated, in the case of a finite type matrix A, we shall
normalize the standard invariant form (.,.) on g(A) by the condition
(6.4.2)
(oIQ) = 2 if Q Edt,
and shall call it the normalized invarian' lorm. We deduce from (6.4.1)
and Proposition 6.4 the following:
COROLLARY 6.4. Let 9 be an aRine algebra of type XJ;>. Then the ratio
of the normalized invariant (arm oE 9 restricted to g to the normalized
invariant form of 9 is equal to r.
o
Note that we have the following description or nand flv:
n = {oo = G;I(.s -D),QI,...,Ot},
n V = {Q6 = K -ao6VtQr,...,o}.

86
Affine Algebra,: the S'ructUf"t Theory
Ch.6
S6.5. Now we turn to the description of the Weyt group W of the affine
algebra g. Recall that W is generated by fundamental reflections rOt rl,
. . ., r l, which ac. t on  · by
ri() =  - {, Qi)Qi,  E ..
As (cS,Qr) = 0 (i = O,....t). we have
w(6) = 6 for all w E W.
Recall also that the invariant standard form is W-invariant.
o
Denote by W tbe subgroup of W generated by rl,..., rt. As ri(A o )
o
= Ao for i = 1,.... i, we deduce that W operates trivially on CAo + C6; it
000
is also clear that . is W-invariant. We conclude that W operates faithfully
o 0
on ., and we can identify W with the Weyt group of the Lie algebra i,
o 0
operating on .. Hence (by Proposition 3.12 e» the group W is finite.
Recall that for a real root Q we have a reflection ra E W defined by
rQ() =  - (, aV)a,  e . .
LEMMA 6.5. Let Q e 6.+' be such that (J := 6 - 00 E A'+ for some Q.
Then
rQ('\) =,\ + (J K)v({3V) - «J,BV) + tlp V I 2 (,\, K))6
for A E . .
Proof. First, we compute mod C6:
rarfJ('\) mod C6 = ra('\ + ('\IcS)2acrl(JQI-2 - (At Q V)a) mod C6
= ,\ - ('\, Q V)o + ('\, K)II(,8V) + (A. Q V)Q mod C6
=  + (, K)v(,BV) mod C6.
To compute the coefficient of 6, we use the equality Ircrrt'()12 = 1'\1 2 .
o
Applying Lemma 6.5 to 9 = 6 - 0000 E 6+ C +e, we get
(6.5.1)
roor,(.\) = .\ + (.\, K)II(9 V ) - «.\,9 V ) + i/ Dv / 2 (.\, K»6.

Ch.6
Affine Algebras: 'he Stncturt Tlatory
87
Motivated by this formula. we introduce the following endomorphism to
o
of the vector space . for Q e .:
(6.5.2)
ta() = ,\ + ('\, K)a - ((Ia) + llo,2(A, K) )6.
In the case when m := (, K) :F 0 we can rewrite this as follows using
(6.2.6):
(6.5.3)
to() = mAo + (X + mer) + 2k(\12 - I  + merI 2 )6.
As (A o , K) = 1, we obtain, in particular:
(6.5.4)
to(Ao) = Ao + Q - !laI26.
Note also that (6.5.2) implies
(6.5.5)
tQ() = ,\ - (AIQ)6, if (, K) = O.
Now we can easily deduce the additivity property of to:
(6.5.6)
tat" = ta+/J'
Indeed, by (6.5.5) it is sufficient to check (6.5.6) for  = Ao. But then,
by (6.5.4 and 5) we have tot(Ao) = ta(Ao + p - i1P1 2 6) = lo(Ao) +
to(/J - !I/J1 2 6) = Ao + 0 - llol26 + /J - tl/J1 2 6 - «/J - tl/J1 2 ,s)Ier),s =
Ao + a + (J -llQ + fJI 2 6 = to+(Ao).
We also have
(6.5.7)
-1 0
tw(a) = wtaw for w E W.
Indeed, wt o (w- I (,\» = W(w-l() + (w- J (,\), K)a - ((W- 1 (A)fcr) +
!l a I 2 {w- 1 (..\), K»6). Now (6.5.7) follows since w{K) = K and (.I.) is
W-invariant.
o 0
Now we introduce the following important lattice M C .. Let Z(W .8 V )
o 0
denote the lattice in . spanned over Z by the (finite) set W · gy, and set
o
M = II(Z(W · 9 Y ».
Here is a description of the lattice M:
(6.5.8)
_ 0
M = Q = Q if A is symmetric or r > 00;
- .
M == II(QV) = v(QV) otherwise.

88
Affine A IgelJm,: the Strwcture TAtorr
Ch.6
0 00
" V ) ·
Indeed, if r = 1, then BY is a short root o(  v I hence W · 8 = (6 .. It 18
well known (cr. Exercise 6.9) that in the finite type case the root lattice is
o
spanned over Z by the short roots, hence in this case M = II(QV), giving
(6.5.8) for r = 1. Equivalently,
o 0
M = Q (resp. M = Zl)
if r = 1 and A is symmetric (resp. nonsymmetric).
o 0
Similarly, if Oar = 2 or 3, then gv is a long root of A v , hence W . IV =
(A V)t, and we get M = Q. Finally for A) one has JI(qV) = i', hence
o
M = ! Z4 it which is equivalent to (6.5.8) in this case also (lee (6.3.6».
Note also the following useful fact:
o 0 0 0
(6.5.9) Q :> II(QV) if r = 1 and Q c v(QV) if r > 1.
The lattice M, considered as an abelian group, operates faithfully on
. by formula (6.5.2). We denote the corresponding subgroup of GL(.)
by T, and c811 it the group of trtJn,/Gtion (formula (6.6.3) below explains
why). Now we can prove
o
PROPOSITION 6.5. W = W  T.
o
Proof. By (6.5.1), to E W for Q = v(8 V ), hence tw(a) E W for w e W by
o
(6.5.7). Now it follows from (6.5.6) that to E W for all Q e M. Since W
o
is finite and M is a free abelian group, W n T = 1. It follows from (6.5.7)
that T is a normal subgroup in W. Finally rao = t.,<,v)r" and therefore
o
the subgroup of W generated by T and W contains all ri (i = 0,... It) and
hence coincides with W.
o
o
Since t.,(,V) = rao r , and T is generated by wt.,('V)W- 1 (w e W), we
have
(6.5.10) det. w = 1 Cor wET.
Rtmark 6.5. A vertex i of the diagram SeA) is called 'pecitJl if 9<') :=
6 - OiQi is a positive root. For example, i = 0 is a special vertex. Denote
by Wet) the subgroup in W generated by aU rj (j :F ;), and denote by
M{i) the lattice spanned over Z by the set II(W(i)(9(i)V». Then the same
argument as above shows that T = {to I Q E M(i)} and W = K'(i) t( T.
Most of the results of this chapter hold if we take a special vertex i in place
of the vertex O.

Ch.6
Affine Alge6ra,: 'he Struc'u Theory
89
56.6. For 6 e R set
: = {E .I (,K) = a}.
t.
Note that o = E RQi and that the hyperplanes : are W-invariant.
.=0
Furthermore, the action of W on o is faithful by (3.12.1).
Consider now the affine space  mod ReS. Since the action of W on o
is faithful, its action on ./RcS  o and thus on i mod ReS is also faithful.
The affine action of W on the affine space i mod R6 has the following
o
simple geometrical meaning. We identify i mod R6 with . by projection,
thus obtaining an isomorphism from W onto a group of affine transforma-
o
tiOhS War of ... We denote this isomorphism by af, 80 tbat
af(w)(I) = w() for  E ;.
o
Now we describe the action of W., on .. It is clear that
(6.6.1)
o
af(w) = w for w E W.
Furthermore,
(6.6.2)
o
af(rao)() = r,() + v(OV) (E it).
Indeed, iflJ e i, then (PtK) = I, hence (p,aX) = (p,K-a o 9 V )
1 - (Jo(p, BV), and rao(JJ) = p - (1- 00 (P. gV) )Qo = IJ + (1 - aO(IJ, 8'1) )00 19
= IJ - (1',6'1)(8") + 1I(8 V ) mod ReS. So, af(rao) is a reflection in the hyper-
plane
o
9 = 1 (i.e.,{ E. I (18) = I}).
o
Also (6.6.2) implies a((t,,(,V»() = af(rao)(r,(» = ,\ + 1I(8 V )(A e .).
Hence, by (6.5.6 and 7), we obtain
(6.6.3)
o
af(ta)() =  + Q for ,\ e ., Q e M.
So, the group Wat is none other than the so--called affine Weyl group of g.
Introduce the jund(JmenttJl alcove:
o
Cal = { E . I (Ala.)  0 for 1  i  I, and (18)  I}.

90
Affine AlgebrtJ.: 1ae S,,,,c'u Theory
Ch.6
o 0
PROPOSITION 6.6. a) Every point of. is W -equivalent mod M to a
unique point of Cat.
o 0
b) The stabilizer of every point o(Car under the action o(W on ./M is
generated by its intersection with {r" rOt'.' ., r Ol }'
c) For every  E i one has
af(W A ) = (War )r,
and W,\ n T = e, where W,\ denotes ehe stabilizer of .
o
Proof Consider the projection map 'K : i -+ .. Then it is clear that 1f' is
surjective and that af(w) 0 1r = 1r 0 w for w e W. Furthermore, (cr. 53.12):
,..-1 (C.,) = C V n i.
But by Proposition 3.12b), C V n i is a fundamental domain for the ac-
tion of W on i C Xv. This together with (6.6.1 and 3) proves a). We
also deduce (rom the above that aC(W A ) = (War)r. Now b) follows Crom
Proposition 3.128). Finally W n T = 1, since af(W) = (W.r)r contains
no nontrivial translation.
o
S6.1. As we have seen, the root and coroot lattices, the root system
and the Weyl group of an affine a)gebra can be expressed in terms of the
corresponding objects for the "underlying" simple finite-dimensional Lie
algebra.
For the convenience of the reader, we give below an explicit construction
/
of all these objects for all simple finite-dimensional Lie algebraa.
Let Rn be the n-dimensional real Euclidean space with the standard
basis tll, . . . ,tin and the bilinear form:
(tJiIVj) = 6 ij .
All the lattices below will be 8ublattices of R" with the inherited bilinear
form (.(.). All indices are assumed to be distinct.
At : Q = QY = {L kiVi E W+ 1 I k i e z, L: k i = O},
i i
A = {Vi - Vj},
n = {01 = v1 - v2,02 = V2 - v3,... ,Ot = Vt - Vt+11,

Ch.6
Affine Alge6ras: 'he SIruc'.re Theory
91
(J = VI - Vl+IJ
W = {all permutations of the Vi}.
Dt : Q = QV = {E i,v, E W f i, E l, E i, E 2l},
i .
A = {%Vi :I: tJj},
n = {Ol = VI - V2,... ,Qt-l = Vt-l - Vi,al = Vl-l + Vt},
(J = V1 + V2,
W = {all permu tatioDs and even number of sign changes or the Vi}.
Bt : Q = {Li,v, E Hli i. E l}, QV = Q(Dd,
i
L1 = {:tVi :t Vj t :%l1i} ,
n = {Qt = VI - V2.... ,Ql-l = Vl-l - Vt,Qt = Vt},
(J = VI + V2.
W = {all permutations and sign changes of the Vi} = AutQ.
1
C,:  = {"J2(Vi Vj), :i:V2 V i} ,
1
 = {(:i:Vi' :i:Vj), :i:V2v,} ,
1 1
n = {a I = yI2 (vI- V2),...,a'-1 = v'2 (v t - l - vl),al = V2 v t},
D = V2Vl,
W = W(Bt).
1
G 2 : Q = "J3 Q (A 2 ), QV = Q(A 2 ),
1 1
d = {(Vi - vJ)' :i: v'3 {v i + vJ - 2v.t)},
1 I
n = {a I = (-VI + 2V2 - V3), Q2 = 'J3(V I - V2)}.
1
8 = y'3 ( VI + V2 - 2V3),
W == :I: {all permutations of the Vi} = Aut Q.
F4: Q = {kiVi E JR4 I all k i E Z or all k i E  + Z}, QV == Q(D 4 ),
.
1
d = {:i: Vi, :i: v. :i: v J ' 2 ( :i: v I :i: V2 :f:: v3 :i: V4)},
1
n = {a I = V2 - V3, a2 = V3 - fl4, a3 = V4, Q4 = -(VI - V2 - V3 - V1)}.
2

92
Affine Alge6ra,: tlae Structure TAeor,
Ch.6
(J = VI + V2,
W = AutQ.
Ea : Q = QV = {EIt,v, E R' I all k, E Z or all It, E 4 + z,  It, E 2Z},
i .
11 = {::i:v, ::i: Vb !(::i:vl ::i: V2 ::i: .. · ::i: va) (even number of minuses)} I
2
n = {QJ = V2 - V3. 02 = V3 - V4, Q3 = Vof - 111.
Q = Va - Ve,O's = Ve - "'r,O's = 117 - tla,
Q7 = !{Vl - 112 - ... - 117 + t1S),Q8 = 117 + Vs},
(J = VI + V2,
W = AutQ.
E1 : Q = QV = {E Itivi e II' I all k. e l or allie, e  + Z, L Ie. = OJ,
i i
11 = {Vi - Vj, l{ ::i:vl :!: · · · :!: v) (four minuses)},
n = {Ql = "2 - "3.02 = V3 - "4,03 = t14 - V5,
04 = Vs - "s,Qa = tis - V7,08 = tl7 - tis,
1
07 = 2{-Vl - V2 - Va - v. + Vs + Ve + V7 + va)},
(J=V2- V l,
W = AutQ.
E6 : Q = QV = {It 1 VI + · .. + Itsvs + J21e 7 V 7 E R 1 1 aU Iti E Z
1
or allit i E 2 + l,lt l + ... + Its = O},
11 = {Vi - Vj (i, j  6), «£1 VI + · · · + £6V6) :i: V2V7 )(e, = ::i: I,
L ti = 0), ::i:v'2v7),
II =:: {QI = VI - V2,02 = V2 - V3, 0 3 = V3 - V4,04:::: V4 - Vs,
as = Vs - V6,06 =  (-Vl - V2 - V3 + V4 + Vs + V6 + V2 V 7)} ,
(J = \1'2"7.
W )( {%1} = AutQ.
In order to prove these facts we first check directly that n is a basis
of Q over Z, and that the matrix (2(Oiloj)/(Oilai») is the Cartan matrix
of the corresponding type. Furthermore, one easily checks that 11 = {o I
(010) = 2} for At, Dt and Et, and A = (reap. C) {o I (alo) = 1 or r} for
Bt., G2, and F. Crespo Ct.), where r = 2 for Bt.ICt.1 F.t and r = 3 for G 2 .

Ch.6
Affine AIge6ra.: lAe S'",cf.re TJaeorr
93
Using Proposition 5.10 a), b), we lee that d is the set of all roots. The
computation of W Collow8 easily from Corollary 5.10 and Remark 5.9 b).
Remark 6.1. It follows from the above discussion that if A is of finite type,
then AutQ = Aut(A) k W. except for A of type C4. In the latter cue, W
is a group of index 3 in AutQ (cC. Exercise 5.8).
We can list now the lattices M for all affine algebras:
XI): M = QV(Xt)i
Al: lif = Q(Dt)j
Dl: M = V2Q(B t )i
E2): M = Q(D 4 )j
D1 3 ): M = Q(A 2 );
A): M == Q(Bt).
56.8. Exercises.
o 0
6.1. Check that the square length (0'10') of a root (t from 4t (resp. A.) is
equal to 2r (resp. 2r/B, where, = max Oji/O;j).
eij o
o
6.B. Let r = 1. Show that h is the Coxeter number of the root system t:.
and that h V = 4>(9,9)-1 = 1 + (p19), where 4> is the Killing form of 9 and
o . .
p is the halC 8um of the roots from A+. Show that (z,JI) = 2h Y (zIJl) for
o
Z, JI e g.
o
[Note that «(J + 2 P, 8) is the eigenvalue of the Casimir operator associated
to the Killing form 4>, hence it equals 1, and use (6.2.3).1
6.9. Let A be a Cartan matrix of type XJ;> from Table Aft" r, let t = rank A,
and Jet h be the Coxeter number. Let o be the finite root system of type
X N. Check that
rlh = IAol.
6.-1. Let A be of type Ail), i.e., A = (:2 -.}). Then
A+ = {(i - l)o + il, io + (i - l)al' io + iall where i = 1,2,...).
6.5. Let A be of type Al), t > I, i.e.,
2 -1 0 ... 0 -1
A= -1 2 -1 ... 0 0
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
-1
o
o
. . .
-1
2

94
Affine Alge6ra.: the S,,,,c,. TlaeorJ
Ch.6
Show that
+ = {k(oo + · .. + ai-i) + (k :i: 1)(0; + · .. + aj-l) + k(aj + ... + OL)}.
where k = 0,1,2,.. .; 0  i  j  l + 1, and only + is allowed if k = O.
6.6. Let A be of type A2), i.e., A = (!1 -;4). Then
A+ = { 4nQ o + (2n - l)Qi. 4(n - I)Qo + (2n - 1)Qi,(2n - 1)00 + no-1,
(2n - 1)00 + (n - l)ai, 2noo + n01; n = 1,2,.. .).
o
6. 7. Let Ot 81,. . . ,.i be the reflections in the vector space  with respect
to the hyperplanes 8 = 1, 01 = 0,. .., Ql = 0, respectively, and let W ca be
o
the group generated by '0. . . . t'l. Show that W o = W  JI-l (M) and that
the map Ii  rOt defines an isomorphism t/J : W G --+ W. Show that the
image under t/J of 8 translation by Q E v-I (M) is to.
o 0
6.8. Let cJ denote the vector space of affine linear functions on . Consider
o
the isomorphism t/J : o --. CI defined by
1P(Qi)=iii for i= l,...,l;
1/J( 6) = 1.
o
The action of W G on  defined in Exercise 6.7 induces a linear action of
o 0
W Q = W on ca. Show that the morphism (t/J. t/J-l) : (W ca , f}CJ) --+ (W, o) is
equivariant.
6.9. Show that for a finite or affine type matrix A, the set of short real
roots Ae(A) span the lattice Q over Z.
(The set Q' = Ze is W-invariant. We can assume that A is not symmet-
ric. If A 1:- A), then there exists a short simple root Q and a long simple
root (J such that (QIP) 'I 0; then r(Q) = Q + {J E Q'. Hence, (J e Q' and
Al C Q'. SO n cArt C Q' and Q = Q/. The same argument works in
the case A) as well.]
6.10. Let r = 1 and rEM. Show that
l(I.,) = 2: 1<-rla)l.
.
oE6+
[Use Exercise 3.6.]

Ch.6
Affine Alge6ral: the Structure Theory
95
6.11. Denote by r n the lattice in Rn defined by the same conditions as the
root lat tice of Es in H8. Show that if n is divisible by 8, then r n is an even
unimodular lattice.
6.1£. Show that an integral lattice Q (i.e., (zlll) E Z if z, y E Q) is spanned
over Z by its vectors of square length 2 if and only if Q is an orthogonal
direct sum of root lattices of type Al, Dt, E6 t E7, E8.
6.13. Show that the group of all automorphisms of  preserving the bilin-
ear form (.1.) and fixing K is the semidirect product of the group of all
o
orthogonal &utomorphisms of  (fixing K and d) and the group of all t a ,
o
Q E ..
56.9. Bibliographical Dotes and comments.
The study of affine algebras was started by Kac [1968 B) and Moody
{1969]. The material of this chapter is fairly standard. The exposition is
based on papers Macdonald [1972J, Kae [J 978 A], Kac-Peter80n [1984 AJ.
The"quadratic n action (6.5.2) "explains" the appearance of theta functions
in the theory of affine algebras (see Chapter 12). Exercise 6.10 is due to
Haddad and Peterson.

Chapter 7. Affine Algebras as
Central Extensions of Loop Algebras
57.0. In this chapter we de.cribe in detail a "concrete" construction of
all "nontwisted" affine algebras. It turDS OU that such an algebra 9 can
be realized entirely in terms of an "underlying" simple finite-dimensional
Lie algebra 9. Namely, it. derived algebra [g, g] is tbe universal central
extension (the center being I-dimensional) of the Lie algebra of polynomial
maps from ex into g. The corresponding algebra of "currents" plays an
important role in quantum field theory.
At the end of the chapter we give an explicit construction of finite-
dimensional simple Lie aJgebraa.
S1.1. Let l = C[t, ,-1] be the algebra of Laurent polynomials in t.
Recall that the residue of a Laurent polynomial P = E c.t' (where all
iEI
but a finite number of c are 0) is defined by Res P = C-l. This is a linear
functional on l defined by the properties:
Re.t- I = 1;
D_  - 0
naJ dt - ·
Define a bilinear C-valued function VJ on l by
dP
v>( P, Q) = Res -;uQ.
Then it is easy to check the following two properties:
(7.1.1 )
(7.1.2)
tp(P, Q) = -'(J(Q, P),
tp(PQ, R) + ",«(JR, P) + rp(RP, Q) = 0 (P,Q, R e l).
S7.2. The affine algebra associated to a generalized Cartan matrix of
type X?) (from Table Aft' 1) is called a nontwi!ted affine alge6ro. Here
we describe an explicit construction of these Lie algebras. Note that the
generalized Cartan matrix A of type X?) (where X :: A, B,. .. ,G) is
96

Ch.7
Affine Alg6nJl ., Central Ezten.ion6 of Loop AIge6ra6
97
nothing else but the so-called extended Cartan matrix of the simple finite-
o 0
dimensional Lie algebra 9 := g(A), whose Cartan matrix A is a matrix of
finite type Xt (obtained from A by removing the Oth row and column).
Consider the loop alge6ra
o 0
.c(g) := t, 0c g.
This is an infinite-dimensional complex Lie algebra with the bracket [ , ]0
defined by
o
[P @ z, Q @ 1/]0 = PQ 0 [%,11] (P, Q e £; Z,JI e g).
It may be identified with the Lie algebra of regular rational maps CX -+ i,
so tbat the element E(,i  Zl) corresponds to the mapping z  I: zi Zi .
i i
Fix a nondegenerate invariant symmetric bilinear C-valued form (.,.) on
g; such a (orm exists (e.g., by Theorem 2.2) and is unique up to a constant
muJtip)e. We extend this form by linearity to an £-valued bilinear form
o
(.1.), on £( g) by
(P 0 zlQ 0 y), = PQ(zly).
Also, we extend every derivation D of the algebra (, to a derivation of the
Lie algebra £(g) by
D(P @ 2:) = D(P) 0 z.
Now we can define a C-valued 2-cocycle on the Lie algebra £(g) by
do
t/1(a, 6) = Res( di , 6)c.
Recall that a C-valued 2-cocycle on a Lie algebra 9 is a bilinear C-valued
function '" satisfying two conditions:
(Co 1)
(Co 2)
1!J(o, b) = -.,(6. a)
tj1([a, 6], c) + .p«(b. c], a) + ,p([c, oj, 6) = 0 (0, b, C E g).
It is sufficient to check these conditions (or a = P@z, II = Q@1J, c = R@z,
o
where P, Q, R E l, and z, 1/, z E g. We have
t/J(tJ,6) = (zlll)( P, Q).
Hence, (Co 1) follow8 from (7.1.1) and tbe symmetry of (.1.). Property
(Co 2) (ollows (rom (1.1.2) and the symmetry and invariance 0£(.1.). Indeed,
the left-hand side of (Co 2) is
([:t.y]lz)cp(PQ. R) + ([II. z]lz)cp(QR. P) + ([zJ z]ly)<p(RP,Q)
= ([z, y]lz)(cp(PQ, R) + cp(QR, P) + cp(RP,Q» = o.

98
Affine Algebra, ell Central Ez'enlionl of Loop AIge6nJI
Ch.7
Denote by ,C(g) the extension of the Lie algebra leg) by a I-dimen-'
sional center, &88ociated to the cocycle.p. Explicitly, ,C(g) = .c(g) $ CK
(direct sum of vector spaces) and the bracket is given by
(7.2.1)
o
[a + >'K, II + pK] = [a. bJo + ,p(a, b)K (0, II E £(g); >., JJ E C).
Finally, denote by £(8) the Lie algebra that is obtained by adjoining to
leg) a derivation d which acts on leg) as t 1; and which kills K (see 17.3).
A 0
In other words, £(;) is a complex vector space
A 0 0
£(8) = £(9) ED CK  Cd
with the bracket defined as follows (z, 'II E g; . 1', 1 t 1-'1 E C):
(7.2.2) [t m 0 Z $,\K E9 pd. t n @ 11 e lK  #Jld]
= (t m + n @ [z, Y] + IJnt n 0 Y - JJlmt m  z) E9 m6 ml - n (zly)K.
We shall prove that '(g) is an affine algebra associated to the affine matrix
A of type Xl).
ST.3. Here we check that d is a derivation of the Lie algebra £(g). More
- 0
generally, denote by d, the endomorphism of the space £( g) defined by
(7.3.1)
d
d,I,C<i) = _t,+1 d' ; d,(K) = 0,
80 tbat do = -d.
PROPOSITION 7.3. d, is a derivation of £(g).
Proof. Since D := d, is a derivation of £(g), we deduce that
D([a + )'K, b + pK]) = D({a,6Jo) = (D(a), 610 + [a, D(h)]o.
But
[D(a), b] = [D(a), 6]0 + t/J(D(a), b)K.
Hence. one has to check that
(7.3.2)
1b( D( a), 6) + 1t'( a, D( b» = o.

Ch.7
Affine AIge6nJ6 tlI CentnJl Erten$;On6 0/ Loop Algebras
99
Set (J = P  %, 6 = Q  y; then the left-hand 8ide of (7.3.2) is:
(zly)(cp(D(P), Q) + VJ(P, D(Q»)
= (zlll) (-cp(Q, D(P» + tp{P, D(Q»)
= (zly) Res ( _t'+l + t'+1) = O.
dt dt dt dt
o
Note that
o := EI1 Cd j
JEI
is a Z-graded 8ubalgebra in derl(g) with the following commtltation rela-
.
tlons:
[cia, dj] = (i - j)di+ i .
This is the Lie algebra of regular vector fields on C X (= derivations of £).
The Lie algebra D has a unique (up to isomorphism) nontrivial central
extension by a I-dimensional center, say Cc, called the Vinaloro tJIge6ra
Vir. which is defined by the following commutation relations (see Exercise
7.13):
(7.3.3)
[d i , d j ) = (i - j)di+j + l2(i 3 - i)6 i .- j c (i,j E Z).
The semidired product of Lie aJgebras Vir +£(9) defined by (7.3.3), (7.2.1),
(7.3.1), and the equation [c, l(g)] = 0, plays an important role in the
representation theory of affine and Virasoro algebras and in the quantum
field theory.
.
o 0 . 0
ST .4. Let A C  be the root system of the Lie algebra 9, let {Ql t . . . , at)
be the root basis, {HI,..., Ht} tbe coroot basis, Ei, Fi (i = 1,... ,t) tbe
o
Chevalley generators. Let 8 be the highest root of the finite root system d
(see Remark 4.9). Let 9 = E9 90 be the root space decomposition of i.
.
aE4uO
RecaJJ that (Qla) :F 0 and dim 90 = 1 for a E  (there are no imaginary
roots). Let  be the ChevaJley involution of i. We choose Fo e 9, such
that (Folc:,(Fo» = -2/(616), and set Eo = -c:,(F o ). Then due to Theorem
2.2 e) we have
(7.4.1)
[EOt Fa] = _gv.

100
Affine Algt6ra, G' Central Ezten,ioft' of Loop Algebra,
Ch.7
The elements E. (i = 0. ..., I) generate the Lie algebra g, since in the
o 0 0 0
adjoint representation we have 8 = U(g)(E o ) = U(n+)(Eo) (recall that 9
is simple).
Now we turn to the Lie algebra £(8). It is clear tbat CK is the (1-
dimensional) center of the Lie algebra l(g>t and that the centralizer of 0 in
£(9) is a direct 8um of Lie algebras: CK (BCdEl) (1  g). In particular, 1  9
is a 8ubaJgebra of £(9); we identify 9 with thi8 8ubaJgebra by z ....... 1  z.
Furthermore,
o
 :=  + CK + Cd
is an (t + 2)-dimensional commutative 8ubaJgebra in £(9). We extend
o. o.
 E  to a linear function on  by setting (At K) = (, d) = 0, 80 that 
is identified with a 8ubspace in .. We denote by 6 the linear function on
 defined by 61_ = 0, (6. d) = 1. Set
+CK ·
eo=,@E o , /o=,-I@Fo,
e i = 1 @ E i . Ii = 1 0 Fi (i = 1,. . . ,I).
We deduce from (7.4.1) that
(7.4.2)
[eo./O] = (98) K - 8 v .
Now we describe the root system and the root space decomposition of ,C(g)
with respect to :
o
 = {j6 + i'. where j e z, 'Y E } U {j6, where j E Z\O},
,C(g) =  $ ($ £(9)0)' where
oEA
o J 0
l(g)j'+'Y = t 0 tI-y,
o . 0
l(g)jl = t J 0 .
We set
n = {oo:= 6-B,Ol,...,Qt}.
2
n V = {a:= (919) K-109 V , a'{:= 10Hl,...,al:= 10Ht}.
Note that our (J is the same as the one introduced in 6.4 (for T :::: 1). Then
Proposition 6.4 a) implies
(7.4.3)
A = «(OJ , or»:J=O'

Ch.7
Affine Alge6rtJ, G' Centrol Ez'en,ioft' 01 Loop AIge6rtJI
101
In other words, (, U, flV) is a realization or the affine matrix A we started
with. (Indeed, nand n V are linearly independent, i.e., (1.1.1) holds and
2n - rank A = 2(t + 1) -I = t + 2 = dim, i.e., (1.1.3) holds.)
Now we can prove our first realization theorem.
THEOREM 1.4. Let g be a complex finite-dimensional simple Lie algebra,
.. 0
and let A be its extended Cartan matrix. Then £(9) is the affine Kac-
Moody algebra associated to &he affine matrix A,  is its Cartan subaIgebra,
n and flv the root basis and the coroot basis, and eo, ..., el, 10, ..., It.
... 0
the Chevalley generators. In other words, (£(g),, D, n V ) js the quadruple
associated to A.
Proof We employ Proposition 1.4 a). Some of the hypotheses of this propo-
sition have already been checked. The relatioDs (1.4.2) are clear. As for
relations (1.4.1), they evidently hold when it j = 1 t ..., I because Ei t F i
(j = 1 t .. ., t) are Chevalley generators or g. The relations [eo t /d = 0 and
(ei, /0] = 0 for i = 1, ..., t hold since 8 is the highest root or g. This
together with (7.4.2) proves all the relations (1.4.1).
Furthermore, leg) has no ideals intersecting f) trivially. Indeed, if i
is a nonzero ideal of £(9) such that i n  = 0, then by Proposition 1.5,
... 0 .
i n 'c(S)o :F 0, for Bome a e A. Hence I' @  E i for some j E Z and
000
x E g-y, x t= 0, I E  U O. Taking Y E Q_..., such that (xly)  0, we obtain
o
{t j @ x, t- j  y] = j(xly)K + {x, yJ E  n i. As (Xt yJ E , we deduce that
o
j = O. But then I  0) and hence (x, yJ is contained in  n i and is different
from zero. This is a contradiction.
Finally, it remains to show that ei. I. (i = 0, ..., l), and  generate
... 0 A 0
the Lie algebra £(9). For that. purpose we denote by Ll(9) the 8ubaJgebra
'" 0
of £(g) generated by them. Since Eit Fi (i = 1, ..., I) generate the Lie
o 0 ... 0 ... 0
algebra iJ, we obtain that 1@9 C £1 (9). Furthermore, 'Eo E £1(S); since
[t @ z,l @ II] = t @ [z, y] for , Sf E i and since i is simple, we deduce that
t @ i c £1(9). Since [t @ z"f: @ Sf] = ,'+1 @ [,JlJt it follows by induction
i 0 A 0
on I: that t 0 9 C £1 (g) for all k  o. A similar argument shows that
o A 0
t i @ g C (,1 (9) for all  < O. completing the proof.
o
The following important corollary or Theorem 1.4 is immediate.
COROLLARY 7..4. Let g(A) be a nontwisted alfjne Lie algebra a{rank t+ 1.
Then the multiplicity of every imaginary root of g(A) is t.

102
Affine Algebra. tI. CeninJl Ezten.iora, of Loop Alge6na.
Ch.7
RmGrk 1.4. Given a simple finite dimensional Lie algebra 8, the Lie aI.
gebra £(g) is usually referred to in the literature as the Gffine .lge6,..
tJlsocia'etl '0 g, or the GffinizG'iora or g.
S1.5. One can alao describe explicitly the rest of the notions introduced
in the previous chapters.
The normalized invariant Corm (.,.) of " (introduced in 56.2) can be
described 88 (ollows. Take the normalized invariant form (.1.) on i and
A 0
extend (.1.) to the whole l(,,) by
(7.5.1)
(P@zIQ@v) = (Res ,-IPQ)(zlv), (z,V E i, P,Q e .c)j
(CK + Cdl£(a» = OJ (KIK) = (did) = OJ (Kid) = 1.
It is clear that this is a nondegenerate symmetric bilinear form. We check
the only nontrivial case of the invariance property:
«(d, p  z]IQ  y) = (dl[P @ z, Q 0 y]).
The left-hand side is (tf-  %IQ' y) = (Res Q)(zIJl); tbe right-hand
aide i. (dlPQ @ [z, 1/] + (Res Q)(zly)K) = (Rea Q)(z'y).
Finally, the restriction of (.1.) to  coincides with the form given by
(6.2.1). Indeed, for both forms, (dJd) = 0, and hence it is sufficient to
compare them on one element. We have (QoIQo) = (6-916-9) = (619) = 2.
Note also that the element K of £(8) is the.D the canonical central ele-
ment and that the element d is the scaling element.
57.6. Let i = - EB  $ + be the triangular decomposition (1.3.2) of the
Lie algebra a. Then the triangular decomposition of £(a) can be expressed
88 (ollows:
A 0
£(8) = n_ EB  EB R+, where
ft_ = <,-IC[C- 1 ] 0 e+ + » + C[t- 1 ] @ _I
0 0 0
n+ = <'Crt] 0 (n_ + ») + Crt] 0 n+.
The Chevalley involution WI of lei) can be expressed in terms of the
ChevalJey involution  of i as follows:
w(P(t)   + >'K + JJd) = P(t- 1 )  (z) - >'K - pd,

Ch.7
Affine Alge6ral III Central EztenlioRI of Loop Alge6na.
103
where P(I) E £, Z E 8; .-\. p E C. Indeed. we obviously have w(e.) = -Ii.
W(/i) = -ei (i = 1, ..., I) and wi" = - ide Furthermore, w(eo) = w(t @
Eo) = t- 1 c:.(Eo) = _,-1 @ Fo = -/0 and similarly, w(/o) -= -eo.
Analogously the compact involution "'0 or leg) can be expressed in terms
of the compact involution :'0 of g:
wo{P(t) 0 z +)'K + pd) = P(t- 1 ) @c:,o(z) - XK - jjd,
where for PC'> = E Cj,J we set 1'(') = E CjC J and a denotes the complex
conjugate of a E C. Hence the compact (orm of £(g) = (space of polynomial
maps from the unit circle to the compact form of g) + iRK + iRd.
Finally, we have
(P(t) 0 zIP(t) 0 z)o = -(pet) 0 zIP(c- J ) 0 c:,o(z)
= Re8(t- 1 P(') P( t-I»(-%Io(z»
= E ICil' (zlz)o.
j
One deduces that the Hermitian form (.1.)0 is positive-definite on the
subspace .e(9) of l(g), using the fact that it is positive-definite on 9
(see Remark 7.9 d) below). A more general approach will be developed
in Chapter 11.
5T. 7. We briefly explain here the physicists' approach to affine alge-
bras. Let 9 be a simple finite-dimensional Lie algebra with the normalized
invariant bilinear form (.1.). Let z. be a basis of g, and let
[z., zJ) = E liZI:, where li E c.
.
We denote the element t n 0 Z of .e(9) by z{n). Then the elements zn) (n E
..
l) and K form a basis of £(8) and obey the following commutation relations
(cr. (7.2.1»:
(1.7.1)
[ (m) (n) ] _ ,. (m+n)  ( I )
z, ,Zj - L..J lij%' + mUm,_R Zi Zj K.
,
Physicists associate to z e 9 tbe generating series:
z(z) := E z{n)z-n-l,
Rei

104
Affine Alge6na. CI. Cen'ral Ezten,ion. of Loop AIge6ral
Ch.7
called the CUrTent of z. Here z is a formal parameter. In order to write
down the commutation relations between currents, we need to introduce
the formaI6-functiora:
-1  ( %1 ) "
6(Z1 - Z2) = Z2 LJ - ·
tiE I %2
Its basic property is
(7.1.2)
Res. a =0 /(%1)6(%1 - Z2) = /(%2).
Here, for I(z) = E IJzl, one defines Res,=o I(z) = 1-1, It is straight-
lEI
forward to check that
(7.7.3)
(Z(ZI)' 11(%2)] = [z, ](%1)6(%1 - %2) - K(zfll)61 (%1 - %2).
In this form the algebra £(g) is called tbe current algebra.
The following lemma is very useful for calculations:
LEMMA 7.7. One has the following equality of formal power series in Z[l
and zr 1 whenever both sides make sense:
(7.7.4) /(%1.%2)6(%1 - %2) = 1(%2,z2)6(%1 - %2),
(7.7.5)
I(ZI,Z2)6a(ZI - Z2) = I(Z2,Z2)6a(ZI - Z2) -1;.(ZI,Z2) l'a:':a6(Zl - Z2).
Proof. MuJtiply both .ides by zr and check the equality of the Re'.t=o
using (7.7.2).
o
One also easily checks the following relation:
(7.7.6)
d
dm((z» = zm(zd; + (m + 1»(:).
Given two Vir-modules V and "'1, and a sequence of operators Fj : V 
t j E l, the generating series F(z) = E 1';:-J-A is called a primary
jEI
field of conformtJl weigA' A if
d
(d m , F(z)] = zm(zd; + A(m + l»F(z), m E l.
Equation (7.7.6) shows that currents are primary fields or conformal
weight 1.

Ch.7
Affine Alge6rtJ 4' Central E:rftn,ion, 01 Loop Alge6ral
105
57.8. As we have seen, the (nontwisted) affine algebras can be described
entirely in terms of the corresponding simple finite-dimensional Lie alge-
bras.
In this section we give an explicit const.ruction of the "simply-laced"
simple finite-dimensional Lie algebras, i.e., those or type At, DI., and EI.
(l = 6,7,8). The remaining cases will be treated in 17.9.
Let Q be the root lattice or type A" Dt, or E" and let (.1.) be the bilinear
symmetric form on Q 8uch that (see 56.7):
J1 = {o e Q I (oJo) = 2}.
Let e : Q x Q --+ {:f:l} be a function satisfying tbe bimuJtiplicativity
condition
f:(a + a',fJ) = €(Q,{J)£(o',{J),
(7.8.1)
E(a,fJ+p') = £(,J3)!(QtfJ') for Q,Q'.{J,{J' E Q,
and the condition
(7.8.2)
£(o,oQ) = (_1)1(0 1 0) (or Q E Q.
(Recall that (ola) E 2Z for a E Q.) We call such a function an asymmetry
function. Since (alP) E Z for a, fJ E Q, replacing a by a + (J in (7.8.2), we
obtain
(7.8.3)
£(a,p)£({J,) = (_1)(0 1 -") for Q,{J E Q.
An asymmetry function £ can be constructed 88 follows: choose an ori-
entation of the Dynkin diagram, Jet
. i
£(Oi, c:rj) = -1 if i = j or if 0-+0
i J i j
£(Qi,Qj)= 1 otberwise, i.e., 0 0 or 0+-0,
and then extend by bimultiplicativity. It is clear that then (7.8.2) holds for
all 0 E Q. (More generally, choose a Z-buis Pl,... ,PI. or Q, let E(Pi,Pi) =
(-I)i(PiIP;), £(Pi,pj) = (_I)(6 i I6;) iri < j and = 1 ifi > j, and extend by
bimuJtipJicativity. Thul, £ "breaks the symmetry" of (.'.».
Now let  be the complex hull of Q and extend (.f.) from Q to  by
bilinearity. We shall identify  with . using this form. Take the direct
sum of  with I-dimensional vector space& CEo. Q E :
(7.8.4)
g =  ED (E!)CE o ).
oE6

106
Affine Alge6ro, ca. Cen'ral Ez'en,ion, of Loop Alge6nJ'
Ch.7
Define a bracket on " 88 follows:
(7.8.5)
[h, hi] = 0
[h, Eo] = (hIQ)E Q
[Eo, E-o] = -0
(Eot E] = 0
[Eo, E] = £(0, fJ)Ea+,
if h, h' E  t
if h E  tOe At
if Q e A,
if a, {J E A, 0 + 11   U {O}t
if a,p,Q + P E 4.
Define the symetric bilinear form (.1.) on g extending it from  as followl:
(7.8.6) (hIEo) = 0 if h E ,Q E ,
(EoIE,B) = -6 ot -(J if 0:, {j E .
PROPOSITION 7.8. IfQ. the root lattice of type Al, Dt, or Et and t .
an asymmetry function on Q x Q, then 9 is the simple Lie algebra of eype
Ai, DI., or EI., respectively, the form (.1.) being the normaJj.ed invariant
form.
Proof. Let 1:, JI. z e g be from either  or one or the Ea. If one or these
elements lies in , the Jacobi identity trivially holds. Let now % = Eo.
y = E, z = E.,. If all Q + p,a + -y,fJ + 1   U {OJ, the Jacobi identity
trivially holds. ThuI, we may assume that a + {J E 4 U to}.
Note that given a.p e 4 we have:
(7.8.7)
o % (J E  iff (0113) = TI,
(7.8.8)
£(0,0) = -1,
£(0, {J)E(P, 0) = (_l)(ol).
If a+{J = O. consider four cases: 1) both 0%1  L\U{O}, 2) 0+1 = 0 or
0-"'( = 0, 3) 0 + 7' e 4, 4) a -1 E A; the Jacobi identity trivially hold.
in cases 1) and 2), and reduces to conditions £(,,0)1:(1 + 0,-0) = (010)
in case 3) and £(-0,1)£(-0 + 1.0) = (010) in case 4), which hold. due
to bimultiplicativity or E. (7.8.7) and (7.8.8). Thus, we may aaaume that
allOt + p,o + "'(, {J + 7' e . i.e., that (alP) = (011) = (PI-r) = -1. Then
10 + fJ + 11 2 = 0, hence 0 + {J +., = 0 (tbil is the place where we use that
(.1.) is pOIitive-definite). The Jacobi identity reduces to the condition:
£(0.11)(01 + (J) + £(fJ.-r)(P + 1) + £(1. 0 )(0 + 1) = o.
or
£(0, fJ)(Ot + fJ) - £(fJ, -Q - (J)o - t( -Q - P. o){J = O.

Ch.7
Affiftt Alge6na. a. Cen'nJl Ez'en.ion. 01 Loop Alge6ra,
107
which holds due to bimuJtipJicativity of £, (7.8.6) and (7.8.7). Thus, 9 is a
Lie algebra.
Let (see S6.7)
n = n V = {al,.'. ,at}, ei = Eat' Ii = -E_ Oi .
It is easy to see that (g, ,fi, n V ) is the quadruple associated to the matrix
A of type At, D l , or E " respectively, hence, by Proposition 1.4 a), " is of
type At, Dt, or Et., respectively.
It remains to check that (.1.) is invariant; this is straightforward.
o
57.9. In this section we give an explicit construction of the non-simply-
Jaced simple finite-dimensional Lie algebras, i.e., those of type B" Ct. F4,
and G 2 .
Consider .the following (simply-laced) root lattices Q(XN) and their au-
tomorphisms jj of order r = 2 or 3. (We use the construction of Q(XN)
given in 56.7.)
Case 1. Q(Dt+l) : jj{Vi) = Vi for 1 S i S t. ji(Vl+l) = -V'+l;
Case 2. Q(A 2 1- 1 ) : p(11i) = -V2t+l-i;
Case 3. Q(E 6 ) : lI(Vi) := -V7-i (or 1 S i  6, 1i(V7) = V7;
Case 4. Q(D 4 ): ji(Ql) = Q3, ji(Q3) = a-t, ji(Q.c) = Ql, ii(a2) = 0'2.
We denote here by (.1.)' the normalized invariant form on Q(XN), by
il' the set of elements of Q(XN) of square length 2, by n' = {o).. · , oN}
the set of simple roots, etc. Note that in all cases, j."I(ll') = il', hence
II, E Aut(XN) is the diagram automorphism of Q(XN).
In all cases, fix an orientation of the Dynkin diagram S(XN) which is
invariant under 71, and let £( a, /1) be the corresponding asymmetry function
(defined by (7.8.4)). Let
g(XN) = 'E& ( $ CE)
oE6'
be the corresponding simple Lie alsebra, as defined in 57.8. It is clear
that the automorphism jj of Q(XN) indutes an automorphism p of S(XN)
defined by
(7.9.1)
1'(0) = ji(o). p(E) = a)'

108
Affine Alge6nJ' G' Cen,,.1 Ezten,iora, of Loop Algebra.
Ch.7
This automorphism bu the following characteristic: property:
(7.9.2)
p(e) = e(i)' p(/;) = f(i)' i = 1, ... ,N.
Such an automorphism is called a tl;tJgram tJutomorpAi,m or g(XN).
Introduce the following notations:
l = {o = a' e A' I ji(o') = o'};
A, = {Q = r- 1 (ji(a') + ... + pr(Q') J 0' e 'JJi(Q')  Q'};
6. = 6.. Ut; Q = Z;
E (j) j r;tl + + rj' } - 2 " / ·
Q = 17 Dji(a') · · · 1] "(Q')' W Ieee Tf - exp 1rt r.
Eo = E if a e At. Eo = EO) if 0 E A,;
V U ) = $ CE); (j) = {h E ' 1 p(h) = rI h}i
oE6,
g(j) = (j) + v(j), j = 1, . . . , r - 1;
 = (O); 9 =  $( EB CEo).
oEA
We can now state the result.
PROPOSITION 7.9. Let (XN, r) = (Dt+l' 2), (A2l-1' 2), (Es. 2), or (D.., 3).
Then
a) g(X N) = 9 ED g(l) (resp. = 9 EB g(l) ED g(2)} jf r = 2 (resp. r = 3), where g
is the fixed point set of JJ and g(l) (resp. g(j), j = 1 or 2) is the eigenspace
of JJ with eigenvalue -1 (reap. exp 27rij/3).
b) 9 is the simple Lie algebra of type Bt, C' I F 4 , or G 2 , respectively, and
its commutation relations are IS (ollows.'
(7.9.3)
[h,h'] = 0 j(hth' e.
[h, Eo] = (hIQ)' Eo jf h E . Q E L\,
[Eo. E-o] = -o(lesp. - rQ) jf Q e III (resp. QEd,),
[Eo. EtJ] = 0 i(o.{J E d, Q +11  dU to},
lEa, EtJ) = (p + l)€{Ck', {J')Eo+p if a. p, Q + {J E At
where p is the maximal positive integer such that Q - p{J E A.
c) The ormaljzed bilinear form (.1.) on 9 is given by the following formulas
(hlh') = (hlh')/I, (hIEa) = 0 ;{h,h' E ,Q E A,
(EoIEtf) = -6 o .-IJ (resp. - r6 a ,-I1) j(Q,/3 e l (resp. Ed,).

Ch.7
Affine AI,e6rtJ6 tJl Central Ezlen,;onl 01 Loop AIge6rtJI
109
d) d is the set of roots of 9 with respect to the Cartan Bubalgebra , A,
(resp. l) being tbe set of short (resp. long) roots, and Q is its root lattice;
A V = dt. U r,.
e) The sets of simple roots n (numbered as in Table Fin) are these:
Case J: Q1 =Q,...,Qt-l = Ql_IJ O l = !(Qt+ Q l+l);
Case 2: Ql = l(Qi + Qt-l)t. .., Ot-l = l(o-l + Qt+l)' Ot = Q;
Case 3: Ql = a, Q2 = Q, 03 = i(o + Q), 04 = i{Ql + Q);
Case 4: Ql = Q,Q2 = l(Q + Q + a).
f) Letting fit. = n n t., 0, = n n " we have: n v = fit U rD,.
g) (g, g(j») = g(j), and the g-module g(j) is irreducible with highest weight
8 0 (= -lowest weight) listed below:
Case J: (Jo = 01 + Q2 + ... + Qt;
Case 2: 6 0 = Ql + 202 + .. · + 2al-l + Q/.;
Case 3: 9 0 = Ql + 2 Q 2 + 3 0 3 + 204;
Case 4: 8 0 == 0'1 + 2 Q 2-
Proof The proof of a), c), d), e), f), aod g) is straightforward; the first
part of b) follows from d), e), and Proposition 1.4 a). All commutation
relations (7.9.3) except (or the last one are clear.
If one of the Q, {J E  is Jong and Q + IJ E , we clearly have:
(7.9.4)
[Ea. E] = £(Q'tP')Ea+fJ,
and Q - {J i 4, hence the last formula in b) holds in this case. It remains
to consider the case when both Q and fJ are short and Q + fJ E A. First,
let r = 2; then p = 1, 0 + /J E I., and we bave to show that
(7.9.5)
(E a , EIJ] = 2E(a', 1J')Ea+lJ.
This follows from the following fact:
(7.9.6) if r = 2 and Q',P' E '\AI., Q' + P' e t!.', then a' + Ji(P');. '_
Since the Z-span of o',{3', J1. (a'), and p, (fJ') can be only one of the root
lattices Q(A n ), n = 2,3, or 4, it suffices to check (7.9.6) in these cases only,
which is straightforward. Hence) we have: [EQ) E.6] = [E, + E(Q'» E, +
E;.<.6')] = c(a') fJ')E, +.6' + c( JL ( Q'), JL (fJ')) (Q' +.6') = 2c(0', fJ')E, +(J"
Finally, let r = 3. Then either 0 ' + (J' E d"p = 1, and exactly one
of Q' + Ti(fJ') , Q' + p 2(p') lies in d', or Q' + P' E A"p = 2, and none of

110
Affine A/ge6N' 4. Central Ez'en,;on, 01 Loop Alge6nJ'
eh.7
0' + p({J') , Q' + ji2(p'} lies in d. Hence. in botb easel the last equation of
(7.9.3) holds.
o
Remark 7.9. a) (7.8.5) is a special case of (7.9.3) for r = 1, since in the
simply-laced case p = o.
b) Commutation relations (7.9.3) can be extended to the whole 8(XN) as
follows. Let EiO) = Eo = E if 0 E At and let aU) = ,,; 1'(0') + ... +
'1 rj pr(o') if Q E A,. Then (0 S i.i  r - 1):
[h(i) I h(j)] = 0
[h(i) t Ei/>] = (h(I)I()' E+J)
[E), E] = -a(i+J)(resp. - a)
[E), E)] = 0
[E), )] = (p + l)t:(a', (1)::)
where p is as in (7.9.3).
if h(i) e (i), h(J) e (J).
if h(') e (I), Q E A,
if 0 E d,(resp. Q e 4l),
if cr,{J E A, a + 11  4 U to},
ifotfJ,Q+{JEd.
c) The Chevalley involution wand the eompact involution "'0 of g(XN) are
given by (0 E 6):
w(Ea) = E-o, w(Q) = -0;
wo(Eo) = E- oJ '-10(0) = -0.
d) It (ollows from c) and Proposition 4.9 that the Hermitian form (.1.)0 is
positive..definite on g(Xt). where XI. is of finite type.
57.10. For applications in Chapter 8, we shall now take care of the
remaining diagram automorphism:
Case 5: Q(Au) : j7(Vi) = -V2l+2-i.
In this case there is no ji-invariant orientation; we consider tbe orienta-
tion o-tO-+.. · ---. 0 instead. Then the diagram automorphism p of g(A2l)
is defined by
(7.10.1)
pea) = j.i(o), p(E) = (_l)1+ht 0 Ei.co).
Let
dt (resp. 6,)
= {l(o' + j.i(a'» I a'  j.i(a') and (o'Ij.i(a'»' = 0 (reap. -;: On,

Ch.7
Affine Algell,.., a' Cen,,.1 Ez'en.;oft' of Loop Alg6rtJ,
111
4 = dl. U A"
Eo = E, - (_l)hto EJ,(o') if a € At,
Eo = J2(E, - (_l)htCt (a'» if () E 6.,
 = {h E ' I p(h) = h}. (1) = {h E ' I p(h) = -h}.
g=(EaCEo),
aE6
g(1) = (1) $ ( E C(E:- + (_l)htCt Ej;(o») $ ( E CEo).
oE6' aE4'
aJf p( a) 0'="( a)
Then we obtain the following extension of Proposition 7.9.
PROPOSITION 7.10. a) g(A2t) = ,,g(l).
b) 9 is the simple Lie algebra of type BI., and its commutation relations
are given by (7.9.3).
e) The normalized bilinear form (.1.) on 9 is given by
(hlh') = 4(hlh')/It (hIEa) = 0 jf h, h' E J Q E At
(EarIEtJ) = -46O'._(resp. - 86cr,_) ita e di (resp. e L\,).
d) Same as Proposition 7.9 d) (with r = 2).
e) The set of simple roots n is this:
1 ( I I ) I e ' I ) 1 ( " )
Ql = 2 QI + Q2l ,QM= 2 Q2 + Q21.-1 ,..., QI. = '2 QI. + QI.+1 ·
f) Letting nl = n n I., n, = n n" we have n v = 2Dl U 4n,.
g) (8, g(l)] = g(l) and the g-module g(l) is irreducible with highest weight
(= -lowest weight) 9 0 = 2 Q l + 2Q2 + ... + 2 o t.
o
Remark 7.10. Letting EO) = Ea, EI) = E, + (-l)htorEJ;(o') if a EAt,
El) = v'2(E, + (-l)htoE(a'» if a E 6." 6. m = (a E 6.' I ji(a) = a},
El) = E if a E Am, formulas (7.9.3) extend in the same way as described
in Remark 7.9.

112
Affine Alge6na, a, Cenlral Ez'en,ion, 0/ Loop Algebra,
Ch.7
51.11. A generalized Cartan matrix of infinite order is called an infinilt
affine motriz if everyone of its principal minors of finite order is positive.
By Theorem 4.8a), it is clear that a complete list of infinite affine matrices
is the following (see Exercise 4.14):
Aoo, A+ oo , Boo, Coo and Doo.
Let A be an infinite affine matrix, and let g'(A) be the associated Kac-
Moody algebra (defined in Remark 1.5 for an infinite n). These Lie algebras
are called infinite rank affine algehra,. For these Kat-Moody algebras we
have d = A r ,; given Q E 4+, we denote by QV the element of ' such
that QV E (90,8-0] and (Q, OV) = 2, and we put A v = {oV E 'IQ E
a} to be the set or dual roots. Here we give an explicit construction of
these Lie algebras, which generaJizes the usual construction of classical
finite-dimensional Lie algebras. We also construct certain completions and
central extensions of them, which play an important ro)e in the theory of
completely integrable systems (see Chapter 14).
Denote by 9 1 00 the Lie algebra of all complex matrices (a e ; )'.i EZ, such
that the number of nonzero O;j is finite t with the usual bracket.. This Lie
algebra Bets in a usual way on the space COO of all column vectors (ai)iEZ,
such that all but a finite number of the aj are zero. Let Eij E gl«J be the
matrix which has a 1 in the i, j-entry and 0 everywhere else, and let Vi E Coo
be the column vector which has a 1 in the i-tb entry and 0 everywhere else,
so that
Eij ( Vj) = Vi.
Let A = Aoo; then g'(A) = ,1 00 := {a E glooltra = O}. The Chevalley
generators of g'(A) are 88 follows:
fi = Ei.l+l t Ii = Ei+l.i (i e Z).
so that
n V = {Qr = Ei.i - Ei+l.4+1 (i e Z)}
is the set o( simple coroots. Then ' consists o( diagonal matrices, and n+
(resp. n_) of upper- (resp. lower..) triangular matrices. Denote by li the
linear (unction on ' such that li(E jj ) = 6 ij (j e Z). Then the root system
and the root spaces of g/(A), attached to nonzero roots. are
A = {li - lj (i 1: j, i,; e l)}j Gfi-fj = CEij,
fi - €j being a positive root if and only if i < j.

Ch.7
Affine Alge6ra, ca, Central EzlenlioRI of Loop AIge6ral
113
The set of simple roots is
n = {Oi = li - (i+l (i E Z)}.
The set of positive dual roots is
(7.11.1)  = n V U {Qr + at+l + ... + oJ, where i < jj i,j E I}.
The description for A = A+ oo is similar, replacing Z by l+.
The remaining infinite rank affine algebras. of type Xoo, where X = B,
e or D, are 8ubaJgebras or the Lie algebra gloo. and consist of matrices
which preserve the bilinear form X, i.e.,
g'(Xoo) = {a e g/(Aoo)IX(a(u), v) + X(u,a(v» = 0 for all u, v e COO}.
These bilinear forms are as follows:
B(Vi,Vj) = (-1)i6 i ,_j (i,j e Z)
C(Vi,Vj) = {-1)'6;,_;+1 (i,j E Z)
D(Vit Vj) = 6i,-j+l (i,j e Z).
We describe below the Chevalley generators ei. Ii (i E Z+), the set of
simple coroots n v , the root system , the set or simple roots fit and the
set of positive dual roots + of g'(A), where A = Boo, Coo or Dca. In all
cases, ' consists of diagonal matrices and n+ (reep. "_) consists of upper-
(resp. lower-) triangular matrices.
Boo : eo = EO.l + E-l,O. ei = Ei,.+1 + E-i-l,-i,
/0 = 2(E 1 . 0 + E O .- 1 ), Ii = Ei+l.i + E_ i .- i - 1 (i = 1,2,. ..);
n V = {Q6 = 2(E_l.-1 - E I,I ), err = Ei,i + E-.-l.-i-l
- Ei+l. i +l - E-i,-i (i = 1,2,. .. )};
 = {:!:fi % lj (i  j, i,j E Z+), :tfi (i E Z+)};
Sfi-fj =' C(EiJ - (-I)'+J E-j,-i), i  j, i,j E Z;
U={ er O=-ll,Qi=€i- l i+l (i=1.2,...)};
A = n Y U {4r + 4r+1 + ... + 4; (i < j, i,j E Z+),
Q + 2Qr + · .. + 2ar
+ Or"'1 + ... + OI;(i < j, i,j e Z+)}.
Here the li are viewed restricted to ', 80 that (i = -l-i.
Coo : eo = Eo. I J ei = Ei,i+l + E-i,-i+l'
/0 = El,O, Ii = Ei+l,i + E-i+lt- i (i = 1.2, ...);
n Y = {4 = Eoo - Ell, 4' = E... + E_.._i - E i +1.i+1
- E-i+l.-i+l (i= 1,2,...)};

114
Affine AI,e6"" G' Cen'ral E%ten,ion, 0/ Loop Alge6ra.
A= {%l.:i:tj. %2£; (ijJ i,j= 1,2,...)};
82ci = CE,,-i+ll 8 C i- C i = C(E'J - (_l)i+J E-J+1.-i+l)i
n={OO=-2fl, Q.=l,-l,+1 (i=lt 2 ....));
d = n V u {or + crr+l + .. · + or (i < j. i,j e Z+).
y v v v ( . ... E l )}
2Qo + · · · + 2ai + ai+l + · · · + QJ · < J, ., J +.
Ch.7
Here the (, are viewed restricted to ', 80 that f, = -(-i+l'
Doc : eo = EO.2 - E-l.1' ei = E..I+l - E-.,-I+l'
/0 = E 2 .0 - EI,-l. I, = Ei+l.' - E-i+l.-i (i = 1,2....);
n Y = {o = Eo,o + E-l.-1 - E2.2 - E I . 1 ,
or = Ei,f + E_.._, - Ei+I,'+l
- E-i+l.-i+l (i= It 2 ....)};
A= {:J:fi:tfJ (ij; i,;= 1.2,...)};
9f.-fj = C(Eii - E-i+l.-i+l);
n = {Go = -(I - (2, Oi = f. - (i+1 (i = 1,2,... )};
d = n V u {or + Or+l + .. · + or (1 S i < j),
o + Q + · · · + oJ (j = 2t 3, · · · ),
Q + Qi + 2Q
+ ... + 2Qr + Q'+1 + ... + 01(1  i < j)}.
Here the £. are viewed restricted to '. 80 that li = -£-1+1.
51.12. Now we turn to the description or a completion and its central ex-
tension for an infinite rank affine algebra. More generally, let A = (Oij )i.jl
be an infinite generalized Cartan matrix. such that every row (and hence
column) contains only a finite number or nonzero entries. Let g' (A) = e g
Q
be the associated Kac-Moody algebra. We denote by I(A) the 8ubspace of
n g consisting of the expressions of the form u = r: 00eQ, where aQ e C,
a a
eo e g, and such that the set {jlj = ht Q with 0 0 1: O} is finite. It is clear
that we can extend the bracket (rom 8 / (A) to g(A) by Jinearity. The Lie
algebra g(A) contains1j := n cor, the completed Cartan 8ubalgebra.
lEI
Denote by 9100 the Lie algebra of all complex matrices (aij)iJEI, such
that Oij = 0 for Ii - jl sufficiently large. with the usual bracket. It acts in
a usual way on the vector space COO of columns (Oi)iEI, where all but a
finite number of the a, are zero.

Ch.7
Affine Algebral ell Centrol Ezten,ion, of Loop Algebral
115
If A is an infinite affine matrix of type Xoo = Aoo (resp. Boo, Coo or
Doo), we denote g(A) by Zoo. Then, clearly, Zoo is isomorphic to iioo (resp.
the subalgebra of 91 00 t which consists of matrices preserving the bilinear
form B, C or D). A completed infinite rank affine algebra, denoted by Zoo,
is the central extension of Zoo defined 88 follows.
The Lie algebra 0 00 has a 2-cocycle t/J defined by:
1/J(Eij, E j ;) = 1 = -1/,J(Eji, Eij) if i  0 and j  I,
(7.12.1)
t/J(Eij, Emn} = 0 otherwise.
One easily checks that this is a cocycle (see Exercise 7.11). Then if A =
Aoo (resp. BOOt Coo or Doo), we put r = 1 (resp. i, 1 or i), and let
Zoo = Zoo El) CK be the Lie algebra with the following bracket:
[aQ)'\K,6$JJK]=(a6-6a)EBrtjJ(a,h)K (o,6€zoo; '\,pEC).
The elements t!i, Ii e g'(Xoo) C Zoo (defined above) are called CAe-
valley generators of zoo, and 6 = 1j + CK is called the Cartan IuhtJlgtbrtJ.
,.."
The elements of the set flv = {ci = Q + K, &r = Q for i :F O} are calJed
;mple corootJ. It is easy to see that we have the usual reJations:
[oj, aj] = 0, [e.,lj] = lJ;j&'t, [a;', ej] = Oijej,
[ai, Ij] = -Oij!jt (ad ei)I-0 i iej = 0, (ad /i)l- Cl i j fj = o.
Remark 7. It. The Chevalley generators {ei,I.}iel generate a subalgebra
(E9 CQr) e ( E9 gOl) of Zoo 1 which is isomorphic to g'(A). This'-follows
tEl aE4
from Corollary 5.11b). Note that the principal gradation of g'(A) extends
I in a natural way to a gradation of Zoo 1 called its principal gradation.
We &)so have the following expressions for the canonical central element:
a oo :
K =  a Y .
 I'
iEZ
K = aX + 2Ear;
;1
hoo :
Coo :
K = E Qr;
i > O
K = oX + o + 2 E or ·
i2
d oo :

116
Affine Algebra, cal Centrol Eztn,iora' of Loop Algebra,
Ch.7
57.13. Exerei.el.
7.1. Let p be a Lie algebra.  Cpa finite-dimensional diagonalizable
subalgebra, p = ED Po the root space decomposition. Show that every
Q
C-valued 2-cocycle .p on JJ is equivalent to a cocycle .po (i.e., ,,(z, y) -
1/1o(z,)I) = f([z. y)) (or some I e p. and all Z,II e p) such that tPO(JJa, pp) =
o for 0 + {J  O.
[p operates on the space of a11 2-cocycles in a natural way, so that  is
diagonalizable. Show that an eigenvector with a nonzero eigenvalue is
equivalent to 0.]
7.1. Let p be a Lie algebra with an invariant symmetric bilinear C-valued
(orm (.'.) and Jet d be a derivation of p such that (d(z)ly) = -(zld(II» , z,
11 E p. Show that tIJ(z,JI) := (d(z)ly) is a 2-cocycle on p. Let p' be the
corresponding central extension. Show that d can be lifted to a derivation
of the Lie algebra ;', so that we obtain the Lie algebra ;; = p' + Cd.
7.3. Let p be a Lie algebra with a nondegenerate invariant bilinear C-vaJued
form (.1.). Let R be a commutative associative C-algebra with unity, and I
a linear functional on R. Extend the form (.(.) to the complex Lie algebra
p = R @c P by
(rl @Pllr2P2) = !(rl r 2)(PllP2).
Let a be a derivation of Rj extend it to a derivation D = a @ 1 of p. Show
that
(D(z)ly) = -(zJD(y»
if and only if Ila(R) = o. Apply the construction of Exercise 7.2 to p with
R = £, p = 9 a simple finite-dimensional Lie algebra, fer) = Res r, a = .Jr,
and show that one obtains & nontwisted affine Lie algebra £(g).
7../. Let R be as in Exercise 7.3, and let 9 be a Lie algebra. Show that
der(R  8) = (der R) 0 I. + R  der g.
[Choose a basis ri of R and write: d(lz) = E ridi(z), z E g. Show that
i
d i E der g. Replacing d by d - E ri  d i , we can assume that d( 1  z) = 0,
i
Z E g. Replacing g by ita associative envelope. we have d(P  z) = d(P @
1)(1  z) = (1  z)d(P  1). Deduce that d e (del R) @ 1.]
1.5. Deduce from Exercise 7.4 that for a finite-dimensional simple Lie al.
gebra gone haa:
der £(g) = D + ad £(8).

Ch.7
Affine AI,e6nJ, II. Central E%.tra.iora. of Loop AIge&ra.
117
7.6. Let p be a Lie algebra with a nondegenerate invariant bilinear form
(.1.) and Jet Ul,U2,... be its orthonormal basis. Let,p be a C-valued 2-
cocycle on p such that for each i only & finite number of t/J( Ui, uJ) are
nonzero. Show that 1/J is of the form described in Exercise 7.2.
7.7. Prove that every 2-cocycle on the Lie algebra £(g) is equivalent to a
cocycle )..p, where  e C. and 1/J i. described in 57.2.
(Use the action of do on the space of 2-cocycles and apply Exercises 7.8.
7.4, 7.3.]
1.8. Show that £(9) is the universal central extension of the Lie algebra
leg) (cr. Exercise 3.14).
[Use Exercise 7.7.)
7.9. Show that a C-valued 2-tocycle on the complex Lie algebra
C[tl,tll,...,t.,t;l]C I, where I is a finite--dimensionalsimple Lie al-
gebra, is equivalent to a cocycle aD, where D = E  8' , with E  = 0,
i · i'
defined by
OD(P0z,Q0J1) = {coefficient at ('1... 1 .)-1 in D(P)Q)(zJg).
1.10. Consider the bilinear form
(,,) = tr(%JI) if 9 = at ft or .Pn t and
(zl.,) = 4 tr(zy) if I = 'On, n > 4.
Check that 191 2 = 2, 80 that the "normalized" cocycle t/J on l(l) is given
by
t/J(o, 6) = Rea tr( 6) ir I = .t n or 'Pn,
t/J(o, 6) = 4 Restr( 6) ir I = 'On, n > 4.
Show that the normalized standard bilinear rorm on £(g) is given by
(016) = constant term of tr(06). ir , = .t n or 'Pn,
1
(016) = 2 constant term or tr(06) ir .. = 'On, n > 4.
o
7.11. Let g= Slt+l(C). We keep all the noation of Exercise 1.13, and
o 0 0 0 0
add 0 on the top: 9. , 0, and n V . We have £(9) = 81t+.(£). Show that

118
Affin Algebral al Central Ezlen,ion. of Loop Algt6rtJ,
Cb.7
the space !(g) = ,It+l(£) $ CK  Cd with the bracket
[o(t) +)'K + JJd,Ol(t) + >'l K + Plcl]
dOl (t) da( t) da(t»
= (O(t)OI (t)-01(1)0(1)+ pi dl -pl t dt ) + Res tr( dt 01(t) K.
(1). 0
is the affine algebra of type At . Set  =  e CK EB Cd, extend lj from 
to  by li(K) = li(d) = 0, and define 6 e . as in 57.4. Set
eo = tEt+lt1'
/0 = ,-lE I ,l+l,
t
o = K - E or ·
1=1
o 0
Set n = {OOI fi}, n Y = !o I nY}. Show that (, 0, n Y ) is the realization
oC the matrix (oC type A/»:
( 2 -2 ) .
A = _ 2 2 If t = 1 ,
and
2 -1 0 . . . 0 -1
-1 2 -1 . . . 0 0
A= 0 -1 2 . . . 0 0 if l > 1.
........................... .
0 0 0 . . . 2 -1
-1 0 0 . . . -1 2
The root space decomposition of £(g) is
£(9) =  ED (
o
EB l' E.j) ED (EBt').
ij=t....,ttl .el
.j. .EI ,o
Show that £(9) = g(A), with the Chevalley generators e" I. (i = 0,.. . ,i).
Show that the set {(I - £j + ,6 (i  j, . E Z); ,6 (, e Z\O)} is the
root system or g(A). Show that its subalgebra n_ (reap. n+) consists of
all matrices from ,tt+l(C[t) (reap. ,tt+l(C[t- 1 )) such that the entries on
and under (resp. over) the diagonal are divisible by , (resp. ,-1). Show
that the Chevalley involution w of l(g) is defined by
w(o(I)) = -(transpose of 0(t- 1 » if a(l) e £(9),
w(K) = -K, w(d) = -d.

Ch. 7
Affi1J AIge6rtJI 01 Cenlral Ezleft,;oftJ 01 Loop Alge6rtJ$
119
7.1£. Introduce the CoJlowing basis or the Lie algebra 81 2 (£):
L 3 . =  (' _.), L 3 .+1 =  (
L:h-l = (t ) (8 E Z).
Then [L.,Lj) = CijLi+i (i.j E Z), where
c.j = -1, 0, or 1 according 88 j - i == -1, 0. or 1 mod 3.
i' )
o '
7.13. Show that the Lie algebra D does not admit a nonzero invariant
bilinear form. Deduce from Exercise 7.1 that every C-valued 2-cocycle on
D is equivalent to a cocycle t/J such that 1j1(d;, d j ) = 0 whenever i + j :F
O. Denote by Cr the vector space (over C) of such cocycles. Show that
dim C; = 2 and that the cocycles .p1 and 1/J2, defined by
tPl(d., d j ) = 6i.-jj; 1P2(d i . d j ) = 6i._jj3
form a basis of C. Deduce that Vir is the universal central extension of D.
7. Ii. Let E-, (1 E A). or (i = 1, ... J l) be a Chevalley basis of a finite-
dimensional simple Lie algebra S, i.e.,
[E7' E_...,J = 1 v ; [1 V , .,,'Vj = 0; [",V, Eo] == ({3, iV)E{j;
[EtJ, E-y] ::::: nJ3,...,EI3+'Yt where np,..., E Z, nl3. = -n_..,.-fJ.
(cf. 57.8 and 57.9 for an explicit construction of such a basis.) Then the
elements
E.,+ltl = t i  E.., (k E Z, "1 E A);
Ei'l = I" fi) Q (k E Z\{O}. i = 1,. ..,i);
0: 't (i = 1, . . . , l) ; k and d,
"
form a basis of £(g). Show that the Z-span of this basis is closed under
the bracket, by writing down all the commutation relations. This is the
"
Chevalley basis of .c(g).
7.15. Let g be a simple finite-dimensional Lie algebra and V a finite-
dimensional g-module. Then we may in a natural way define an £(g)-
module V = £, c V. Fix A E C and define an action  of the affine
algebra £( g) on V by:
d
1r/.C(,) unchanged; 1r(K) = OJ 1r(d) = ''di fi) 1 + A1v.
Show that this is an integrable l(g)-moduJe.

120
Affin Alge6rt1' I' Cta'nJl Ez'en,ion. 0/ Loop Alge6ra.
Ch.7
7.16. Let g be a simple finite-dimensional Lie algebra, (.1.) the normalized
000
invariant form,  a Cartan subalgebra, + 8 system of positive roots, P
o
their half-sum, (J the highest root. hoose a basis U1,. . . ,Ut of  and the
dual basil u 1 , . . . t u t . For each 0 e + chooee root vectors eo and e_ Q such
· . A 0
that (tole-a) = 1. Let n be the Cuimir operator or g. Finally, let .c( g)
be the nontwisted affine Lie algebra aasociated to 8, and set p = h v Ao + P
(where h V is the dual Coxeter number of l(;». Show that
(p, or> = 1 (i = 0, . . . ,l)
and that the Cuimir operator n for the affine algebra £(8) (see Chapter
2) can be written II followl:
.
a = 2d( K + h Y) + ()
I.
+ 2 E {E (t- n  _o)(tn  a) + E(t- n  u')(t n  u.)}.
n>l · .=1
- aE6
n
7.17. Let Ln = E Z Vi be the standard lattice in the Eucildean space Rn .
'=1
Define on Ln x Ln a bimultiplicative function E by letting
£( vJ, v,) = 1 if j  , and = -1 if j > }.
Show that by restricting to root lattices Q(At) C L'+l and Q(DI.) C L,
(described in 16.7), we obtain an asymmetry function.
n
7.18. Let iLn = E ZUi, where u, = lVi, and define on <iLn) x <iLn) a
1:1
birnultiplicative function € (extending the one from Exercise 7.17) by:
£( U j , Ui) = 1 if j S '. and = e. if j > i.
Show that by restricting to the root lattices Q(E 8 ) and Q(Er) (c !L s ), we
obtain an asymmetry function. Similarly, taking the lattice L6+Zu in R7,
where u = V1' and extending E by E( U, Uj) = £( u, u) = e¥ 1£( Uj, tl7) = I,
we obtain an asymmetry function for Q(Ee).
1.19. Let p be & Iubalgebra or £(g) or finite codimension. where 8 is a
simple finite-dimensional Lie algebra. Then there exists a nonzero Q e l
such that p ::> Ql @c 8.
(Show that for a subspace Y of finite codirnension in 1:., the space y2
contains a nonzero ideal of l.]

Ch.7
Affine AlgebrtJ' ., Ctntml E%tn6ion. 01 Loop Algebra,
121
1.10. Show that the adjoint representation of the affine Lie algebra" =
a1n(l) + CK + Cd induces the adjoint representation of the loop group
G = S Ln (£) on 9 given by tbe foDowing formula:
.
(Ad a)(z +"\K + I'd)
= (aza- 1 - pta'a- 1 ) + (.\ + Restr(a'za- 1 - 4Pt(a 1 a- 1 )2)K + pd),
where a E G, a' denotes the derivative of a by t. % E ,I n (£), and >-. #J E C.
[Use the (ollowing facts: a) Ad CI preserves (.1.); b) [a,] = -a';
c) (Ada)z == aza- 1 mod CK; d) (Ada)K = K.]
o
7.11. Let G be a connected simply-connected complex algebraic group op-
e
erating faithfully on a finite-dimensional complex vector space V. Thi8
o 0
action extends to the action or the loop group G := G(£) on V := £@c v.
o 0 . 0 0
Let 9 be the Lie algebra G,  a Cartan 8ubalgebra, QV C  the dual root
lattice; let V = E9 VA be the weigbt space decomposition with respect to
A
o 0
. Given -, E QV J define t E EndL V by:
,ey (VA) = t(,ey) VA.
.
Show that t" E G, thus giving an injective homomorphism I : QY -. G.
Show that, via the adjoint action of G on £(g), we have:
I(-r) . (t i @ eo) = Ct+(o.oy) @ eo (eo Ego);
l(t-r)I, = t_..,.
(Thus, we obtain a canonieal embedding of the group of translations T
.
into G.) Similarly, define an injective homomorphism I : pY .-. Ad G
o. 0
(where pv = {h e  I (, h) E l (or all Q E A}). Show that the group
- 0 -0 _
T := I(P Y ) acts on leg) by the same formulas. Show that the group T
---.... -
normalizes Wand that W := W t< T preserves QV and Q. Show that
- --
the group W+ = {w E W I w(4+) C A+} acts simply transitively on
(AutA).Q.
1.tl. Let 9 be a simple finite-dimensional Lie algebra and let R be a conr
mutative associative algebra with unity. Let n denote the space of all for-
mal differentials over R, i.e., expressions of the form Jdg, where /, 9 e R,
with relation d(lg) = Jdg + ,4/. Then the univenal central extension or
the complex Lie algebra R c 9 is
o -+ n/dR -t ill := lit $ (OkldR) ..... SJR .-. 0,

122
Affine Alge6rta, al Cen'rol Ezten,ion, of Loop Alge6N11
Ch.7
where the bracket on 8R is defined by:
[rl @ 91 t r2 0 92] = rl r2  [,1,92) + ('1192)"1 dr2 mod dR
(This is a generalization of Exercise 7.9.)
7.!3. Let T(z) = E d n z- n - 2 be the generating series for Vir. Show that
nEI
[d rn . T(z)l = zm(zl; + 2(m + l»T(z) + mr zrn- 2 c. (T(z) is called the
energy-momentum tensor. It is & primary fieJd if and only if the "conformal
anomaly" c is 0.)
7.14. Consider the representation of the Lie algebra t) (= representation or
Vir with c = 0) on the .pace UQ, of "densities" or the form P()tO(dt)',
where a,{J are some numbers and P(t) E £. Show that in the basis v. =
t i + a (dt)J1 this representation looks 88 follows:
dn(v,,) = -(k + Q + fJ + pn)tJ n +".
Show that F(z) = E FtZ-t-A, where Ft : V - VI, is a primary field of
iEI
conformal weight A if and only if the map F : UA-l.I-A 0 V -+ VI defined
by F( Vi @ V) = F (V) is a Vir-module homomorphism.
7.15. Let 9 be a simply-laced simple Lie algebra as constructed in 17.8.
Given Q E A, let
Ro = (exp - ad E_o)(exp - ad Ea)(exp - ad E-o).
Show that Ral = ra, Ra(E) = -£(Qt{J)Er.() if (alP) :F 0 and
Ra(E) = E otherwise.
7.96. Let £(0,/1) be a 2-cocycle on Q witb vaJue. in {:t:l}, i.e.,
E(a, (3)€(a + (3, f) = E({3, "()e(o, (j + 1') for a, (J E Qt
€(O, 0) = 1 (and hence c(O, Q) = E(a, 0) = 1 for a E Q);
suppose that in addition
£(0, -0) = (_l)!(olo) and £(0./3)£(/3,0) = (_l)(oIJJ), 0,/3 E Q.
Then Proposition 7.8 still holds.

Cb.7
Affine Algebr(J tJ CentrtJl Ez'en;on of Loop Algebras
123
7. £7. Let Q be tbe root lattice of a simply laced simple Lie algebra. Let
B : Q x Q --+ Z be a biJinear form such that
B«(),j3) + B(I3» 0) = (QIP).
Then £(a,l3) = (_l}B(a. tJ ) is an asynunetry function. Write B(Q,IJ) =
(RaIP). Then the conditions on B are equivalent to R(Q) c Q. and R +
R. = 1. Show that ifw E AutQ is such that Q C (l-w)Q., then R = l-w
satisfies these conditions. Show that the Coxeter element has the above
property.
7.£8. Let V = V_ EB V+ be a vector space over C (in general infinite di-
mensional) represented as a direct Bum of two subspaces. Let gl.(V) denote
the Lie algebra o( endomorphisms or V which have the form a = (: ::)
with respect to the above decomposition, where 03 : V- --+ V+ has a finite
rank. Show that
f(a, b) := trv_ a2 b 3 - trv_ b 2 a 3
is a 2-cocycle on the Lie algebra 91. (V). Show that the restriction of this
cocycJe to the 8uba]gebra of finite rank endomorphisms is trivial. Show
that for V = m CVi. V_ = E CVil V+ = E CVi, the cocycle / restricted
iE I i>O iO
to 9 1 00 coincides with the cocycle tP defined in S7.12.
7.£9. Let V = Crt. t- I Jn be the natural stn(C[t, t- 1 J)-module. This ex-
tends to the module over the Lie algebra of differential operators a =
sf"{C{t,c 1 ,  ]). Set V+ = (tCft))", V_ = (Cft-I]n C V. Show that the
restriction of the cocycle f of Exercise 7.28 to the subalgebra Q C gl.(V)
is given by the following formula (f E Z+ j m, m' E Zj a, a' E gfn (C)):
I ( l+rn ( d ) 1 /.'+m' ( d ) l' ' )  ( ' )( l) l t l t " ( m+l )
t dt a,t dt a = Um,-m' traa - .. Hi'+! .
S1.14. Bibliographical notes and comments.
Except for the explicit formula for the central extension, the realjzation
of nontwisted affine Lie algebras was given by Kae [1968 B] and Moody
[1969]. The formula for the cocycJe has been known to physicists for such
a long time that it is difficult to trace the original source. Proposition 7.B
(in the form given by Exercise 7.26) is due to Frenkel-Kac [I 980J.
The Virasoro algebra (first studied by Virasoro (1970]) plays a promi-
nent role in the dual strings theory (see. e.g.. Mandelstam [1974], Schwartz
[1973J, [1982]). Mathematicians started to develop a representation the-
ory of the Virasoro algebra quite recently (Kac (1978 B]. (1979). {1982 BJ,

124
Affine Algebra, 61 Cn'ral Ezfen,ion. oj Loop Alge6nJ'
Ch.7
Frenkel-Kac [1980}, Segal (1981), Feigin-Fuchs [1982J, 11983 A, B), (1984 A,
B}t and others). A survey of some of these results may be found in the book
Kac-Raina {1987}. .
This has recently become a topic of interest among physicists in con-
nection with statistical mechanics and a revival or the dual strings theory.
The conCormally invariant field theory, which i. tbe Coundation of both of
these remarkable developments. originated in papers by Belavin-Polyakov-
Zamolodchikov [1984 At B]. The notion of a primary field, briefty discussed
in 57.1, is a key notion of this theory. A surveyor recent developments in
string theory may be found in the two volume book by Green-Schwartz-
Witten {1987].
Exercise 7.2 is due to Kuper8hmid [1984] and Zuckerman (unpublished).
The first published proof of Exercises 7.7 and 7.8 that I know is'in Garland
[1980]. Exercise 7.14 is due to Garland [1978]. In this paper Garland
studies in detail the I..form of the universal enveloping algebra of an affine
l,ie algebra. Exercises 7.28 and 7.29 are taken from Kac-Peterson [1981].
Exercise 7.19 is due to R. Coley [1983]. Exercise 7.20 is taken from Frenkel
(1984], Segal [1981], and Kac-Peterson (1984 B]. Exercise 7.22 is due to
Kassel [1984). The rest of the material of Chapter 7 is fairly standard.
There has been recently a number of papers dealing with the groups
associated to affine algebras. Such a group is a central extension by C.
of the group of polynomial (or analytic, etc.) maps of C)( to a complex
simple finite-dimensional Lie group. The torresponding "compact form't
is a central extension by a circle of the group of polynomial (or ana-
lytic, or Coo, etc.) loops on a connected simply-connected compact Lie
group. Thus. there is a whole range of group. associated to an affine Lie
aJgebra (or rather a certain completion of it). The group of polynomiaJ
maps iS t naturally, the minimal associated group; this is a special case of
groups discussed in S3.15. Various aspects of the theory of loop groups may
be found in Garland [1980], Frenkel (1984], Pressley (1980J, Segal (1981J,
Atiyah-Pressley (1983], Goodman-Wallach [1984 AJ, Kac-Peterson [1984
B], [1985 B], [1981}, Freed (1985]. Pressley-Segal (19861, Mitchell [1981J,
[1988J, Kazhdan-Lusztig [1988], Mickelsson [1987], and others.

Chapter 8. Twisted Affine Algebras
and Finite Order Automorphisms
S8.0. Here we de.cribe a realization of the remaining, "&wi8ted" affine
algebras. This turns out to be cloeely related to the Lie algebra of equiv-
ariant polynomial maps from ex to a simple finit&*dimensional Lie algebra
with the action of a finite cyclic group. As a aide result or this construc-
tion we deduce a nice description of the finite order automorphillJl8 or a
simple finite-dimensional Lie algebra, and in particular the cl....ification of
symmetric spaces.
SS.1. Let 9 be a simple finite-dimensional Lie algebra and let tT be
an automorphism of 9 satisfying (fm = 1 for a positive intege m. Set
( = exp 2:'1 . Then each eigenvalue of (f has the (orm (i , j e Z/ml. and
since tT is diagonalizable, we have the decomposition
(8.1.1) 9 = ffi Ii,
jEl/ml
where gj is the eigenspace of (f for the eigenvalue f. j . Clearly, (8.1.1) iI a
Z/ml-gradation of g. Conversely, if a Z/ml-gradation (S.I.I'is given, tbe
linear transformation of 9 given by multiplyinl the vectors of gj by (J is
an automorphism l1' of 9 which satisfiea (Tm = 1.
Let  o be a maximal ad-diagonalizable subalgebra of the Lie algebra go.
We first prove the following:
LEMMA 8.1. a) Let ( . , .) be a nondegenerate invariant bilinear form on ,.
Then: (giIGj) = 0 jf i + j  0 mod m, and Ii and 8j are nonde,enerately
paired jf i + j E: 0 mod m.
b) The cen"traJizer J of  o in 9 is a Cartan Bubalgebra of'g.
c) Go is a reductive subalgebra of g.
Proof. Given z E Ui, JJ e Sjt we have (zly) = (O'(%)lu(y» = (i+j(zIJl)
(the form (.,.) is Aut g-invariant being a multiple of the Killing form),
which proves the first part of a). The second part Coilowl since (.1.) is
nondegenerate.
In order to prove b), note that
J =  + E 10,
a
125

126 Twisted Affine Algebra, Gntl Finite OnJer Aatomorplai,m.t Ch. 8
where  is a Cartan 8ubalgebra of 9 containing o, go are root spac.es with
respect to  and the summation is taken over Q e A such that QIo = o.
It follows that J =  + " where I is a C7-invariant semisimple 8ubalgebra
(::: the derived 8ubaJgebra of I), whose intersection with 90 is triviaJ. This,
in particular, proves c). Thu8 (8.1.1) induces a Z/ml gradation 8 = j.j
such tbat '0 = {O}. N umbering the elements of Z/ ml by the corresponding
integers in the set N m = {O, 1,.. . t m - I} and defining 'Q = '6 if b E N m
and a ==" mod m, we shall prove 'n = I-n = 0 by induction on n.
We know '0 = 0; let n > 0 and z e 8n. Then (ad Z)"'i C 'nr+i. Select a
positive integer r such that n(r-l) < m- i  nr. Then nr+i = m+C with
o  t < n, so by the inductive assumption, an,.+i = 'e = o. Thus ad zl. is
nilpotent; similarlYt ad yl. is nilpotent if II E '-n. But ['n, '-n] C .0 = 0,
so adz and ady commute on , and by the nilpotency, tr.( ad z ad 11) = o.
Now Lemma 8.1a) (applied to .) implies _ = 0, proving b).
o
It follows from Lemma S.tb) that o contains a regular element. say z, of
g. Hence, the centralizer  of z in g is a (7-invariant Cartan subalgebra, and
the sum of eigenspaces of adz with positive eigenvalues (we say that a e C is
positive if either Rea> 0 or Rea = 0 and 1m a > 0) is a maximal nilpotent
D'-invariant subalgebra n+. Let + be the corresponding set of positive
roots. Thus, tr induces an automorphism of +j let p be the corresponding
diagram automorphism of g. It is clear that t1JJ-l = exp(ad h), where
h E o. Since Cartan subalgebras or 9 are conjugate, we have proved the
following
PROPOSITION 8.1. Let 9 be a simple finite-dimensional Lie algebra, let
 be its Cartan 8ubalgebra and Jet n = {O'l'..' t Q} be a set of simple
roots. Let tr E Aut 9 be such that (1m = 1. Then t1 is conjugate to an
automorphism of g of the form
(8.1.2)
21ri
J.l exp(ad -h), h E o.
m
where JJ is a diagram automorphism preserving  and U', o;s the lixed
point set of p in , and (Q, h) E Z, i = 1,. . . , N.
o
S8.2. We associate a 8ubaIgebra £(s, u, m) of £(9) to the automorphism
iT of g 88 fQllows:

Ch. 8 Twi,Ied Affine Alge6ra.t tJnd Fini'e Onltr Automorphi,ml 127
(8.2.1) £(g,O',m) = E9£(g,D',m);, where £(g,D',m); = Ii @ 9; mod m.
lei
The decomposition (8.2.1) is clearly a Z-gradation of .l(g, iT, m).
Note that .l(g, 0', m) is the fixed point set of the automorphism u of £(g)
defined by
u(t j 0 z) == «(-it j ) 0 O'(z), (j E Zt z E g).
lIenee, £(g, tT, m) may be identified with the Lie algebra of equivariant
maps (with respect to the action of Z/ml):
(C X ; multiplication by £-1) -+ (g; action of (1).
Recall the Lie algebra leg) == .l(g) EB CK' 6) Cd' defined in 57.2 (here
we write K' and d' instead of K and d Cor reasons which will become clear
later on). We set
£(g, (7, m) = £(g, 00, m) ED CK' EB Ccf.
This is a subalgebra of £( g). which is the fixed point set of an automorphism
if of leg) defined by ul.C(..,m) = i, u(K') = K', u(4) = 4. Its derived Lie
algebra is leg, 0', m) := Leg, 0', m)E9CK'. Note also that £(g, I, 1) = £(g),
leg, It 1) = £(9) and £(g, 1,1) = £(g).
Setting degK' = degd' = 0 together with (8.2.1) defines a Z-gradation
of £(g,a,m):
(8.2.2)
l(g,D',m) = ffil(g,C1,m)j.
jEI
S8.3. The structure of the Le algebra £(g,tT, m) for arbitrary tr will be
studied in the folJowing sections.
Here we consider SOUle very special examples, which give us an explicit
construction of all twisted affine algebras, i.e., those listed in Tables Aft' 2
and Aff 3.
Let 9 = ' $ ( EB CE) be a simple finite-dimensional Lie algebra of
oE4
type X N, as constructed in 51.8 and 7.9 with the normalized invariant form
( .).). Let jj be an automorphism of the Dynkin diagram of 9 of order r
(= 1, 2 or 3), and let #J be the corresponding diagram automorphism of g.
Case O. For X N == At, B l, · · · , Es and r = 1 let n = {(}' 1 , . . . , Qt} be the
set of simple roots of 9 enumerated as in Table Fin. Let E i = E;, F, =
-£'-0 1 ' Hi == Qi (i = 1,. .. t l), Eo = E9' Fo = -Eo, Ho = (J = 8 0 .

128 Twi.Cetl Affine Alge6,... Inti FiniCe Onler A.tomorpAi,m, Ch, 8
Cues 1-5. Let DOW XN = Dl+ 1 , A2l-1' Ee. D4 or Au with the ordering
of the index let 88 in Table Fin, and let r = 2, 2, 2, 3 or 2 respectively
(theBe are, clearly, aU the cues when JJ  1).. We have the correeponding
Z/rZ-gradatioRI (described by Proposition. 7.9&) and 7.10a) in a slightly
different notation):
(8.3.1)
g = go €a gT if r = 2, and g = go E9 91 Ea g2 if r = 3;
(8.3.2)
' = o €a T if r = 2, and I = ii $ I €a  2 if r = 3.
Here and further, S E Z/rZ stands for the residue of s mod r. Let 0' =
{0 1 ' · · · , oN} E  be the set of simple roots of 9 (enumerated as in Table
Fin) and let E = Ei' F/ = -E'-ai (i = 1,..., N) be its Chevalley
generators and H: = av simple coroots. Introduce the following elements
0° E  and Eit t Hi (i = 0, . . . ,l) E g:
Case 1: XN = Dt.+l, r = 2:
SO = Q + · . . + o;
Hi = H:(l S i  i-I), Ht. = H; + H;+I' Ho = _0° -11(0°);
Ei = EH1  i  1- 1), E, = E; + E;+1,Eo = E,o - E'JI('o);
F i = F/{I  i  t - 1), F, = FI + F:+ 1 , Fo = -Eo + E!,r('O).
Case 2: XN = A2l-1, r = 2:
8° = Q + ...+oU-2;
Hi = H: + HU-i (1  i  t - 1), Hi = H l , Ho = _9° - ji(9o);
E i = E: + E;,_i (1  i  t - I), E, = E;, Eo = E,o - E;;(,o)j
Fi = F/ + F'-i (1  i  1- 1), Ft = F:, Fo = -Eo + Efr<,o).
Case 3: XN = Ee, r = 2:
eO = Q + 2o + 2Cr + Q4 + Q + Q;
HI = Hf + H; t H 2 = H 2 + H4' H3 = H3t 114 = H, Ho = -0° - j1(80);
E 1 = E 1 + E, E2 = E 2 + E41 E3 = E, E4 = E.., Eo = E90 - E J.I (90)i
Fl = F{ + FSI F2 = F 2 + F41 F3 = F31 F4 = F 6t Fo = -E;o + E;;(6 0 ).
Case 4: XN = D4, ,. = 3. " = exp 2wi/3:
8° = 0'1 + Q + Q;

Cb. 8 Twi,'etl Affine Alge6ra, and Finite Order A.'omorpAi,m, 129
HI = H + H + H, H 2 = H, Ho = _9 0 - jI(60) - ji2(8°);
El = E + E; + E, E2 = , Eo = E,o + rl E;;(,o) + '1E'_jtJ(,o)i
, , F ' D 1:'1 C1 E ' E ' 2 E '
Fl = Fl + F3 + 4' r2 = .c2 t .cO = - 60 - 11  (80) - 1] ", '(80).
Case 5: XN = A2l, r == 2:
8° = o + · · · + Ol;
Hi = HI + Ht-i+l (1 < i < l- 1), Ho = 2(H; + H;+I)' Ht = -0°;
E i = E + Et-i+l (1 < i < i-I), Eo = v'2(E + E+I)m, Et = E80.
F; = F: + Ft-i+l (1  i < l - 1), Fo = V2(F; + F , + 1 ), Ft = -Eo.
Let 0 0 = :(j1(90) + .. · + ji"(SO» in Cases 1-4, and 9 0 = gO in Case 5.
Let £ = 0 in cases 0-4, and E = t in case 5, and let I = {OJ 1.... .l}\{E}.
PROPOSITION 8.3. a) The elements Ei (i = 0,... .t) generate the Lie
algebra g.
b) Elements E;, F. with i E I are Chevalley generators of the Lie algebras
80' elements Q; = 2Hi/(H; J Hi), i E 1, being simple roots.
c) [Ee, FrJ = He, (Ee I Fe) = r/oa, and (9 0 16 0 ) = 2ao/r, where 00 = 1 in
Cases 1-4 and 00 = 2 in Case o.
d) The representation of 90' on "r is jrreducjble, and js equivalent to the
representation on 9_ 1 ,
e) Fo (resp. Eo) is the highest (reap. lowest) weight vector of the go-module
gr (resp. g_ I) with weight 8 0 (resp. -80) (i.e. Do + ai is not a weight). The
types of g o and the decompositions 9 0 = E Gia, are listed in the following
'El
table:
r 9 Go a,
2 Au, t > 2 Bt -,- 2 2 2
0-0-".-0::>0
2 A 2 l- J , I  3 C l 1 2 2 1
o-o-'..-oo
2 Dt+!, l2 BI. 1 I 1 1
O-O-...-oo
2 A 2 Al 2
0
2 E6 F. 1 2 3 2
0-0:::>0-0
3 D. G 2 1 2
o=t-o
Proof follows inunediately from the results of 551.9 and 7.10.
o

130 Twi,ted Affine Algt6rtJ, Gntl Finite Order Au'omorpAilm. Ch. 8
Remark 8.9. Letting () = -9 0 and 0. = I, we caD write:
l
EOiai = O.
'=0
where the at are given by Table Afr.
The restriction of (. I .) to o = ' n g o is nondegenerate, and hence
defines an isomorphism v :  o --. ij. Let i (8 = 0, ..., T - 1) be the set
of nonzero weights of  o on 9i, and let gi = E9 gi,Q be the weigllt
QE.U{O}
space decomposition. Proposition 8.2 implies that (ala) :F 0, dim 91,0 = 1
and (g7,o, g-7,-0] = Cv- 1 (a) if a e .,.. A
Now we turn to the Lie algebra £(8, p, r), which we denote by £(g,)
for short.. Set  = o + CK' + Cd' and define 6 E . by 61O+CKI = 0,
(cS,d') = 1. Set , = t @ E f , It = ,-I fg) Fe, ei = 1 @ E., I. = 1 @  (i E I).
Then we have:
[e., 'i] = 1  Hi (i E I); [ef,I,] = roo 1 K' + 1 @ He.
A
We describe the root system and the root space decomposition of £(9, JJ)
with respect to :
(8.3.3)
A = {j6 + 1t where j E l, 1 EAT. j ==. mod r, , = 0,..., r - I}
U {j6, where j e Zt j # O};
(8.3.4)
£(g, p) =  $ (E9 £(g,p)o),
aE4
where
(8.3.5)
£(g,p),,+'1 = "  gJ',-y. .c(g,p)" = " 0 91,0.
We set
(8.3.6) n = {Qf := 6 - 90, Qi (i e I)},
(8.3.7) n V = {Q := roO' 1 K' + 10 Hf' Qr := 1  Hi (i e I)}.
Using Proposition 8.3 we obtain that if g is of type X Nand r (= 2 or 3) is
the order of Jlt then the matrix
A = ((aJ' ar»fJ:o
is of type XJ;> and the integers 00, .... at are the labels at the diagram of
this matrix in Tables Aff2 and Aff'J.
Now we can state the second realization theorem. Its proof is similar to
that of Theorem 7.4.

Ch. 8 TW;$ttd Affine Algebra.! and Finite Order A utomorphi8ms 131
THEOREM 8.3. Let 9 be a complex simple finite dimensional Lie algebra
of type XN = Dt+ll A 2 1- 1 , ESt D. or Au and kt r = 2, 2, 2, 3, or 2,
respectively. Let p be a diagram automorphism of 9 of order r. 1 Then the
Lie algebra £(g, p) is a (twisted) affine Kac-Moody algebra. g(A) associated
to the affine matrix A of type X};) (rom Table Aif r (r = 2 or 3),  is its
Cartan subalgebra, A the root system, n and n v the root basis and the
coroot basis, and eo, .. . , el, lOt . . . , II. the Chevalley generators. In other
words, (£(9,1'),, n, n V ) ;s the quadruple associated to A.
o
We can summarize the results of Theorems 7.4 and 8.3 88 follows. Let
A be an affine matrix of type xt), let 8 be a simple finitedimen8ional Lie
algebra of type X N and let JJ be a diagram automorphism of 9 of order
r (= 1, 2 or 3). Then the Lie aJgebra leg, p) is isomorphic to the affine
Lie algebra g( A). Note that £( 9, JI) is isomorphic to 8' (A) and £( 8, JI) to
g'(A)/CK.
COROLLARY 8.3. Let g(A) be .an affine algebra oE rank t -t- 1 and let A be
of type X};'). Then the multiplicity of the root jrfJ is equal to t, and the
multiplicity of the root 56 for s  0 mod r is equal to (N - f)/(r - 1).
Proof If r = 1 t then multj6 = l (for j 'F 0) by Theorem 7.4. If r = 2 or
3, then by Theorem 8.3, mult .6 (for. 1= 0) is equal to the multiplicity of
the eigenvalue exp 211'is/r of p operating on ', which gives the result.
o
The Chevalley involution w, the compact involution "'0 and the triangu-
lar decomposition of the Lie algebra £(9,1') c £(g) are induced by those
from leg).
The normaJjzed invariant form on l(gt p) is given by
(8.3.8)
(P 0 z J Q @ y) = r- 1 Res(,-I PQ)(z' y) (z, y E g, P, Q E £);
(CK' + Cd' Il(lI,p» = 0; (K'IK') = (d'Id') = 0; (K'Id') = 1,
where ( .1 .) is the normalized invariant form on 9. The proof is similar to
that of (7.5.1).
It is also easy to see that K = rK' is the canonical central element, and
that d = aor-1d' is the scaling element.
1 For r = 3 there are two auch automorphism. which u-e equivalent. We choose one of
them.

132 Twis'ed Affine Al,e6N' tJnd Fini1e Onltr A,,'omorpl&i,m, Ch, 8
o
Warning. The Lie algebra 8 0 i. isomorphic to the Lie algebra 9 introduced
in 56.3 in all cases except A) i in the latter case 8 is or type CI. whereas
80 is of type Bt.
58.4. Here we preeent another application or realization theorems.
PROPOSITION 8.4. Let g(A) be an alfine algebra.
a) Set t = CK + E S". Then t is isomorphic to the infinite-dimensional
'EI
,o
Heisenberg algebra (= Heisenberg Lie algebra of order 00; see 52.9) with
center CK.
b) The Hermjtian (orm (zlu)o = -(wo(z)ly) is positive semidelinite on
g'(A) with kernel CK.
Proof. By the realization theorem, g'(A)/CK  .c(g, p). The gradation of
9 which corresponds to IJ induces the gradation of the Cartan subalgebra
' of 9 (see 58.2): ' = ED j. We obtain the (ollowing isomorphism:
jEl/rl
tICK  ED' @ "modr.
'EI
,o
It follows that tICK is a commutative subalgebra. It is easy to see that
the restriction of the cocycle 1/J to this 8ubalgebra is nondegenerate. This
proves a); b) follows from the remarks at the end of S7.6.
o
The sub algebra t is called the homogeneoul Hei,en6erg ,ubalgebrtl of the
affine algebra g(A). It plays an important role in representation theory of
affine algebras.
58.5.' Let g be a simple finite-dimensional Lie algebra t let m be a positive
integer and let t = exp. Let -yet} : CX _ Aut 9 be a regular map (we
view Aut; 88 an' algebraic group over C)j we may regard -yet) as an element
of Autl(;} (by pointwise action of '}'(t) on aCt) E l(g». We shaJi view
tr E Aut; 88 an element of Autl(g) by letting = cr(tt 0 a) = tt @ tr(a).
The following lemma follows from an equivalent de6niton of leg, cr, m)
(u m = 1):
l(g.C1,m) = {a(t) e £(g) J(1(o(£-lt» = a(')}.
LEMMA 8.5. Let Q,{J E Aut 9 be such that Q'''' = fJ'" = 1, and let -yet) E
Aut £(g). Then
1(t).c(", Q. m) = l(StP, m)

Chi 8 Twi,'ed Affine Algebra, Gnd Finite Order A u'omorplailms 133
j{ and only jf
(8.5.1)
fJr(£-lt) = 1(1)0 for all t E C X .
o
If a,p E Aut 9 and r : CX -+ Aut 9 (a regular map) are such that (8.5.1)
holds, we write Q 2. {J. The (ollowing relations are immediate:
(8.5.2)
-t
Q 2. P implies fJ T--. Q;
(8.5.3)
-, t:I "'. R. J 771 R
Q -+ fJ -+"'1 Imp Y Q -. fJl,
in particular
(8.5.4)
Q ..!. fJ implies Q !Z g{Jg-l for 9 E Aut g.
PROPOSITION 8.5. Let (T be an automorphism ofa simple linite-dimensional
Lje algebra S o{the form (8.1.2). Denote by t" the regular map CX --+ Aut 9
such that t" on ga is an operator of multiplication by ,(a,h). Then
(8.5.5)
(1&(£(", p, m)) = £(g, (1J m).
Proof foUows immediately from Lemma 8.5, since t h I' = pt".
o
Remark 8.5. It follows from Propition 8.5 that the isomorphism class of
£(g, 0", m) depends only on the connected component of Autg containing
tr. It is not difficult to show that this statement still holds if 9 is replaced
by an arbitrary finite-dimensional algebra (not necessarily a Lie algebra).
We can now prove the following theorem.
THEOREM 8.5. Let g be a simple finite-dimensional Lie algebra of type
XN and let tr be an automorphism of 9 such that (Tm = 1. Let r be the
least positive integer such that (1" is an inner automorphism; then r = 1) 2
or 3. Let ( .1.) be the normaljzed invariant form on g. Let J& be a diagram
automorphism of 9 of order r. Choose a Cartan Bubalgebra ir of the fixed
point set gC1 of 6. Let A be the aIline matrix of type XJ;>. Then there
A
exists an isomorphism  : £(8, (7', m) -+ g(A) sucb ehat:

134 Twilled Affine Alge6rtJ' Gntl Finite Order A ulomorphilm, Chi 8
(i) the bj/inear form on £(g, fT, m) defined by (8.3.8) induces the nor-
malized invariant (orm on g(A);
(ii) the Z-gradation (8.2.2) of £(g, (1', m) induces a Z-gradation of
g(A) of type. = ('0,"" .t), where .j are nonnegative integers which
satisfy the relation
(8.5.6)
l
r Ldj'j = mj
J=O
(Hi) ct( 1  o + CK' + Cel') is the Cartan Bubalgebra of g(A)j
(iv) m(K/) is the canonical central element K of g(A);
(v) m- 1 <f1(d o ) = ao1d+u- !(ulu)K, where d is the scaljngelement
o
of g(A) and u E  is defined by (ait u) = rSi/m (i = 1,..., t).
Proof. Due to Proposition 8.1, Lemma 8.5 and (8.5.4), we may assume that
(T is of the form (8.1.2). Note that r divides m (otherwise om + 6r = 1 Cor
some a," E Z and hence t1 = (err)' would be an inner automorphism). By
Proposition 8.5 we have an isomorphism
t"" : £(g,, r)  C(g, (1', m).
In other words, according to Theorems 7.4 and 8.3, we have an isomor-
phism, denoted by t,p for short,
'P : g'(A)/CK  £(g, iT, m).
The Z-gradation (8.2.1) of C(g,(1,m) induces, via Y'-l, the Z-gradation of
type " = (,, · · . J ,) of g'(A)/CK, where 'i = deg ei = - deg Ii' Here and
further in the proof, Cor z e g'(A) we denote by z the coset % + CK. Note
l
that degg, = L di':' On the other hand, multiplication by t r increases
i=O
the degree in l(S, (I, m) by m. Since degii = 0 and J r , = t r o, we deduce
l
that degir' = m = r E GilL proving (8.5.6) (or ,'.
i=O
Te isomorphism t(J-l can be lifted to a (unique) linear isomorphism
.' : l(g, (1, m) - g(A), which satisfies (iv), (v) with 'i replaced by ., and
the following condition (and hence (iii»):
(8.5.7) .,-I(Qn = Ip() + r(di'/dnK' (i = 0,... ,t).

Ch.8
Twisted Affine Alge6ra, Gnd Finite Order A utomorphisms 135
One easHy checks that this is a Lie algebra isomorphism.
Furthermore, it is clear that the Z-gradation (8.2.2) of £(g, (1, m) induces
via the isomorphism ' the gradation of type s' of g(A). 1"he property (i)
()f ' is also straightforward. It remains to show that the s can be made
non-negative. For that pick v E  (the Cart an subalgebra of g(A)) such that
(Oi, v) = s; note that 6(v) = m/r > o. Hence,  Lemma 3.8, Proposition
3.12b) and Proposition 5.Sb), there exists w E wad (see Renlark 3.8) such
that (Oi,W(V)) :::: Si E Z+. Now  = WOl' is tile desired isonlorphism.
o
58.6. We deduce from Theorem 8.5 a classification of finite order auto-
morphisms of a simple finite-dimensional Lie a]gebra g of type X N. Let p
be a diagram automorphism of 9 of order r. Let E i , Fi, Hi (i = 0, ... S
l) be the elements of 9 introduced in 58.3 and let Qo, ..., Ot be the roots
attached to the Ei. Recall that the elements E i (i = 0, . . . ,l) generate g
(by Proposition 8.3a) and that there exists a unique linear dependence
t
E a,a. = 0 such that the 0, are positive relatively prime integers (see
i=O
Remark 8.3). Recall also that the vertices of the diagram xt) are in
one-t(>one correspondence with the E. and that the ai are labels at this
diagram.
In order to derive an application to finite order automorpbisms, we need
the fo))owing fact.
LEMMA 8.6. Every ideal of the Lie algebra C(g,p) is of the form
P(tr).c(g,p), where P(t) E {,. In particular, a maximal ideal is of the
(orm (1 - (at)r).c(g,p), where a E CX.
Proof. Let i be a nontrivial ideal of £(9, p) and z =  t i 17,.(1) @ aT.I E i,
i.I
where 0 5 j < r is 8uch that 1 == j mod ,., PT (t) E £, P-i-  0 and
J,' J.'
OJ.. E Qj are linearly independent. We show that Q(,r)PJ. (t)£(g, p) C i
J.-
for all Q(t) E £.
Let  o be a Cartan 8ubaJgebra of 80; we can assume that z is an eigen-
vector (or ad o with weight 0 E i. I( 0 ¥ 0 , taking [z, t i  a _ .,.]
with a-I of weight -0, instead of z, we reduce the problem to the ca.:e

136 Twi6ttd Affine AIge6nJ6 Grad F;n;t Order A "tomorplai,m, Ch. 8
Q = 0 and ] = 0, i.e. t (Jl.. E o' Let 1 e i be a root of 80 such
that (1. OT,)  O. Then the element II = lIz, Q(''') 0 e.,J. e_.,] E I,
where e%' is a root vector with root %11 has the following form: 11 =
Q(''')(P h + tPrhr+.'. + ,r-l Pr-l @h'-I). where P = f},,(t), F; e l,
h e /, h ;: O. and the h r e 8T have zero weight with respect to . Since
[lItej] e i (or all root vectors ej e 8o, we conclude that Q(tr)p C i and
therefore Q(tr)Pl(,. p) C I.
For z E 91 let to = {P(t) e £1 tJp(t)  Z E f}. It follow. from above
that io is an invariant ideal (with respect to transformation' -+ ,,-It) of
£', independent or] and z, and that i = io£(g,p). Since all ideals of {,
are principal and 10 is invariant. we deduce that t = p £( at #J) for some
P E C[C" t '-"J.
o
THEOREM 8.6. Let. = ('0, · · . "l) be a sequence of nonnegative relatively
l
prime integers; put m = r E 0i';. Then
i=O
a) The relations
(8.6.1)
(E ) 2..i, t im E ( · 0 t)
iT , ;r j = e J i J = t...,
define (uniquely) an m-th order automorphism t1'jr of 9.
b) Up to conjugatjon by an automorphism of 8, the automorphisms iT,;,.
exhaust all m-th order automorph.ms of 8.
c) The elements iT.;,. and (1.,;,.1 are conjugate by an automorphism of g jf
and only if r = r'. and the sequence, can be transformed into the sequence
" by an automorphism of the diagram xt>.
Proof. We use Theorem 8.5 and the covering homomorplailm 'Ptl : £(g, 0') -to
8 defined by , ..... 1, i.e., tI(E  @ gi) = E (I)gi. It is clear that
Kercpo = (1- ,m)£(a.u).
To prove a) note that the root space decomposition (8.3.4) of C(G, IJ) in-
duces a gradation £(g,p) = £(8,IJ)/CK' = EBla. Define the automor-
Q
phism i, of .c(1I, JA) by:
i.(e o ) = fEt"'e o if eo e lOI where Q = EkiQi.

Ch.8
Twilled Affine Alge6ra. Gntl Finite Order A .'omorplai,m, 137
If £(g,l1) = EB £(g, lJ.)j is the gradation of type 5, then £(9, p)j and
jEZ
C(g,J.L)j+m lie in the eigenspace of u, with eigenvalue exp21rij/m. Since
t r C(g, J.L)j C £(g, JL)j+m, we deduce that the ideal (1 - tr)C(g, Ii) is U ,-
invariant and hence q induces the automorphism of g with the properties
described in a).
Let now tT be an m-th order automorphism of,. Theorem 8.5 gives U8
an isomorphism
.: £(8,tT,m)  £(g,p)
such that the Z-gradation of £(9, CT, m) induces a Z-gradation of type. of
£(g,p) with Ii E Z+ satisfying (8.5.6). Denote by Tea the automorphism
of £(g, ,,), which corresponda to changing t to at, (J E C X . Then, since by
Lemma 8.6, any maximal ideal of £(g,p) is of the form (1- (0')")£(8,1£),
we have the (ollowing commutative diagram (or a 8uitable automorphism
tP of 9 and 0 e C x :
Leg, m, 0") ! £(g, J.L)  £(g, J.L)
'l -l
g
tJ1
--+
9
Hence VJ(1t/J-l = (11,,., provin& b).
The proof of c) also uses the covering map. Suppose that (7 = U .f;r and
(7' = q ,,';r' are conjugate, i.e., TUT- 1 = u' for some T E Aut g. Note that
by Propostion 8.6b) (see below), ii (the Cartan subalgebra of 8") is the
Cartan subalgebra of gl1 and gl1'. Since r(gl1) = gl1l, r(ii) is another
Cartan subalgebra of 8 11 '. Let rl be an inner automorphism of 8 111 such
that Tl(T(O») = . Replacing T by TIT, we may assume that T leaves
o invariant and that the sets of positive roots of ga ahd ga' correspond
to each other under T (by Proposition 5.9 applied to ,,41). The extension
f of T given by f(t i @ a) = t J 0 T(a), is an isomorphism of Z-graded Lie
algebras: .c(g, 0', m) -+ .c(8, 0", m), which maps positive roots onto positive
roots. Hence the sequences _ and .' correspond under an automorphism of
the diagram XJ;>.
o
Given a nonzero sequence. = (.0, . . ., -t) of nonnegative integers and a
number r = I, 2 or 3, we caD the automorphism tT,;r of 9 defined by (8.6.1)

138 Twied Affine Alge6ral tJnd Fini'e Order A,,'omorpAi,m. Ch.8
the tJuiomorpAi,m o/type (; r). Let g = ED Gj('j r) be the Z/ml-gradation
J
associated to it. Here are some of its properties.
PROPOSITION 8.6. a) r is the least positive integer (or which 6=;,. is an
inner automorphism.
b) Let ii, ..., i" be all the indices (or which Bit = ... = 'i, = O. Then the
Lie algebra go{,; r) is isomorphic to a direct sum o{the (t- p)-dimensional
center and a semisimple Lie albra whose Dynlcin diagram is the Bubdja-
gram of the affine diagram xt consisting of the vertices it, . . ., ill'
c) Let )1,... ,jn be all the indices for which Sjl = .., = Sj" = 1. Then the
go(s;r)-module gT(s;r) (resp. g_T(s;r)) is. isomorphic to 8 direct sum of
n irreducible modules with highest-wejghts -ajl' · · · , -OJ,, (resp. Qjl' . . . ,
Ct j" ) .
Proof. a) follows from the (easy) fact that a finite order automorphism
(T of g is inner if and only if there exists a Cartan subalgebr& which is
pointwise fixed under tI. b) is immediate from the isomorphism g 
£(g,o-)o and Corollary 5.12b). To prove c) note that the go('ir)-module
9r( 8; r) is isomorphic to the l( g, (1 )o-module £( g, (7)1. Furthermore, using
the Jacobi identity, we see that £(g, 0')1 is spanned by elements of the form
[- .. [[ti l , ei2]' ti,] .. . ei,.]. such that .'1 = 1 and 8.. = 0 for t > 1. Using the
Weyl complete reducibility theorem proves c).
o
S8. 7. Later we will need the following reformulation of Theorem 8.5
(which is a generalization or Theorems 7.4 and 8.3).
THEO REM 8.7. Let A be an affine matrix of type xt). Let 9 be a simple
linjte--dimensional Lie algebra of type XN, and let ( .1.) be its normalized
invariant form, Let IJ be a diagram automorphism of order r of 9 and E.,
Fit Hi (i = 0, .... l) the elements of 9 introduced in 58.2. Let 0',;,. be an
automorphism of type (8; r) of g, and let 9 = E9 9j('; r) be the associated
j
t
l/ml-gradation, where m = r E Oil;. Deline the Lie algebra structure on
i=O

Ch. 8 Twisted Affine Algebral tJnd Finite. Order A utomorph;$m$ 139
(E t j @ OJ mod m(S; r» $ CK by
jEI
[(Pl(t) @ 91) EJ) >'l K t (P2(t)  92) EB 2K]
1 dP l (t)
= p}(t)P 2 (t) @ {91192] EB -(Res d P 2 (t)) (9d92)K.
m t
Then this is isomorphic to the derived affine algebra g'(A), with Chevalley
generators f. = t'l @ E" 'i = t-I. 0 Fi (i = 0, ..., l), the coroot basis
Q = (1 @ H.)$ (o,8./a'fm)K (i = 0, ..., t) and canonical central element
K. Extending this Lie algebra by Cd', where d'(P(t) @ g) = t d(t) @ 9
m t
and {d'. K] = 0, we obtain a Lie algebra whjch is isomorphic to the afflne
algebra g(A) with the scaling element d = ao(d' - H - !(HIH)K), where
H E E CHi is defined by (ai, H) = 6i/m (i = 1, ... . l). The normalized
invariant form is defined by
(PI (t)  91I P 2(t) @ 92) = r- 1 (Res t- 1 PI (t)P2(t)(91192).
(CK + Cd'IP(t) @ g) = 0, (KIK) = 0, (tfld') = 0 and (Kid') = 1.
Finally, setting deg t = 1, deg 9 = 0 {or 9 e 9 and deg K = deg d' = 0
defines the Z-gradation of type..
o
The realization of the affine algebra of type XJ;} provided by Theorem
8.7 is called the realization of type 8.
Using Corollary 6.4 and Exercise 6.2, we obtain
COROLLARY 8.7. Let f( 9, p, r) be an affine algebra of type XJ;>. Then
(z, y) = 2rh V (zly) {or z, y E 9, where tP is the Killing form on g. In
particular :
L (Ala)(lJla) = 2rh v PIIJ) (or A,IJ E .,
oEA.
where  is a Cartan subalgebra of 9 and d, C . the root system of g.
58.8. Exercises.
In Exercises 8.1-8.4 we sketch another proof of Theorem 8.5.

140 Twi'etl Affine Alge6rw. tlntl Fini'e Onler A.'omorplai,m, Ch. 8
8.1. Choose a Cartan subalgebra o of go and set  = 1 @  o + CK' + Cd J c
£(g, 0', m). With respect to  we have the root space decomposition; denote
by  the set of roots and by C Q the root space attached to a E d. Denote
by X the restriction of ,\ E ,,* to 1   o and set (All-') = (Xl tl ) for A, JJ E ..
Set l:1 0 = {ci E 6 Jei = O}. Show that ad £a is locally nilpotent provided
that a  6. 0 .
8e. Let Q e d\.. Then
(i) dim La = 1 and (Qlo) ;: 0;
(ii) (or 13 e A, the let of 11 + io e A U {OJ is a Itring {J - po, ..., (J + fa,
where p and fare lome nonnegative integersldeh that p-q = 2(alfJ)/(a-IQ).
(iii) [.c p , IT]  0 if P, ." {J +,., E A, fJ  AO.
8.3. Let o c A be the set of roots of 1  90 C £(g, 0', m) and let 60+ be
a subset of positive roots in 60; let + = Ao+ U {Q E I(a, d) > O}. A
root a E + is called simple if it is not a sum of two members of +. Let
n = {Ql t Q2, · · .} be the set of all simple roots. Then
(i) each Q e £1+ can be written in the form Q = E kia', where i4 e Z+;
i
(ii) n c \AO;
(iii) 01, iii, · .. span o and the bi)inear form ( . , . ) is positive definite on
ERoii
i
(iv) there exists a nontrivial linear dependence of (ij' with nonneg-
ative coefficients;
(v) (or i ;: j we have aiJ := 2(0;lo;)/(Q;loi) E -Z+;
(vi) if Q E + is not simple, then 0 - 0, E +.for some Qi e D.
8.4. Deduce the existence of the isomorphiem .
8.5. Let G be a connected 8emi"8imple complex alsebraic group and let G
be the group of regular maps CX -. G. Let 0 E AutG be such that om = 1;

Ch. 8 Twiltttl Affine Alge6nJa Gnd Finite Onler A.'omorpAi,m, 141
we extend Q to an automorphism of G :> G by letting a(-r(t» = 0(1(£-1,».
--
Show that, given, e G, there exists, E G such that
(8.8.1 )
9 = ,-1 0 (1)
if and on)y if
(8.8.2)
1 0 (').. . a m - 1 (g) = 1.
-
8.6. Taking for granted that any finite order automorphism of G leavea
invariant some connected eemi8imple subgroup G' of maximal rank (lee
IV
Kac-Peterson [1981]), deduce (rom Exercise 8.5 that given g E G t there
.--.,
exists 1 E G satisfyng (8.8.1) if and only if ,satisfies (8.8.2)
8.1. Show that every ideal of the Lie algebra £(g, (1) is of the form
P(tm).c(S,tT), where P(I) E Crt]. This is a maximal ideal if and only if
P(t) = a + bit where a, 6 E CX.
8.8. Taking for granted that all maximal ad-diagonalizable 8ubalgebras or
a Kac-Moody algebra 9 are conjugate (see Peterson-Kac (1983]). show that
every Z-gradation 9 = 6)jEI 9j with finite-dimensional go is conjugate to a
Z-gradation of type, with Ii e Z+. ·
8.9. Let g be a simple finite-dimensional Lie algebra and let ei, Ii (i = 1.
t.
· · · t I) be its Chevalley generators. Let IJ = E 0iQi be the decomposition
i=l
of the highest root of " via simple roots. Define an involution tTi of 9 by
i(ei) = -ei, tTi(/i) = -Ii, (Ti(eJ) = ej, tTi(fj) = Ij for j  i. Let p be a
diagram involution of g. Show that every involution tr of 9 is conjugate to
one from the following lilt: a) CTi for 0i = 1; b) C1i for 0i = 2; c) p; d) p oa,
for 0i = 1 or 2 and p(i) = i. Show that the Ii)module 9r is irreducible if
and only if (f is of type b), c) or d), and that t1 is inner if and only if it
is of type a) or b). Show that in the case a), 81 = V_I EJ) "1, where V:!:I
are irreducible Uomodules, and 8 = V-I 6) SifEB Vl is a Z-gradation of ".
show that we thus obtain all up to conjugation Z.gradatioDs of the fOfm
9 = 9 - I $ 80 $ 9 1 ·
8.10. Show that an automorphism tT of order 2 or 3 of a simple Lie algebra
is determined (up to conjugacy) by the isomorphism class of the fixed point
subalgebra of IT. Give an example of two non-tonjugate automorphiams of
order 5 of A 2 with isomorphic fixed point subalgebras.

142 Twi,tetl Affine AlgelN' Grad Finil' Onler A.'omorpAi6m8 Ch. 8
8.11. Show that the minimal order or a regular automorphism (i.e., an au-
tomorpbism with an abelian fixed point let) or a simple finite-dimensional
Lie algebra g is the Coxeter number h ( = (height of the highest root)
+1). Show tbat such an automorphism is conjugate to tbe automorpbism
of type (1,1,..., Ii 1). Show that every regular automorphism of 8 of order
h + 1 is conjugate to the authornorphism of type (2,1, . . . t 1; 1).
8.JI. An automorphisn1 t1 of order m of a aimpJe finite-dimensional Lie
algebra" is called 'fja,iraCioraal if (or the U80ciated Z/mZ-gradation 9 =
EB 8j one has: dim G. = dim GJ if (it m) = (j, m). or, equivalently, the
j
characteristic polynomial or " on g bu rational coefficients. Show that the
automorphism8 of type (1.1,.. ., 1; r) and (2,1.. . . ,I; 1) are quasirational.
Classify all quasirational automorphi8D11 of A 1 and A, up to conjugation.
8.19. An automorphism tT of order m of 9 is called rational if (1. is conju-
gate to tr for every k such that k and m are relatively prime. Show that
a rational automorphism is quasirational. Find a counterexample to the
converse statement.
8.1.. Show that the automorphismB of Exercise 8.11 are rational.
8.15. Let JJ be an automorphism of the Lie algebra 9 = .t,,(C) defined by
,..(EiJ) = -( _1)'+J E n + 1 -J,n+l-i.
Show that J.-L is the diagram automorphism of g. Show that the subalgebra
£(g, JL) c £(g) is an affine Lie algebra of type A2), K being the (resp.
twice the) canonical central element if n = 2t (resp. n = 2£ - 1). Describe
the root space decomposition.
8.16. Introduc:e the following basis of the Lie algebra .c(..t 3 (C) , 1') (, E Z):
t 2 . 0 0 0 t 2 , 0
L8, = 0 0 0 L 8 ,+1 = 0 0 t 2 . t
0 0 _t 2 , 0 0 0
La. +2 = (  0 ) I ( t2+1 0 ) I
0 £8.+3 = 0
,2'+ 1 0 _t 2 ,+1

Ch.8 Twi,'ed Affine AIge6rtJI dntl Finite Order A .tomorpAi,m, 143
( ,2'+ 1 0 o ) L.,+I = ( t 2, + 1 0
L.,+4 =  -2t 2 ,+1 o , 0 _t 2 ,+1 ,
0 ,2' + 1 0 0
L 8 ,+8 = ( 0 ,2'+ 1 ) L - ('2+2 0 ) .
0 o , 0
8.+7 -
0 0 0 t 2. + 2
Show that [Li. Lj) = dijLi+j (i,j E Z), where d;j E Z depend on i, j mod
8; compute them.
8.17. Construct a "Cheval ley basis" of l(g,p) (cr. Exercise 7.14).
58.9. Dibliographical Dotel and comments.
The realization of the CCcenterless t ' twisted affine Lie algebras was given in
Kac [1968 A, B); 80 was the application to the classification of synunetric
spaces. The application to the classification of all finite order automor-
phisms was found in Kac [1969 A]; a detailed exposition of this is given
in Helgason [1978). The present prooC of Theorem 8.5 is simpler than the
original proof. It resulted from conversations with G. Segal and D. Peter-
son.
Exercises 8.1-8.4 are taken from Kac [1969 A] and Helgason [1978].
-
As B. Wei.sfeiler pointed out, Exercise 8.6 means that Hl(Z/ml,G) =
{I}; note aJso that HO(Z/mltG) = GO. Steinberg (1965] proved that
H 1 (Zjml,G(C«t») = 1. Exercises 8.12-8.14 are taken (rom Kac [1978
A].
Levstein [1988] has classified the involutions of all affine algebras. For a
classification of finite order a.utomorphisms and conjugate linear automor-
phisffiS see Bausch-Rousseau [1989L Kac-Peterson [1987}t Rousseau (1989}.
There is an intriguing connection of the material of Chapter 8 with
invariant theory. Namely, given a Z/ml-gradation of a simple Lie algebra
g = ED gj corresponding to an m-th order automorphism t1, we get a go.
j
module 81; the corresponding connected reductive linear group G acting on
the space gl is called a (T-group. These groups have many nice properties
(see Kac [1975], Popov [1916], Vinberg [1916], Kac.-Popov-Vinberg [1976),
but the most remarkable thing is that cr-groups almoet exhaust all "nice"
irreducible linear groups (see Kac [1980 D] for a review of these theories).

144 Twilted Affine AI,e6ra, Inti Finite Order A.tomorpAi,m, Ch. 8
I believe that this is an indication or a deep connection between the theory
of infinite-dimensional Lie algebru and groupe, and invariant theory. We
have already dilcuued one aspect or such a connection in 55.14.

Chapter 9. Highest-Weight Modules
over Kac-Moody Algebras
9.0. In this chapter we begin to develop the representation theory of
Kac-Moody algebras. Here we introduce the so-called category 0, which is
roughly speaking the category of restricted -diagonalizable nlodules (the
precise definition is given below). We study the "elementary" objects of
this category, the so-called Verma modules, and their connection with ir-
reducible modules. We discuss the problenls of irreducibility and complete
reducibility in the category O. We find, as an application of the represen-
tation theory, the defining relations of Kac-Moody algebras with a sym-
metrizable Cartan matrix. At the end of the cllapter we discuss the category
CT for t}1e infinite-dilnensional Heisenberg algebra and the Virasoro algebra.
59.1. As in Chapters 1 and 2, we start with an arbitrary complex
n x n matrix A and consider the associated Lie algebra g(A). Recall the
triangular decomp06ition:
g(A) = n_ e EDn+.
We have the corresponding decomposition of the universal enveloping al-
gebra:
(9.1.1)
U(g(A» = U(n_) @ U()  U(n+).
Recall that a g(A)-module V is caned -diagonalizable if it admits a
weight space decomposition V = E9 VA by weight spaces VA (see 13.6). A
AE.
nonzero vector from VA is called  weigh' tJtctor of weight . Let P(V) =
{.A E .IVA  OJ denote the set of weights of V. Finally, for  E . set
D('\) = {" e .IJ.C S }.
The category 0 is defined 88 follows. Its objects are g(A)-modules V
which are -diagonalizable with finite-dimensional weight spaces and such
that there exists a finite number of elements  1 t . . . ,. E . such that
6
P(V) C U D(.\a).
i=l
The morphisms in 0 are homomorphisms of g(A)-modules.
145

146
Highes'. WigAt Modu/el over Kac-Moody Algebra,
Ch.9
Note that (by Proposition 1.5) any 8ubmodule or quotient module of a
module from the category 0 is also in O. Also, it is clear that a sum or
tensor product of a finite number of modules from 0 is again in o. Finally,
remark that every module from 0 is restricted (see 52.5).
S9.2. Important examples of modules from the category 0 are highest-
weight modules. A g(A)-module V is called a highe,t-weight module with
highelt weight A E . if there exists a nonzero vector VA E V 8uch that
(9.2.1)
(9.2.2)
n+(VA) = OJ h(VA) = A(h)VA for h E ; and
U(g(A»(VA) = v.
The vector VA is called a A;ght6t-Wt;gAt tJtctor. Note that by (9.1.1), con-
dition (9.2.2) tan be replaced by
(9.2.3)
U(n_)(VA) = V.
It follows from (9.2.1 and 3) that
(9.2.4)
v = EB V; VA = CVAi dim V < 00.
ASA
Hence a highest..weight module lies in 0, and every two highest-weight
vectors are proportional.
A g(A)-module M(A) with bighest weight A is called a Venno module
if every g(A)-module with highest weight A is a quotient of M(A).
PROPOSITION 9.2. a) For every A e . there exists a unique up to isomor-
phism Verma module M(A).
b) Viewed as a U(n_)-modu/e, M(A) is 8 free module olrank 1 generated
by a highest-weight vector.
c) M(A) contains a unique proper maximal submodule M'(A).
Proof If M 1 (A) and M 2 (A) are two Verma modules, then by definition
there exists a surjective homomorphism of g(A)-modules .p : M 1 (A) --+
M 2 (A). In particular, y,(M.(Ah) = M 2 (Ah and hence dimM1(Ah 
dim M2(A). Exchanging M 1 (A) and M 2 (A) proves that ,p is an isomor-
phi8m.
To prove the existence of a Verma module, con8ier the left ideal J(A)
in U(g(A» generated by R+ and the elements h - A(h) (h E ), and set
M(A) = U(g(A»/I(A).

Ch.9
H;glae3t.. Weigh' Modules over KGc-Mood, Alge6rtJ
147
The left multiplication on U(g(A» induces a structure of U(g(A»-modu1e
on M(A). It is clear that M(A) is a g(A)-module with highest weight A,
the higbest-weight vector being the image of 1 E U(g(A»). If now V is a
g(A)-moduJe with highest weight At then the annihilator of VA C V is a
left ideal J 1 which contains J(A). Hence, V  U(g(A)/J l and we have an
epimorphism of g(A)-modules M(A) --. V,which proves a)_
b) follows from the explicit construction of M(A) given above and the
Poincare-Birkboff-Witt theorem. c) follows from the fact that a sum of
proper submodules in M(A) is again a proper submodule (since. by Propo-
sition 1.5, every 8ubmodule of M(A) is graded with respect to the weight
space decomposition and does not contain M(A)A).
o
Remark g.e. One can also obtain M() via the construction of an induced
module. Let V be a left module over a Lie algebra a, and suppose we are
given a Lie algebra homomorphism 1/J : a --. b. Recall that the induced
b-module is defined by
U(b) 0U(a) V := (U(b) 0c: V)/ E C(6.p(a) 0 v - I> 0 a(v»,
0,&."
where the summation is over II E U(b), a E a, v E V, and the action of II
is induced by left multiplication in U(b). Define the (n+ + )-module C
with underlying space C by n+(l) = 0, h(l) = (,h}l for h E . Then
M() = U(g(A» U(n++) C.
S9.3. It follows from Proposition 9.2c) that among the modules with
highest weight A there is a unique irreducible one, namely the module
L(A) = M(A)/M'(A).
Clearly, L(A) is a quotient of any module with highest weight A.
To show that the L(A) exhaust all irreducible modules from the category
0, as well as for Borne other purpoeea, we introduce the following notion.
Let V be a g(A)-moduJe. A vector v e VA is ca))ed primitive if there exists
a submodule U in V such that
v tI. U; n+(v) C U_
Then  is called a primitive weight. Similarly, one defines primitive vectors
and weights for a g'(A)-module. A weight vector v such that R+(V) = 0 is
obviously primitive.

148
High,'- Wi,,,, Motlul, otJr Kac-Mood, Alge6ro,
Ch. 9
PROPOSITION 9.3. Let V be a nODlero module from the category O. Then
a) V contains a non.ero weight vector v such that n+(v) = o.
b) The following conditions are equivalent:
(i) V is irreducible;
(ii) V . a bjgbest-wei,he module and any primitive vector of V is a
highest-weight vector;
(iii) V  L(A) for some A E I)..
c) V is generated by its primitjve vectors u a g(A)-module.
Proof. To prove a), take a maximal ,\ E P(V) (with respect to the ordering
). Then one can take t1 to be a weight vector of weight A.
Let V be an irreducible module; then a weight vector t1 is primitive jf
and only if ft+(tJ) = O. Take a primitive vector t1 of weight "\. Then U(g)(v)
is a lubmodule of V, hence V = U(g)(t1) and V is a module with highest
weight . In particular, P(V) S ,\ and dim V = 1. Hence every primitive
vector is proportional to 11, which proves the implication (i) => (ii) of b).
If V is a highest-weight module and U c V is a proper 8ubmodule, then
U contains a primitive vector by a). This proves the implication (ii) :} (iii)
and the assertion b).
Let V' be the 8ubmodule in V generated by all primitive vectors. If
V'  V, then the g(A)-module V/V' contains a primitive vector t1 by a).
But a weight vector in V which is a preimage of v i8 & primitive vector.
o
Thus we have a bijection between . and irreducible modules from the
category 0, given by A 1-+ L(A). Note that L(A) can also be defined as an
irreducible g(A)-module, which admits a nonzero vector VA such that
(9.3.1)
n+(VA) = 0 and h(VA) = A(h)VA for h e .
Remark 9.3. A module V (rom the category 0 is generated by its primitive
vectors even 88 a "_-module. Indeed, a weight vector v e V is not primitive
if and only if 11 e U(n_)UG(n+>v (= the lubmodule generated by "+(11».
Here and further Uo(g) denotes the augmentation ideal gU(g) or U(g).
We have the (ollowing "Schur lemma." . ,
LEMMA 9.3. End'(A) L(A) = CIL(A)'
Proof. If CI is an endomorphism of the module L(A) and VA is a highest-
weight vector, then by Proposition 9.3b), we have a(tJA) = AVA for some
.\ E C. But then a(u(vA» = .\U(VA) for every U E U(g), hence a = .\h(A).
o

Ch.9
Hig},e8'- Wei,h, Mod.le. over Kac-Mood, Alge6na,
149
S9.4. Let L(A). be the g(A)-module contragredient to L(A). Then
L(A). = n(L(A)).. The lub.pace
A
L.(A) := $(L(A)",.
A
is a 8ubmodule or the ,,(A)-module L(A).. It is clear that the module
L.(A) is irreducible and that for  E (L(A)Ar one has:
R_(V) = 0; h(v) = -(A, h)t1 Cor h E .
Such a module is called an imtl.ci6le mod..le with lowe,' weight -A.
As in 59.3 we bave a bijection between . and irreducible lowest-weight
modules: A...... L · (-A).
Denote by 1f,\ the action of g(A) on L(A). and introduce the new action
1I'A on the space L{A) by
(9.4.1) ""A(g)v = "A(W(g»v t
where WI is the Chevalley. involution of g(A). It is clear that (L(A), X"l)
is an irreducible g(A)-mo<tule with lowest weight -A. By tbe uniqueneu
theorem, this module can be identified with L.(A), and the pairing between
L(A) and L.(A) gives us a nondegenerate bilinear form B on L(A) 8uch
that
(9.4.2)
B(g(), y) = -B(z,w(g)(» for all 9 E SeA), ,7I E L(A).
A bilinear form on L(A) which satisfies (9.4.2) is called a cORtrat1G';CltI'
bilinear form.
PROPOSITION 9.4. Every g(A)-module L(A) carries a unique up to COD-
stant factor nndegenerate contravariant bilinear form B. . This form iB
symmetric and L(A) decomposes into AD orthogonal direct Bum 0{ weighe
spaces with respect to thjs form.
Proof. The existence of B was proved above, the uniqueness follows from
Lemma 9.3. The symmetry follows from tbe uniqueness. The fact' that
B(L(A)" , L(A),,) = 0 if  ;: II fOUO"8 from (9.4.2) for g e . Z E L(A)",
II E L(A)".
o
A more explicit way to introduce the contravariant bilinear form is the
(ollowing. Let V be a highest-weight '(A)-module with a fixed higheet
weight vector VA. Given v e V. we define ita ezpecl"t;ora "",., (v) E C
by
v = (V)VA + L VA-ch where VA-a e VA-o.
oEQ+\{O)

150
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Ch.9
Extend the negative Chevalley involution -eM to an anti-involution w of
U(g(A». Due to (9.1.1) we immediately lee that
(9.4.3)
(W(O)VA) = (OVA), a e U(g(A».
It follows that ("(o)o'v,\) is symetric in a, 0' E U(g(A», hence the formula
B(aVA' a' VA) = (W(O)O'VA)
gives a well-defined symmetric bilinear form on V, which is contravariant,
and normalized by B(VA, VA) = 1.
S9.5. The underlying statement of most of the complete reducibility
theorems is contained in the following lemma.
LEMMA 9.5. Let V be a ,,(A)-module {rom the category O. 1((or any two
primitive weights A and p of V the inequality   #J implies A = IJ, then
the module V is completely reducible (i.e., V decomposes into a direct sum
or irreducible modules).
Proof. Set V O = {v E Vln+(v) = OJ. This is -invariant, hence we have
the weight space decomposition VO = EB V).o, where all elements from L
AeL
are primitive weights. Let,\ eLand v e Vf. v  O. Then the g(A)-module
U(g)(v) is irreducible (and hence isomorphic to L(». Indeed, if this is
not the case, then by Proposition 9.3b») we have U(n_)(v) n V: ¥- 0 for
some p < # This contradicts the assumption of the lemm.8. Therefore,
the g(A)-submodule V' of V generated by V O is completely reducible.
It remains to show that V' = V. If this is not tbe case, we consider the
g(A)-module VIV ' . Then there exists a weight vector v E V of weight JJ
such that v  V' but e.(v) e V' and  0 for some i. But then, since V is
from the category 0, there exists ,\ E L such that   1J + Qi, and hence
 > 1', which contradicts the assumption of the lemma.
o
S9.6. Unfortunately, a module V E 0 does not always admit a compo-
sition series (i.e., a sequence of 8ubmodules V ::> VI ::> V2 :> ... such that
each /Vi+l is irreducible) (see Exercises 10.3 and 10.4). However, one
can m&nage with the following substitute for it.

Ch.9
Higlae.'. Weigh' Module. over Kac-Mood, Algt6rtJ,
151
LEMMA 9.6. Let V E 0 and  e ".. Then there exists a filtration by a
sequence of Bubmodules V = ve ::> -1 ::> .. · :> VI :> Vo = 0 and a subset
J C {I, . . . , t} such that:
(i) jf j E J, then 1tJ!1tJ-l  L(Aj) (or some >'j  ;
(ii) jf j  J, then (Y}Iy}-I)" = 0 for every I'  -\.
Proof. Let a(V,.-\) = E dim V". We prove the lemma by induction on
,,A
a(V, ). If a(V,) = 0, then 0 = J'O c Vi = V is the required filtration
with J = I. Let a(V,) > O. Chooee a maximal element p  P(V) luch
that p  , choose a weight vector 11 e VI" and let U = U(g)(v). Then,
clearly, U is a highest-weight module. Proposition 9.2c) implies that U
contains a maximal proper 8ubmoduJe U. We have
o cUe u c V; u/U';::!. L(p), p? .
Since a(17,) < a(V,) and a(V/U,) < a(V, ), we use induction t get
a suitable filtration (or U and V/U. Combining them we get the required
filtration of V.
o
Let V E 0 and p e .. Fix  e . such that #J   and construct a
filtration given by Lemma 9.6. Denote by [V : L(IJ)] the number of times
JJ appears among {j Ii E J}. It is clear that (V : L(#J)] is independent of
the filtration furnished by Lemma 9.6 and of the choice of ; this number
is called the multiplici', of L(p) in V. Note that L(p) haa a nonzero
multiplicity in V if and only if IJ is a primitive weight of V.
19.7. Now we will introduce and study the Cormal characters of modules
(rom O. For that purpose,' we define a certain algebra £ over C. The
elements of £ are series of the form
E cAe(),
AE.
where CA E C and CA = 0 for A outside the union of a finite number of
sets of the form D(J.l.). The sum of two such series and the multiplication
by a number are defined in the usual way. £ becomes a commutative
associative algebra if we decree that e('\)e(J.L) = e('\ + p,) and extend by
linearity; here the identity element is e(O). The elements e() are called
formal exponentials. TIley are linearly independent and are in one-to.one
correspondence with the elements A of . .

152
Highe,'. Weigla' Mod.le, over Kac-Mood, Algebra,
Ch.9
Let now V be a module from the category 0 and let V = ED VA be ita
AE.
weight space decomposition. We define the formlJl cAartJcltr of V by
ch V = E (dim VA)t(.\).
E'.
Clearly, ch V e £.
First, we prove the following
PROPOSITION 9.7. Let V be a g(A)-module (rom the category O. Then
(9.7.1)
ch V = E [V : L(.\)] ch L(.\).
E'.
ProD/. Denote by t/J the map which 8880ciatel to each V E 0 the difference
(V) E £ between the left- and right-hand sides 0(9.7.1}. Then (L('\) =
0, and given an exact sequenee of modules 0 -.  -+  -+  -+ 0 we
have tP(V 2 ) = () + (V3)'
Using Lemma 9.6, .we deduce that given ,\ E . there exist modules
,.
Ml"" t M r e 0 such that (Mi),. = 0 when #I  A, and (V) = E (Mi)'
i=l
In particuJar, for every  e . the coefficient at e() in t/J(V) is zero.
o
Let us compute the Cormal character or a Verma module M(A). Us-
ins Proposition 9.2b) and the Poincare-Birkhoff-Witt theorem, one can
COh8tru ct a basil or the apace M ( A) A u folJow,. Let PI, P2, . .. be aJI
the positive roots or the Lie aJgebra g(A), and let t-,.l. be a basis of
8-,1.(1 S i, S multp, = m,). Let VA be a highest-weight vector of M(A).
Then the vectors
nit' "1._, n.1 "3."'3 ( )
t,_I.l · · · e_I.,"1 e_",1 · · · e_'2,m2 · .. VA ,
Iuch tbat (nl.l + · .. + Bl,m, ){Jl + (n2.1 + ... + n2,m2)P2 + .. · = A - ,\ and
n'J e Z+. form a buis of M(A)A' Therefore
ch M(A) = e(A) n (1 + e( -0) + e( -20) + ... )multc:r.
oE6+
Hence, we have
(9.7.2)
chM(A)=e(A) II (l_t(_o»-multcr.
QE6+

Ch.9
Higlae,'. Weight Module, otJer Kac-Mood, Alge6na,
153
59.8. Assume now that A is a symmetrizable matrix and let (.1.) be
a bilinear form on g(A) provided by Theorem 2.2. Then tbe generalized
Casimir operator {} acts on each module from the category 0 (see 52.6).
LEMMA 9.8. a) If V is a g(A)-module with highest weigh& A, then
o = <fA + pl2 - 'pf2)lv.
b) If V is a module from the category 0 and v is a primitive vector with
weight , then there exiBts a 8ubmodule U C V such that II  U and
O(v) = (I + pl2 -lpI 2 )v mod U.
Proof. (ollows immediately (rom Gorollary 2.6.
[J
PROPOSITION 9.8_ Let V be a g(A)..module with highest weight A. Then
(9.8.1)
ch V = E c.\ ch M(;\), where c.\ E Z, CA = 1.
<A
1+,IJ;IA+,12
Proof. Using (9.7.1), it suffices to prove (9.8.1) for V = L(A). Set B(A) =
{A < AUA + pJ2 = IA + pI2}, and order the elements of this set, .xl, A2'...
so that the inequality ",  Aj implies i < j. Then the proof of Proposition
9.7 arId Lemma 9.8 imply the following system of equations:
ch M(;\i) = L CiJ ch L(;\j).
jEB(A)
The matrix (Cij) or this system is triangular with ones on the diagonal.
Solving this system proves (9.8.1).
o
59.9. Now, usinS the Casimir operator. we can investigate irreduc-
ibility and complete reducibility in O.

154
Hj91a,'- Weigh' Module, ovr Kac-Mood, Alge6rtJ,
Ch.9
PROPOSITION g.9. Let A be a symmetrizable matrix.
a) If 2(A + piP) :f- (,81,8) for every (J e Q+, P  0, then the g(A)-module
M(A) is irreducible.
b) If V is a ,,(A)-module from the category 0 such that (or any two
primitive weights  and IJ ot V, Buch that  - IJ = {J > 0, one has
2( + pffJ)  (PIP), then V is completely reducjble.
Proof Proposition 9.3b) implies that if M(A) is not irreducible, then there
exists a primitive weight ;\ = A - /3. where p > O. But then Lenuna 9.8a)
gives: 2(A + piP) = (PIP), proving a).
To prove b) we may 888ume that the g(A)-module V is indecomposable.
Since, clearly, 0 is locally finite on V, i.e. t every v e V lies in a finite-
dimensional O-invariant lubapace, we obtain that there exists a E C such
that O-a[ is locally nilpotent on V. Hence Lemma 9.8b) impJjesl;\+pJ2 =
Jp+pJ2 for any two primitive weights.-\ and 1'. Now b) follows from Lemma
9.5.
o
S9.10. Here we consider the subalgebra g/(A) = [g(A), SeA)] of g(A)
instead of SeA). Recall that g(A) = g'(A) + t and that ' = E Corr =
. i
g'(A)n. Recall the free abelian group Q and the decomposition U(g'(A»
= ED U (see 52.6 for details).
oEQ
A g'(A)...module V is called a laight,'-wt.iglat module with laight., wtight
A E ('). if V admits a Q+-gradation V = ED VA-a such that U(VA-a)
oEQ+
C VA-o+, dim VA = 1, h(v) = A(h)v for h e (')., v E VA, and V =
U(g'(A)(VA). In other words, this is a restriction of a highest-weight
module over SeA) to S/(A). In the same way 88 in 59.2. we define the
Verma module M(A) over g'(A) and show that it contains a unique proper
maximal graded 8ubmodule M'(A). We put L(A) = M(A)/M'(A). This
is, of course, a restriction of an irreducible g(A)-module to g'(A).
LEMMA 9.10. The g'(A)-module L(A) is irreducible.
Proof Let V C L(A) be a nonzero g'(A)-submodule. We choose v =
m
E tJi E V such that Vi e L(Ah;! Vi  0 and Eht(A - ..\.) is minima},
'=1 i
Ie ..\.  A for some i, then tj(v"J :F 0 for some j (by Proposition 9.3b).
Hence t1 e L(A)A and V = L(A).
o

Ch, 9
Highe,'" Weight Modu/t, OfJer Kac-Mood, AIge6rtJI
155
We shall sometimes describe A e . by its ItJ6e (A, or) (i = 1,..., n).
If AJ M E . have the same labels, they may differ only off '. Then it is
clear from Lemma 9.10 that the modules L(A) and L(M) when restricted
to g'(A) are isomorphic as (irreducible) g'(A)-modules, and the actions of
elements of g(A) on them differ only by scalar operators. Note that
(9.10.1)
dim L(A) = 1 if and only if AI." = o.
Indeed, if AlI}I = 0, we can consider the I-dimensional g(A)-module C which
is trivial on g'(A) and is defined by h --+ (A, h) on ; by tbe uniqueness
of L(A) (see 19.3), g(A)-modules L(A) and C are isomorphic. The "only
if' part (ollows from the computation: 0 ::; e./i(v) = (A,or)v if tJ is a
highest-weight vector.
If A is a symmetrizable matrix, then g'(A) carries a symmetric invariant
bilinear form (.1.) which is defined on ' by (2.1.2), and wbose kernel is c
(by Theorem 2.2). We put (.811) = (II-I ()1)I,,-I(,) for for /3, -, E Q (see
S2.6). We can also define p E ('). by (p, Qj) = lai; (i = 1,..., n).
In the sequel we will use the following version of Proposition 9.9 for the
Lie algebra g'(A), where A may be infinite.
PROPOSITION 9.10. Let A be a symmetrizable matrix, possibly infinite.
a) If2(A+p, 11- 1 (.8)) #:- (.aJP) (or every fJ e Q+ \{O}, then the g'(A)-module
M(A) is irreducible.
b) Let V be a g'(A)-moduJe such that the following three conditions are
satisfied:
(i) (or every v E V,ei(v) = 0 (or all but a finite number o{the ei;
(ii) (or every v E V there exists k > 0 such that ei l ... ei.(v) = 0 whenever
8> k;
(iii) V = Ea V", where VA = {v E Vlh(v) = (, h)v (or all h E '};
E(').
(iv) i(  and JJ E ('). are primitive weights such that  - JJ = .al' (or
some fJ E Q+ \{O}, then 2( + p, ,,-1(,8»)  (,8IP).
Then V is completely reducible. i.e., is isomorphic to a direct sum of g'(A)-
modules of the form L(A), A E (')..
Proof To prove the proposition, we employ the operator 0 0 instead of n
(see 52.5). It is clear that 0 0 is locally finite on V as long as conditions
(i) and (ii) of b) hold. Furthermore, (2.6.1) implies the following fact. Let
v E V be such that (0 0 - a/v)i(v) = 0 for some k E l+ and a E C, and
let v' e UIJ(v) ({J E Q); then (see (3.4.1»

156
HigAe,'. Wei,A' Module. otler Kac-Mood, AIge6,..,
Ch.9
(9.10.2)
(0 0 - (d + 2( + Pt 11- 1 (Il») - (1lIP»Iv )'v' = o.
To prove a), suppose that J is a proper nonzero 8ubmodule of M(A). Take
a nonzero element v' = E Vi E J. where Vi e U. (v), V is a highest..weight
I ·
vector of M(A) and Pi e Q+\{O}, such that EhtPi is minimal. Then
i
"+(Vi) = 0 and therefore OO(Vi) = O. Note &180 that Oo(v) = O. Applying
(9.10.2) to V = M(A) and v' = Vi, we arrive to a contradiction with the
hypothesis or a).
To prove b). put VO = {v e Vln+(v) = O} and V' = U(g'(A»V o .
Note that VO i. graded with respect to the decompoeition (iii). Using
(9.10.2) and (iv) we show that every V e VO n VA generates a g'(A)...
module L(). Indeed. first we show that the g'(A)-module U(g/(A»(v)
is Q+-graded. In the contrary case, there exists a linear combination of
nonzero vectors E Vi = 0, where Vi E Ul3i (V), Pi E Q + and Pi #- Pj for
i
i =F j, with E ht,8, minimal. By the same argument as in proof of a)
i
we arrive at a contradiction with (iv). Second. we show that this mod-
ule is irreducible. By Lemma 9.10 it suffices to show that it has no Q+_
graded ideals. In the contrary cue, there exists a nonzero v' E U( v),
where,8 E Q+\{O}, luch that ft+(v') = O. Again, (9.10.2) gives us a
contradiction with (iv). Hence the g'(A)-module V' is completely re-
ducible.
Now we prove that V' = V. Suppose the contrary; then there exists
a vector II e V \ V' such that n+ (11) C V' and (0 0 - a/v)i (11) = 0, for
some k E Z+ and a e C. Since. clearly, Oo(v) E V', we have a = 0 and
hence O ( tJ) = O. But then there exists ,8 E Q + \ {O} and u e U such that
n+(u(v» = 0 (by (ii». Since Oo(u(v» = 0, using again (9.10.2), we arrive
at a contradiction with (iv).
o
COR.OLLAR.Y 9.10. Let A be a symmetrizable matrix with nonpositive real
entries and let V be a g'(A}-module satisfying conditions (i), (ii) and (iii) of
Proposition 9.10b). Suppose that for every weight  o(V one has {, on >
o for all i. Then V is a direct sum of irreducible g'(A}-modules, which are
free of lank 1 when viewed as U (ft_ )-modules.
Proof. We may assume that A = (Oij) is a symmetric matrix (replacing

Cb.9
Higlae,'. WeigA' Module, over Kac-Mood, Alge6N'
167
the elements or by proportional onea). For Il = E iai e Q+ \ to} and a
.
weight A of V we have
2(A + P, v-I ({l») - (PIP) = 2 E i(At an
i
- E Oijkikj - E aii(l- k,) > o.
iJ .
Now we apply b) and then a) of Proposition 9.10.
o
S9.11. Here we give an unexpected application of the results of the
representation theory developed in tbis chapter, to the defining relations of
the Lie algebras g(A) with a symmetrizable Caftan matrix A. As before,
(.'.) denotes a bilinear form on g(A) provided by Theorem 2.2. Let j(A) be
the Lie algebra introduced in 51.2, so tbat g(A) = i(A)/r and r = f_ ED f+.
Set t a = go n f.
PROPOSITION 9.11. The ideal t+ (resp. t-J is generated as an ideal in ii+
(resp. n_J by those fo (resp. t- OI ) for which a E Q+ \U and 2(plo) = (ala).
 -
Proof We define a Verma module M(-\) over j(A) by U(i(A»/J. where
j is the left ideal generated by "+ and h - (h) (h E ) (d. 59.2). Let
- ---
M'() be the unique (proper) maximalsubmodule of M(). Then we have
an isomorphism of i(A)-modules
(9.11.1)
,.
AI'CO)  E]1M(-ai).
i=l
This is due to the fact that iL is a free Lie algebra (see Theorem 1.2b» and
hence U(iL) is Creely generated by /1, ... I Jra. The isomorphism (9.11.1)
gives us the fo))owing isomorphism of j(A)-modules:
(9.11.2)
U(g) U(i) AI'(O)
n n
 U(g) U(i) (EI) M( -0.»  ED M( -a;).
.=1 .=1
Here and further we write U(g>,. .. in place of U(g(A»,... for ahort. Let
.. : JCA) - g(A) be the canonical homomorphism. We define a map

158
H.gAe,'. Wei,Ia' Moll./e. over KGc-Mood, Alge6na.
Ch.9
.Al : t_ - U(g) @U(i) M'(O) by .Al(a) = 1 @ a(ii), where ii is a highest-
weight vec.tor of M(O). This is a g-module homomorphism; indeed, (or
z e j, (J e t_ we have
l([Z,G]) = 1  (ZG - Gz)ii = 1r(z) 0 o(v) - 1'(0)  z(v)
.
= I"(z) @ a(ii) = %(;\1(0» since 1f(a) = O.
Similarly, we get 1 ([t_, t_]) = 0, so that we have a g(A)-module homo-
morphism:
n
.A: t_/(t_, t-J - ED M( -Oi)
.=1
by (9.11.2). More explicitly,  is delcribed as follows: write a E t_ in the
form (I = EUi/i, where Ui e Uo(n_); then (a + [t_,t_]) = LW(Uj)Vi,
i i
where Vi is a highest-weight vector of M( -0,). We deduce that  is in-
jective. Indeed, ;\(0 + [f_, t_]) = 0 implies w( Ui) = 0 for all i, hence
Ui e t_Uo(ii_), hence 0 e t-Uo(n_) n t_. Therefore, (J e [t_, t-J by the
following general fact: given a Lie algebra n and a 8ubalgebra t C ft, one
has
(9.11.3)
t n rUo(n) = [t, r]. .
Using the Poincare-Birkhoff-Witt theorem, we reduce (9.11.3) to another
fact about an arbitrary Lie algebra t:
(9.11.4)
t n U o (t)2 = [t, r).
This follows by passing to U(t/[t, r]), which is a polynomial ring.
As a result we have an imbedding .A : t_/[t_, L] - $M( -Oi) in the
category 0 of g(A)-modules. Now let -0' (Q E Q+) be a primitive weight
of the g(A)-module t_/[t_ t f_J.
Note that Q  n since no Ii lies in t. Using the embedding .A we
conclude by Lemma 9.8 that I - Qi + pl2 = I - Q + pl2 for some i and
therefore 2(plo) = (010) (since 2(ploi) = (oiIQi»' Applying Remark 9.3
we deduce that t_ is generated 88 an ideal in R_ by those t- a for which
o e Q+ \n and 2(pI0) = (0)0). This completes the proof of the proposition
for t_. The result for t+ follows by applying the involution w of g(A).
o
In the case of a Kac-Moody algebra we deduce the following theorem.

Ch. 9
Higlae'- Weigh' Modu/t over Kac-Mood, Algt6nJ,
159
THEOREM 9.11. Let g(A) = i(A)/f be a Kac-Moody algebra with a sym-
metrizable Cartan matrix A. Then the elements
(9.11.5)
(9.11.6)
(ad ) 1-.'. '-J,. - ( -. 1 )
el "ej, IrJ 1,)= ,...,n,
(ad Ii) 1- OJ j I;, i  j (i J j = 1,. . . J n)
generate the ideals t+ and t_, respectively.
Proof. Denote by 81 (A) the quotient or g(A) by the ideal generated by
all elements (9.11.5 and 6) (these relations hold by (3.3.1». We have the
induced Q-gradation 81(A) = ED g. Let r' denote the image of f:J: in
oeQ
91(A). Suppose the contrary to tbe statement of the theorem: t'+  0 (the
case of r_ is obtained by applying w). Choose a root a of minimal height
among the roots Q E Q + \ {O} such that t'+ n g :F 0, and let Q = E ki Qi. It
i
is clear that r+ n g must occur in any system of homogeneous generators
of r+.
It follows from the proofs of 53.8 (which used only the relations (3.3.1»
that there is r, E Aut SheA) such that r,(g) = gi(Q) and ri(t'+) = t'+
for all i. Hence ht r,(a)  ht 0- for aJI i = It..., n, and 'hence (ala-.) 
o for all i, where (.1.) is a standard Corm. Therefore (0'10')  O. But
2(pla) = 2 E ki(plai) == L ki{Oiloi) > 0 and we arrive at a contradiction
i i
with Proposition 9.11.
o
Theorem 9.11 implies the following definition of a Kat-Moody algebra in
terms of generators and relations. Let A = (Qij )'j:::::l be a symmetrizable
generalized Cart an matrix and let (, fi, n V ) be a realization of A (see
Sl.t). Then g(A) is a Lie algebra on generators e., Ii (i = 1,..., n),  and
the following defining relations:
[ei,lj] = 6ijQr , [h, ei] = (0;, h)ei t [h, Ii) = -(Oi J h) Ii,
[h, h'] = 0 (h t hiE ) t ( ad e i ) 1- CI ij e j = 0 t
(ad li)1-4,j Ij = 0 (i,j = 1, _.., nji  j).
Furthermore, g'(A) has the presentation described in the Introduction
(see SO. 3) .

160
Higlae,'" Weig'" Mod.le. over KGc-Mood, Alge6ra.
Ch.9
59.12. Given below are some other important applications of Proposi-
tion 9.11.
PROPOSITION 9.12. Let A = (aiJ) be a .ymmetrisable n x n-matrix, g(A)
the &.9Bociated Lie algebra, A+ the set o(poeitive roots, g(A) = n_ EB(Bn+
the triangular decompOtJitioD, etc.
a) If OiJ are nonzero real numbers of the lame .igD {or all i, j = 1,..., n,
then n+ (reap. ft-J g a free Lie algebra on generators
e 1 t · · · ten ( reap. /1, · · · , In)'
b) Let L C 4+ be such &hat
(i) (0111) are non.ero real number. of the .&me 8;gn (or all OJ fJ E L;
(ii) a,p eLand Q - fJ e + imply ellae Q - fJ e L.
Let ni be the 8ubalgebra of n generated by EB ga; set £ =
(JeL
E Cv- 1 (Q), and
GEL
(9.12.1 )
gL = n $ L $ n.
For 0 e L set gio = {z e g:l:ol(zlll) = 0 for all II e [n, n]}. Let 1
be a (possibly infinite) set containing Q E L witb multiplicity dim g:, and
consider the matrix B = «QltJ»(JIEl and the associated Lie algebra 8'(B).
Then gL is a Lie algebra isomorphic to a quotient of g'( B) by a central
ideal (which lies in the Cartan aubalgebra of 8'(B»). Furthermore, (9.12.1)
;. induced by the triangular decomposition of ,'(B), and n (resp. n) is
a free Lie algebra on a basis of the 'pace EB g: (resp. EB g(J)'
oEL oEL
Proof. First we prove a). Changing the elements of the dual root basis by
proportional ones, we may aasume that the matrix A is symmetric and has
positive entries. But then for Q = E kia, e Q+ we have
i
2(pI0) - (010) = E a.i(k, - kl) - E aijkikj < 0
 'j
ir a  0 and Q  n. Now we can apply Proposition 9.11.
To prove b) we may U8ume that L is finite. We use induction on ILl
and the statement a) to prove b) and also to prove that (.,.) give. a noo-
degenerate pairing of n and n.
o

Ch.9
Higlae.- Wei", Mod.le. over KGc-Mootlr Alge6ra,
161
COROLLARY 9.12. a) Let A be a symmetrj.able matrix and let 0 E 4+(..4)
be such that (alo)  O. Then ED gto M a free Lie algebra OD a basi. of
.>1
the space ED g, where g = - {z E g.I(zly) = 0 for all y (rom tbe
t>1
Bubalgebra generated by g-, .. ., g-(i-l)O)'
b) Let A be a geDerali.ed CartaD matrix and let Q E A+(A) be AD
isotropic root. Then CII-l(o) (& (ED 8;0) is an infinite Heisenberg Lie
jO
algebra.
Proof a) (oilowl from PropOlitioD 9.12 b) by letting L = {iali e Z+}.
I( a is an imaginary root of an affine Lie algebra, then b) amount. to
Proposition 8.4. Applying PropOIitioD 5.7 and Lemma 3.8 proves b) in the
general case.
o
59.13. Following is an application of the results of 59.10 to representa-
tion theory of the infinite-dimensional Heisenberg algebra, (= Heisenberg
Lie algebra of order 00; see 12.9). Recall that this is a Lie algebra with a
basis Pi, 9i(i = 1 t 2, - . .) and c. with the following commutation relatioDs:
(9.13.1)
[Pi, qi] == c( i == 1, 2, . . .), all the other brackets are zero.
This is a nilpotent Lie algebra with center Ce.
It is well known that Cor every a e cx t the Lie algebra. has an irre-
ducible representation (T., called the canonicol commutation re/a'ion, rep-
re,en'Gtion, on the space R = C[Zlt %2,. ..] of polynomials in infinitely
many indeterminates %" defined by:
a
O'a(Pi) = a- a ' O'a(Qi) = Xi, Ua(C) = aIR.
Xi
Here and further the operator of multiplication by a polynomial P is de-
noted by the same symbol. We denote this a-module by R II -
Introduce the Collowing commutative 8ubalgebru of .:
.+ ='ECpj;
J1
.- = ECqj.
.> 1
J_
A vector v of an .-module is called a t1Gcu..m vector with eigenvalue  e C
if '+(11) = 0 and c(v) = v.

162
Higlae,'. Weiglaf Module. over Kac-Mootl, Alge6nz"
Ch. 9
Note (cf. 52.9) that. can be viewed as the Lie algebra g'(O)/tl. where
tl C f is a central ideal, 10 that n+ (reap. R_) is identified with a+ (resp_
1_). Since (R,t1'.) i. a Cree U(._)-module of rank 1, it i. nothing other
than a Verma moduJe over a, the vector 1 being the highest-weight vector
= vacuum vector. Now the general PropOIitioD 9.10 givea, in our special
situation, the following result, which can be viewed 88 an algebraic version
of the Stone-yon Neumann theorem.
LEMMA 9.13. a) Let V be an .-module such that c = oIv, where 0  0,
wbich has a vacuum vector Va  0, such that V = U(,-)(vo). Then the
I-module V is isomorphie to R..
b) Let V be an I-module such that c is diagonalizable with nonzero eigenval-
ues and such that for every v E V there exists N such that Pit. . . Pi" (v) = 0
whenever n > N. Then V is isomorphic to 8 direct sum of s-modules of the
form Rat a -:f; O.
Proof. We can assume in b) that c = a/v with a 1: o. V may be viewed
as a g'(O)--module for which or = a/v for all i. But then for every weight
,,\ and for {J E Q we have (;\, v-I (p)) = ahtp. Since (PIP) = 0 and p = 0,
we have 2( + p, II- J (P») = 2ahtp  (PIP) Cor (J e Q+ \{O}. Now a) and b)
Collow (rom Proposition 9.10 a) and b).
o
COROLLARY 9.13. Let V be an irreducible .-module which has a nonzero
vacuum vector with a nonzero eigenvalue. Then the .-module V is iso-
morphic to R>..
The Lie algebra - is often extended by a derivation do defined by
[do,q;] = mj9j, (do,pj) = -mjpj,
where mj are some pOIitive integers. The Lie algebra a = (. + Cd o ) e Qo,
where Go is a linite.dimensional central ideaJ. is called an olcillator tJIge6rn.
Given" E C and ,\ E 00' we can extend the '-module Ra &0 the a-module
R. . A as follows:
. .
do "+ E mjzj 0 8. , 0..... (-',o)! (or a E CIa.
· ZJ
J
Letting '0 == Cc + Cd o + Clo, we have the trianglur decomposition
t = 1_ EB 'Q e '+.
The following proposition is immediate by Lemma 9.13.

Ch.9
Hight,$t- Weight Mod.lt, over KdC-Mootl, Alge6,.,,6
163
PROPOSITION 9.13. Let V be an a-module such that .0 is diagonalizabJe
and c has only nonzero eigenvalues.
a) If there exists Vo E V, Va  0, such that
a+(vo) = 0, U(I_)Vo = V,
than V is isomorphic to an a-module Ra.',>,
b) If for every v E V there exists N such that Pi. . . . Pin (v) = 0 whenever
n > N, then V is isomorphic to a direct sum of a-modules R C1t b,)., a =F o.
Note that the monomial z{a . . . zi,.- E R....>. is an eigenvector of do with
eigenvalue E mll;ll. Hence, (or the a-module R = t6.>' with d  0, we
It
have:
(9.13.2)
00
trR qd o = q& II (1- qmi)-l,
j=1
IIere J as usual, for a diagonalizable operator A on a vector space V with
eigenvalues 1, 2t... counting the multiplicities, one defines try qA =
E qAt.
i
As in 59.4, it is easy to see that tbe .-:module Ril carries a unique ("con-
travariant") bilinear form B such that B(I, 1) = 1 and Pn is an operator
adjoint to tin, provided that Q E R. It is also easy to see by induction that
distinct monomials are orthogonal with respect to B and that
(9.13.3)
B( ..... i. i. i. ) - Ei. II I.. I
"'1 ... Zn t %1 ... Zn - CI j..
j
As in !9.4, B can be written also in the following form:
(9.13.4)
B(P. Q) = (p(a al . a a2 .. .. )Q(%l. %2. .. · ») (0).
Note finally that for the a-module R II ,',). the operaton do and , e .0 are
selfadjoint with respect to B.
59.14. The following basic notions of the representaion theory or the
Virasoro algebra Vir (see 57.3), very similar to that of Kac-Moody algebras,
will be used later.
Define the trianga/tJr decompolition of Vir 88 follows:
Vir = Vir - $ Viro $ Vir +. where Vir:i: = €a Cd:i:j. Viro = Cc $ Cdo,
;>0

164
Higlae.'. Wei", Mod.le. over Kac-Mood, Alge6ra.
Ch.9
Given c, h e C, define a Vir-module V with highest weight (e, h) by the
requirement that there exists a nonzero vector v = Ve,1a 8uch that (cf. 59.2):
Vir+(v) = 0, U(Vir_)v = V, do(v) = hv, c(v) = cv.
(Here and further we follow the Ulual physicists practice to denote the
eigenvalue of a central element by the lame letter. hoping that this will not
cause confusion.)
The definition of the Verma module M(c,Ja) over Vir, 81 well as the
proor or its existence and uniqueness are given in the lame way as in 19.2.
It is clear that c acta on M(c, h) as cl. The number c is called the
conformal cntral charge. As in 59.7, we easily show that the elements
(9.14.1) d_ J . .. · d_ J2 d_ J1 (tie."), where 0 < it  j2 S · .. ,
Corm a buis or M(c,h). Since (do.d_ n ] = nd_ n . we see that do is di-
agonalizable on M(c, h) with spectrum Ia + Z+ and with the eigenspace
decomposition
(9.14.2) M(e,h) = Ea M(e,h)"+j,
JEI+
where M(c. h)Ia+J is spanned by elements of the form (9.14.1) with
jl + · · · + in = j.
It Collows that
(9.14.3)
dim M(c, h)A+J = p(j).
where p(j) is the cl&88ical partition (unction. (Thus, the Koetant partition
function for the Virasoro algebra is jU8t the claaaieal partition function.)
Equation (9.14.3) can be rewritten 88 follows:
co
trM(e.") qtl o := E dim M(e t h)9 = 9" II (1 _ qi)-I.
 j=l
As in 59.7. the series try ,4 0 is called the lormal claGrGcferofthe Vir-module
V.
As in 59.2, one shows that there exists a unique irreducible Vir-module
L(c, h) with highest weight (c, II).
The Chevally involution w of Vir is defined by
(9.14.4) w(d n ) = -d_n,w(c) = -c.
The contravariant bilinear form B on a Vir-module is defined as in 59.4.
In other words this is a symmetric bilinear form with respect to which d n
and d_ n are adjoint operators. Its existence, uniqueness and construction
are established in the lame way as in 19.4.

Ch, 9
HigAe,'- Weiglat Mod.le, over K(Jc-Moody Alge6rtJ,
165
59.15. Exercises.
9.1. Show that every nonzero homomorphism M(l) -+ M('\2) is an im-
bedding.
9.t. Prove that for V e 0 there exists an increasing filtration (in general
infinite) by submodules 0 = Vo C VI C . .. such that +1/ is a higheat-
weigbt module.
9.9. Let V be a module with highest weight A. Then ch V =
E c ch M('\), where CA = 1, CA E Z, and C = 0 unless A is a primi-
A<A
ti;e weight of a Verma module M(p), where p is a primitive weight of V.
Show also that if CA -F 0, then [M(A) : L('\)]  o.
g.. Let A be a generalized Cartan matrix of finite type t and V E 0 a
finitely generated module over g(A). Show that V admits a Jordan-Holder
series, i.e., a filtration by 8ubmodules V = V o J VI :> ... :>  = 0 such
that all the modules /+1 are irreducible. Describe tbe Jordan-Holder
series (or Verma modules over ./ 2 (C).
9.5. Let g'(O) be the derived algebra of the Lie algebra ,,(0) associated to
the n x n zero matrix. Given c) t . . . ,C n E C, the following formulas define
the structure of a g'(O)-module on the space V = C[ZI, .. . ,znJ:
ei -+ Ci !JIB , Ii -+ multiplication by Zi, Otr -+ Ci Iv .
uZi
Prove that V  M(A), where (A, err> = Ci (i = 1,. .. , n). Show that t' is
irreducible if and only if all Ci are nonzero.
9.6. Let M(O) be the Verma module with highest weight 0 over g(A).
Then if v is a highest-weight vector. Ji(V) is killed by R+, and one haa an
imbedding M( -Oi) C M(O).
9.7. Assume that A is symmetrizabJe. Show that M( -0;) is an irreducible
module if 2(pl-r) :F (,,1,,) for all nonzero" e Q+ \D.
9.8. Using Exercise 9.7. show that if 2(pl,,)  (11,,) for all nonzero 1 e
Q+ \fit then the 8ubmodule E M( -0,) C M(O) is. in fact, a direct lum
i
1'1
ED M( -Oti). Deduce that in this cue n (I-e( _O/»multcr = 1- E e( -0/.).
· QE4+ '=1
9.9. Use Exercise 9.8 to 8how that if A is 8ymmetrizable and 2(pJ)  (111)
for all nonzero 1 E Q+ \0, then the subalgebra ft+ (resp. ft_) of ,(A) iI a
(ree Lie algebra on generators ei (resp. I.) 'i = 1,..., n. (This is a special
C88e of PropC8ition 9.11, but this alternative proof is simpler.)

166
Hi,Ae,t.. Weight MDJ.lt, over KGc-Moo, AI,6",.
Ch.9
9.10. Prove that the Lie algebra on generators tit Ii (i = 1,..., n) and
h and defining relations (ej,/j) = 6i j h, [h,ei] = e., [h,/i] = -Ii (i,j =
1 t · · · , n) is simple, and therefore e 1 t ..., eft generate a free Lie algebra.
9.11. Prove that if A is an indecomposable 8ymmetrizable generalized
Cartan matrix of indefinite type, then g(A) contains a free Lie algebra of
rank 2, and hence has exponential growth, i.e.,
Jim )ogdirngj{l)/Ijl > o.
J ..... 00
- -
9.1 I. Let ; be a Lie algebra. a an ideal in "," = P / G. Show that the
following sequence or p-modulee is exact:
o -. a/[a, 0] -+ Uo(p)/aUo(p) -+ Uo(lJ) -+ o.
9.13. Using Exerci 9.12 construct the following exact sequence of g(A)-
modules:
t1
o -"f_/[t_. L] - E!)M(-Oi) - M(O) - C - 0,
.=1
where C is viewed 88 a trivial module.
9.14. Let:F be the space of all functions on . which vanish outside a finite
union of seta of the form D(). Then one can define product (convolution)
of two functions I. 9 e F by: (I. g)() = E /(1')
E'.
g(A - IJ). Define the delta function 6(p) = 6A.IA. Show that the map
e(.-\) .... 6). gives an aJgebra isomorphism t'::"F.
9.15. Let A be an affine matrix of type AP>. Show that the g'(A)-module
M( -p) hu an irreducible quotient which is not isomorphic to L( -p).
[Let v be a highest-weight vector of M(-p). Show that v - (1112 + I2I1)v
generates a proper lubmodule of M(-p).]
9. J 6. Suppose that we have two filtrations ' and " of V E 0 satisfying
conditions of Lemma 9.6. Then there exists a bijection t1 : J' -+ JII such
that Ai = A: U > for all j e J'. Furthermore. if v,. is a primitive vector of
V. then JJ = Ai where j is the leut integer such that v,. e Vj.
9.17. Consider the aaeociative algebra a on generators o,,(n e l). and
commutation relations
(am, a,,) = m6",._".

Ch. 9
HigAe'f- Wigla' Mod./e, over Kac-Mood, Alge6ral
167
Let V be an a-module such that 00 ........ pI and for every v E V, Qn( v) = 0
for all but & finite number of n > O. Let
Lo = (2 + p2)/2 + l>LjOj,
j>O
Ln = 4 E O-jOj+n + i..\non for n :F o.
jEZ
Show that
3
m -m 2
[Lm. L n1 = (m - n)L m + n + 6 m ,-n 12 (1 + 12 )1,
producing thereby a representation of Vir with conformal central charge
c=1+122.
9.18. Consider an associative algebra II on generators am, a and 6m (m E
Z) and the following defining relations:
[0;', on] = 6m.-n,[6m,6n] = m6m.-n,
all other brackets are O.
Let V be a b-module such that 60 ....... 0 and for every 11 E V, an (v) =
0, a(v) = 0 and 6n(v) = 0 for all but finitely many n > O. The sign
:: used below of normal product of generators, 88 usual, means that the
generators with non-negative indic should be moved to the right of the
rest of generators. Let
Hn = E : an_jaj :1
jEI
N = Eja_jaj +  Lh_jh j .
j>O j>O
Given numbers J #J e C, let
1 2 (
"'>.,p(K) = 2 P - 2, "'>.,p(d) = -HI "'>.,p(e n» = an,
"'>.,p(ln» = (n(p2 - 2) - )a; - L: Hja_j : -p E6ja_JI
JEI lEI
"'",p(h(n» = 6n.oI + 2Hn + ph".
Show that "''',p is a restricted representation of the affine algebra £( a12( C»
on V.

168
HigAe.t. Wei,At Mod./e. otJer KGc-Mootl,l AI,e6nJ'
Ch.9
9.19. Ie in Exercise 9.17 (reap. 9.18) there exista a a... (reap. &..) cyclic vector
10) e V such that an fa) = 0 for n > 0 (reap. (I" (0) = 0 = 6n 10) for n  0
and G: 10) = 0 for n > 0) then the character of V is equal to tbe character
of the Verma module M(t + 12A2, i(A2 + p2»(resp. M( il'2 -  - 2)A o +
AA 1 », but V is not necel8arily isomorphic to a Verma module.
9.10. In Exerciee 9.18 let 6,. = 0 for all n and let p = O. Show that if
there exist. a b.cyclic vector 10) E V such that an 10) = 0 for n  0 and
0: 10) = 0 for n > 0, then
ch V = e«- - 2)Ao - Al) n (1- e(_Q»-l,
oEA+.
9.11. (Open problem) Let V be an irreducible module over the Virasoro
algebra such that do is a diagoDalizable operator whose eigenvalues have
finite multiplicitiee. Then either these multiplicities are < 1 (then V is an
irreducible 8ubquotient of a module UQ, discussed in Exercise 7-.24), or
else V is a highest- or a lowest-weight module, i.e., there exists 11 :F 0 such
that di(v) = 0 (or all i > 0 or (or all i < O.
59.16. Bibliographical Dotes and comments.
The category 0 of modules over a finite-dimensional semisimple Lie algebra
was introduced and studied in Bernstein-Gelfand-Gelfand [1971], (1975],
[1976]. There is a vast Jiterature on the subject, which is 8ummarized in
the books Dixmier [1974] and Jantzen [1979], [1983]. The first nontrivial
results on Verma modules were obtained by Verma [1968]. One of the main
techniques of the theory is the Jantzen filtration of a Verma module. The
study of tbis filtration is based on a formula due to ShapovaJov [1972] for
the determinant of the contravariant form on M()I lifted from L(A).
The study of the category () and the highest-weight modules over Kac-
Moody algebras was started in Kac [1974]. There have been several devel-
opments of thil in the papers Garland-Lepowsky [1976]t Lepowsky (1979],
Kac-Kazhdan [1979], Deodhar-Gabber-Kac [1982], Rocha-Wallach [1982],
Ku [1988 A-C), [1989 A, B] and others. Again, the basic tool is the formula
for the determinant of the contravariant form, proved in Kac-Kazhdan
{1979J (generalizing Shapovalov's formula) and Jantzen's filtration.
This technique wu applied by Kac [1978 B]. [1979J to initiate a study
of highest-weight module. over the Virasoro algebra by computing the de-
terminant or the eontravariant form on Verma modules M(c. h). There
have been several published proofs of the determinant formula: see Feigin-
Fuch. (1982), Thorn [1984], Kac-Wakimoto [1986J. and other.. The com-
putation of chL(c, h) consist' of two steps, based on this formula. First,

Ch.9
HigAe.t- We;,A' Mod./e. over KtJc-Mootl, Alge6rtJ.
169
one finds all possible inclusions of Verma modules, and all cases when
[M(c, h) : L(c, hi)) # 0, using the Jantzen filtration (see Kac (1978 B]). Sec-
ond, one shows that [M(c l h) : L(c, hi)] $ 1. This more difficult fact, speciaJ
for the Virasoro algebra, was conjectured by Kac (1982 B} and proved by
Feigin-Fuchs (1983 A, BJ, (1984 A, BJ. The explicit character formulas for
all L(c, h) are easily derived from these facts: see Feigin-Fuchs (1983 BJ,
[1984). (The cases c > 1 and c = 0 were examined previously by Kac [179]
and Rocha-Wallach [1983 AJ.) Using the determinant formula, Kac (1982 BJ
proved that L(c, I) is unitacizable for c > 1, h > o. By a detajJed anaJysis
of the determinant formula, the critical strip 0 < c < 1 was examined by
Friedan-Qiu-Shenker [1985] who found the (discrete) set of possible places
of unitarity. The unitarizability for this set was proved by Goddard-Kent-
Olive [1985J, (1986), Kac-Wakimoto (1986J and Tsuchia-Kanie (1986 BJ.
The solution to the problem of integrability of unitary L(c, h) was given by
Segal [1981), Neretin [1983J and Goodman-Wallach {1985J (see Pressley-
Segal [1986] for all account of this). As a result, the only problem from Kac
[1982 B) that remains unsolved is Exercise 9.21. Kaplansky-Sa.ntaroubane
(1985J solved this problem in the case when the spectrum of do is simple
and Chari-Pressley (1988 B] in the case when V is unitary..
Tile remarkable link between the theory of highest-weight modules over
the Virasoro algebra and conformal field theory and statistical mechan-
ics was discovered by Belavin-Polyakov-Zamolodchikov 11984 AJ, [1984 B).
Conformal field theory bas become by now a huge field with many remarkable
ramifications to other fields of mathematics and mathematical physics: see
the reviews Goddard-Olive [1986J, Ginsparg [1988], Furlan-Sotkov-Todorov
[1988}, Goddard [1989], and many others. The basics of the representation
theory of the Virasoro algebra may be found in the book Kac-Raina (1987).
One of the main problems in the theory of Verma modules is to compute
the multiplicities fM(A) : L(A')}. In the finite-dimensional case, Kazhdan-
Lusztig (1979J carne up with a remarkable conjecture, which was soon af-
ter proved by Beilinson-Bernstein [1981] and Brylinski-Kashiwara [1981].
Kazlldan-Lusztig conjectures were generalized by Deodhar-Gabber-Kac
[1982] to arbitrary Kac-Moody algebras. Quite recently these conjectures
were proved by Casian (1989J and by Kashiwara [1989J. (Kazhdan-Lusztig's
{1980) cohomologicaJ interpretation of the multiplicities was extended to ar-
bitrary Kac-Moody algebras by Haddad [1984J.)
'The exposition of 9.1-9.8 is a simplification of that in Deodhar-Gab-
Ler-Kac (1982J. A more complete statement than Proposition 9.9 a) can
· 1"his problem has been sotved recently by O. Mathieu uClassification of Harish-
Chandra modules over the Virasoro algebra" and by C. Martin and A. Piard "In-
decomposable moduJes over the Vjrasoro algebra and a !:onjecture of V. Kac. u

170
Higlae,'. WeigAt Modulel over KClc-Mood, Alge6ral
Ch.9
be found in Kac-Kazhdan (1979J. The exposition of 9.9, 9.10 and 9.12 is
based on Kac-Peterson [1984 A). The results of 9.11 on defining relations
are due to Gabber-Kac (1gB!]. A simple cohomological proof of Theorem
9.11 was found by O. Mathieu (unpublished).
Exercise 9.2 is due to Garland-Lepowsky (1976). Exercise 9.4 is taken
from Bernstein-Gelfand-Gelfand {!97!J. Exercises 9.6-9.10 follow Kac
[1980 CJ. Exercises 9.12 and 9.13 are taken from Gabber-Kac [1981].
Exercise 9.17 is taken from Chodos-Thorn [1974]; its special case goes
back to Virasoro (19101. Exercises 9.18 and 9.20 are due to Wa.kimoto (1986).
These "free field" constructions of representations of the Virasoro algebra
 A
and the affine algebra .c(Sl2(C)) (and, more generally, .c(stn(C))) play an
important role both in the study of representations (Feigin-Frenkel (1988),
{1989 A-C]) and in the conformal field theory (Feigin-Fuchs (1984 A],
Dotsenko-Fateev [1984], Felder [1989], Bernard-Felder [1989], Bershadsky-
Ooguri (1989J, Distler-Qiu [1989]), and others.
One last comment concerns Exercises 9.10 and 9.11. The main result
of the paper Kac 11968 B) is the following. Let 9 = €a gj be an infinite-
jEZ
dimensional Z-graded Lie algebra which satisfies the following conditions:
( ' ) _ I . log dim Q j · d . I . II f . ,
1 Imj-too log 1;' < 00, I.e., 1m OJ grows po ynomla y as J -+ 00;
(ii) there are no nontrivial graded ideals;
(Hi) 0-1 + go + 01 generate 9 and the go-module 9-1 is irreducible.
Then 9 is isomorphic either to g'(A)/CK, where A is an affine matrix,
with the gradation of type (0, . . . , 1, . . . , 0), or to one of the simple Z-graded
Lie algebras of the polynomial vector fields on en: W nt Sn, Hn, and Kn.
It is actually proved that a Lie algebra satisfying (i)-(iii), which is outside
this list, contains a subalgebra of Exercise 9.10 with n :: 2, and hence has
exponential growth, as shown by Kac (1980 C). (By a different method,
exponential growth was proved in the rank 2 hyperbolic case by Meurman
[1982]). My conjecture is that if one drops condition (Hi), the only algebra
which should be added to the list is the "centerless" Virasoro algebra i) (see
7.3) and its subalgebra ()+ = E Cd j . A special case of this problem when
j > -1
dim gj < 1 has been solved by Mathieu [1986 A] (who showed that in this
case the list consists of the Z-graded Lie algebras defined in Exercises 7.12,
7.13 and 8.16). Quite recently Mathieu [1986 B] has solved this problem
in the case dim gj <canst... He has shown that any Z-graded Lie algebra
9 = $9j with only trivial graded ideals, is either g'(A)/CK with gradation
of type s, or i) or n+.
.. Recently Mathieu completely solved the problem in the paper uClassification of simple
graded Lie algebras of finite growth."

Chapter 10. Integrable Highest- Weight Modules:
the Character Formula
510.0. The central result of this chapter is the character formula for an
integrable highest-weight module L(A) over a Kac-Moody algebra) which
plays a key role in further considerations. We also study the region of
convergence of characters, prove a complete reducibility theorem and find
a product decomposition for the "q-dimension" of L(A).
SIO.1. Let g(A) be a Kac-Moody algebra of rank n and let  be its
Cartan subalgebra. Set
p = { E .I(A, Qi) E Z (i = 1,..., n)},
p+ = {E PI(,Q') > 0 (i = 1,... ,n)}',
p++ = {E PJ(,a) > 0 (i = 1,... ,n)}.
The set P is called the weight lattice, elements from P (resp. P+ or P++)
are called integral weights (resp. dominant or regular dominant integral
weights). Note that P contains the root Jattice Q.
Let V be a highest-weight module over g(A), and v a highest-weight
vector. It follows from Lemmas 3.4 b) and 3.5 that the module V is inte-
grable if and only if I/"(v) = 0 for some Nt > 0 (i = 1,. .. ,n) (see S3.6 for
the definition of an integrable module).
LEMMA 10.1. The g(A)-module L(A) is integrable jf and only jf A E P+.
Proof. Formula (3.2.4) implies the "only if' part and the following formula:
edi(A.o)+1(v) = 0 if (A,an E 1+,
where v is a highest-weight vector. It follows (since rej, /i) = 0 for j # i)
that if the vector li(A.O)+l(v) is nonzero, it is a primitive vector of L(A),
which is impossible (by Proposition 9.3b). Hence, for A E P+, we have
(10.1.1) li(A,orJ+1( v) = 0 (i = 1,..., n),
which proves the "if" part (by the remark preceding this lemma).
o
Denote by P(A) the set of weights of L(A). It is clear that peA) c P if
A E P. The following proposition follows from Lemma 10.1 and Proposition
3.7a).
171

172 Integrable Higlae.t- Wtight Modulel: t/ae ChaNcter Formula Ch, 10
PROPOSITION 10.1. If A E P+, then
multL(A)'\ = multL(A) w() (or w e w.
In particular, P(A) is W -invariant.
o
COROLLARY 10.1. If A e P+ then any e peA) is W-equivaJent to a
unique JJ E P+ n P(A).
Proof Take pew · A such that ht(A - p) is minimal; Lemma 3.12 b)
implies the uniqueness.
o
IIW
510.2. We let the Weyl group W act on the compJex vector space f, of
aJl (possibly infinite) linear combinations of formal exponentiaJs by
w(E cAe(.\» = L cAe(w(.\» (w E W).
 A

The space E contains e as a subspace. However, product of two eJements
-
PI, P2 E E doesn't always make sense, but if it does, then W(PI P 2) =
W(Pl)W(P2)' Proposition 10.1 says that
(10.2.1)
w(ch L(A)} = ch L(A) for w E Wand A E P+
Consider now the element (cf. 59.7)
R= IT (l-e(-a»multOtEE.
QE4+
Fix an element p E . 8uch that (cf. S 2.5)
(p, Q r) = 1 (i = 1,. . . , n).
For w E W set l(W) = (_I)l(w). By (3.11.1) we have
€(w) = det. w.
Furthermore, one has
(10.2.2)
w(e(p)R) = f(w)e(p)R for w E W.

Ch. 10 Integrable HigAe3t- Weight Module,.. the Character FoullJ 173
Indeed, it is sufficient to check (10.2.2) for each fundamental reflection
ri. Recall that by Lemma 3.7, the set A+ \{Qi} is ri-invariant and, by
Proposition 3.7. we have mult ri(Q) = mult a for Q E A+. Hence,
ri(e(p)R) = e(p - a.}ri(1 - e( -ai»ri II (1 - e( _a»mult Q
aEA+\{Qi)
= e(p)e(-a;)(I- e(a;» n (1- e(_a»multQ
crEA+\{ai)
= -e(p)R = €(ri)e(p)R.
510.3. From now on we assume that the generalized Cartan matrix A
is symmetrizable. Let (.1.) be a standard invariant bilinear form on g(A);
recall that (alfai) > 0 (i = 1,... t n). The (ollowing is the key fact in the
proof of the character formula, complete reducibility theorem and other
. results.
LEMMA 10.3. Let A,  E P be such that   A and A +  E P+. Then
either (A + ;\. or) = 0 (or i E 8upp(A - ) or (AlA) - (AI) > O. In
particular, jf A E P++.'\ E P+ and'\ < A, then (AlA) - (I'\) > o.
Proof. We have  = A - {J, where {J = E kiOi, k i  O. Hence (AlA) -
i
(,\),\) = (A + ,\111) = E ik;(o;JQ;)(A + .Ql). Since (0;'0.) > 0 for aJI i,
i
the lemma follows.
o
(10.4.1)
SlO.4. Now we can prove the (ollowing fundamental result of our repre-
sentation theory.
THEOREM 10.4. Let g(A) be a symmetrizable Kac-Moody algebra, and
let L(A) be an irreducible g(A)-module with highest weight A E P+. Then
E l(w)e(w(A + p) - p)
ch L(A) = wEW .
n (1 - e( _Q»mult a
OE4+
Proof. Multiplying both sides of (9.8.1) by e(p)R and using (9.7.2) we
obtain
(10.4.2)
e(p)Rch L(A) = L c;\e(A + p),
A<A
1+pI2 : IA+pI2

174 Integrable Highe.'- Weight Mod.lt,: 'he Characler Fonnu/G Ch, 10
where c'A = 1, CA E Z. By (10.2.1 and 2) the left-hand side of (10.4.2) is
W-skew-invariant (an element L e l is called W-.kew-invariant if w(L) =
l(w)L). Hence the coefficients in the right-hand side of (10.4.2) have the
following property:
(10.4.3) CA = (w)c" if w( + p) = p + P for some w E W.
Let  be such tbat CA  0; then by (10.4.3) we have Cw{A+p)-p  0 for
every w E W; hence, it fo)lows from (10.4.2) that w( + p)  A + p. Let
p E {w( + p) - pew e W)} be such that ht(A - p) is minimal. Then,
clearly, p + p E P+ and 'p + pl2 = IA + p12. AppJying Lemma 10.3 to the
elements A + p E P++ and p + p, we deduce that JJ = A. Thus, CA -1 0
implies that  + p = w(A + p) for some w e Wand in this case, CA = l(W)
(aee (10.4.3».
But A + P E P++, hence, by Proposition 3.12b), the equality w(A + p) =
A + p implies that w = 1. Hence, finally, we obtain
e(p)Rch L(A) = E c(w)e(w(A + p»J
cuEW
which is (10.4.1).
o
Now set A = 0 in (10.4.1). Since L(O) is the triviall-dimensionaJ module
over g(A) we deduce the following "denominator identitytt:
II (1- e(_a»multG = L: c(w)e(w(p) - p).
OE+ wEW
Substituting (10.4.4) in (10.4.1) we obtain another form of the character
formula:
(10.4.4)
ch L(A) = E c(w)e(w(A + p»1 E c(w)e(w(p».
wEW wEW
Of course, in the case when g(A) is a finite-dimensional (semisimple) Lie
algebra, (10.4.5) is the classical Weyl character formula, formula (10.4.4)
being the Weyl denominator identity.
Finally, remark that in the proof of Theorem 10.4 we used only the
fact that L(A) is an integrable highest-weight module (and never used its
irreducibility). This fact is guaranteed by relations (10.1.1). Therefore, if
V is a g(A)-module with highest weight A € P+ such that (10.1.1) hold,
then ch V is given by formula (10.4.1) and hence V = L(A). Thus we have
(10.4.6) L(A) = M(A)/)U(n_)Ji(A,On+1VA) if A E P+.
(10.4.5)
.

Ch. 10 In'gm61e Higlae61. Weigh' Mod./ea: 'Ae ClatlnJcter FOMna/a 175
This can be stated also as foJlows:
COROLLARY 10.4. Let A be a symmetrizable generalized Cartan matrix
and let A e P+. Then the annjhilator in U(g(A» of a highest-weight
vector of the g(A)-module L(A) is a left ideal genera,td by the elements
ei,/i(A.o rH l and h - (A, h), where; = 1,. .. ,n and h E . In particular, an
integrable highest-weight module over g(A) js automatically irreducible.
o
In .the finite-dimensional case, (10.4.6) is a result of Harish-Cbandra.
Another corollary is the following
PROPOSITION 10.4. Let A be a symrnetrizabJe generalized Cartan matrix
and let A e '. be such that (A. Qr> E Z+ (or i = 1,..., n. Then the
g'(A)-module L(A) is characterized by the properties that it is irreducible
and that there exists a nonzero vector v E L(A) such that
(10.4.7)
o(v) = (A, Qj)v and e;(v) = 0, i = 1,.. ., n.
In particular. the definitions of a g'(A)-module L(A), A E P+, given in
the Introduction and in Chapter 9, are equivalent.
Proof. Let V be an ir.reducible g'(A)-module which has a nonzero element
v satisfying (10.4.7). Considering the gradation of V by eigenspaces of
Qr, we see that the element li(A.oJ+l(v) generates a proper 8uhmodule of
V and hence is zero. Due to (10.4.6), we have a surjective g'(A)...module
homomorphism L(A) --+ V Lemma 9.10 completes the proof.
o
SlO.S. Consider the expansion
(10.5.1) II (1 - e(Q»- multo = E K({J)e({J) ,
oE4+ I'E".
defining a function K on . called the (generalized) partition function (K
in honor of Kostant). Note that K(P) = 0 unless P E Q+; furthermore.
K(O) = 1, and K({J) for {J E Q+ is the number of partitions of P into a
sum of positive roots, where each root is counted with its multip)icity. The
last remark follows from another form of formula (10.5.1):
E K({J)e({J):: II (1+e(Q)+e(2a)+...)muha.
EQ+ aE4+

176 Integrable Hight,'- Weigh' Mod.lt,: 'he Character Formula Ch. 10
Note that (9.7.2) ean be rewritten 88 follows:
( 10.5.2)
multM(A)  = K(A - ).
We proceed to rewrite formula (10.4.1) in terms of the partition function.
Inserting (10.5.1) into (10.4.1), we have
L(multL(A) )e() = E €(w)e(w(A + p) - p) E K({1)e(-p)
SA tuEW JJE.
= E E €(w)K(p)e(-fJ + w(A + p) - p)
....EW E.
= L I: f(w)K( w(A + p) - ( + p»e().
wEW '\E.
Comparing the coefficients at e(A), we obtain the multiplicity formula:
(10.5.3)
muItL(s\)  = L €(w)K(w(A + p) - (+ p».
WEW
In the finite-dimensional cue, this is Kostant'8 formula.
Warning. The series e( -p) E K({J)e(J3) is not W-skew-invariant,

though this is the inverse of the W -skew-invariant element e(p )R. This
,.""
is pOSBibJe since the inverse in £ is not unique.
SI0.6. Here we adopt a less formal point of view toward the characters,
replacing the formal exponential e() by the func:tion e on  defined by
).(h) = e().,h) for hE". We define the chaNcier chv of a g(A)-module V
from the category 0 to be the function
h ..... chv(h) = E multvp)e()..1aJ I
AEP(V)
defined on the set Y(V) of the elements he" such that the series converges
absolutely. Note that
chv(h) = try exp h for h e Y(V).
Let us introduce some additional notation. Define the complezified Tit!
cone Xc by
Xc = {z + iylz E X, , e .} .

Ch. 10 Integra6/e HigAe.t. Weight Module$: tAe CAtJrtJcttr Formula 177
Set
Y = {h E I E (multQ)I-(a,,,), < OO}I
aEA+
YN = {h E IRe(Qi,h} > N (i = 1....,n)} for N e R+.
Note that Y lies in Xc (by Proposition 3.12 c). We also have
Xc = U w(Yo).
wE W
LEMMA 10.6. Let V be a highest-weight module over g(A). Then
a) Y(V) is a convex set.
h) Y(V) ::> Y n Yo.
c) Y(V) :> }Jogn-
Proof a) is clear (rom the convexity of the function leAl_ Furthermore,
since V is a quotient of a module M(A), we deduce the following estimate
from (10.5.2):
(10.6.1)
multv   K(A - A).
which gives
L (multv )Ie('\''') 1 < le(A.A) 1 E K(p)le-(.8. A ) I.
E. EQ+
But (10.5.1) implies that for h E Yo we have
E K(p)le-(.8. A ) I = II (l-le-(a,A)n- multa.
PEQ+ oE4+
The product part of this formula converges if hEY, proving b). Part c)
follows from b) by the estimate (1.3.2).
o
For the proof of the prop08ition below we need the following
Remark 10.6. Let T C Xc be an open convex W-invariant set. Then
T C convex hull ( U w{Tn Yo»).
tuEW
Indeed, To:= U w(Yo\Y o ) is nowhere dense in Xc. Hence every h e T
weW
lies in the convex hull of T\To = U w(T n Yo).
wEW
o
For a convex set R in a (reaJ) vector space denote the interior of R in
metric topology by Int R.

178 In'egrtJ6/ Higlae,'. Weigh' Module.: 'he Charac'er Fomaula Ch. 10
PROPOSITION 10.6. Let g(A) be a Kac-Moody algebra and let L(A) be
an irreducible g(A)-moduJe with highest weight A e P+. Then
a) Y(L(A» is a solid convex W ..invariant set, which for every z e Int Xc
contains tz (or all sufficiently large t e R....
b) chL(A) is a holomorphjc function on lot Y(L(A).
c) Y(L(A) :) Int Y.
d) The series E l(w)ew(A+p) converges absolutely on Int Xc to a halo-
wEW
morphic function, and diverges absolutely on \ lot Xc.
e) Provided that the Cartan matrix A is symmetr;zable, chL(A) can be
extended (rom Y(L(A» n Xc to a meramorphic {unction on IntXc.
Proof. Set T = lot Y; it is clear that T is open, convex (see the proof of
Lemma 10.6a), and W-invariant; by Lemma lO.6b). we have Y(L(A» ::>
Y n Yo. Furthermore. Lemma 10.6a) and Proposition 10.1 imply that
Y(L(A» is a convex W-invariant set. Now c) follows from Remark 10.6.
In order to complete the proof of a) we have to show that X' := {z E
IntXcltz E Y(L(A» for sufficiently large t e R+J coincides with IntXc.
But again, X' is W-invariant and convex and contains Yo by Lemma lO.6c).
Again, we can apply Remark lO6.
The convexity of leA I implies that the absolute convergence is uniform
on compact sets, which proves b).
In order to prove d) remark that w(A+p)-(A+p) are distinct Cor distinct
w E W by Proposition 3.12b). It is clear Crom the proof of Proposition
3.12d) that (A + p) - w(A + p) E Q+. Hence we have for h e Yo:
I L t( w)e(III(A+p)-(A+P),h) I S L le-(a.h), < 00.
wEW aEQ.
Thus the region of absolute convergence of the series in question contains
Yo and is clearly convex and W-invariant. Now we apply Remark 10.6. It
follows that the series in question converges absolutely on Int Xc. On the
other hand, let h e \ lot Xc. Then the set do := {O' E A+I Re(Q, h)  O}
is infinite by Proposition 3.12c) and f), and for every Q E AO J we have
le(".(A+p),h) I  le(A+P. h ) I, proving the divergence at h.
Finally, e) Collows from d) and (10.4.5) (or rather (lO.6.1))
o
We shall give a more precise description of Y(L(A) in Chapter 11.
Denote by Oe the full subcategory of the category 0 of g(A)-modules
V such that chv converges absolutely on YN Cor some N > O. Also denote

Ch. 10 Integra6/e Highest- Weight Modu/e8.. the Character Fonnulo 119
by Ec the subaJgebra of the series from E which converge absolutely on
YN (or some N. By Lenuna 10.6, every highest-weight module lies in Oc,
We have a homomorphism tP of £e into the algebra of functions, which are
holomorphic on YN for some N, defined by f/J : e('\) .-. e. Applying y" to
both sides or formulas (10.4.5) and (10.4.4), we obtain on Y:
(10.6.2)
chL(A) = L f(w)ew(A+p) / L f(w)ew(p).
wEW wEW
II (1 - e-o)mult ° = L f(w)ew(p)-p.
QE+ tvEW
(10.6.3)
S10.T. Here we prove the complete reducibility theorem.
THEOREM 10.7. Let A be a symmetrizable generalized Cartan matrix.
a) Suppose that a g'(A)-moduJe V satisfies the following two conditions:
(10.7.1) for every v e V there exists N such that
ei a.. · ei. (v) = 0 whenever k > N.
(10.7.2) for every v E V and every i there exists N
such that It' (v) = o.
Then V js jsomorphic to a direct Bum of g'(A)-modules L(A) such that
(A, or) E Z+, (or all i.
b) Every integrable g(A)-module V (rom the category 0 is isomorphic to
a direct sum o(modules L(A), A E P+.
Proof. Thanks to Proposition 9.10 b) and Remark 3.6 in the case of state-
ment a) (resp. Proposition 9.9b) in the case of statement b», it suffices to
check that if ,\ and #J are primitive weights and {J E Q+ \ {OJ is such that
,\ - JJ = 13, then
2('\ + P. v- 1 ({J) :F (PIP).
Since the module V is integrable, we have by (3.2.4):
(10.7.3) (tQ;')  0 (i = 1,...,n) for every primitive weight.
But then we can write
2< + P, ,,-1(,8») - (,81,8) = < + ( - ,8) + 2p, ,,-1(,8»)
= ( + p + 2p, ,,-1(11» > 0,
using (10.7.3) and the fact that (P, ,,-1(,8» > 0 for aU P E Q+ \ to}.
o
.

180 Integrable Highet. Weight Module!: the CharCJcter Formula Ch. 10
COROLLARY 10.7. a) A g(A)-module V from the category 0 is integrable
if and only jf V is a direct sum of modules L(A) with A E P+.
b) Tensor product of a finite number of integrable highest-weight modules
is a direct sum o{modules L(A) with A E P+.
Theorem 10.7 contains as a special case the classical Weyl complete
reducibility theorem for finite-dimensional representations of semi-simple
Lie algebras.
SlO.S. Let 8 = (Bl,...,In) be 8 sequence of integers. In 51.5 we intro-
duced the Z-gradation of type .:
8(A) = ED 8/(6).
jEI
A particular case of this is the gradation of type I = (1,. ..,1) called
the principal gradation. Note that if 8i > 0 (i= l....,n), we have
dimgj(') <00.
Similarly we have the Z-gradation of type 8 of the dual Kac-Moody
algebra
g(' A) = €a' 8j ( 8 ) ·
jel
Fix elements , e . and h' E  which satisfy
("or) = 'i, (h'.Qi) = Ii (i = 1,...,n).
Note that  I = P and hi = pV are the elements p for the Kac-Moody
algebras g(A) and g(' A), respectively.
Warning: JI(p V )  2p/(plp).
Provided tha.t all Ii > 0, the sequence , defines a homomorphism F, :
C[(e( -(1). · · · , e( -on)]] -+ C[[q]] by
( 10.8.1 ) F, ( e ( - Q i» = q' i (i = 1,.. . , n ) .
This is called the 'pecitJ/iztJtion 01 type ,. Note that
( 10.8.2)
F,(e(-o)  q(h',o).
PROPOSITION 10.8. Let g(A) be a symmetrjzable Kac-Moody algebra.
Then
dim gj( I) = dim 'gj( I).

Ch. 10 In'egrab/ Highe,'- Weigh' Module: the Charac'er Formula 181
Proof. Note that both sides of identity (10.4.4) are elements from the al-
gebra C[le(-al),...,e(-a n )]]. Applying the homomorpbism FI to both
sides of this identity, we deduce
( 10.8.3)
II (1- qi)dim 'j( 1) = L l( w)q(p,p")-(w(p),p").
j > 1 wEW(A)
Similarly for g(t A) we have
( 10.8.4)
II (1 - qi )dim -,;( 1) = E l( w)q(p" ,p)_(w(pV),p).
jl tuEW('A)
Since W(A) and W(' A) are contragredient linear groups, the right-hand
sides of (10.8.3 and 4) are equal.
o
Comparing (10.8.3 and 4). we also deduce
(10.8.5)
II (1 - q(pV,o»muho = II (1 _ q(p,o»multo
QE+ oEA.
Remark 10.8. In the sequel we use the specialization of type, when some
of tbe 8, are o. Then F, is not defined everywhere and we have to check
that F, is defined on a given power series.
SIO.9. The specialization or type I is called the principtll IpecitlliztJtion.
The following proposition gives a product decomposition of the principally
specialized character.
PROPOSITION 10.9, Let g(A) be a symmetrizable Kac-Moody algebra.
Let A E P+ and set . = «A, or),.... (A, Q). Then
(10.9.1)
F. (e( -A) ch L(A» = II (1 _ qi )dim -,;(.+ 1 )-dim -,;( 1).
.>1
J_
Proof, By (10.4.5) we bave
( 10.9.2)
E l(w)(w(A + p) - (A + p»
e(-A)cb L(A) = wEW E f(w)e(w(p) _ p)
UlEW
For ,\ E P++ set
N). = E f(w)e(w(A) - A).
tuEW

182 Integra6/e Highe't- Weight Modult,.. the Ch(Jrdcter Formula Ch. 10
Note that N). E C[[e( -(1), . . ., e( -On)]] (by Proposition 3.12 b), d». We
have
F. (N).) = E l(w)q(>,.pYI-(tcI(>'),,"') = E l(W)q(>'.,Y-Ul(p V »
EW wEW
= F,.( E l(w)e(w(pV) _ pV»,
"'EW
where r = «(A,an,...,(A,Q». Applying identity (10.4.4) for g('A) we
obtain
F. (N).) = F,.( IT (1 - e( _o»mult a).
aE6 Y
+
Hence
(10.9.3)
F.(N).)= II (l_q()..a»mu!ta.
QE6
So, by (10.9.2 and 3) we have
n 1 - q(A+p,a) mutt Q
(10.9.4) F. (e( -A) ch L(A» = ( 1 _ (p,a) .
oE q
It is cler that (10.9.4) is an equivlent form of (10.9.1).
o
StO.l0. Let V = $ VA be a g(A)-module with highest weight A.
SA
Again, fix a sequence of nonnegative integers I = ('1,. . ., In). Let
deg(A - L k.o.) = E ki'"
, i
Then setting
J';(.) =. EB VA
A:del >'=J
defioea the gradation oj V of type ,:
V = EB (.).
jEI+
Note that dim \.)(,) < 00 if all 'i > O. The gradation of type 1 of V is
called the principal gradation. Provided that dim \.'; (,,) < 00, we have
F,(e( -A) ch V)  L: (dim \.)(.»qi.
Jel,+
We call the formal power series E dim \'J ( l)qi the q-dimenlion of V and
.>0
J_
denote it by dim, V. Then Prop08ition 10.9 and formula (10.9.4) can be
reetated as follows (here we use also Prop08ition 10.8):

Ch. 10 Integra61e Hight". Weight Module,: 'he Characttr Formula 183
PROPOSITION 10.10. Consider the principal gradatjon L(A) =
ED Lj( I). Then under the hypotheses and notation or Proposition 10.9,
jEI+
one has:
(10.10.1)
dim f L(A) = n (1 - qi)dim 'Ij('+ . )-dim Ij( I)
Jl
_ ( 1 _ q(A+P,O) ) mult Q
- II 1 _ qep,o)
oE6 V
+
o
The following corollary is a classical result of Weyl.
COROLLARY 10.10. Let A be a finite type matrix, so that g(A) is a sim-
ple finite-dimensional Lie algebra. Let A e P+. Then L(A) is finite-
dimensional and
dim L{A) = II (A + P, o)/(P. 0).
Q'EV
+
Proof Let q tend to 1 in (10.10.1) and apply I'Hopital's rule.
o
510.11. Exercises.
10,1# Show that
ch L(mp) = e(mp) II (1 + e( -0) + · .. + e( _ma»multCJ
oEA+
[Replacing e( -0;) by e( -em + 1)0;) we deduce from (10.4.4):
1: c(w)e«m + l)w(p» = e«m + l)p) II (1 - e( -em + l)o»mu1to.]
tueW arEA+
10.t. Show that p - w(p) (w E W) is equal to the sum of all positive real
roots a such that w- 1 (Q) < 0; the number oCsuch roots equals .t(w).
[See Exercises 3.6 and 3.12.]

184 IntegrtJ61e Higlae,'. Weight Module,: tlae Character Formula Ch. 10
10.3. Show that w(p) - p (w E W) are primitive weights of the module
M(O) over a Kac-Moody algebra g(A); deduce that this module has no
irreducible 8ubmodules if dim g(A) = 00.
10.4. I..Iet w be the Chevalley involution of g(A) and let V be a g(A)-module
from the category 0. Set VW = {f E V.lf(V) = 0 for all but finitely many
}. The algebra g(A) acts on VW by (z./)(tJ) = -/(w(z).v)(z e SeA), t1 E
V, I E VW). Show that VW E 0, ch V = ch VW, L()  L('\)w. Show that
M(O)W has no proper maximalsubmodules if g(A) is an infinite-dimensional
Kac-Moody algebra.
10.5. Show that the adjoint representation of an affine Lie algebra is not
completely reducible.
10.6. For  e . such that (,or) = .i let FA stand for F,. Show that
Fp(N).) = F).(N p ), , P E ..
10. 7. Let A = (OiJ) be a symmetrizable generalized Cartan matrix and let
n be a Lie algebra on generators ei (i = 1,..., n) and defining relations
(ad ei)l-tJ i ;ej = 0 (i  j). Setting deg ei = Oi defines a gradation n =
E9 n Q , where Q+ is the semigroup generated by the Qi in the free abelian
aEQ+
group Q on O'l,...,Qn. Define automorphism& ri ofQ by ri(Oj) = OJ-
a.j 0., and let W be the group generated by rl,. . . , r n . Extend the action
of W to the lattice Q ff) Zp by ri(p) = P - 01 and put ,(w) = p - w(p) for
w E W. Show that 8(W) is the sum of all Q e Q+ such that n Q  0 and
-w- 1 (o) e Q+. Show that
II (1- e(o»dim ft . = E €(w)e(I(w».
QEQ+ wEW
10.8. Let g(A) be a Kac-Moody algebra with the Cartan matrix A =
(:. -;0), where ob  4. For a given pair of nonnegative integers m and
n define the sequences tJj(m, n) and 6j(m, n) (j E Z) by tbe (ollowing
recurrent formulas (j e Z):
0J-l(m, n) + OJ+l(m, n) = o6 J (m, n) + m
6 j _ 1 (m,.n) + 6J+t(m, n) = hOj(m, n) + n
and oo(m,n) = ol(m,n) = 6o(m,n) = 6 1 (m,n) = O. Show that for  E.
such that ("\, or) = m, (, o) = n. one baa
E €(w)e(w('\) -,\) = E(-lY e( -oj(m. n)ol - 6 j +1(m. n)o2)'
tUEW(A) lEI

Ch. 10 Integrable Higlae,'. Wig/a' Module,: t/ae Charocler Formula 185
10.9. Show that if we set u = e(-QJ), v = e(-a2), the identity (10.4.4)
(or A = (_2 2 2) turns into the famous Jacobi triple product identity:
00
11 (1 - u n v n )(1- u n - 1 v n )(1 - u n v n - 1 )
n=1
=  (_1)m u m(m-1)/2v m (m+1)/2.
mel
and if A = (':1 "), the identity (10.4.4) becomes the important quintuple
product identity:
00
II (1-u 2n v n )(1 - u2n-lvn-1)
n=1 (1- u 2n - 1 v n )(1_ u" n -"v 2n - 1 )(1_ u4nv2n-1)
= E(u3m-2mv(3m2+m)/2 _ u3m-4m+1v(3m2-m)/2).
mEI
10.10. Let L(A) be an integrable A1)-module, so that k, := (A, on E Z+
(i = 1,2); put 8 = k 1 + k 2 + 2. Check that if i 1 =F k 2 , then
II (1- ,2n-1) dilTlt L(A) = II (1 _ ,n)-1,
nl n>1
"_0,:1:<'1+1) mod,
(For (i 1 . k 2 ) = (2,1) or (3,0) the right-hand side appears in the celebrated
Rogers- Ramanujan identities.)
In Exercises 10.11-10.20, A is a finite type matrix, so that g(A) is a
simple finite-dimensional Lie algebra. Denote by 9 its highest root and Jet
h = ht 8 + 1 be the Coxeter number. Let G be the associated complex Lie
group.
10.11. Show that {L(A)}AEP+ exhaust, up to isomorphism, all irreducible
finite-dimensional moduJes over g(A).
10.1 t. Set E = L: t!,. Define constants C1, . . . , C n by p v = L: CiQr , and put
i i
H = 2 p v and F = 2Eci'i. Show that {E, H, F} form a standard basis of
;
"2(C) (this is the so-called principal 3-dimeosional subalgebra). Using the
representation theory of .12(C) show that for A E P+ the expression
n 1 - q(A+p.a)
1 - ,(p,o)
aEA
is a polynominal in q with positive coefficients do = 1, d 1 J . . . , d m , and that
this sequence is unimodal, i.e., it increases up to d[m/2], and di = d m - I .

186 In'egra6/e Highe,'. W eight Module,: the ChartJcter Fonn.la Ch. 10
JO.J. Show that the sequences of the coefficients of the following polyno-
mials are unimodal:
a)  where U]! := (1 - qi)(l - qi-I) .. . (1 - q)/(l - qy i
b) (1 + q)( 1 + q2) . . . (1 + 'I").
[Apply Exercise 10.12 to the k-th 8ymmetric power of the natural repre-
sentation of An and the spin representation of Bn].
10.1-1. Let r be a real number such that Irl  0,1,..., h - 1, and let L(A)
be an irreducible g(A)-module with highest weight A E P+. Show that
· V I ) II .  (A+p.Q) / . "(p,a)
trL(A) exp(2rlp r = 81n sin.
r r
OE4+
Prove a similar formula for trL(A) exp(21riv- 1 (p)/r).
10.15. Let m = h (resp. m = h + 1) and let L = Q (resp. L = P). For
w E W  mL put ((w) = ((w'), where w' is the canonical image or w in
w. Show that for ,\ E p. either (, Q) == 0 mod m for some 0 E A v or
else there exists a unique element w e W t< mL 8uch that WA() = p. Let
A E P+, and let, e Z be relatively prime to m. In the case m = h + 1,
MBume that. is divisibJe by det A (which divides h). Show that
o if (A + p, 0) == 0 mod m
trL(A) exp(28ipV 1m) = for some Q E A v t
t(WA+p)(= :i:l) otherwise.
[We have
trL(A) exp(2..,i p v 1m)
= L (U1)e2.i( ",:- ,w(pV» / E (U1)e2.i(-!.,UI(pV» I
cuEW tuEW
hence, this is unchanged (resp. changes only the sign) when A+p is replaced
by A + p + mo (0 e L) (lesp. by rj(A + p), j = 1,... ,t).]
2ih .
10.16. Let trA+1 = exp h + 1 pV E G and let C be the cyclic group gener-
ated by trA+1' Show that the order of C is h + 1. Let h + 1 be a prime
integer. Deduce from Exercise 10.15 that the G-module L{A) (A e P+)I
restricted to C I is a direct sum of several copies or the regular representa-
tion of C plus at most one I-dimensional or h-dimensional representation
of C.
[Show that C is defined over Q.)

Ch. 10 lntegrob/e HigAe$t- Weight Modulel: 'At CAtJfTJc'er Formula 187
10.11. Let h + 1 be a prime number and let A E P+. Show that
dim L(A) == 0 or :i: 1 mod (h + 1),
(in particular, dim L(A) == 0 or :1:1 mod 7, 13, 13, 19, 31 if A is of type
G2, F, E fJ . E7, Ed, respectively), where 0, 1 or -1 appear according to
Exercise 10.15 (in the case m = h + 1).
10.18. For A of type A"-l' p a prime number, and the element
exp(2'1ri p v /p) E G one has 8tatementa similar to those of Exercises 10.16
and 10.17. Furthermore, sbow that:
trL«II-l)P) exp(2ipV Ip) = (;) (the Legendre symbol).
Deduce the quadratic reciprocity law. Compute tr t11a+ 1 in the spin repre-
sentation of Bt. Deduce that 2¥ E (-1) mod t for an odd t.
10.19. Let M C . be the lattice spanned by long roots, C»(.f.) the Killing
form on g(A), v :  ..... . the corresponding isomorphism, 9 = (OI8)-I,
and set !- = 2 for type Bt, Cl, F4,  = 3 for type G2 and k = 1 for
the rest of the types. Show that exp 4".;v- 1 (p) is an element of order
I:g, which coincides with exp(21ri p v /h) if i = 1. Show that for  E P,
either 2(IQ) E Z for some Q e A or else there exists a unique element
w E W  gM such tbat w(,\) = 1I- 1 (p). Deduce that for  e P+ one has
o ir(A+pIQ) == 0 mod g
trL(A) exp 4iv-l(p) = (or some 0 E d
f( w A + p ) otherwise.
10.20. Show that the conjugacy cJ888 of the eJement iTA = exp 27ri p V /h in G
consists of all elements tr such that Ad 6 is a conjugate of the automorphism
of g(A) of type ( I ; 1). Show that the conjugacy class of the element (1...+1
in G consists of all regular elements of order h + 1, and that Ad /7h+1 is an
automorphism or g(A) of type (2,1,...,1; 1).
10.£1. Let R be a countable-dimensional commutative associative algebra
over C with a unique maximual ideal m. Show that any z E m is algebraic.
f z is not algebraic then elements (z - )-1 for .\ E CX are linearly
independent elements of R.]
10.22. Consider the canonical filtration U(g(A») = U U j of the universal
.>0
J_
enveloping algebra of a Kac-Moody algebra g(A), and Jet Gr U(n(A» be
the associated graded algebra. Let AnnA be the annihilator of a highest-
weight vector of L(A) in U(g(A) and let R A = Gr U(g(A))jGr Ann". Using
Exercise 10.21, show that R A has a ullique maximal ideal if and only if
A E P+.

188 Infegra6/e Higlae,'. Wei", Module... ,Ae Character Formula Ch. 10
10.13. (Open problem). CJassify irreducible integrable modules over a
Kac-Moody algebra.
SlO.12. Bibliographical Dote. and CO tJIrn entl.
Theorem 10.4 88 we)) u other results of 5il0.1-10.5 are due to Kac [1974];
the exposition cloeely f0110wl this paper. Formula (10.4.1) is usually re-
(erred to &I the Weyl-Kac character Cormula. The results of 510.6 are due
to Kac-Peterson [1984 A] (see al80 Looijensa [1980]).
The first version of the complete reducibility theorem, which is Theorem
10.7 b), wu obtained in Kac [1978 A]. In it. present form, Theorem 10.7 is
taken from Kac-Peterson [1984 A]. Note that this refinement is important
(or the proof of a Peter-Weyl type theorem in Kac-Peterson (1983].
The exposition of 1510.8-10.10 follow8 Kac [1978 A]. The trick employed
in the proof or Proposition 10.9 goes back to Weyl and may be found in
many textbooks (e.g. Bourbaki [1975], Jacobson [1962J). The only new
thing is the use or the dual root system. The results of 5510.8 and 10.9
were generalized by Wakimoto [1983].
Exercises 10.3 and 10.4 are taken from Deodhar-Gabber-Kac [1982J.
Exercise 10.10 is taken from Lepowsky-Milne [1978]. This observation
eventually led Kac, Lepowsky and Wilson to a Lie algebraic interpretation
and proof of the Roger8-Ramanujan identities, as announced at the 83rd
AMS summer mting in 1979, abstract 768-17-1, and at the 175th AMS
meeting in 1980, abstract 775-A14 (see Lepowsky-Wil80D [1982] for an ex-
position or these results). For further development see Lepowsky-Wilson
[1984J, Misra [1984 A,D}, (1987], (1989 A.D]t Meurman-Primc [1987],
Lepow8ky-Primc [1985], Prime [1989], and others.
Exercise 10.12 goes back to Dynkin, Exercise 10.13 is due to Hughes
and Stanley (see Stanley (1980». The fact that trL(A)(exp 27ri p v /h) and
trL(A)(exp 47riv- 1 (p)) is 0,1 or -1 is due to K08tant [1976] (his proof is
more complicated). The rest or the material of Exercises 10.14-10.19 is
taken from Kac (1981]. Exercise 10.22 is e88entially due to Feigin-Fuchs
[1989] .
Theorem 10.7 implies that Hl(g'(A), L(A)) = 0 for A E P+. Dufio (un-
published) and Kuular [1986 AJ showed that Hk(g'(A), L(A)) = 0, k > 1,
for every A E P+, 1\ ::f. O. Kumar (1984J showed that H*(g'(A), C) is iso-
morphic to the cohomology of the associated topological group G(A); Kac-
Peterson [1984 B] computed previously the Poincare series of G(A).
The complexified Tit. cone Xc is related to the theory of singularities of
algebraic surfaces. Looijenga [1980] constructed a partial compactification
of the space of orbits of W t( 27riQv acting on the interior of Xc, which

Ch. 10 Integrable Highe.'" We;,A' Mod.le,: 'Ae CAtJnJc'er Form.la 189
I>lays an important role in the d(formation theory of singularities. Further
connections of the theory of Kac-Moody algebras and roups to the theory
of siIlgularities may be found in Slodowy (1981], [1985 A, B).
Chari (1986J and Cl1ari-Pressley [1987], [1988 A], [1989] have made a
significant progress in classification of irreducible integrable modules over
affine algebra.s. In particular, they showed that suel} a Inod ule with finite-
diluensional weight spaces is either L(A) or £.(A) (A E P+) or is of the
tYI)e considered in Exercise 7.15.
In a remakal)le recent developn1ent, KUlnar (1987 A] and 'lathien [1987]
proved Demazure'. character formula for an arbitrary Kac-Moody alSe-
bra, u8ing the geometry 0( Schubert varieties. (In the finite-dimensional
case tbis is due to Demazure (1974) and Andersen [1985J.) A corollary of
this is the Weyl-Kac cbaracter Cormula for an arbitrary (not necessarily
symmetrizable) Kac-Moody algebra.
The integrable module L(A) over a symmetrizable Kac-Moody algebra
g(A) can be deformed to a module over the quantized Kac-Moody algebra
Uf(g(A», and it remaiRs irreducible for generic values of fJ (Lusztig [1988]).
A remark on terminology. The integrable highest-weight modulea are
80metimes called standard modules in the literature. I object to this term
since first, it carries no information, and second, it is already used in the
representation theory of Lie groups for completely different representations.
Also, the theory of modular invariant representations by Kac-Wakimoto
[1988 BJ, [1989 A] show8 that not only integrable L(A) are of interest.

Chapter 11. Integrable Highest-Weight Modules:
the Weight System and the Unitarizability
111.0. In this chapter we describe in detail the set of weights P(A) of
an integrable highest-weight module L(A) over a Kac-Moody algebra g(A).
We establish the existence of a positive-definite Hermitian form on L(A),
invariant with respect to the compact form e(A) of g(A). Fina)JYt we study
the decomposition of L(A) with respect to various subalgebras of g(A) and
derive an explicit description of the region of convergence of ChL(A).
S 11.1. Fix A E P +. From the results of Chapter 3 one easily deduces
the following statement.
PROPOSITION 11.1. Let  E P(A), Q E 6,'" and m, = multL(A)( + 'a).
Then
a) The set of t E Z such that  + ter E P(A) is the interval
{t e ZI- p $ t $ q}, where p and q are nonnegative integers and
p_q=(,QV).
b) For eo E go\ {OJ the map eo : L(A)+lo - L(Ah+(Hl)O is an injection
jf -p  , < !(q - p); in particular, the function t t-+ m, increases on this
interval.
c) The {unction t ...... m, is symmetric with respect to t = !(q _ p).
d) If both ,\ and  + a are weights, then gQ(L(A)) ¥- o.
Proof By Lemma 10.1 and the resuits of S3.6, the proposition holds for a
simple root Q. Applying Proposition 3.7a), we deduce that it holds for an
arbitrary real root Q.
o
511.2. Fix A e P+. Recall that P(A) is W-invariant (by Proposition
10.1 ).
An element ,\ E P is called nondegenerate with repect to A if either
,\ = A or else ,\ < A and for every connected component S of supp( A _ ,\)
one has
(11.2.1)
S n {iJ(A, Qi)  O} ¥- 0.
190

Ch. 11
The Sirucl.re olln'egra6le Higlae,'.Weigh' Module,
191
LEMMA 11.2. Every weight  of the g(A)-module L(A) is nondegenerate
with respect to A.
Proof Suppose that  E P(A)\ {A}. Let S be a connected component of
8upp(A - ,\). Denote by "_(5) the 8ubalgebra of n_ generated by the fa
such that j E S. Then
(11.2.2)
L(A),\ C U(n_)n_(S)L(A)A.
If (11.2.1) were false, then n_(S)L(A)A = 0 (or some S and hence, by
(11.2.2), L(A),\ = 0, a contradiction.
o
Now we can describe explicitly the set of weights of the g(A)-module
L(A).
PROPOSITION 11.2. Let A E P+. Then
a) P(A) ::: W · { E P+IA is nondegenerate with respect to A}.
b) Il{il(A,or):::: O} C S(A) is a disjoint union of diagrams oEfinite type,
then
P(A) = W · { E p+ 1,\  A} .
Proof The inclusion C in a) and b) follows from Corollary 10.1 and Lemma
11.2. The other inclusion in b) follows from that of a). Indeed, if" E
p+ and  = A - fJ, where fJ E Q+, then (fJ'JQ) < 0 for i such that
(A, Qr) == 0, which implies that  is nondegenerate with respect to A under
the hypothesis of b).
It remains to show that if IJ = A - Q E P+, where Q = L kiQi,
i ·
k,  0, E k, > 0, and S n {il{A, fin 1:- O} 1:- . (or every connected comp<r
i
nent S of supp OJ then JJ E P(A). Let
no = {-y E Q+lr S Q and A -1 E P(A)} .
The set Oa is finite and the union oCsupports of its elements has a nonempty
intersection with each connected component of supp fi. Let Ii = L m,fi,
i
be an element of maximal height in nQ. It follows from Proposition ILIa)
that
(11.2.3)
8upp{1 = SUPPQ.

192
The Struc'tJre of In'egra6le Highe.'" Weigh' Module!
Ch. 11
Suppose that {J :F Q. Then
(11.2.4)
A - (J - 01 ;. P(A) if ki > mi.
Set S = {j E S(A)lk j = mJ}. Let R be a connected component of
(suppo)\S. Since JJ E p, we deduce from (11.2.4) and Proposition 11.1&)
(11.2.5)
(,8,or>  (A,ar> and (aJar>  (A,Qr) if i E R.
Set
pi = E miai,
'Ell
a ' = I)k i - mi)ai.
ieR
Then (11.2.3 and 5) imply
( 11.2.6)
(11.2.7)
(fJ',aj)  0 if i E R,
(Q', 0')  0 ifi E R.
It follows from (11.2.7) that R and hence S(A) are not of finite type. In
particular, for every  E P(A) there exists 0; such that  - Q. e P(A).
(Otherwise, dim L(A) < 00 aDd dim SeA) < (0). Hence S 'I 0 and, by the
properties of p, we can chooee R 80 that it is not a connected component
oCsuppa. But then, in addition to (11.2.6), we have: (fJ/, a,/> > 0 for some
j E R. Hence R is a diagram of finite type. This is a contradiction.
o
511.3. We proceed to study the geometric properties oCthe set of weights
P(A) for A E P+.
PROPOSITION 11.3. a) P(A) coincides witb the intersection of A + Q wjth
the convex hull of the orbit W · A.
b) If '\, #J e . are such tbat  - p e Q and p lies in the convex hull of
W · A, then multL(A)(p)  mult£(A)('\).
Proof. First we prove by induction on ht(A - ) that a weight  of L(A)
lies in the convex hull of W · A. Ie  = A, there is nothing to prove. If
 < A, there exists i such that  + at e P(A). Take the maximal s such
that p := ,,\ + 'i E P(A). Then JJ lies in the convex hull of W · A by the
inductive assumption; since A lies in the interval (p, ri(p)] (by Proposition
11.1») it also lies in the convex hull of W · A.

Ch. 11
The S'rucl.re o/In'egru6Ie Higlae,'. Weight Modulel
193
Let now  = E cww(A) E A + Q, where c. > 0 and E CVJ = 1. Then
w W
( 11.3.1 )
A -.\ = L c.,(A - w(A» E Q+.
trI
Replacing  by w() with minimal ht(A - ), we may assume that  E P+.
Finally it is clear from (11.3.1) that  is nondegenerate with respect to A.
Hence, by Proposition 11.2a), ;\ E P(A), proving a).
To prove b) we may assume that  E P+. Then we apply a) to L()
to obtain JJ E P(A). We prove b) by induction on ht(A - p). If  = p,
there is nothing to prove. Otherwise, IJ + a, e P('\) for some i. Let, > 0
be such that IJ + BQi E P(A) but JA + (8 + l)Qi  peA). By a), p + 'Oi
lies in the convex hull of W · ,,\ and in P(A). Hence, by the inductive
assumption, multL(A)() < multL(A)(p + sa;). On the other hand, p lies in
the interval (JJ+SQ., ri(JJ+BQi)] and hence multL(A)(P+SQi)  multL(A)(p)
by Proposition 11.1 b), c). Combining these inequalities proves b).
o
S 11.4. In the rest of the chapter we will assume that A is a symmetriz-
able generalized Cartan matrix; we fix a standard invariant bilinear form
( .,.) on g( A).
PROPOSITION 11.4. Let A E P+ and >.,Jl E P(A). Then
a) (AlA) - (Ip)  0 and equality holds jf and only jf >. = p. E W · A.
b) IA + pl2 -I + pl2 > 0 and equality holds if and only jf A = A.
Proof Since both ( .1.) and P(A) are W-invariant, we can assume in the
proof of 8) that  E P+. Since {J := A -  and PI := A - JJ lie in Q+, we
have (AlA) - (fp) = (AI) + (AIP1)  0 (cr. the proof of Lenuna 10.3). In
the case of equality we have (AlP) = (IPl) = O. Since ,\ is nondegenerate
with respect to A (by Lerruna 11.2), we deduce that (J = 0, i.e.,  = A. But
then (AlP!) = 0 and, by the same argument, p = A, proving a).
To prove b) we write:
(A + piA + p) - ( + piA + p) = (AlA) - ('\I») + 2(A - Ip)  0,
since (AlA) - (I'\)  0 by a) and A -  E Q+. Clearly, equality occurs if
and only if A = .
o

194
The S'ructure olln'egra6le Hight"" Weigh' Moclultl
Ch. 11
SII.S. Let V be a g(A)-module. A Hermitian form H on V is called
con'nltlanClni if
H(g(z), I/) = -H(z,wo(9)(II» for all, E g(A), Z,Jj E V.
For example the Hermitian form ( ".)0 OD I(A) is contravariant (see 12.7).
LEMMA 11.5. Let A e .. Then the g(A)-module L(A) carries a unique,
up to a CODstant (actor, Dondegenerate contravariant Hermitian (orm. With
respect to this form L(A) decomposes into &lJ orthogonal direct sum of
weight spaces.
Proof. Denote by G(A). the real 8ubalgebra of g(A) generated by tit Ii
and ., and let L(A). = U(g(A).)VA. Then H is the Hermitian extension
of B (see Proposition 9.4) from L(A). to L(A).
o
As in 59.4, it is easy to see that the fomula
(11.5.1) H(gvA, g'VA) = (wo(g)g'VA)
where g,g' E U(g(A» and "0 is the extension of -Wo to an antilinear anti-
involution of U(g(A». defines a contravariant Hermitian (orm on L(A)
(provided that A e .), normaJjzed by H(VA' VA) = 1.
Note that, by definition, the operators 9 and -wo(g) are adjoint opera-
tors on L(A) with respect to H. Thus, the compact form f(A) is represented
on L(A) by skew-adjoint operators.
The g(A)-module L(A) is called unitarizable if the Hermitian form H
defined by (11.5.1) is positive definite.
111.6. We proceed to prove the p08itivity of ( . ")0 on n_ + n+ and of
H on L(A). First we prove tbe inequality
(11.6.1) 2(pI0) > (Qlo) if Q E Ai- \n.
If a E dm, this is clear, since then (a la)  0, but (pia) > 0 for all
Q > o. If Q E L\\n. then a" E (+)"'\nV, and henc£
2(pla)/(alo) = (p,OV) > 1. This and (Qlo) > 0 imply (11.6.1).
By analogy with the "partial" Casimir operator 0 0 , we define an oper-
ator 0 1 on ft_ 88 Collows:
rh(z) = E E[et[e}tz]_] (z e "-).
oE6+ I
Here, as before, {e)} and {e} are dual basea of 90 and 9-0 with respect
to the bilinear form ( .1.), and the "minus" subscript denotes the projection
on n_ with respect to the triangular decomposition. Now we are in a
position to prove the crucial lemma.

Cb. 11
Th SIrucfure 01 Integro61e Higla,t- Weighf Modu/t
195
LEMMA 11.6. 1£0 E + and % E 8_0, then
01(Z) = (2(pIQ) - (QJo)z.
Proof We calculate in M(O) the expression Ooz(,,), where 1J is a highest-
weight vector, in two different ways. By (2.6.1), we have
(11.6.2)
OOZ(I1) = (2(pIQ) - (oJQ»z(v).
On the other hand. by the definition of 00, we have
Ooz(v) = 2 L E eei)z(v)
tJE4+ i
= 2 E Le[e),z](v).
EA+ i
Putting S = {P E + 111 < Q}. we may write
Ooz(v) == 2 E L e[e), z](v)
fJES i
= E E([e,[e),zJJ + [e)rzJe + e1re)rZJ)(V).
ES i
Using (2.4.4), we obtain
Oo(v) == E l)er[e),zJ}(tJ).
/lES i
Comparing this with (11.6.2) gives
«2(plo) - (olo»z) (v) = E E(e, (e), z]_](v).
E4+ I
As M(O) is a free U(n_)-module (see Proposition 9.2 b», the lemma fol-
Jaws.
o

196
The Struc'ure of Inlegro6/e Higlae,'. Weight Module,
Ch. 11
S 11.7.
result.
THEOREM 11.7. Let g(A) be a symmetrizable Kac-Moody algebra. Then
a) The restriction of ehe Hermitian form (.1')0 to every root space 90
(Q E d) is positive-definite, i.e., ( . J ')0 is positive-definite on n_ Et) "+.
b)Every integrable highest-weight module L(A) over g(A) is unitarizable.
Conversely, jf L(A) is unjtarizable, then A e P+.
Proof. We first prove a). Using Wo, it 8uffices to show that (.1.)0 is
positive-definite on CJ-a with 0 e A+. We do it by induction on ht Q. The
case bta = 1 is clear by (2.2.1). Otherwise, put S = {P E A+IP < o} and
use the inductive assumption to choose, (or every {J E Sf an orthonormal
basis {e} of g- with respect to ( .1. )0. Then, setting e) = -wo(e)1
we have (e) le) = 6;j . Now we apply Lemma 11.6 with this choice of
e) and e (i  (the choice ror the p E l1+ \8 is arbitrary). For It E 9-« we
have
Now we are in a position to prove the following fundamental
(2(plo) - (oIQ»(zlz)o = (01(2)lz)0
  [ (i) { (i) ]]
= L.J L.J( e_, ell . Z Iz)o
fJES i
 ,, ([ (i) 11[ (i) J
= L.J L..J efJ . z elJ t % )0.
PES i
By the inductive 8S8umption. the last 8um is nonnegative; using
(11.6.1), we get (zlz)o > O. Since (.1.)0 is nondegenerate on 9-0, we
deduce that it is positive-definite, proving a).
Using Lemma 11.5, one has to show for b) that the restriction of H to
L(Ah is paeitive-definite. We prove this by induction on ht(A - .\). Let
.\ E P(A)\ {A} and v E L(Ah. Thanks to a), we can choose a basis {e)}
of g(t such that {-wo(e»} is the dual (with respect to (.1.» basis of
9-cr. Then we have
n = 211- 1 (p) + E UiU' - 2 E Ewo(e»e),
i oE6+ i
and hence:
(11.7.1)
O(v) = (.\ + 2pl.\)v - 2 E Lwo(e»e)(v).
QE+ i

Ch. 11
T/a S'ruc'ure of Integrable Higla,'- Weiglat Mod.le,
191
Computing H{O(v), v) in two different waye by making use of Coronary
2.6 and (11.7.1), and equating the results we obtain
(IA + pf' -1.\ + pI2)H(v, v) = 2 E E H(e)(v),e)(v».
oE+ i
By the inductive assumption, the right-hand side is nonnegative.
Using Proposition 11.4 b) we deduce that H(v, v)  o. Since H is Ronde-
generate on L(A) t we conclude that it is positive-definite.
To prove the converse, note that by (3.2.4), we have:
o $ H(ll'VA, Iii VA) = H(/ i i - 1 VA,eifi'VA)
= k«A, or) + 1 -1:)H(ll:-1VA' lii-l vA )
.
= · · · = II j«A, Qn + 1 - j).
j=l
Hence (A, or) E z+.
o
Warning. The restriction of ( .1 .)0 to b and even ' is in general an
indefinite Hermitian form: the matrix «orI Q l)o) is a symmetrization or
the matrix A. In fact it is positive-definite (reap. positive-semidefinite) on
' if and only if A is of finite (reap. affine) type.
As was just mentioned, if A is a matrix or finite type, then the restriction
of (.1.)0 to  is positive-definite and hence, using Theorem 11.7a), the
Hermitian form (. '.)0 is positive-definite on g(A), 80 that g(A) carries a
positive-definite t(A)-invariant Hermitian form. Thus, Theorem 11.7 is a
generalization of a classicaJ result of the finite--dimeusional theory.
S11.8. We deduce from Theorem 11.1 b) another complete reducibility
result. For that we first prove
LEMMA 11.8. Let h e Int Xc. Then (or every r E R 'he number of
eigenvalues (countjng multiplicities)  of h in L(A) (A E P+), such that
Re A > r, is finite.
Proof. This follows from Proposition lO.6d).
o

198
The SC",c'"re ollft'gra6It Higla,'.. Weight M od./e,
Ch, 11
PROPOSITION 11.8. Let a C g(A) be an Wlo-invariant Bubalgebra which is
normaljzed by an element hElot Xc (i.e., [h, oj C a). Then with respec&
to a, the module L(A) (A E P+) decomposes into an orthogonal (with
respect to H) direct Bum of irreducible h-invariant 8ubmodules.
Proof. Put al = a + Ch. By Theorem 11.7b) and Lemma 11.8, L(A)
decomposes into an orthogonal direct sum of finite-dimensional eigenspaces
of h. It follows, using Theorem 11.7 b) and the wo..invariance of Gl, that for
every Gl-submodule V C L(A), the 8ubspace VJ.. is also an Gl-submodule
and L(A) = V e VJ.. Hence L(A) decomposes into an orthogonal direct
sum or irreducible Gt-modules.
Let U C L(A) be an irreducible DJ-aubmodu)e. It remains to show
that U remains irreducible when restricted to a. Let UA denote the -
eigenspace of h in U and let o be the eigenvalue or h with maximal real
part. Let CIA denote the A-eigenspace of ad h in a; we denote by Go (reap.
G+ or CI_) the sum of all a A with He,\ = 0 (resp. > 0 or < 0). Then
a = CI_ EB Go 6:) G+ J and it is clear that U).o i8 an irreducible Go-module and
that {z E UAla+(z) = O} = 0 if Re,\ <. Re o. Hente U is an irreducible
a-module.
o
S11.9. Fix 0 e +, and set
nar) =E99*i ar i gear) = n)$CV-l(a)$nar).
.> 1
J_
Then gear) is a subalgebra of g(A) (by Theorem 2.2e).
It follows from Proposition 5.1 and Chapter 3 that if Q is a real root,
then gear)  "2(C) and module L(A) restricted to gear) decomposes into a
direct sum of irreducible finite-dimensional modules. If Q is an imaginary
root. then gCar) is an infinite--dimensional Lie algebra as described in 59.12.
Now we can describe the restriction or L(A) to g(o) for Q e Arn.
PROPOSITION 11.9. Let a E Arn and A E P+. Introduce the followinl
two subspaces of L(A):
L(A)O) = EB L(Ahi
:('\la)=O
L(A)O) = E9 L(A).
A:(IQ»O
a) Considered as a g(a)-module, L(A) decomposes into a direct sum of
submodules L(A) = L(A)ar) e L(A)<;).

Ch. 11
The S'ruc'ure 0lInttgm61t Higlae,t. Weight Modulel
199
b) L(A)o> = (z e L(A)lg(o)(z) = OJ.
c) L(A)<,:> ;s a free U(no»-modu1e on a basis of the subspace
{z E L(A)<':>lno)(z) = O}.
d) The g(a)-module L(A) is completely reducible.
Proof. Recall that. by Prop08ition 9.12 b), we can view g<o) 88 a quotient of
a Lie algebra g'(B). Using Proposition 11.8, L(A) decomposes into a direct
sum of -invariant irreducible g<or)-submodules. Each of these 8ubmodules
is clearly generated by a nonzero vector v" e L(A)" such that no)(v,,) = O.
If ('\Ia) = 0, then Cv is g(a)-8table by (9.10.1). If (IQ) > 0, then Cv
generates a Verma module over 8(0) by Proposition 9.10 a).
To complete the proof note that
(11.9.1)
('\Ia) > 0 if  E peA) and Q E Am.
Indeed, by Corollary 10.1 and because Am is W-invariant, we may assume
that  E P+. But then (11.9.1) is obvious.
o
Statements a), b) and c) of Proposition 11.9 imply
COROLLARY 11.9. Let a E  and A E P+. Let  be a weight of the
g(A)-module L(A). Then either
a) (rQ) = 0; then A - kct is not a weight of L(A) (or k # 0; or else
b) (IQ)  0; then ('\IQ) > 0 and one has the following properties:
(i) the set of' e I such that  - ta E P(A) is an interval [-p, +00),
where p  0, and t ...... multL(A)( - to) is a nondecreasing (unction
on this interval;
(ii) jf z E g-o, % :F 0, then z : L(A)A-'Q --+ L(A)A-('+l)a is an injection.
c)l{ A E P++ and v is a wejght vector oE weight , then tIJe map n_ -+ L(A)
defined by n t-+ n( v) is injective.
S 11.10. Here we use the results of the preceding section to describe
explicitly the region of convergence of chL(A).
PROPOSITION 11.10. Let A be an indecomposable sgmme'rizable gener-
alized Canan matrix, Gnd let L(A) be on irnd.cible g(A)-moduie with
laighe' weigh' A E P+ 1 ,ucla f1atJ (A, Ql) -F 0 for "orne i. Thera the

200
The StrwctU1'e 01 Integra hie H;gJael'. Wtight Module,
Ch. 11
region Y(L(A) C  of (lb,o/ute convergence of chL(A) i, open and coincide,
tuitla the ,et
Y = {h E'" L (muJtl1')/e-<o,Ia)/ < oo}.
oEA+
Proof. By Proposition 10.6 c) it suffices to show that Y(L(A» C Y and
that Y is open.
The inclusion in question is obvious if A is of finite type. If A is of affine
type, we have
(11.10.1)
Y = {h E IRe(6th) > OJ.
This folJows from the description of the root system A(A) given by Propo-
sition 6.3 and the fact that the multiplicities of roots are bounded (by l)
by Corollaries 7.4 and 8.3. Now the inclusion in question follow8 since
multL(A)(A - ,6) :F 0 for all. e l+ by Corollary 11.9b(i). Finally, if A
is of indefinite type, then, by Theorem 5.6 c), there exists l1' E l1m such
that 8UPP a = S(A) and (0, Qr> < 0 for all i. But then A - Q e P{A)
by Proposition 11.2a). Moreover t by Proposition 11.1b) and Corollary
II.ge), for every nonzero tJ e L(A)A-o the map t/J : ft_ -+ L(A) defined
by tP(lI) = Y( 1)) is injective. This completes the proof of the inclusion
Y(L(A» c Y.
To see that Y is open, note that
(11.10.2)
Q E l1 im implies {h E Int Xla(h) = O} = 0.
This is clear since we can assume that h e C and then apply Proposition
3.12a) and f). Using Proposition 5.1a), we see that Y is the region of
convergence of the infinite product B := no EA ,..(l _ e-o)muICQ'.
+
For h E IntX consider the meromorphic function 1(:) = B(:h)-I. By
a standard property 0( Dirichlet series with positive coefficients (see e.g.
Serre [1970]), the set {t e Rlth e Y} of convergence or the series obtained
by multiplying out the product B( : h) -1, which represent8 1(:), is an open
segment (c, +(0). It follows that if th E Y, then (t - t)h E Y for some
t > O. But the argument in the proof of Proposition 10.6a) shows that
Y is convex and that with every h' it contains ,'h' for sufficiently large
" > o. Hence Y n. is open in ., 80 that Y = (Y n .) + i. is open
in .
o

Ch. 11
TA St",c'wre ollnttgm6/e Higlae61- WeigAl Mod.le.
201
Slt.l1. In this seetion we deduee a specialization formula or the de-
nominator identity (10.4.4). Let o be a subspace of  such that
(11.11.1)
o nIntXc 'F e.
Let  1-+ X denote the restriction map . -+ o; denote by p tbe homomor-
phism of £ to the eompleted group algebra of o defined by p(e(» = e(X).
Put
Ao = {o E Alii = O}; Ao+ = 60 0 n A+; Ro = 11 (1- e(-o».
QE 60+
By Proposition 3.12, the set Ao is finite; hence &0 C AN. It ia clear
that Ao satisfies the usual axioms for a finite root system (see Bourbaki
(1968]). Denote by W o the (finite) subgroup of W generated by reflections
r Q (Q E Ao), and let Po (reap. p) be the half-sum of roots from J1 o + (resp.
+). Define a polynomial D(;\) on . by
D() = II (, oV)/(POI OV).
oEAo...
LEMMA 11.11. For'\ E . we have
p( JlOl L (w)e(w(») = D(A)e(A).
tuEW o
Proof. Let Do C Ao+ be the set of simple roots or the root system 60.
Define the homomorphism F : C(t:(-a); Q e Do] --+ C[q] by F(e( -0» = 9
for all Q E fio. Then
(11.11.2) p(/) = lim F(f); F(e(-a») = q(p,o).
'....1
But by (10.9.3) we have
(11.11.3) F( E f(w)e(w(.\) - .\») = II (1 - q(,QY).
-EW o oEAD+
Formulas (11.11.2 and 3) togeher with (10.8.5) prove the lemma.
o
Now we can deduce the specialization formula:
(11.11.4) II (1- e(_o»multo
oE6+\60
-
-
E €(w)D(w(p»e(w(p) - 71).
tIIEWo\W

202
TAe Structtjre o/ln'egra6Ie Hi,ht". Weight Module,
Ch. 11
Here and rurhert Wo\W denotes a let or representatives of left coset. of
Wo in W. Indeed, dividing both sides of (10.4.4) by Ro. we deduce that
the leCt-hand side of (11.11.4), with 0' replaced by Q, is equal to
ROI E t(w)e(w(p) - p) = E t(l1) ( R0 1 E t(u)e(u(t/(p» - p».
wEW wEWa\W uEW.
Applying p and using Lemma 11.11, we 8et (11.11.4).
Now we consider a very special case of identity (11.11.4). Fix a Ie-
quence of nonnegative integers, = (.1.... t In) such that the lubdiagram
{i e S(A)I'i = O} C S(A) is a union of diagrams of finite type. Fix an
element h' E  such that:
(Qi, h') ='i (i = I, . . . . n).
The Iubspace Ch' of  satisfies condition (11.11.1) by Proposition 3.12.
Define A E (Ch")* by (A, h S ) = 1, and set q = e( -'x). Then p(e( -0,» =
q 8 i (i = 1,..., n). In other words, p is nothing else but the specialization
of type s. Now (11.11.4) can be written as follows:
(11.11.5 )
II (1 - ri)dim 'je,) = E t(w)D,(w(p»q(p-wC,).Ia').
jl tuEW'
Here D,(.\) = n (.\,OV)/(p"OV), where ,+ = {o E +
oE4.+
1(0, h') = O} and p, is the half-sum o( roots (rom ,+; W' is a system o(
representatives ofleft cosets o( the subgroup W, generated by r a , 0 E '+I
in W, 80 that W = W,W'j SeA) = Eagj(') is the Z-gradation of SeA) o(
J
type a.
511.12. In this section we shan discuss the unitarizabiJity problem (or
the OIcinator and the Virasoro algebras (cr. 559.13 and 9.14).
A contravariant Hermitian (orm H on a G (reap. Vir)-module V is a
Hermitian (orm with reaped to which the operators p", and 9", are adjoint
and the operators c and Q E Go. are self adjoint (resp. d", and d_ n are
adjoint and c self adjoint). As in 511.5, it is clear that the a-module R...,).
(resp. the Vir-module L(e, h» admits a contravariant Hermitian form if
and only i( a, 6 e R and .\ e Go. (reap. c, heR), and that this (orm is
uniquely determined by the condition
H ( 1, 1) = 1 (reap. H ( v C ,It J vel") = 1).

Ch. 11
The S'ructure ollra'egra61e Hight".. Weigh' Module,
203
RecaJJ that a module is called unitarizable if H is positive-definite.
It is immediate by (9.13.3) that the CI-module R..6.,. is unitarizable if
and only if a > O.
The unitarizability problem of the Vir-modules L(c, h) is quite non-
trivial. We shall make here only a few simple remarks (which will be used
in Chapter 12).
PROPOSITION 11.12_ a) If the Vir-module L(c, h) is unitarizable, then
h > 0 and c > o.
- -
b) If V = L(O, h) is unitarizable, then h = 0, and hence V is the trivial
l-dimensional Vir..moduJe.
c) Let V be a unitarizable Vir-module such that do is diagonaljzable with
finite-dimensional e;genspaces and with spectrum bounded below. Then
V decomposes into an orthogonal direct sum of unitarizable Vir-modules
L(c, h), and the spectrum of do is non-negative.
Proof Let tJ be the highest-weight vector of a unitarizable Vir-module
L(c, h). Then H(d_ 1 (v),d_ 1 (v» = H(v,d 1 d_ 1 (v» = H(v,2d o (v» +
H(v,d_ 1 d 1 (v) = 2h. Hence unitarizability implies h  o. Looking at
H(d_j(v),d_j(v» as j -.. 00 similarly implies c  0, proving a).
To prove b), consider the 8ubspace Un of L(O, h) spanned by d_ 2n (v)
and d 2 n( v). Since H is positive-definite on Un we have: detu. H  0,
which, after a simple calculation gives 4n 3 h 2 (8h - 5n) > 0 for all integral
n > o. This implies h = O.
To prove c) note that if U is a Virsubmodule of V, it is graded with
respect to the eigenspace decomposition of do (by Proposition 1.5), hence
V = U El) U 1.. Since U.l, the orthocomp1ement to U, is clearly also a V ir-
submodule, we deduce that V is an orthogonal direct sum of unitarizable
irreducible V ir-modules Va such that do is diagonalizable on Ui with specter
bounded below. Thus the Ui are highest...weight modules L(c, h) and the
positivity of the spectrum of do on V follow8 from a).
o
511.13. In conclusion of this chapter, we shall sketch a theory of gener-
alized KGc-MooJy Glge6raa. This is a Lie algebra g(A) associated to a real
n x n matrix A = (Qij) satisfying the following properties (cr. S 1.1):
( CI / ) either a.. - 2 O r a.. < O .
II - .. _ ,
(C2') aij < 0 if i =F j, and aij E Z if aii = 2;
(C3') Qij = 0 implies Gji = o.

204
The s,,,.c,u of Inltgrahle Highe,'. Weight Module,
Ch. 11
Let nre (resp. nim) = {a; e n J 0;; = 2 (reap.  O)}, let W be the
subgroup of GL(.) generated by the fundamental reflections r. 8uch that
Qi e nre (cf S3.7), and Jet C = {h e . , (ai.h)  0 if aii = 2}, C V =
{e .I (,Qr)  0 if (Jii = 2}.
As in 13.3 we have:
(11.13.1) (ad ei)I-0 ij ej = 0; (ad li)I-Gi j Ii = 0 if Oii = 2 and i -1= j;
( 11.13.2 ) (e. I e j] = 0, (Ii, 'j) = 0 if O;j = o.
A -diagona)jzabJe 8(A)-module V ia called ;ntegrdhlt if ei and Ii are
locally nilpotent when OJ; = 2.
All the results of Chapter 3 hold with obvious modifications for gen-
eralized Kac-Moody algebras-one just restricts attention to the i with
aii = 2.
The case of i with aii  0 is treated using the following
LEMMA 11.13.1. Let V be an integrable g(A)-module, let ail  0, let
 E . be such that (, Qr) < 0 and let v E VA be a nonzero vector such
that fi(V) = o. Then e{(l1)  0 tor all j = 1,2,....
Proof. This follows from the formula (cf. (3.6.1):
Jie1(V) = -j«(,an + (j - 1)oii)e1- 1 (v).
o
COROLLARY 11.13.1. If aii < 0, Q E L\+ \{Oi} and supp(a: + Cti) is con
nected t then 0 + jo; E L\+ for a}} j. E Z+.
Proof. 'We may assume that a - ai " 1.\+. Note that (ai, a'f) < 0 for all
j. Since supp(o + Qi) is connected we conclude that (at Qi) < 0 and apply
Le n1 m a 11.13.1.
o
All the results of Chapter 4 hold for generalized Kac-Moody algebras,
except that there is one additional affine matrix-the zero 1 x 1 matrix (cr.
52.9).
In order to describe the set ohoots , let re = W(n re ), im = \re,
K = {a E Q+ I Q E -C Y and suppa is conneded}\ U jn im .
.>2
J_

Ch. 11
TAt S'rvc'.re of In'egra61 Higlae,'" Weight Mod./el
205
Then we have
(11.13.3)
Am = U w(K).
wEW
The proof is the same 88 that of Theorem 5.4 using Corollary 11.13.1. All
other results of Chapter 5 hold for generalized Kae-Moody algebras with
obvious modificatioDs.
From now on in this section we will assume that g(A) is a generalized
Kac-Moody algebra with a symmetrizable Cartan matrix.
THEOREM 11.13.1. g(A) is a Lie algebra on generators ei, I. (i = 1, . . . , n)
with defining relations (1.2.1), (11.13.1) and (11.13.2).
Proof. Let 91 (A) be the quotient of g(A) by the ideal generated by all
elements (11.13.1 and 2), and let dl be the set of roots of gl(A). By the
argument proving Lemma 1..6, the support of any 0 E  1 is connected, and
since dl is W-invariant, we obtain, using (11.13.3), that 41 = 4. Now the
theorem follows from Proposition 9.11 and Lemma 11.13.2b) below.
o
LEl\ttMA ] 1.13.2. a) Let a = 2: kiQi be such that tlle k i are positive
iEI
integers ClIJd Q' E - C V , Then 2(p' 0) > (Q' I 0) with equaJjty jf and only jf
(Qi 100i) < 0 81Jd (Qi 10j) = 0 when i f= j and i,j E I, and (Qj '0;) == 0
\vlJen k i > 1.
b) Inequality (11.6.1) holds.
Proof. 2(pla) - (ola) = E ki(Q.IQi - 0). If 0, E ore, then (QiIQi) > 0 and
i
(Q;J - Q) > 0, so k;(;J()i - Q) > O. If 0; e Dim J then k;(Q;IQ; - 0) > 0
sinc Q - Qj e Q+. Hence 2(plcr) - (oJQ)  0, and if the equality holds
then all the Qi with i E 1 are in nim and satisfy (0; )Q; - a) = o. This
completes the proof of a).
To prove b) note that we can keep on strictly reducing (pia) by funda-
mental reflections while keeping a positive until either Q e n or Q E -C Y .
In the second case Q E Dim by a) since supp Q is connected.
o
It is clear by Lenuna 11.13.2 b) that Theorem 11.7 a) holds for general-
ized symmetrizable Kac-Moody algebras. It turns out that this property
characterizes them. More precisely, let A be a symmetrizable matrix, pO&-
sibly infinite but countable, satisfying (Cl')-(C3'), and let g/(A) be the ..
80ciated algebra, i.e. the quotient of the Lie algebra defined in Remark 1.5

206
The S'",c'.re o/ln'egra6Ie Highe,'. WeigA' Module,
Ch. 11
by relations (11.13.1 and 2). Let c (c ') be the center of g'(A) and let 9 be
a central extension or g'(A)/c (described explicitly by Borcberds [1989 A).
Define a Z-gradation of 9 by deg( center) = 0, deg ei = - dg Ii = 'i, where
II is a collection of positive integers with finite repetitions. Then 9 has the
following properties:
(gl) 9 = E\)8j is a Z.graded Lie algebra, [So, 90] = 0 and dimgj < 00;
J
(g2) 8 has an antilinear involution w which is -Ion 801 and w(gm) = g-m;
(93) 9 carries an invariant bilinear form ( .1. ) such that (9i I gj) = 0 unless
i + j = 0;
(g4) the Hermitian form (z I 11)0 = -(w(z) 111) is positive definite on 9j if
j  O.
THEOREM 11.13.2. Any Lie algebra 9 satisFying (91-4) can be obtained
(rom a generalized Kac-Moody algebra g'(A) as described above.
Proof. The argument proving Proposition 9.12 b) shows that 9 can be ob-
tained from some g'(B) 88 described above, where B is a symmetric matrix.
It follows from the proof of Theorem 11.7a) that we have the inequality
(11.6.1). This implies that DB satisfies (Cl')-(C3') for an invertible diag-
onal matrix D (see Borcherds [1988] for details).
o
We turn now to the study of g(A)-modules L(A) with A satisfying
(11.13.4)
(A, Qr> e Z+ if (Iii = 2; (A, Qr)  0 for all i.
We denote by P+ the set of such A 'B.
LEMMA 11.13.3. Propitjon 11.4 holds (or a generalized Kac-Moody al-
gebra.
Proof. The proof of part a) of Proposition 11.4 for a generalized Kac-
Moody algebra needs no modifications. The proof of its b) part is different.
Let A E P(A). It is clear that (A I OJ) > 0 for all Oi E n im . We can act
on ,\ by fundamental reflections each time strictly increasing IA + pJ2 until
.A E C V , so we may assume that (Alai) > 0 for all G:i E IIre. Thus:
('Qi)  0 for all Qi E n.
If A  A, let i e 8upp(A-A). If a. e U re , then, of course (a.IA+A+2p) > O.
If a4 e U im we have: (ailA + A + 2p) = (ailA) + (aIJA + a.)  0 since
(Alai)  0 and A+ai = A-fJ, where p e Q+. Hence (A-A I A+A+2p)  0,

Ch. II
TAt Struc'.re olln'egra61t Higla,'. Weight Module,
207
which is equivalent to IA+pI2_1+pI2  O. Equality implies that (AIQi) = 0
for any i E supp(A - ), i.e. tbat A = .
o
It is clear by Lemma 11.13.3 that Theorem Il.7b) holds for generalized
eymmetrizable Kac-Moody algebras.
Finally, we find a character formula.
THEOREM 11.13.3. Let A E P+ 1 and let SA = e(A+p) Ec(s)e(-s), where
s
. runs over all sums of elements (rom fitm and £(.) = (-I)m jf. is the
sum of m distinct pairwise perpendicular elements perpendicular to A, and
E(') == 0 otherwise. Then
chL(A) = L £(W)w(S/t.>/e(p) IT (l_e(_o»multCi.
wEW oEA+
Proof. Following the argument of S 10.4 we find that
( 11.13.5)
e(p) II (1 - e( _o»multo ch L(A) ::' L c"e(.\ + p),
oE6+ 
where both sides are antisymmetric under W, the CA are integers and  E
A - Q+ satisfy I + pl2 = IA + p12.
Let SA be the sum of the terms on the right of (11.13.5) for which
A + p E C V . If A + p  Int C V , then E t:(w)e(w(A + p» = O. hence the
tuEW
right-hand side of (11.13.5) is equal to E £(W)W(SA). and it remains to
tuEW
evaluate SA.
If e("\ + p) occurs in SA, we write  = A - E kiQi where ki are positive
iEl
-integers. Then fA + pl2 = IA + pl2 implies:
E ki(AIi) + E ki(.\ + 2pli) = o.
iEI iEI
But (Alai)  0 and ( + 2pJQi)  0 (the second inequality is proved in the
same way 88 in Lemma 11.13.3). It follows that
(11.13.6)
(Aloi) = 0 and (+ 2pfoi) = o.
Since (A + pJOi) > 0 and (plai) > 0 for Oi E n re , it follows that Qi E n im
for i E I. The second equality of (11.13.6) can be written as follows:
L)k. - 6 ij ){nil o j) = 0, j E 1.
iEI

208
The S,,..c,. olln'egra61e Haglae,'. WeigAt Mot/ule.
Ch. 11
Hence (ailoJ) = 0 Cor i,; e I unless i = j and i, = 1.
If e(A - L kiCti) occurs in ch L(A)t then (Alai) :F 0 for some it hence the
i
only terms on the right-band side of (11.13.5) which give a contribution to
SA are thoee coming from e(A + p) n (1 - e(_n»muJta. If (o.IQj) = 0
QE6+
for i 1:- j, then the coefficient or e(A + p - E ',0,) in this expression is 0 if
some of the k i are greater than 1 and is (-1 )Ei; otherwise.
o
Since chL(O) = 1. we deduce the following
COROLLARY 11.13.2. Let S = e(p) Et(.)e(.), where, runs over all Burm
,
of elements (rom nim and t(,) = (_I)m if, i5 the Bum of m distinct
pairwile perpendicular e/ement6, Gnd £(8) = 0 otAerwi,e. Then
II (1- e( _o»multo = e- P E t(w)w(S).
oE4+ WEW
o
!11.14. Exercilel.
11.1. Show that if A E P+ and A e P(A)\ {A}, then
E feW) multL(A)( + p - w(p» = o.
wEW
(This formula allows one to compute multL(A) by induction on
ht(A - ).
[Rewrite (10.4.5), multiplying by the denominator, 88:
E e() E feW) multL(A)( + p - w(p» = E c(w)e(w(A + p) - p),
 wEW wEW
and note that, by Proposition 11.4, the equaJity  + p = w(A + p} implies
 = A.]
11. e. If A is of finite, affine or strictly hyperbolic type, and A E P+, then
every dominant   A is nondegenerate with respect to A.
11.3. Show that if A E P+, then peA) n P+ = { E PI  w() for every
w E Wand A is nondegenerate with re8pet to A}.
11.. Show that if A e P+, then the let or uymptotic rays for the set of
rays through {A - pJp E peA)} lies in Z.
[Use Exercise 3.12 and Lemma 5.8.]

Ch. 11
The S'ruc'..re o/ln'egra61e Higlae.'- Weigh' Mod./e.
209
11.5. Let A, A', ... e P+. Show that the U(g(A»-submodule generated
by VA @ VA' (8) · .. of tbe g(A)-module V = L(A) 0 L(A')  . .. coincid
with the eigenspace of {} corresponding to the eigenvalue CA+A'+..., where
CM := 1M + pl2 - 'pI2. Moreover, CA+A'+... > CM for every M such that
L(M) C V.
11.6. Let g(A) be an infinite-dimensional Kac-Moody algebra and let A,
M E P+, AI'  o. Show that L(A) 0 L(M) 1J L(O).
11.7. Let g(A) = ED gJ be a Z-gradation of type (61.it... t 6 n .i) of a sym-
jEI
metrizable Kac-Moody algebra 8(A). Prove that the go-module g_1 is
isomorphic to tbe go-module L( -Oi).
[See the proof of Proposition 8.6c).]
11.8. Let A be a symmetrizable generalized Cartan matrix and
QEd;m. Show that 11'( Q) := lim loa muJt no exists. Show that
noo n
t/J(Q) = 0 if and only if (£tIer) = 0 and that .p(Q) > 0.48 if (aler)  O.
Show that Y C Int Xc coincides with (nt Xc if and only if A is a direct
sum of matrices of finite and affine type.
11.9. Let 9 and gO be Kac-Moody algebras with symmetrizable Cartan
matrices. Let  and o be their Cartan 8ubalgebra8 t Wand WO their WeyJ
groups, etc. Let . : gO --. " be a homomorphism such that
1r(o) C; ,..(Int X) n Int Xc  Ii (:1:0) n . (Z) = e.
Let A E P+. Show that the g-module L(A) i8 isomorphic as a gO-module
to a direct sum of integrable highest-weight modules LO(p) with finite mul-
tiplicities, which we denote by (A : p). For Jl = w(p' + pO) - po, where
w E WO and p/ E p, set (A : p) = l(w)(A : IJ'); for all other IJ E o. set
(A : 1') = o. We define AOt Wo, Po. D()t etc. t for o 88 in 511.11. Show
that .
11'- ( IT (1 - e(_o»multo) E (A : p)e(l')
aE 4+ \40+ lie '0.
E f(w)D( w(A + p»e(:,,-(w(A + p) - p» IT (1 - e{ _o»muUo CIr.
weWo\W QE6
11.10. Let m be asubalgebraof J such that m+n_ (resp. m+n+) has finite
codimension in g(A) and let A E P+. Show that dim {v E L(A)lm(v) = O}
< 00 (resp. dimL(A)/mL(A) < (0).
[Let U+ = (exptadeil tEe, i = l,,,..n). Show that UU+(ei)
l
-
-
span n+ .]

210
The S'",cl.re olln'lnJ6Ie Biglet.,. Wei,la, Module.
Ch, 11
11.11. Let g( A) be a .ymmetrizable Kac-Moody algebra. Set
R= n (l_e(_a»multo i F = -JoIR.
oE6+
Fix an orthonormal buil {Ui} of , and define derivations 8i of t by
8i(e(» = (. ui)e(). Show that
(8.F)2 -a1 F = R- 1 al Rj E a1 R = (plp)R.
I
Deduce that
E«aiF)2 -a1 F) = (pip).
I
J1.11. For fJ E Q+ set e, = E n- 1 muJt(,8/n). U8ing Exercise 11.11 show
n>l
that under the hypotheses or thie exercise one bas
(PIP - 2p)c# = L (fJ'IJ1")C#IC#".
""EO+
' +#"=
(This formula allow8 one to compute the multiplicities of roots by induction
on the height, thank. to (11.6.1). Note that if (l e K, then, due to Exercise
5.20, all summands on the right-hand side are Don-positive.)
11.13. Consider the following generalized Cartan matrix of order m where
m> 3:
-
A=
2
-1
-1
_ 1 _ 1
. . .
2 0
o 2
-1
. ..' 0
. . .
o
...................... .
-1
o
o
. . .
2
Set (J = 20 1 + 02 + Q3 + · · · + Om e A(A). Show that
multp = 2,"-2 - (m - 1).
11.14. Let A : VI -+ V 2 and B : V 2 ....... V l be linear maps of finite-
dimensional spaces. Show that trAB = trBA. Let V be a module from
the category 0 over a Kac-Moody algebra with a symmetrizable Cart an
matrix.. Deduce that for eo E go: and e_ o E S-o such that (eole- o ) = 1,
Q E +j, one has
trv.. e_ot o = ECA + jata) dim V+Jo'
J1

Ch. 11
The S'",ct.re of In'egrable HigAt,t- Weight Module,
211
Let V be a module with highest weight A. Deduce the folJowing general-
ization of Freudenthal's formula:
(IA + p('_I..\ + p12) dim VA = 2 E E(multa)(A + jQIQ) dim V,Hjo'
aE+j1
n
11.15. Let V = ED L(Ai), where A. E P+ satisfy (Ai,Qj) = 6ij (j =
i=1
1,. . ., n). Show tbat the group associated to the g(A)-module V is a central
extension of every group G. constructed in Remark 3.8.
J 1.16. Show that if a module L(A) over a generalized Kac-Moody algebra
is unitarizable then A E P +.
11.17. Let A, A' e P+ and IT e W be such that M := u(A) + A' E P+.
Show that
multL(A)(M + p - w(A' + p)) = 61.w.
Deduce that the multiplicity of L(M) in L(A) @ L(A') is 1.
(Use Proposition 11.4 and the outer multip)icity formula: multiplicity of
L(M) in L(A) 0 L(A') is E £(w) multL(A)(M + p - w(A' + p)).]
wEW
11.18. Using Exercise 9.17, show that the Vir-module L(c, h) with c  1
and h > (c - 1)/24 is unitarizable.
11. J 9. Show that the fixed point set of a diagram automorphism of a gen-
eralized Kac-Moody algebra is a central extension of a generalized Kac-
Moody algebra.
11.tO. Prove an analogue of Corollary 10.4 for generalized Kac-Moody
algebras.
11.£1. Let g(A) be a generaljzed Kac-Moody algebra. Show that + =
A+(A) is a subset of Q+ \{O} defined by the following properties:
(i) cr, E A+, 20i r;. A+, i = l,...,n;
(ii) if Q E A+ t ai E Dim and supp( Q + ai) is connected t then a + Q, E A+;
(iii) if Q E A+t Q, E ore and a f. Qit then (Qt ri{a») n Q+ C A+;
(iv) if Q E + then 8UPP 0' is connected.
J J.tt. Let S'(A) be a generalized Kat-Moody algebra. Show that the
U(g'(A»-module ED L(A) is faithful. (Use Exercise 11.20 to show that
AEP+
U(n_) acts faithfully. Then 0 < H(u_v;\, U_fl;\) = (wo(U_)U_fl;\) for some
A E P+t depending on u_ E U(n_). Hence U(n+) also ads faithfully.]

212
The Structure 01 Integrable Higlae6t- Weight Module!
Ch. 11
SI1.15. Bibliographical Dotes and co mm ents.
Proposition 11.2 was stated without a proof in Kac-Peterson [1984 A].
Propositions 11.3 and 11.4 are due to Kac-Peterson [1984 A). or course,
Proposition 11.4 is a standard fact in the finite-dimensional case. PropOli-
tion 11.3 a) in the finite-dimensional case, is due to Steinberg (see Bourbaki
(1975). It seems that Proposition 11.3 b) was not previously known even
in the finite-dimensional case.
Lemma 11.6 and the positivity Theorem 11.7 are due to Kac-Peter80D
[1984 B). The proof of Theorem 11.7 b) is a direct generalization of that
of Garland [1978] in the affine case (as lOon as Theorem 11.7 a) is estab-
lished). Theorem 11.7 ie important since it allows one to apply powerful
Hilbert space methods, see e.g. Goodman-Walla(h [1984 AJ, Kac-Peterson
[1984 B]. (Note that the Hilbert completion of L(A) is tbe space of a unitary
representation of the "compact Corm" of the group associated to a Kac-
Moody algebra.) Jakobsen-Kac [1985], [1989] classified the unitarizable
highest-weight modules over affine algebras for all natural choices of anti-
linear involutions and "Borel 8ubalgebrae"; it turned out that there exists
only one essentially new family of unitarizable highest-weight representa-
tions, that of the Lie algebra of maps of the circle with a positive-finite
meaaure into Bu(n, 1). There is an interesting connection of these repre-
sentations when n = 1 to the representations constructed in Exercise 9.20
(cf. Bernard-Felder (1989], Feigin-Frenkel [1989 OJ).
The material of iil1.9-11.11 is taken (rom Kac-Peterson [1984 AJ. The
proof of Proposition 11.12 b) is taken from Goddard-Olive [1986]. Most of
the material of 511.13 (on generalized Kac-Moody algebras) is taken from
Borcherds [1988 AI. The new material is the description of the root system
and Theorem 11.13.1 (that filJs a slight gap in his paper).
Exercise 11.1 in the finite-dimensional case is attributed to Racah. Ex-
ercises 11.4, 11.8 and 11.9 are taken from Kac-Peter80D (1984 A], and Ex-
ercises 11.5 and 11.6 are from Kac-Peterson [1983]. Exercise 11.10 is due to
Peterson-Kac [1983]. This pr<?perty has important applications to the con-
Cormal field theory. A similar "admissibility" property plays  prominent
role in representation theory of p-adic groupe.
Exercises 11.11. 11.12 and 11.14 are due to Peterson [1ge2], [1983). The
proofs indicated here were communicated to me by Peterson. These recur-
rent multiplicity formulas are very convenient for computations or root and
weight multiplicities. Another formula for root multiplicities, which is also
a formal consequence of identity (10.4.4), was found earlier by Berman-
Moody [1979].
Exercise 11.13 is taken from Kat [1983 A]. It gives some nontrivial

Ch. 11
The S'",c'.re o/ln'egra6Ie HigAel'. Weigh' Module.
213
evidence in support of the conjecture mentioned in 55.13. Exercise 11.15
is taken from Petel'8on- Kac [1983]. Exercise 11.17 is taken from Kac-
Wakimoto [1986]. This is a special case of the Parthasaraty-Rao-
VaradarajaD conjecture proved recently by Kumar [1988] and Mathieu
[1988]. Exercise 11.19 is taken from Borcherds [1988 A]. The set of roots
discussed in Exercise 11.22 is related to quivers with tadpoles (Kac (1983]).
The multiplicities of aJI roots of an indefinite-type Kac-Moody algebra
are not known explicitly in any lingle case (in some easel explicit for-
mulas are known for low "level" roote, lee e.g. Feingold-Frenkel [1983].
Kac-Moody-Wakimoto [1988]). Nevertheleaa Borcherds [1989 BJ, [1989 C)
managed to find the root multiplicities of some generalized Kac-Moody
aJgebras intimately related to the Monster simple group, and used this to
establish the modularity properties of the MODster.
Given below are a few computations of the root multiplicities done
on a computer by R. Gr088. using Peterson's recurrent formula. Here
(k i , k 2 .. ..) denotes the root 0 = i 1 0 1 + i 2 Q 2 + . . .. We may assume
that Q E -G v and that (a J a) :== ! E aijkik j =F o. If A is symmetric of
rank 2, we can assume that k 1 > k 2t since then mult{k 1 , k 2 ) = mult(k2' k 1 ).
In 1'able H 2 (resp. H 3 ) all such roots are listed with k 1 < 21 (resp k 1 < 14).
A part of Table H3 was computed by FeiI?-gold-Frenkel [1983].

214
The St".cture o/lra'egra6It Higlat.,.. Weigh. Module.
A = [ 2 -3 ]
-3 2
TABLE H2
cr - (ala) muJt a Q -(alo) mult (I
( I, 1) J J (15, J I) J9 23750
( 2, 2) 4 1 (16, 1 t) 151 25923
( 3, 2) 5 2 (J3, 12) 155 30865
( 3, 3) 9 3 (14,11) 164 4511
( , 3) 11 .c (13, 13) 169 55Q3
( 4, 4) 16 6 (IS, 12) 171 606S4
( 5, 4) 19 9 (16, 12) 116 7.c
( 6, 4) 20 9 (17, 12) 179 84121
( 5, 5) 25 16 (18, 12) 180 875-C1
( 6, S) 29 23 (14, 13) 181 91257
( 1, 5) 31 27 (15, 13) 191 135861
( 6, 6) 36 39 (14, 14) 196 165173
( 1, 6) 41 60 (16, 13) 199 185528
( 8, 6) 44 73 (17, 13) 205 233487
( 9, 6) 4S 80 (IS, 14) 209 271860
( 1, 7) 49 107 (18. 13) 209 271702
( 8, 7) 55 162 (19, 13) 211 29947
( 9, 7) 59 211 (16, 14) 220 «>9725
(10, 7) 61 240 (15, 15) 225 492420
( 8, 8) 6.. 288 (11, 14) 29 569358
( 9. 8) 71 449 (18. 14) 236 732180
(10, 8) 76 600 (16, 15) 239 81521.
(11. 8) 79 720 (19, 14) 241 874650
(J2. 8) 80 758 (20, 14) 244 972111
( 9, 9) 81 808 (21, 14) 245 1 00699
(10, 9) 89 J267 (17, 15) 251 1242438
(II t 9) 95 17 (16, 16) 256 1476973
(12. 9) 99 2167 (18, (5) 261 1152119
(10. 10) 100 2278 (19, 15) 269 2298090
(13. 9) 101 2407 (17, 16) 271 245
(11, 10) 109 3630 (20. 15) 275 2808958
(12, 10) 116 5130 (21, 15) 279 32017
(11. II) 11 6559 (22, 15) 281 3426450
(13. 10) 11 655S (18, 16) 284 3183712
(14. 10) 124 75s.4 ( 1 7, 1 7) 289 4.56:155
(15, 10) 125 7936 (19, 16) 295 511212
(12. 11) 131 10531 (20, 16) 304 717527
(13,11) 139 15204 (18, 17) 30S 7453376
(12. 1) 14. 19022 (21, 16) 311 9005900
(14, 11) 145 19902
Ch. 11

Ch. 11
The S'ructure of In tegra6/e Higlae,'- Weight Module$
215
2 -2 0
A = -2 2 -1
o -1 2
TABLE Ha
Q -(Qlo)multQ 0 -(010) muJt  a -(010) multo
( 2, 2, 1) I 2 ( 8. 8, 4) 16 297 (13, 13, 3) 30 6818
( 3, 3, 1) 2 3 (10. 10, 2) 16 297 (10. 12, 5) 31 8326
( 3, ., 2) 3 5 (8, 9, 3) 11 385 (II. 12, .f) 31 8322
( ., 4, 1) 3 5 ( 1 0, 11, 2) 11 385 (10, 12, 6) 32 10111
( 4, 4, 2) 4 7 (9. 9, 3) 18 490 (11, 13, 4) 32 10108
( 5, 5, 1) .. 1 (11,11,2) 18 .90 (12, t 2, 4) 32 10107
( 4. 5.2) 5 11 ( 8, 9, 4) 19 627 (13, 14, 3) 32 10096
( 6, 6, 1) 5 11 (11,12.2) 19 626 (10, 13, 6) 33 12266
( 5. S,2) 8 JS ( 8, 10. 4) 20 792 (14, 14, 3) 33 12246
( 7, 7, J) 6 15 (9. 9, ) 20 792 (11 t 12, 5) :w 14821
( 5, 6.2) 7 22 ( 9, 10, 3) 20 792 (11, J 2, 6) 35 17892
( 8, 8, 1) 7 22 (12, 12, 2) 20 791 (12, 15,5) 35 17893
( 5. 6,3) 8 30 ( 8, JO, 5) 21 1002 (12. 13, .c) 35 11886
( 6, 6,2) 8 30 (10, 10. 3) 21 1002 (14, 15, 3) 35 17861
( 9. 9, 1) 8 30 (12, 13, 2) 21 1001 (11, 13, 5) 36 21525
( 6, 6, 3) 9 42 (13. 13. 2) 22 1253 (12. 12, 6) 36 21526
(6, 7. 2) 9 2 ( 9. 10, 4) 23 1574 (12.14,4) 36 21514
(10, 10. 1) 9 42 (10, 11, 3) 23 1574 (13. 13. 4) 36 21515
( 7, 7, 2) 10 56 (13, 14. 2) 23 1571 (11. 13, 6) 38 30993
(11,11.1) 10 56 ( 9, to. 5) 24 1957 (II, 1-4, 6) 39 370&1
(6. 7, 3) 11 17 ( 9, t I, 4) 24 1957 (12, 13, 5) 39 37080
( 7. 8, 2) 11 77 (10, 10. 4) 24 1957 (13, 14, 4) 39 37053
(12,12.1) 11 77 {II, 11.3) 24 1956 (11,14,7) 40 44258
( 6, 8, 4) 12 101 (14, 14, 2) 24 1953 (13, 13, 5) 40 ..4241
( 7, 7, 3) 12 101 (10, 10, 5) 25 2434 (1, 14, 4) 40 44217
( 8, 8, 2) 12 101 (14, 15, 2) 25 2429 (13, 1 S, 4) 40 44219
(13. 13, 1) 12 ]01 ( 9, ] 1, 5) 26 3007 (12, 13, 6) 41 52753
(8, 9, 2) 13 135 (11,12,3) 26 3005 (12, 14, 5) -41 52741
(I". J4. J) 13 J3S ( 9, 12, 6) 27 3712 (13, J3, 6) 42 62719
( 1, 8. 3) J 176 (10. II, ..) 27 3713 (12, 14, 6) .... 88255
(9. 9, 2) 14 176 (12, 12,3) 27 3110 (13. 14, 5) 44 88230
( T. 8. 4) IS 231 (10. 12, 4) 28 4557 (12,14,7) 45 104456
(8, 8, 3) 15 231 (II, II, 4) 28 4557 (14, 14, 5) ..5 104415
( 9, 10, 2) IS 231 (10, t 1, 5) 29 5593 (12, 15. 6) 5 104450
( 1, 9, 4) 16 297 (12, 13, 3) 29 5587 (13, 14, 6) 41 145513
(11, 11,5) 30 6826 (13, 14. 7) 48 1 11355
(14, 14, 6) 48 171337
(14. 14, 7) 49 201527

Chapter 12. Integrable Highest-Weight Modules
over Affine Algebras.
Application to '1-Function Identities
Sugawara Operators and Branching FUnctions
S12.0. In the last three chapters we developed a representation theory
of arbitrary Kac-Moody algebras. From now on we turn to the special case
of affine algebras.
We show that tbe denominator identity (10.4.4) for affine algebras is
nothing else but the celebrated Macdonald identities. Historically this was
the first appJication of the representation theory of Kac-Moody algebras.
The basic idea is very simple: one gets an interesting identity by com-
puting the character of an integrable representation in two different ways
and equating the results. In particular, Macdonald identities are deduced
via the trivial representation. Furthermore, we show that specializations
(11.11 5) of the denominator identity turn into identities for q-series of mod-
ular forms, the simplest ones being Macdonald identities for the powers of
the Dedekind '1-function.
We study the structure of the weight system of an integrable highest-
weight module over an affine algebra in more detail. This allows us to write
its character in a different form to obtain the important theta function iden-
tity. This identity involves classical theta runctiODS and certain modular
forms called string functions, which are, essentially, generating functions
for multiplicities of weights in "strings." Furthermore, we consider branch-
ing functions, which are a generalization of string functions when instead
of the Cartan 8ubalgebra a general reductive 8ubalgebra is considered.
Finally, we introduce one of the most powerful tools of conformal field
theory, the Sugawara construction and the coset construction, in relation
to the study of general branching rules.
In the next chapter all this will be linked to the theory of modular forlm
and theta function8.
112.1. Let g(A) be an affine algebra ortype xt> (here X = A, H. C, D,
E, F or G; r = 1,2 or 3; N as in Table All' r). We keep the notation of
Chapter 6. Since the multiplicities of the roots of g(A) are known (by
216

Ch. 12 'lntgfYI61t HigAe,'. Weight Modu/e6 over Affine AIge6rtJI 211
Corollary 8.3), we can write down the denominator identity (10.4.4) ex-
plicitly.
Recall that the left-hand side of (10.4.4) is
Rtert = IT (1 - e( -0 )) mu Ita .
oEA+
Introduce the following polynomials in z (cr. Proposition 6.3):
L(z) = (1- z)' II (1- ze(o» for A of type X?>j
o
oE"
L(z) = (1- z)'(I- z")L-' II (1- ze(a» II (1 - z"e(o»
o 0
QE6, oEA l
for A of type XJ;> with 00r = 2 or 3, where , = N - It ;
r-
L(z) == (1- z)' II (1 - ze(a» II (1 - ze(i(6 - 0»)(1- z 2 e(a»
o 0
aEA. QE6,
for A of type A).
o
Set R = n (1 - (-a». Then, by Proposition 6.3 and Corollary 8.3
o
oE+
we have
(12.1.1)
Rlelt = kIT L(e(-n6».
nl
Furthermore, the right...hand side of (10.4.4) is
Rnah' := E l(w)e(w(p) - p),
wE W
where p is 88 defined in 16.2. By Proposition 6.5, we can write w = uta,
.
where U E W, Q eM. Using the equalities (p, K) = h v and JpJ2 := JpJ2, we
deduce from (6.5.3):
(12.1.2)
utQ(p) - P = u(p + hYo) - p + 2Y (lpl2 -I;; + h V aI 2 )6.

218 /ntegnJ6/e Highe.'. Weigh' Module. oller Affine Al,ebnJ' Ch. 12
Hence we obtain
(12.1.3) Rrilht=e( : 6) E(E f(w)e(w(p+hVa)-p»)
oEM 0
wEW
X e( - 2V Ip + h V a/ 2 6).
Comparing (12.1.1 and 3) we deduce
(12.1.4)
1-1 2 0
e( - ;;.v 6)R II L(e( -n6»
n1
= E ( E l(w)e(w(p + hVa) - p») e( - 2V IP + h V 01 2 6).
oEM 0
wEW
These are the Macdonald identities.
Example 1'.1. Let A be of type A1 1 ). Then n = {ao,o1}; 4+ =
{(n-l)QO+ nQ l; nao+(n-l)al; nao+nQl(n = 1,2,...)}; multo: 1
o
for a E d+; M = ZO'I; P = lOt; (Qllcwl) = 2; h V = 2; W = {:l:l}. Put
u = e( -00), v = e( -Ql). Then using the expression for Rlen in (10.4.4)
and that given by (12.1.3) Cor Rrilht, we obtain:
(12.1.5) IT (1 - u"vn)(l- u"-1 v ")(1 - U"V"-1)
n1
= L(-I)J u !J(J+1)v!JU- 1 ).
lEI
This is one of the forms of the Jacobi triple product identity.
For  e . set
I: l(w)e(w(" + #1) - p)
o
O (  ) WEW
X = n (1 - e( -Q»
aEl+
Note that it (,Qr) E Z+ for i = 1,...,1, then X() is nothing other
than the formal character of the I-module !(X) (where a is a simple finite-
dimensional Lie algebra with the root system A). Dividing both sidea of
o
(12.1.4) by R we obtain another form or Macdonald identities:
(12.1.6)
(12.1.7)
1-, 2 1
e( - ;;'v 6) II L(e( -n6» = E X(h V o)e( - 2h v I p + h V aI 2 6).
nl oEM

Ch. 12
Integrable HigAe,'. Wei,A' Mod.le, over Affine AIge6rtJI 219
Consider now the particular case of a nontwisted affine algebra g or type
X?). In this case a ia of type X,. Recall that one hu the "strange" formula
of Freudenthal-de Vries:
( 12.1.8)
1P12 dim 9
2h v = 24 ·
Later (see formula (12.3.7» we will prove a much more general "very
strange" formula. Recall also that M is tbe lattice spanned over Z by
long roots of g. Setting q = e( -6), we can rewrite (12.1.7) as follows:
(12.1.9) qdiml/24 n «1 - q")t II (1 - q"e(o»)
n>1 0
- aEA
_  O (h V ) IJ+""012/2I1v
- L.JX QIl ·
oEM
512.2. The specialization of type (1 to, . . . to) is called the 6a,ic special-
ization; we denote it by F. Note that
(12.2.1)
F(e( -0» = q(a.d).
o
In particular, F(e(a») = 1 if Q E A. Hence, by Lemma 11.11 we obtain
( 12.2.2)
o 0 0 n
F(x(» = d(), where d('\) = (,\ + P. o)/(p, a:).
o
oE4 V
+
Introduce the Euler product
(12.2.3)
00
'P(') = IT (1 - In).
n=1
and the Dedekind '1-(unction:
( 12.2.4)
'I = 91/24cp(q).
Applying F to both sides of (12.1.9) and u8ing the equality dim J = t+ ,AI,
we obtain Macdonald's q-runction identities
( 12.2.5)
"dim 1 = 1: d( h v Q )qI1+" y 01 2 /2,. y .
GEM

220
IntegrtJ61e Highe,'. Weigh' Module, over Affine Algebra,
Ch. 12
S12.3. Here we derive a generaJization of identity (12.2.5) using an
arbitrary speciaJization of type I. We keep the notation of f8.2. Let 9
be a simple finite-dimensional Lie algebra of type XN, let p be a diagram
automorphism of g of order rand g = ED gj the corresponding Z/rZ-
gradation. Introduce elements E., Fi' H (i = O. . . . ,t) of 9 as in S8.2.
Given a nonzero sequence of nonnegative integers, = (so.... t It), we
t 2 ·
set m = r E Oi'; (where 0; are given by Table Air r). Let € = exp :" .
i=O
Recall the definition of the automorphism 6,;r of type (6; r) of g:
(J;r(Ej) = €si Ej, O's;r(Fj) = E-; Fjt O's;r(H j ) = Hj.
Let" = $ e;<'j r) be the associated Z/ml-gradation. Put
J
d;(,; r) = dim g;(.; r) (j E Z).
where J e Z/mZ denote jmod m. By the last assertion of Theorem 8.7, we
have
LEMMA 12.3. Let S(A) be the af1jne algebra of type XJ;> and let g(A) =
ED gj(6) be its Z-gradation ot type.. Then
;EI
dim 8j(') = d;<,; r).
o
Hence, the left-hand side of (11.11.5) is
(12.3.1 )
Rtert = II 0 - q1 r',<,;r) ·
.>1
J_
Recall that the right-hand side of (11.11.5) is
Rncht = L t(w)D,(w(p»q(p-tu(p},A.t.
tuEW'
We rewrite Rri,ht in terms of the Lie algebra ,. Let A, be the sub-
set of ! which consists or linear combinations of the roots from the set
{ Q; I.i = 0 (i = 0, · · . , t)} . Let W, be the subgroup of the Weyl group W
generated by reflections in the roots from t:..,. Let W' be the set of rep-
resentatives of left. coseta of W.T in W, 80 that W = W,TW'. We may

Ch. 12
Integra61e Hi,At". WeigAt Moll./e. otler Affine Al,e6nu 221
o
chooee W' = TW'. Then the set {faw}, where 0 E M. w E W', is a let
of representatives of left cOletti of W, in W. Using (6.2.8) and (6.5.3) we
obtain
( 12.3.2)
h V
faw(p) = h V Ao + w(p) + h V 0 - ('2(010') + (w(p)lo»6.
Hence we bave
(p - tQw(p). h')
m h V
= (p, h') - (w(p), h') - h Y (0', h') + ;-( 2(0'10) + (w(p)JO'».
We define -" e o by
(12.3.3)
(')', fOi) = r.i/m (i = I,..., t).
Then the last formula can be rewritten as follows:
(12.3.4) (p - fow(p), h') = 2; r (Ih v 0' + w(p) - h v 1,1 2 - Iii - h v 1,1 2 ).
Using (12.3.1 and 4) we deduce from (11.11.5) the fol)owing identity:
(12.3.5) q11-Av'Y.12 II (1 - tf)"J<,;r)
.> I
J_
= E feW) L: D,(w(p) + h V O')9ltu(P)+AY(o-'Y.)I:t.
WEW; oEM
Note that the 1]-function identity (12.2.5) is a special case of this identity
for s = (1, 0, . . . , 0), T == 1.
As in that special case, one taD express the first factor in the left-hand
side of (12.3.5) entirely in terms or the d j (.; r). Namely, one haa the fol-
lowing "very strange formula," which is a generalization of the "strange
formula" (12.1.8):.
( 12.3.6)
m-l
2r lp-hv'Y,12 = dimg- 4 :Ej(m-j)dj(.;r).
j=l
The proof of this formula uses identity (12.3.5) and some elements of the
theory of modular faro1S. It wil) be given in Chapter 13.

222 Infegrd61e Hi,Ia",. Weiglaf Module. oller Affine Alge6ra. Ch. 12
512.4. Recall that the center of an affine algebra g(A) is I-dimensional
and is spanned by the canonical central element (see 56.2):
l
K = Eoror.
i=O
It is clear that K operates on a g(A)-module L(A) by the Icalar operator
(A. K}IL(A). In particular, (A, K) = (A. K) for every  e P(A). The
number
(12.4.1)
t.
.:= (A,K) = Lor(A,on
4=0
( 12.4.2)
is called the level of A E ., or of the module L(A).
If A e P+ 1 then the level of A i. a nonnegative integer; it is zero if
and only if all the labels (A, Qj) of A are zero. Hence, by (9.10.1), an
integrable g(A)-module L(A) baa level 0 if and only if dim L(A) = 1; if
L(A) is integrable and dim L(A) ;: I, then the level of L(A) is a positive
integer.
Note that the level of p is equal to
I.
(p, K) = E or = h v ,
i=O
the dual Coxeter number.
Let Ai (i = 0, . . . t t) be the !-ntlomenCGl weiglaf,:
(Ai,QJ) = 6ij. j = 0,... ,t, and (Ai,d) = o.
Note that
(12.4.3) Ai = Ai + or AOt
where Ao = 0 and At, . . . , Al are the fundamental weights of g. Note also
that
t
(12.4.4) P+ = EZ+A; +CcS.
1=0
It i. clear that the level of A Crom P+ i. 1 if and only if A == Ai mod C6 and
i is such that or = 1; in particular. the level of Ao is always 1. A"glance at
Table Aff gives that if A is symmetric or r > I, then
(12.4.5) level(A;) = 1 if and only if i E (Aut S(A» · O.
Finally, tbe following observation. which (ollows Crom Propositions 3.12
b) and 5.8 b), is useful:

Ch. 12
In'egra6/e Higlae'- We.gh' Module, over Affine Alge6ra, 223
LEMMA 12.4. If Re(A, K) > 0, chen (W · ) n {p E . I Re(p, or) 
o (or all i} consists of a single element. In particular, jf  E P has a
positive level, then &he set P+ n W · ,\ consists of a single element.
o
We let P'(resp. P) = { E P(resp. P+) I (, K) = k}.
512.5. We collect here some facts about the weight system P(A) of an
integrable module L(A) over an affine algebra g(A), proved earlier in the
general context or Kac-Moody algebras.
PROPOSITION 12.5. Let L(A) be an integrable module of positive level i
over an affine algebra. Then
a) P(A) = W · { e P+I  A}.
b) P(A) = (A + Q) n convex hull of W · A;
c) If ,p E peA) and JJ lies in the convex hull of W .'\, then multL(A)}J 
multL(A) .
d} peA} lies in the paraboloid { E ."II2 + 2k(AIAo) < fAJ2;
(A, K) = k}; the intersection or peA) with the boundary o(this paraboloid
is W · A.
e) For .,\ e P(A) the set of' E Z such that  - t6 e peA), is an interval
[-p,+oo) with p  0, and'...... multL(A)(A -f6) is a nondecreasing (unction
on this interval. Moreover, j{z e 9-6, z -I 0, then the map z : L(A).\_,,, -+
L(A).\_(t+ 1)6 is injective.
f) Set n) = m 9-nli then L(A) is a (ree U(n)}-module.
n>O
Proof. a) follows from Proposition 11.2 b), while b) and c) are special
cases of Proposition 11.3 a) and b). d) folJows from Propa;ition 11.4 a)
and formula (6.2.7). e) follows from Corollary 11.9 b). f) is a special case
of Proposition 11.9 c).
o
S 12.6. We continue the study of the weight system P(A) of an integrable
module L(A) of positive level k over an affine algebra.
A weight  e peA) is called maximal if +6  peA). Denote by max(A)
the set of all maximal weights of L(A). It is clear that max(A) is a W-
invariant set (since P(A) is W-invariant and 6 is W-fixed) and hence, by
Corollary 10.1, a maximal weight is W-equivalent to a unique dominant

224 lra'eg,.61e Hi,Ae,'. We;,A' Mod./e. otJer Affine Alge6,... Ch. 12
maximal weight. On tbe other hand, it foilowl from Proposition 12.5 e)
that for every p e peA) there exists a unique  E max(A) and a unique
nonnegative integer n Iuch that p =  - n6, i.e.. we have
(12.6.1)
peA) = U { - n61n E Z+} (disjoint union).
Emax(A)
Here is a description or dominant maximal weights.
PROPOSITION 12.6. Tbe map  ..... I defines a bijection from
max(A) n P+ onto ItC.J n (A + (1). In particular, the set of dominant
maximal weight. of L(A) i. finite.
Proof. Straightforward using Proposition 12.5.
o
The following lemma describea explicitly the weight system of certain
particularly important highest-weight modules.
LEMMA 12.6. Let A be an affine matrix of type XJ;), where X =: A, D or
E. Let A e P+ be of level I. Then
(12.6.2)
max(A) = W · A = T · A,
(12.6.3) peA) = {Ao + llAI2 6 + a - <llal 2 + .)6,
where Q e 1'+ Q, . e Z+}.
Proof. Since level(A) = 1, A is a fundamental weight Ai mod C6. Using
(12.4.5), one easily sees that i is a special vertex, and hence W · A. = T. A.
(see Remark 6.5). Renee (12.6.1 and 2) imply (12.6.3), using (6.5.2) and
(6.2.6).
To prove (12.6.2) recall that T = {'ola e (J} (see S6.5). Let  e
max(A)j we have  = A - p, where fJ E Q. Since fJ = 7J mod C6, we
have: '1<) = A mod C6 by (6.5.2). This prove. (12.6.2).
o

Ch. 12
Inte,ra6/e Hight.'" Weight Mod.le, over Affine Alge6ra.
225
S12.T. Let A E P+. Irrollows from Proposition 11.10 and (11.10.1) that
chL(A) converges absolutely to a holomorphic (unction in the region
Y = {h E  I Re(6, h) > O}.
In fact, Y is the region of convergence of thL(A) if level(A) > o. Note also
tbat for any highest-weight module V over an affine algebra, thy converges
absolutely in the domain Yo (see Lemma 10.6 b).
For  e max(A) introduce the generating series
00
a = E multL(A)( - ncS)e-n'd'/J.
n=O
'fhis series converges absolutely in the region Y since it is majorized by
-A h
e C L(A).
Since W n T = 1 Cor  E P(A) (see Proposition 6.6 c» and W(6) = 6,
we deduce using (12.6.1):
(12.7.1)
chL(A) = E e.\a = E (E e'(.\) ) at.
Emax(A) Emu(A) aET
A mod T
This simple formula reduces the computation of the character of L(A)
to the computation of the (unctions a. The following case is especially
simpJe.
LEMMA 12.7. Let A and A be &!J jn Lenuna 12,8. Then
e-llA1'1 chL(A) = a: E eAo+'Y-lhl'l.
.,EQ+A
Proof. Use (12.6.3) and the fact that every at is W-conjugate to a = a:
due to (12.4.5) and Lemma 12.6.
o
The function a: jIJ be calculated explicitly at the end of this chapter.
We proceed to rewrite character formulas (10.4.5) and (12.7.1) in terms
of theta (unctions. For  E . such that level() = k > 0 set
112
e = e--sr' Le,(A),
leT
(12.7.2)
A.\ = E t(w)etll().
o
wEW

226 Infeg",61e HigAe,'. Wei", Mod./e, Dtler Affine AIge6ral Ch. 12
Usinl (6.5.3) we obtain
(12.7.3)
9A = e Uo E e-!ihl'Hi'Y,
,.EM+,-aI
which is a cl881ical theta function (see Chapter 13 for details). It is clear
that this series converges absolutely on Y to a holomorphic function. Note
also that 8A depends only on  mod M + C6.
Using theta (unctions, we can rewrite the denominator identity (10.4.4)
in yet another Corm. Recall that (P. K) = h V (the dual Coxeter number)
o
and p = p+ h V Ao (formula (6.2.8». Using the decomposition W = W  T,
we get
E f(w)etl/(p)-p = e- P E feW) E e'otll(p) = e- P + : C , A,.
tuEW 0 oEM
"'EW
Hence. (10.4.4) can be rewritten as (ollows:
y 1;12
(12.7.4) Ap = L f(w)9 h vAo+tu(P) = e h Ao+1-;rv' II (l_e- o )multa.
WE oEA+
Introduce the following number for A E P'::
( 12.7 .5 )
IA + pl2 Ipl2
mA = 2(k + h Y ) - 2hY '
For reasons which will become clear later we call this number the modular
aRomtJ/, 01 A, and we introduce the nOMnG/izttl character
-mA' h
X A =e C L(A)'
For a weight A E P(A) introduce the number
(12.7.6)
1,\1 2
m A =m A --
. 2 ·
It is clear that mA.'\ is a rational number.

Ch. 12
Integm61e HigA,'- Weigh' Module, over Affine Algebras
227
For  E . set
(12.7.7)
c = e-mA.' E multL(A)(A - ncS)e- n '.
nEI
Just as the series Q t this series converges absolutely to a holomorphic
function on Y. Note that c = -mA.' a if ,\ E max(A).
The (unction c is called the .tring junc'ion of .;\ E .. Note that
(12.7.8)
c(;\) = c for w E w:
o
Since W = W  T, we use (6.5.2) to obtain
(12.7.9)
A A . 0
cw()+i1'+G' = c.,\ for  E  , W E W, 1 E M, a E C.
Note also that
(12.7.10)
c = cZ+ClI for Q e C.
o
Using W = W K T and (12.7.2) we can rewrite the chara.cter formula
(10.4.5) as follows:
(12.7.11)
X A = AA+,/A,.
On the other hand, we have by (12.1.1) and the definitions of SA and
C A.
A.
X A = L c9;\.
AE P.mod C'
A mod T
Using (12.7.9) we can rewrite this as follows:
(12.7.12)
x" = E C9;\.
AEP.mod <iM+el)
Comparing (12.7.11 and 12) gives tbe theta function identity
(12.7.13)
AA+p = Ap L c9;\.
EP.mod (iM+CI)
We shall use this important identity in the next chapter to study and
compute the string functions.

228
Integrable Highelt.. Weigh' Module.. over Affine AIge6rtJI
Ch. 12
S12.8. We turn now to the Sugawara construction. For the sake of
simplicity, we consider only the case of a non-twisted affine algebra g' =
s'(X1 1 », leaving the general case as an exercise (see Exercise 12.20). Recall
that g' = leg) = C[t, ,-1] @c 9 + CK with conunutation relations
(12.8.1 )
(z(m), y(n)] = [I ](m+") + m6m.-n(zly)K.
Here & is a simple finite-dimensional Lie algebra of type Xl, (zly) is the
normalized (by (6.4.2») invariant bilinear form on &t and z(n) stands for
t n @ z (n E Z, % E ). .
Let {Ui} and {u i } be dual bases or I. i.e. (Ui , u J ) = 6 i j. RecaJI that
o = E u. u' ,
i
the Casimir operator of 9t is independent of the choice of these dual bases.
In particular. E Ui u i = E u. Ui. This proves the following useful formula:
i i
( 12.8.2)
)um), ui(n)] = m6 m ,_n(dim g)K.
,
FUrthermore. note that
(12.8.3)
00
{} = 2(K + hV)d + 0 + 2:E L u-n)ui(n)
n=1 i
is the Casimir operator of tbe affine algebra
9 = g(Xjl» = g' + Cd, where [d, zen)] = nz(n).
Since 0 is independent of the choice of the dual bases {Ui} and {u i },
we can check this for a special choice of these base8. Consider the root
d · o' 0
space ecomp081tion 9 =  ED ( EB Ceo). such that (eo Ie-a) = 1, and
o
aE6
1 t 0
choose dual bases hI,... t ht and h ,..., h of. Then we have: S1 =
L hih i + L e_oe o . Using that [eo, e_ o ) ::::: V-I (a) (see Theorem 2.2e»,
i 0
aE
o 0
and that 2 p= L Q we obtain the form of n given in 2.5:
o
aE6.
t
o = E hih i + 211-1() + 2 :E e_oe o .
i=l .
aE6+

Ch. 12
Integra6/e Higlae.f- Weigh' Mod.lt. ovtr Affine AIge6ral
229
Hence (12.8.3) can be rewritten in the same form as n defined in 2.5t due
to (6.2.5), (6.2.8) and 7.4 (see Exercise 7.16).
In order to perform calculations, it is convenient to introduce tbe re..
6tricted completion Ue(g') or the universal enveJoping algebra U(g'). Con-
sider the category of all restricted g'-modules, i.e. modules V such that for
any v E V, %(J)(v) = 0 for all % E g and all j > 0 (cr. 52.5). Consider all
00
series E Uj with Uj E U(g') such that for any restricted g'-module V and
j=1
any 11 E V, Uj (tJ) = 0 Cor all but finitely many Uj. We identify two such
series if they represent the same operator in every restricted g'-module.
We thus obtain an algebra Uc(g') which contains U(g') (since, by Exercise
11.22, ED L(A) is a faithful U(gJ)-module) and acts on every restricted
AEP+
g'-moduJe. Note that the derivations d n (see 7.3) extend from U(g') to
derivations of Uc(g').
Now we introduce the SugawartJ oprato,., Tn (n E Z):
(12.8.4)
00
To = L UiU i + 2 L L u-n)ui(n)1
i n=1 i
Tn = E E u-m)ui(m+n) if n 1: o.
mEI i
Due to (12.8.2), these operators are contained in Uc(g'). Note also that
these operators are independent of the choice or dual bases or g.
We have the following basic Ie nun a.
LEMMA 12.8. a) For z E 9 and n, m e Z one has:
(z(m), Tn] = 2(K + hY)mz(m+n).
b) Let V be a restricted ,-module and let tI E V be such that n+ (v) = 0
and h( v) = ('\, h)1J {or some  E .. Then
To ( v) = (X J I + 2p)v.
Proof a) We perform the calculations in the semidired product  K Ue(g').
Fi1'8t, note
(12.8.5)
To = -2(K + hV)d+ O.

230 In'egra61e Higlae.'- Wei,AI Moll.le. over AJ1in Alge6na. Ch, 12
Alao, it is straightforward to check using (12.8.2) that
TJ = [djtTo] if j  O.
J
Now we bave. using Theorem 2.6 and (12.8.5):
[x(m) f To] = [x(m), 2{K + h V)do] = 2(K + h v )mx<m).
Finally, for j :F O. we have:
1
[z(m).1JJ = -:(z(m), (dJtTo]] =
J
2(K + Ia V)( [mz(m+J) t To] + (dj, mz(m») = 2(K + Ia V)mz(m+J).
J J
b) Collows immediately fro (12.8.5) and (2.6.3).
o
Now we can calculate [Tm, Tn]. One should be careful about staying
within the algebra Uc(g'). We may assume tbat m > n. Let first m+n #: 0,
m .,. 0, n 1: O. Then we have:
[Tm, Tn] = E E[u-J)u.(m+J), Tn]
jEI ;
= E E ( u -J){u.(m+J) , Tn} + (u -J), TnJui(m+ J »)
JEI i
= 2(K + la V ) E E (m + j)u-J)u.(m+J+n) - jU-Hn)ui(m+J»)
;EI ,
by Lemma 12.8. Replacing j by j + n in the second 8ummand t we obtain:
(12.8.6)
[Tm, Tn] = 2(K + la V ) E l)m - n)u-j)ui(m+n+J)
JEI i
Thus, we have, provided that m + n .; 0, m  O. n  0:
(12.8.7)
[Tmt Tn] = 2(K + 1a V)(m - n)Tm+n.
A similar calculation showl that (12.8.7) holds when m + n fa 0 but m or
n = O.
Let now m+n = 0, m > O. Then the right-hand side or (12.8.6) does not
lie in U e ( 8'). We proceed aa Collows. Since 0 is independent of the choice

Ch. 12
Inttgns61t Hig/at6t- Weight Motl"/, over Affine Algt6rtJ 231
of dual bases we have Li u-j)uiU+m) = E. uj+m)ui(-j) when m  o.
Hence we can write (here and further we drop the sign of summation over
it but assume that it is present):
Tm = L u-j)u'U+m) + E u-i+m)ui(j).
jO ;>0
We have:
v 0
[Tm, T-m] = 2(K + h )mO
+ 2(K + h Y ) I: «j + m)u-j)u'U) - ju-j-m)uiU+m)
j>O
+ ju-j+m)ui(J-m) + (m - j)u-j)ui(j»
= 2(K + hY)(mT o + E ju-J+m)ui(j-m)_ L ju-i-m)u'(j+m».
i>O j>O
Replacing j by j + m in the first summation and j by j - m in the second
summation, we obtain:
[T"" T-m]
= 2(K + hV)(mT o + E (j + m)u-j)ui(j) + E (m - j)u-j)u.(j»
j>-m It>;
-1 m-l,
= 2(K + h Y )(2mTo + E (j + m)u-J)uiU) - E (m - j)u-J)u.(J»
J=-m+l ;=1
m-l .
= 2(K + h Y )(2mT o + E j(m - j)[ui(J)t u-l»))
j=o
m-l
= 2(K + h V )(2mT o + E j(m - j)(dim g)K)
J=o
( 12.8.10)
"'-1
by (12.8.2). Since E j(m - j) = (m 3 - m)/6, combining with (12.8.7),
;=0
we obtain the final formula:
3
(12.8.8) lTm, Tn] = 2(K + hY)((m - n)Tm+ra + 6m.-ra m ; m (dim g)K).
We immediately obtain from (12.8.8) and Lemma 12.8 tbe following
COROLLARY 12.8. Let V be a restricted g'-module such that K is a scalar
operator kI, i  _laY. Let
1
Ln = 2( k + h v ) Tn t n E l,
( i ) - i(dim 8)
C - k + h V '
h _ (A + 2p1A)
A - 2(k + h Y ) if V = L(A).
(12.8.9)
(12.8.11)

232
In'egrtJ61 Hi,Ae,'. Weigh' Mod./e, over Affine Alge6ra,
Ch. 12
a) Letting
cln .... Ln t c.... c(  ) ,
extends V to a module over 8' + Vir (the semidirect Bum defined in 17.3). In
particular, V extend, to a module over 9 (= 8' + Cd) by letting d..... -La.
b) If V is the g-module L(A), then Lo = hAI- d.
o
The number c(k) is called the conformal tJftomalr of the g/-module V.
and the number hA is called the vClcuum tJRoma/, of L(A). Using the
"strange" formula (12.1.8). we obtain from (12.7.5). (12.8.10) and (12.8.11)
the following simple relation between the modular, conformal and vacuum
anomalies for A E . of level :
(12.8.12)
mA = hA - c(k).
I
Another useful property of Sugawar& operators is given by the following
PROPOSITION 12.8. Let V be a restricted g'-module with a non-degener-
ate contravariant Hermitian form (see 111.5). Then the operators Tn and
T-n are adjoint with respect to this form.
o 0
Proof. Let t be the compact form of g, Le., the fixed point set of WOo Then
o
the bilinear form (.1.) restricted to t is negative definite (see Theorem
11.7a». Hence we may choose a basis {Vj} of I such that (Vi I Vj) = -6 ii .
Let Uj = v=r vi j this is an orthonormal basis of J such that wo( Uj) =
-uJ' hence wo(uj(n» = -uj(-n). The proof of the proposition is now
immediate.
o
Remark Je.8. Let s. be "a finite-dimensional vector space over R with a
positive definite bilinear Corm (.1.). Let a be its complexification and
extend ( · I.) Lo g by bilinearity. Viewing g &8 an abelian Lie algebra, we
t f. 0 0 0'. 0
have: B = IJt d = e, Q = Ot QV = 0, P = 1)-, W = {I}, p = O. We let
also h V = O. Then all results of thi8 section (and their proofs) hold in this
case as well. In particular, the conformal anomaly
c() = dim 9
is independent of k, the vacuum anomaly h A = , where k = level(A) 
0, and we let
1
mA = h A - 2ic(k).

Ch. 12
Inttgra6/e HigAelt- Weig"t Mod. It, otJer Affint AigellrtJ6
233
Finally, (or the g-module L(A), A E "-, or Jevel k 1: 0, chL(A)
= eAlcp(e-')dirnl, where cp is defined by (12.2.3), hence
(12.8.13)
IAl d . 0
X A := e- mA ' chL(A) = e- "1JrHA 1'1 am.,
where '1 = e- 6 / 24 cp(e-'). Also, L(A) is unitarizable if and only if A e .
and i is a positive real number. This follows from (9.13.1).
512.9. We shall extend the above construction to the case of a reductive
finite-dimensional Lie aJgebra 9 (it wi)1 be more convenient to use here this
notation instead if g). We have the decomposition of 9 into a direct sum
of ideals:
(12.9.1)
, = S(O) @ 8(1) e "(2) $ · · · ,
where 9(0) is the center of 11 and 8(i) with i  1 are simple. We fix on II a
non-degenerate invariant bilinear symmetric form ( .1. ) 80 that (12.9.1) is
an orthogonal decomposition. We shall assume that the restriction of ( .1.)
to each S(i) with i  1 is the normalized invariant form, and that 9(0) and
the form ( . J .) restricted to it are 88 described in Remark 12.8. We sha))
call such form a normalized invariant form on g.
We Jet
leg) = E!)l(g(i»'
i>O
-
,.",
where £(g(i» = £(9(i) + CKi.
We also let (cf. 17.3):
£(g) = leg) + Cd,
d
where dli('(i) = -';Ii' d(Ki) = O.
The Lie algcl)ras l (g) and 12(g) are caBed affine algebras associated to the
reductive Lie algebra g. The subalgebras £(g(j») (resp. £(9(t») + Cd) are
calle<i components of £(g) (resp. C(g)).
"'" ...
Note that' = 9(0) + L CK. is the center of £(g) and £(g). As before,
il
we identify 9 with the 8ubaJgebra 1  g. Let  be 8 Cartan subalgebra
of g and let 9 = If_ EB 1; ED W+ be a triangular decompalition or 9. The
subalgebra  = 9 + c + Cd is called the Cartan 8ubalgebra of £(g). The
triangular decomposition
£(g) = n_ e  EB n+

234 Integra6/e Higlae,'" Weiglal Module, over Affine AIge6,., Ch. 12
is defined in the lame way 81 in 57.6. For  e ., we denote (as before) its
restriction to ij by X. A. before J define 6 E . by
6li+c = O. (6, d) = 1.
Given A E . t we denote (as before) by L(A) the irreducible £(g)-
module which admits a non-zero vector tlA such that "+(VA) = 0 and
h( VA) = (A. h)VA (or h e . Using uniquenese of L(A), we clearly have:
(12.9.2)
L(A) =  L(.»,
i>O
-
where A(i) denote the restriction of A to (') := f) n l(g(i» and L(A(i» is
the l(g(i»-module with highest weight A(.).
We let lei. the eigenvalue of K i on L(A), be the i-'" level of A. and let
k = (ko,kl... ..). Define c() = Ec(i,), hA = EhA(i ) ' mA = EmA(i).
i i i
Due to (12.9.2), Ch£(A) = n Ch£(A(i) and
,
(12.9.3)
X .- .-mA' ch - llx
A .- ... L(A) - A(i) ·
.

Let V be a restricted !(g)-module such that k i acts as i,I and k.  -hr
(where hr is the dual Coxeter number of l(g(.». Let ') be the Sugawara
,..,
operators for £(g(i», and let (n E Z):
(12.9.4)
L (i) - 1 i) L . -  L (I)
n - 2(lo. + hn 1 ,.., n -  n'
.
The operators L! are called the VirtJ'oro operator, for the g-module V.
Then letting dn ....... L, c ...... c() extends V to a module over leg) + Vir.
Note also the following useful formula (cf. (12.8.5»):
(12.9.5)
.  Oi
Lo=  2(ki+hn -d,
.
where 0, is the Casimir operator for !(g(;».

Ch. 12
Infegrable Highe,'- Weigh' Module over Affine Alge6rtJI
235
512.10. In the remainder of this chapter, we let 9 be a reductive finite.
dimensional Lie algebra with a normalized invariant form ( .1.) and let g
be a reductive subalgebra of 8 such that (.,. )Ii is non-degenerate. Let
9 = EB U(i) and 9 = ED 8(i) be the decompositions (12.9.1) of 9 and g.
i>O i>O
Let ( .1.)" be 8 normali;ed invariant form on g, which coincides with ( .1.)
on 9(0). Due to uniqueness of the invariant bilinear form on a simple Lie
algebra we have for Z,II E 8{.)t ,  1;
( z(,.) , Y( r » = j,,. (z I II)' t
where z(r) denotes the projection of z on g(r) and j,,. is a (positive) number
independent of % and 1/; we let jOr = 1. The numbers j,r (a, r  0) are
called Dynkin indice,.
The inclusion homomorphism t/J : 9 --+ 9 induces in an obvious way the
inclusion homomorphism £(9) --+ £(g). This lifts uniquely to a homomor-
phism  : £(g) -: .cg) by )tting (k,) = J:.rj,,.K,., which extends to 8
homomorphism t/J : £(g) -+ leg) by letting 1/J(d) = d. Here and further the
overdot refers to an object associated to g.
-
Let V be a restricted £(g)-module such that K. acts ask;!, i. 'F -hr.
... - . .
Via 1jJ, this is a .c(g)-module with Ki acting as It i l, where
(12.10.1)
k, = L j,ilcj ·
i
. .
We shall assume that k i  -hi. Let (see (12.9.4»:
L "i - L I - L i
n - n n.
. -
PROPOSITION 12.10. a) The operators Lt' commute with Leg).
b) The map d n ...... Lti, c -+ c(k) - c(k) defines a representation of Vir on
v.
Proof. a) is immediate by Lemma 12.8. Furthermore, we have:
[LiJ, L:.iJJ = (L', L:J (since L: E uc:(l(g»)
= [L. L] - [L. L]
= [ LI L' ] - [ Li LI,i + L ' ]
mJ n m' n n
.. ..."
= (L:a, L:J - [L:'. L:J (since L:' E Uc(£(g»).
.
o

236 Integrable Hi,Ae'- Weigh' Mod"le, over Affine AIge6,., Ch. 12
The Vir-module defined by Proposition 12.10 i. caJled the COltl Vir.
module. . .
Choose Cartan lubalgebras ij and ij of 9 and G 8uch that i) C Tj. Choose
& triangular decomp08ition 9 = W_ ED ii (D n+; then we have the induced
.
. -. .
triangular decomposition g = R_ (D  ED "+, where ii z = if n g. We have
 
the associated triangular decompositions: £(g) = n_ + l) + "+1 £(g) ==
n_ + 6 + "+' etc., and we have: cp(6) C , cp(n+) c n+, etc.
Let P + = {.oX E .I.oX I /In£(lJ(.,) C P +(i) for i > I, and .oX I "ng(O). is real
and ("\, Ko) > O} be the set of dominant integral weights for £(g). Let
p = {A E P + I A (K i ) = k i (i = 0 t 1, . . .) }.
Fix A e P:. Ueing (12.9.2), Theorem 11.7 and 511.12. we see that the
£(g).moduJe L(A) il unitarizable. It folloWI, by Proposition 11.8, that
.
viewed 88 a £(g)-module, the module L(A) decomposes into a direct sum
of £(g)-modules i() with  e p!. Since the eigenspacel of d on L(A) are
finite-dimensional, it foilowl that the multiplicity or occurrence of L(A) in
this decomposition is finite. We denote this multiplicity by multA(; g).
512.11. The following notion will play an important role in the sequel.
The pair M e P and P E P(M)I. n p such that hu = hp is called a
vacuum pair oJ level k (cf. Proposition 12.12b). below). Denote by R. the
set of all vacuum pairs of level k.
PROPOSITION 12.11. Let (M;p) E Rite Then
(12.11.1)
muItM(Pi;) = E muJtL(M)(jj) > O.
iJEP(M)
pl,="
In particular, (MiP) e R. if and only if hM = hp and multM(pi 8)  O.
Proof Let jj E P(M) be such that pi, = p. Let tJ be a non-"zero vector
(rom L( M)p. We have to show that n+ (v) = o. In the contrary case, there
exists P € Q+ \ {OJ such that .muJtM(p + Pi 8) > O. But then
h _ h - '" IpCi) + P(i) + p(i)1 2 - 'PCi) + PCi}1 2
p+/J p -  2( k i + h n
= E «P(i) + l1(i}).+ P<,') + 2p, I P(i» > 0
i 2(i + hr>
e
Hence hA-f < h+I3' which contradicts Proposition 12.12b) below.
o

Cb. 12
Inlegra61e Higlae,'. WtigA' Mod./e. over Affine Alge6,...
231
Remark 1.11. Let M E P+. Since the module L(M) is completely re-
ducible with respect to £(g), we see from (12.9.5) that L8" is diagonalizable
on L(M). It is clear from the proof of Propition 12.12b) that its spectre
is non-negative. Each of its eigenspaces is L.:(g)-invariant (by Proposition
12.10a)). Finally, its zero eigenspace (the vacuum space) decomposes into
a direct sum of £(g)-modules L(p,) such that (M; J.t) is a vacuum pair, with
roul ti plicities given by (12.11.1).
112.12. For A e p and  e 6-. let
6(8) = e-(mA-m)' E multA( - ratSj 8)e- ni .
nEI
.
This series converges absolutely to a holornorphic function on Y, called a
6ranching func'iora.
Note that string functions are euentially special cases of branching func-
tions:
(12.12.1)
 = 6(i),,-I.
This follows from (12.8.13).
The branching function8 have a simple representation theoretical mean-
ing. To explain this. let
.
U(A.) = {v E L(A)Jn+(v) = 0 and h(v) = (, h)v for h E ij}.
Note that. due to Proposition 12.10, the subspace U(A,) is a coset Vir-
submodule (with c = c(k) - c(i». Comparing Corollary 12.8b) with
(12.8.12), we obtain the Collowing interpretation of branching functions:
(12.12.2) hNg) == trU(A.A) q£:.I- i,(c(II)-c(i»,
where q = e-'.
Thus. we have obtained the Collowing decomposition of L(A) with respect
to leg) $ Vir (direct sum of Lie algebras):
L(A) = E9 t() @ U(A.)t
AEP: mod cj
wbich immediately implies an equation for normaJized characters:
( 12.12.3)
X A = 1: XAhte;).
AEP: mod C;
Now we can prove the following important proposition.
( 12.12.4)

238 Integralle HigAe.t. We;g'" Module. otJer Affine Alge',.. Chi 12
PROPOSITION 12.12. a) The module L(A), A e P+t viewed as a coset
Vir-module decomposes into an orthogonal direct Bum of unitariaab/e irre-
ducible highest-weight modules.
.
b) If A E P+ and multA(; g)  0, then hA  h)..
.
c) If ko  0 and i. e Z+ (Ol i > 0, then c(k)  c(k).
Proof. By Proposition 12.8. tbe coset Vir-module L(A), A e P+  unitary,
hence all Y(A,) are unitary. Also, all eigenspaces of L.i on U(A,) are
finite-dimensional and ita spectrum is bounded below. Now 8) follows from
(12.2.3) and Proposition 11.12c).
 .
Let t) E L(A) be a bighest-weisbt vector of a £(g)-submodule L() of
L(A). Using Coro)lary 12.8b) we obtain:
L.i(,,) = (hA - ia"),,.
b) and c) follow now from Proposition 11.12.
o
Now. if 9 is semisimple, then the sum in (12.12.3) is clearly finite. This
is, of course, not the case in general, 88 we have seen on the example 9 = ij.
We shall transCorm this sum to a finite one using the same trick 88 in 512.7.
For this we shall assume that
(12.12.5)
9C O ) n <.t spans 9(0) over C 1
where ct c Tj is the coroot lattice of g. Introduce the lattice Mo =
.;-I(g(O) n i1'), which is a 8ublattice of the lattice M.
We let , = ED (i)' Then we have:  = 6(0) + ', (O) n 6' = Cd, hence
i1
. . . . .. .
. = (o) + ,. and (O) n ,. = C6. Given  E ., we have a decompoeition
(12.12.6)
.\ = (o) + .\(1), where (O) e 6(0)' (1) E 6'.,
which is unique up to adding multiples of 6.
Due to (12.9.3) and (12.8.3) we have for  E p:
(12.12.7)
I" Ol .' ·
X" = X(1)(eA(O)- n . '/,1'-0), where to = dim 9(0).
Here X,,(J) = n X". is the normalized character of the £(g')-moduJe
i> 1 (I)
L((l», where i' is the derived algebra of g.

Ch. 12
Integrable Higlae,'" Weigh' Module, over Affine Alge6rtJ'
239
.
It is cJear that for a E Mo we have:
ia{x A (.) = X.\(I) and 'teA) = 6.
Using this, we can rewrite (12.12.3) in the following form:
(12.12.8)
X A = L 6{g)XA(I)(e1(o,l'1tO),
EP: mod (Ci+ioMo)
_ 1.\(0) r 2 .
where eo = e 2 k O 6 " ein() is the theta function associated to tile
-X(O) L-t
aEMo
lattice Mo. It is clear that the sum on the right is finite. Equation (12.12.8)
is a generalization of the theta function identity (12.7.3). It will be used in
the next chapter to study branching functions.
S12.13. We apply here the developed machinery to calculate the func-
tions a of level I, for affine algebras of type X};), where X = A, D or E
(cf. Lemma 12.7). For this we shall use the following well-known relation:
(12.13.1)
dimg(XN) = N(h V + 1) if X = A,D or E.
Recall also that
(12.13.2)
a = 1 +019+ .... where q = e-'.
Consider the coset Vir..submodule U(A, A) of L(A) (for the twisted case use
Exercise 12.20). We hav the following formula for its conformal central
charge:
( 12.13.3 )
e(l) - e(l) = div9i') - N = O.
Hence, by Proposition 1l.12b) and c), the coset Vir-module U{A, A) is
trivial. Hence, comparing (12.13.2) with (12.12.2), we obtain
A ·
6A() = 1.
Comparing this with (12.13.2) and (12.12.1) in the non-twisted case Crespo

240 1,,'egra6/e HigAe.'" Wei,,,, Mod.le. over Affine Alge6ra. Ch. 12
Exercises 12.20 and 12.21 in the twisted cue), and uling Lemma 12.7, we
deduce
X (r)
PROPOSITION 12.13. Let A e P. Then for aIline algebru of type N'
where X = A, D or E, we have:
00
(12.13.4) CI = II(I- e-n')-multn'.
n=1
(12.13.5) multL(A) A = pA«IAI2 - IAI2)/2), ,!here
Eyt(j)qI := II (1 - q")- multn'
J nl
(12.13.6) e-l IAI2 , chL(A) = E e Ao +.,-iI'rI 2 , / IT (1 - e-n')mult"'.
EQ+A n1
o
RemtJrk 11.13. If A is oftype AI), Dj1), Ejl) or Ag). then EpA(j)qI =
J
cp(q)-l. 80 that pA(j) is the number of partitions of j into positive integral
parts of t different colours; in particular t pAr) (j) = P(j) (r = 1 or 2) is the
classical partition function. .
Incidentally, using (12.13.3) and Proposition 11.12b), we obtain hA =
.
h A . hence the following
COROLLAR.Y 12.13. Under the hypothesis of Proposition 12.13 we have:
(A I A)h v = 2(p I A).
112.14. Exercises.
1.1. Show that setting q = e(-6), z = e(-oo) in (12.1.4) for g(A) of type
Ail) and A2) ,one gets the following classical triple and quintuple product
identities (which are alternative forms of identities from Exercise 10.9):
00
II (1- 9 n )(1 _"n z-I)(1 - 9"-1 z) = E (_I) m q lm(m-l) zm;
n=1 mEI
00
IT (1- q")( 1 - ,n-I: )(1 - ,n z-1 )(1 _ ,,2n-1 z2)(1 _ q2n-1 :-2)
n=1
= 1: qi(3m 2 +m)(z3m - Z3m-l).
mEI

Ch. 12
In'tgna61e Higlae,'. Weigh' Mod.le, over Affine Alge6nJ. 241
11.1. Let 8 be a simple finite-dimensional Lie algebra of rank t, let (, II)
be the Killing form of a, let  be a Cartan 8ubalgebra, A the root system,
Mp the lattice spanned by {o/+(o,o), 0 e AI, !+ a set of positive roots,
p their half-sum. Deduce from (12.2.3) tbe following identity:
"dim; = L d(1)q.('Y+;T,7+1).
7EMp
J t. 3. In the notation of Exercise 12.2, let Q 1 , . . . ,Q/, be the set of simple
roots, h the Coxeter Dumber of 8, and Jet Mq denote the lattice spanned
o
by {ho, Q E A}. Deduce the following identity from formula (12.1.1):
t
n'1(qh.(Gi,Gi»"+l = E d(1)q.(1'+,,"Y+1>.
i=l 7 EM o
I.
[Use the formula E (aj, Qj) = t/h].
i=l .
/1..1. Show that for automorpbisms of .1 2 or type (1,1; I), where. =
0, 1,2,3, the identity (12.3.5) turns, respectively, into the following clas-
sical identities
lp(q)3 = 2)4n + l)q2n2+n
nEI
lp(q)2/cp(q2) = E(_1)n,n 2
nEI
lp(q) = E(_1)n q (3n 2 +n)/2
nEI
cp(q2)' /cp(q) = L q2n2+n
nEI
(Jacobi)
(Gauss)
( Eu)er )
(Gauss)
11.5. Let W be the Weyl sroup of an affine algebra. Show that if W =
.
WW 1 . where WI is the let of repreaentative& of minimal length of left
o
coeets of W in W J then w(p) - peP + for w e WI' Show that WI = {w e
W, A+ C W(4+)}.
[Use Lemma 3.11 a).)
/1.6. We kep the notation of Exercise 12.2 and identifY' with &. via the
Killing form. Let 15+ be the set of dominant integral weights and let i(A)

242 Integrable H;gAe,'. Weigh' Mod.le, DlJer Affin AIge6,., Ch, 12
denote an irreducible a-module with highest weight  E P +. Prove the
following identity:
o
qdim8/24 II «1 - q")t II (1 - q"e(a»)
nl aE6
= "tro (exp411'i p )chi(..\)q(.Hp,.>.+p).
 L(>')
AE P+
[Consider the decomp08ition W = WW 1 from Exercise 12.5. We can write
(cr. S 12.1 )
L l(w)e(w(p) - p)/R = L c(w)cb l(w(p) - p)e(p - p)
tuEW wEW a
=,-.(;,1) E f(w)chl(.\),.(A+p,A+P>,
wE WI
wher e q = e( -6) and A = w(p) - P E P+ by Exercise 12.5. Note that
w(p) = af(w)( p ) (see 6.6). Now we can use Exercise 10.19 to find that
f(W) = tro (exp47rip).]
L().)
J I. 1. We keep the notation or Exercise 12.6. Let at, . . . , al, be .imple root.
of 9 and let , be the number of short root. in this set. Let k be the ratio
of square lengths of a long and a short roots, and let A" A, be the set.
of all short and long roots. Let l' be tbe half-sum or positive dual roots
l
and h the Coxeter number of J. Set a = h(h + 1) E (Qi,Q,), Prove the
i=l
(ollowing identity:
9 0/24 II ((1- ,")'(1 - ,J:n)'-, IT (1 - ,e(a» IT (1 - ,.e(a»))
nl 0 0
oEA. OEAl
= L h! (exp 21f'i p V Ih) c:h L(.\),.(A+;,A+J>.
AE7+ (A)
{The proof is similar to that of Exercise 12.6. Use Exercise 10.15 Cor m = h.
and the hint from Exerciee 12.3].
Je.8. Show that the identities of Exerci&el 12.2 and 12.3 can be written u
follows:
"dim; = E trl(A)(exp 41f'ip) dimL(.\)q.(A+;,A+P>j
AEJJ+

Ch. 12
In'e9rt16/e Highe$t- Weight Module, over Affine Alge6rtJ
243
l
II ,,( qla.(Oi. O i»Ia+1
i=l
= '" tr exp(211'i p Y /h) dim i()q.(A+PIA+P).
 l()
EP+
In particular, the case 9 = .l2 gives another form of Jacobi's identity:
00
cp(q)3 = 1)-I)n(2n + l)q"(n+1)/2.
n=O
Jt.9. We keep the notation or Exercises 12.2 and 3. Let 8=(80,... t8i) be
I.
a set of nonnegative integers, not all of them zero. Set m = .0 + E a,6,.
i:1
l
where the Oi are defined by: E GiQi is the highest root. Define h, E 
;=1
by: (Qi,h.) = 8ilm for i = l,....i, and set CT, = exp21rih,. Prove the
following identity:
qdim 1/24 IT det; (1 - qnD',) = E X:o(D',)qlp+,ol',
nl oEM
where X°).(tT,) = ( E f(w)eVl(+p)/ E f(w)e w (P»(-211'ih,).
tuEdr "'Edr
[Apply the following specialization to formu)a (12.1.7):
().(e( -(0» = 9. 4t,(e( -a,») = e 2fli ,.!m (t = 1,. . . .i)J.
1 t. J O. Deduce from Exercise 12.6 another form of tbe identity of Exercise
12.9:
qdim 1/24 II det; (1 _ qn tT, )
nl
= "tro (exp 411'ip) tro (D' )q.(,\+1.,\+1> ,
 L(A) L(A) ,
AEP+
12.11. Let a be a simple finite-dimensional Lie algebra, and let g' = l(l)
(see 11.2). Given aD irreducible finite-dimensional i-module L() and a
o
non-zero, complex number 6, we can give a 8tructure of a g'-module L(; b)
by letting K ...... 0, t i 0 %  bi Z (z E 8). Show that all irreducible finite..
dimensional g'-modules are of the form i(li 6 1 ) 0 '" 0 !(ni 6n) where
all the hi are distinct. Show that all irreducible integrable Crt]  I-finite
g'-modules are of the (orm F @ L(A), where F is an irreducible finite-
dimensional g'-module and A e P+.

244
In'egrabl Higlae,'. WeighC Module, over Affine Algebras
Ch. 12
1 e.l f. We keep the notation of Exercise 12.11. Every integrable highest-
weight g' -module of level k can be constructed as follows. Take 8 g-module
1(.\) with (.\, DV)  k, and consider it 88 a g+ := CK + (C[t)  g).
module by letting K ....... leI, t i 0 9 ...... 0 if j > O. Then the g'-moduJe
U(g') U('+) 1(.\) has a unique maximal submodule, this 8ubmodule is
generated by the element (F-l»-(A,'Y)+l @ VA, and the quotient by this
submodule is an integrable g'-module with highest weight kAo + .
Jt.19. Let L(A) be an integrable Al)..module of positive level. Show that
multL(A)(A - n6)  p(n), where p(n) is the classical partition function.
11.1.1. Let £(A) be an integrable Ap>-module; set, = (A,O'n, r = (A, an.
Show that aI) dominant maximal weights of L(A) are either A - jCtal where
o S j S [6/2], or A - jQt, where 0 5 j S [r/2].
11.15. Prove that all the weights of the AP> -module L(Ao) are of the form
Ao - k 2 0 0 - (k 2 - k)Ol - 86, where k Ells E l+.
1 e. J 6. Obtain the following decomposition of the tensor square of the A P).
module L(Ao):
L(Ao)  L(Ao) = L 0"L( 2A o - nlS) + E 6"L(2 A o - 0'0 - n6),
n>O n>O
- -
where (In and 6n can be determined (rom the equation
E dnq2n + E 6"q2n+1 = n (1 + q2n-l).
n>O n>O n>1
- - -
[Use the principal specialization to determine an and b n ].
J e. J 7. Let g(A) be an affine algebra, let A E p be such that (A, d) = 0
and let L(A) = e L(A)j be the gradation of type (1,0,... ,0) (the basic
jel...
gradation). Then L(A)j is the eigenspace of d attached to the eigenvalue
- j, and hence it is S.invariant. For .\ E P + put
+A.A(q) = L(multiplicity of L(.\) in L(A)_n)q" I
n>O
-
o
where L(.\) denotes the integrable I-module with highest weight .\. Show
that
t»A  (q) = n-' A  € ( w )q IA+1-tU(J)12 cA
· 'I L..J Ao+A+1-Ul(p).
WE

Ch. 12
Integrable Higlae,'- Weigla' Mod./e. over Affine Alfe6na. 245
Deduce that for A of type XJ;>, where X = A, D or E, one has
.1. 0 .,,(9) = 9 il -\I' n (1- ,P,+Tlo»! II (1 - tI)multil if .\ E M,
o Jl
OE4+
and 4t Ao . A (q) = 0 if   M. More generally, if k = 1 and A is the only
maximal weight in P(A) n P+, then
c)A,A(q) = a(q)qt<'''''-IXI') II (1- q(-\+Plo» if .\ E A + M,
o
oEA+
and -tA,(q) = 0 otherwise.
Je.J8. Show that the partition function K(o) Cor Ail) is given by the fol-
lowing formula:
K(koo o + klod = L(-I)i p (3)«j + I)le o - jlc 1 - !j(j + I»,
.>0
J_
where p(3)(j) is defined by:
LP(3)(j)ql = cp(q)-3.
jEI
[Show that for 0 = k O O O +1e 1 0 1 one has: K(a)+K(rl(a+p)-p) = p(3)(le O )]'
le.19. We keep the notation of Exercise 7.21 and Remark 3.8. Let (V,.) be
an integrable highest-weight module ofp08itive level Ie over an affine algebra
S(A), where A is the extended Cartan matrix of g. Show that the group G"
constructed in Remark 3.8 is a central extension v>: G"  G of the group
G (of Exercise 7.21) by C)(. Show that putting T fr := cp-l(J(QV» n W.,
we get a central extension:
VI 0
1 -+ {:i:1v} -+ T ff -+ T  QV -+ 1.
Show that if 0, /J E T" are such that cp(o) = a, (/J) = p, then:
opo-l/J-l = (-l)(ol.8>lv.
1'.'0. Let 9 be a simple finite-dimensional Lie algebra oftype XN and let
9 = ED9j (..; r) be its Z/ml-gradation of type (.j r). Consider the realization
of type 6 of the affine algebra S'(XJ;» given by Theorem 8.7:
g'{XJ;» = E ,i  S;<.; r) 6) CK,
jEZ

246
Inlegrab/e HigAe.'" Wei", Mod.le. over Affine Alge6ra,
Ch. 12
and u.e notation or thil theorem. Chooee a basis Ul;-J of g-J('; r) and the
dual bui. uta of 8j ('; r), and define tbe Collowing operator:
L';' - 1  ( u(O)u'tO(O) + 2  u-n)ui;R'("» ) _ H + rk (BIB)
o - 2r ( i + h V ) L...J i;O L..J li-JI' 2
. n>O
( dim 9 IPI2 ) k
+ 24 - 2h v r k + h v ·
For j e z. j  0 t let
,., 1 [ '-' ]
Lj' =  d mjt La' ·
mJ
Show that [%(n), Ll;'} = n%(n+mJ), and that
.3 · L
[ I" L '.' ) ( . . ) I" 6 I - I d . (X ) "
L i I I ;' = . - J L i + J + i,-j 12 1m 9 N k + h V ·
12.21. Under the hypothesis of Exercise 12.20, let g be a simple or abelian
subalgebra of g, invariant with respect to the automorphism (1:r, and let
if = iJ j-f be the induced automorphism of a, so that f E aisi = m (Si are
t _ .
not necessarily elatively prime). Let Lj = L'i.l - L'i.l. Show that the L i
commute with £(9, ii, m) and that they satisfy Virasoro commutation re-
lations with central charge c(k) - c(k).
l!.BI. Applying Exercise 12.21 to i = , k = 1, derive the rollowing
formula:
dim g(XN) Ipl2 (r + l)(h V + l)(N - t)
24 - 2h v r = 24r
J I. B3. Show that
6t() = ctqR II (1 - qn)multn' I
nl
where R = t (= IpI2/2h V (h + 1» if r = 1 and R = IpI2/2h Y (h Y + 1) if
r > 1.
11.14. Show that for the ,C(g)-module LeA). A E P, one h
E d(l + ( + hV)-y)qIA+1+(i+AYhI1J2(i+AY)
tr£(A) ,£:-e(6)/24 = 'fEM
'1 dim .

Ch. 12
Integrable Highe,'- Weight Modules over Affine Algebra!
247
S12.15. Bibliographieal Dote. and CO fnD1 ents.
Identities (12.1.4), (12.1.9) and (12.2.5) are due to Macdonald [1972]. His
proof of (12.1.4), which is done in the framework of affine root systems
(= re), is quite lengtllY and does not explain the ((mysterious tt factors
corresponding to imaginary roots. (These factors were explained in Kae
(1974) and Moody [1975}). These identities have been earlier obtained by
Dyson (1972} in the classical case, but he did not notice the connection
with root systems. Identities (12.3.5) were obtained by Kac {1978 AJ and
Lepowsky (1979}.
The study or the series a has been started by Feingold-Lepowsky [1978J
and Kac [1978 AJ. Lemma 12.6 for r = 1 is proved in Kac [1978 A]; its
present proof is taken Crom Frenkel-Kac [19801. The fact that the string
(unctions  are modular forma is pointed out in Kac [1980 OJ. This ob-
servation was inspired by the "Monstrous moonshine" of Conway-Norton
[1919].
The exposition of 512.4-12.7 closely follow8 Kac-Peterson (1984 A).
The construction be operators Tn goes back to Sugawara [1968). The
exposition of 512.8 uses Wakimoto (1986], Kat-Raina [1987J and Kac-
Wakimoto [1988A]
Proposition 12.10 is due to Goddard-Kent-Olive [1985J. This coset con-
struction was used in Goddard-Kent-Olive {1986] (and independently by
Kac-Wakimoto (1986] and Tsuchiya-Kanie (1986 B}) to prove unitarizabil-
ity of discrete series representations of the Virasoro algebra. This construc-
tion has been playing an important role in recent development of conformal
field theory.
The exposition ofSS12.9-12.12 follows Kac-Wakimoto [1988A], [1989A].
Results of S 12.13 were obtained by Kac (1978A] and Kac-Peterson [1984A]
by a more complicated method.
Exercises 12.2 and 12.3 are due to Macdonald [1972]. The form of Mac-
donald identities presented in Exercise 12.8 is due to K08tant [1976] (his
proof is more complicated). Exercise 12.9 is due to Macdonald (unpub-
lished). Exercises 12.13 and 12.16 are taken Crom Kac [1918 AJ. Exercise
12.17 is taken from Kac (1980 BJ and Kac-Peterson [1984 A]. Exercise
12.18 is taken (rom Kac-Peterson [1980], [1984 A). A special case of Exer-
cise 12.19 is treated in Frenkel-Kac [1980]; the general case may be deduced
using the formula (or the central extension via the tame symbol (described
in Garland [1980]). Exercises 12.20-12.24 are taken from Kac-Wakimoto
[1988AJ.

Chapter 13. Affine Algebras, Theta Functions,
and Modular Forms
513.0. We begin this chapter with an exposition of a theory of theta
functions. Using the clusical transformation properties of theta functions,
we show that the linear span of normaJized characters of given level is in-
variant under the action or SL 2 (Z). Using the theta function identities
(12.7.13) and (12.12.8), we show that the string and branching functions
are modular forms and find a transformation law (or these forms. Fur..
thermore, using tbe theory of modular forma, we prove tbe C'very strange"
formula (12.3.6), which in turn, is used, along with the Sugawara construc-
tion, to find an upper bound of orders of poles at all cusps for the string and
branching (unctions. FUrthermore, we study the "high-temperature limit"
of characters and of .tring and branching functions. All this is applied to
find explicit formulas for the weight multiplicities and branching rules for
integrable highest-weight modules.
113.1. We develop a theory or theta functions in the following gen..
eral framework. (Keeping in mind applications to affine algebras t we use
notation which is identical to that used in previous chapters.)
Let t be a positive integer and Jet . be an (l + 2)-dirnensionaJ vector
space over R with a nondegenerate symmetric bilinear form (.J.) of index
(t + 1, 1). We will identify . with . via this form. Fix a Z-lattice M in
. or rank t. poeitive-definite and integral (i.e. (016) E Z for all a, 6 EM).
Fix a vector 6 E . such that (616) = 0 and (6IM) = o.
000
Put . = R I M C .,  = C . .,  = C. . C  and extend
(.1.) to  by linearity. For ,\ e  we denote by I the orthogonal projection
o
of ,\ on . Let
Y = {II e  I (6ItJ) > O} .
o
For 0 e . let '0 denote the automorphism of  defined by (d. (6.5.»:
tOl(v) = fJ + ( t1 16)Q - ((fila) + 1(olQ)(1116) 6.
Note that the automorphism to is characterized by the properties:
a) '0(6) = 6; b) t o (lI) = 11 + (1116)0 mod C6, and c) (.1.) is 'o-invariant. It
follows that toc = ' Q +,.
248

Ch.13
Affine Alge6raa, TAettJ Function" and Modular Form,
249
For a E . Jet PerC fI) = fI + 2rio, fI E . AU the transformations Per and
tp (o,,B E .) of  generate a group N, called the Heisenberg group. More
o 0
explicitly, N = . X . X iR with multiplication:
(0, fJt u)(a', {1, u') = (B + 0', fJ + 13', u + u' + 1fi((alp') - (0'1.8»).
The action of N on  is given by
(13.1.1)
(Q,fJ, u)(1J) = tp(v) + 21"o + (u - 1ri(nIP»6,
so that (a,O, O)(v) = pQ(v), (O,P,O)(v) = t(v), (0,0, u)(v) = II + u6 t and
Y is N -invariant.
Denote by N z the subgroup of N generated by all (Q)O,O), (O,P.O) wit
Q,P E M and (O.O,u) with U E 2".iZ. Then
Nz = {(Q,J1J u) e N I at P e M, U + ",i(afJ1) E 21riZ}.
513.2. Fix 8 nonnegative integer k. A theta (unction of degree I: is
a holomorphic function F on the domain Y such that the following two
conditions hold for all tJ E Y:
(Tl)
(T2)
F(n · v) = F(v) for &11 n E NIt
F(t1+o6) = e'OF(v) for all Q E C.
--
Let Thlt denote the space (over C) of alJ theta functions of degree It. Then
fi = EI1 Thll
iO
is a graded associative C-algebra, called the algebra of theta functions.
.
In order to produce examples of theta functions let M* = { e . I
(Io) E l for aJJ Q E M} be the lattice dual to Mj for a positive integer 
put
Pi = { e  I (AleS) =  and I e M.}.
Given  E Pi t we define the classical theta (unction of degree i with
characteristic X by the series
(13.2.1)
e = e- J.Wl' E e'.('\).
oEM

250
Affine Algebra" Theta Func',on" tJntl Modular Fonn,
Ch, 13
Equivalently, we take ' ==,\ mod C6 such that ('I/) = 0, and let
e = E ef.(').
oEM
Note that this (unction is exactly the one defined by (12.7.2) (which nat-
urally arises in the theory of affine algebras). As in 512.7, we can rewrite
SA in another (orm:
( 13.2.2)
9 = e Uo E ,t.(,.h)e."
'YEM+-a:'
As before, q stands for e-' , and Ao e . is the unique isotropic vector such
that
(AoI6) = 1
and
(AoIM) = O.
It is clear that the series (13.2.2) converges absolutely on Y to a holomor-
phic function; one easily lees from (13.2.1) (or (13.2.2» that SA satisfies
(Tl) and (T2), and hence is a theta (unction of degree k. Note also that
( 13.2.3)
eA+ia+CI' = e for 0 E M, a E C.
.
Choose an orthonormal basis VI t . . . , Vt of . and coordinatize  by
( 13.2.4)
tJ = 2i ( t t,V, - rAo + 1.16 ) J
.=1
, 0
80 that we shall often write (r'%Ju), where z = E z,t}, e , 1',U e C, in
,=1
o
place of t1 E . Then Y = {(T,Z,U) I z E ; T,U E C,Imr > OJ, and we
can rewrite 9A in its cl8S8ical form:
(13.2.5 )
e(r, t, 1.1) = e 2 .' h l: e.. h (,.h)+2.i.(,.,,)
-rEM+,-aI
, ,
(where (")'1%) = E 1i Z i, 1 = E 1i V I). Note also that in these coordinates
;=1 i=l
we have: q = e- I = e 2 .. iT .
Let 'It = {r e c 11m T > O} be the Poincare upper half-plane. Note
.--.... ,..""
that a holomorphic function in T e 1t. lies in Tho. Conversely, if F e Tho,
then F is independent of u and for each fixed r E 1l t F is periodic in z

Ch.13
Affine Alge6ra8, Thet. Fanct;on, lJnd Mod.lar Fonn8
251
o
with periods in M + rM. Since /(M + rM) is compact, we deduce that
F is a fUDction in T. SOt we have proved
-
LEMMA 13.2. Tho is the algebra of holomorphic (unctions in T e 1t.
o
The following multiplication formula is very useful for applications.
PROPOSITION 13.2. Let e A and e" be classical theta functions of degree
m and n, respectively. Then
8,,8 p = E 8.\+p+"0,pO'
EMmod (m+n)M
where
,po = L ql nA -mii-mn(0+,.)\2/2mn(m+n).
'1E(m+n)M
Proof. We may assume that 1;\1 2 = 1141 2 = o. Then
9.\ = E e'G(A) I e p = L e"C#') I and
oEM "EM
e.\e,. = E eC,,(AHG(p»
a."eM
= L q -1.\+c G (p)1 2 /2(m+n)e "+'G(I')
oEM
-
-
E 9 A + CG (p)
aEMmod (m+n)M
E 9.\+p+no
aEMmod (m+n)M
E q-IAHG+,,(p)I'/2(m+n)
"1E(m+n)M
E qlnA-m'Gh(p)J2/2mn(m+n).
.,e(m+n)M
-
-
o
513.3. Let D be the Laplace operator associated with the form (.1.). We
have in coordinates (13.2.4):
(13.3.1) D = -.!.- ( 2.!!..!... _  (  ) 2 )
4",2 Ou Or f=: Oz, ·
Since D(e) = (IA)e'\ we deduce from (13.2.1) that
(13.3.2) D(8A) = o.
---
We put Tho = C, Thi = {F e Th, J D(F) = O} (or k > 0, Th = 'e Th i .
t>o
 -
Note that the subspace (over C) Th of Th is not a subring (see Proposition
13.2).

252
Affine Algtbra., TAt" Fnc';on', Gntl Motlu/Gr Form,
Ch. 13
PROPOSITION 13.3. Tht1 aet teA ,  E p" mod (kM + C6)} is a C-basis
-." -
ofThlt (resp. Tho-basis ofThi) if k > O.
.-...
Proof Let F e Th i . Using F(Pa(v» = F(v) for Q E M and (T2), we can,
for a fixed T, decompose F into a Fourier series:
F = e Uo E GTh)e'Y.
.,e M.
By using (Tl), we obtain that o,.(1)e-. ii - I 1'(7J1) depends only on
"y mod k M. It follows that
( 13.3.3)
F = E c,\(r)e,\.
EP.mod 'M+CI
Furthermore, fix a positive real number Q; then for Q E k- 1 M. we have
e,\(2ia + aAo) = e 21ri (Ilo) L e-ikbb)ca.
EM+.-1X'
Since the characters of the group (k -) M'*) / M are linearly independent, we
deduce
( ) {6.\(a, %,0) , E Pk mod IcM + C6} is a linearly independent
13.3.4
set over C, where the eA are viewed as functions in z.
This completes the proof of the linear independence of the SA over C and
-
over Tho.
Finally, if D(F) = 0, then. applying (13.3.2), we deduce from (13.3.3)
that
"It
o = D(F) = !... E<dc.\/dr)8.\.
1t 
Using (13.3.4), we get dc)../dr = 0, hence the 8).. span Thlc over C.
o
E%Qmple 19.9. Let M = Zo be a I-dimensional lattice with the bilinear
form normalized by (ala) = 2. Then M* = tM and the following classical
theta functions form a basis of Th m :
e ( r z U ) .= e 2 . imu  e 2..im(i2"....,)
n.m J,. L.J '
i:EI+ i:a
n E Zmod 2ml.

Ch.13
Affin Alge6ra6, TAettJ Function" and Modular Forna,
253
513.4. We recall some elementary facts about the group SL 2 (R) and ita
discrete subgroups. The proofs may be found in the book Knopp (1910).
The group S L 2 (R) operates on 1f. by
( a 6 ) . T = aT + 6 .
c d CT + d
For every positive integer n define the princ;pIJI congNJt1Jct Iubgroup
fen) = {(: ) E S£2(Z) 10 E dE Imod n, 6 == c:; Omod n},
and the subgroup
fo(n) = {(: ) E 5£2(Z) I C == Omod n}.
Another important subgroup is
f, ::: {(: ) E 8£2(Z) I ae and 6d are even}.
All these subgroups have finite index in r(l) = SL 2 (Z).
Put 5 = ( 1 ), T = ( :). Then:
(13.4.1) Sand T generate reI); S and r 2 generate r't
(13.4.2) T, ('T)" and -[ generate ro(r) for r = 2 and 3.
Recall that the metaplectic group Mp2(R) is a double cover of SL 2 (R),
defined as follows: M P2(R) = {( A, j) I A = (::) e 5 L 2 (R) and j is a
ho)omorphic function in r E 1t such that j2 = cr + d}, with muJtiplication
(A,j)(A 1 ,jl) = (AA.,j(A 1 · r)j.(T». MP2(R) acts on 1{ via the natural
homomorphism Mp2(R) -.5L 2 (R). We put:
Mp2(l) (resp. Mp(Z» = ((A,j) E Mp2(R) I A E SL 2 (Z) (resp. E f,)}.
Furthermore, we introduce the folJowing action of Mp2(R) on Y (actually,
the SL 2 (R)-action):
(: )'(T,Z,U)= ( :;: ' CT:d ,u- 2(:)d» )'
o
where z E , T E 1{, II e C.
It is clear that the groups Nand SL 2 (R) act faithfully by holomorphic
automorphisms of Y. One checks that Mp2(R) normalizes N; namely:
(13.4.3) «(: ) ,j)(Q,P,u)«(: :) ,jr 1 = (oQ+bP,cQ+dP,u).
Hence, we have an action of the group G := MP2(R) K N on Y.

254
Affine Alg6nJ', Theta Function" and Modular Form.
Ch. 13
One checks direc:tly the following
LEMMA 13.4. The normalizer of Hz in the subgroup Mp2(R) of G is
Mp2(Z) j( the lattice M is even (i.e., all (,,1,,) are even for" eM), and is
Mp(Z) j{ M is odd (i.e., not even).
o
Finally, we define a (right) action of G on holomorphic functions on Y
as follows «A,j) E Mp2(R), n E N)=
F I(AJ) (T, Zt u) = j((r»-t F(A · (r, z, u»; F In (v) = F(n(v».
(At this point the use of Mp2(R) instead or SL 2 (R) is essential to have an
action.) Obviously:
( 13.4.4)
D(F»)n = D(Fln) for n E N.
We prove now the following important result:
-... .-.-
PI,tOPOSITION 13.4. a) Th I(AJ) = Th if the lattice M is even and A E
SL 2 (Z) or j{the lattice M is odd and A e r,.
b) Th I(A,J) == Th jf the lattice M is even and A e 5£2(Z) or jf the lattice
M is odd and A E r , .
Proof. a) follows immediately from Lemma 13.4 and (13.4.4).
Due to Proposition 13.3, in order to deduce b) (rom a), it suffices to
check that
D(e I(T.l» = 0, D(8A I(s,j») = O.
The first equation is clear. The second is immediate from the following
simple formula (recall that j2 = r):
D(j-t e 2..ik(u- h -I I2 /2f'» = 0, where 'Y e .
o
We record also the following two simple transformation properties of
classical theta (unctions of degree k, which Collow directly Crom the defini-
tions:
(13.4.5)
(13.4.6)
e1 = e 2 ..(alv)e for Q E k- 1 M.'
(0,0,0)  ,
8 A I(o,Q,o) = e-ia for Q E ,-1 M..

Ch .13
Affine Alge6ra" Tlae'o Funcfion" tJnd Modular Form,
255
We have the following corollary of Proposition 13.3 and formula (13.4.5):
COROLLARY 13.4. The [unction SA (defined by (13.2.1) is character-
jzed among the holomorphic functions on Y by the properties (Tl), (T2),
(13.3.2), and (13.4.5).
o
S13.5. Denote by n = n(M) the least positive integer such that nM. C
M and n(111) e 2Z (or all 1 E M..
Now we are in a position to prove the following transformation law which
goes back to Jacobi.
THEOREM 13.5. Let'\ E Pi. Then
( 13.5.1 )
9.\(-.!.,':',u _ (:1:» = (-ir)itIM./J:MI-i
T T 2r
x L e- 1fi <Alil>9 p (r,:, u);
"ep.mod (iM+CI)
( 13.5.2)
8,,(r + 1, z, u) = e.'fAI2/leA( T, %. u).
Furthermore, jf A e r(kn) (resp. r(kn) n r,) when M is even (resp. odd),
then
(13.5.3)
8 A J(A.J) = tJ(A,j; k)9A,
where v(A,j;k) E C and 11J(A,jjk)f = 1.
Proof. Using that
9.\1, = CeUOI(o._I--II,o})I, = C 6 .u o l,)!,-I(o.-1- I I.o},
for 9 E G, by (13.4.6) it suffices to prove the theorem for A = o. Note also
that when replacing (.f.) by k(.'.), the theta function e(T, Zt u) of degree
k transforms to the theta function 8 k -l,\(T, z, ku) of degree 1. Hence we
may (and shall) assume that A = Ao.
By Proposition 13.4b), we may write for A = (::) E SL 2 (l) (resp.
E r,) if M is even (resp. odd):
(13.5.4)
9A o l(AJ) = L l(p)8 Ao + p , where I(p) E C.
pe M.mod M

256
Affint Alge6ra" Tlaet4 Function" Gnd Modular Form,
Ch. 13
Fix Q E M.. Since, by (13.4.5), eAol(o,o,o) = eA o ' we get by (13.4.3):
BAo f(.A,j) = SAo I(A,}) I(A,j)-t(a.o.O)(A,j) = aA o r(A,j) l(cla.-eQ.) ·
Hence, applying (A,j)-l(Q,OtO)(A,j) to both sides of (13.5.4), we get,
thanks to (13.4.5 and 6):
e , -  1( ll ) .tri(4e(QJa)+2d(aJ» aA + +
Ao (A,j) - L..t ,-  0  ca.
"EM- mod M
If A = S, comparing this with (13.5.4), we get that 1(1' + Q) = !(IJ) for all
Q,Jj E M. and hence:
(13.5.5)
eAol(SJ) = v(S,j) L eAo+I' , where v(S,j) E C.
IJEM- mod M
If A E r(r), we get: !(p) = f(p + Ctr) = !(p)e 2 "i(olp) for all Q, #J E M..
Hence f(JJ) = 0 unless p E M. This completes the proof of (13.5.3). The
fact that Iv(A,j; m)1 = 1 follows from Corollary 13.5 below. (13.5.2) is
obvious.
Furthermore t as explained in the beginning of the proof, (13.5.5) gives us
that (13.5.1) holds up to a constant factor v(S,j) E C. In order to compute
this constant, notice that (5, j)8 = I and that the rows of the matrix of
the transformation (S,j) in the basis {8,\} are pairwise orthogonal. It
foJ)ows that this matrix is unitary. We deduce immediately that Iv(S, j)J =
IM./kMI-i. Finally, using that 8A(O,i,O) > 0, we deduce from (13.5.5)
that v(S, j) = (-i)itlv(S,j)l.
o
( 13.5.6)
Since the matrix of the transformation (8, j) in the basis {e} is unitary,
and the matrix or the transformation (T, j) (resp. (T 2 , j) for M even (resp.
odd), is (diagonal) unitary. using (13.4.1), we get the following useful result.
COROLLARY 13.5. The matrix oE a transformation (rom Mp2(Z) (resp.
Mp(Z» if M is even (resp. odd) in the basis {8 A hEP.modtM+CI is unitary.
Erample 19.5. Theorem 13.5 gives the following transformation law for
theta functions 8n.m from Example 13.3:
en m ( _.!. !. u + ::. )
· r) T' 2r
= (-iT) t(2m)-! L e- .,,:,,, en' ,m(T, z, u).
n'Elmod 2ml

Ch.13
Affine Algebra" Tlae'tJ Function" Gnd Modular Form.
251
513.6. Here we give a brief account of some facts about modular forms
which will be used in the sequel.
Fix a subgroup r of finite index in r(l), a function X : r -+ Cx with
IX(A)t = 1 for A E r, and a real number i. Then a function / : 1l -+ C
is calJed a modular form of weight i and m.lIiplier ",fem X for r if I is
holomorphicon 1{ and' ( ::tn = x(A)(cr+d)i'(r) (or all A = (::) E r
and T E 1t.
Let I be 8uch a modular form. Then, since r has a finite index in
r(l), T. E r for some positive integer B and hence /( T + B) = e 2rriC f(r)
for some C e R. Set
F(e 2Wi .",) = e-2fliCT"!(r).
Then F is a well-defined holomorphic (unction in z = e 2JriT /' on the punc-
tured disc 0 < IzJ < 1. Hence F has a Laurent expansion F(z) = E On ZR
REI
converging absolutely for 0 < Izi < 1.. Therefore, we have the Fourier
.
expansion
fer) = L on e2 ...(n+C)rf. for r E 1t..
nEZ
We call I meromorphic at ioo if Ora = 0 for n <: 0. holomorphic at ioo if
an :F 0 implies n + C  0, vtJnis.hing at ioo if an :f; 0 implies n + C > O. If I
is holomorphic at ioo, we say that the value of f at ioo is Q-c (interpreted
as 0 if C  l). If no = min{n IOn -F OJ, we let r = I(no + C)/B' and say
that it has zero (resp. pole) of order r at ioo if no + C  0 (resp.  0).
A cusp of a subgroup r of finite index in SL 2 (Z) is an orbit of r in
Q U {;oo}, where a/O is interpreted 88 ioo for a E Qt a  O. Since r( 1)
acts transitively on Q U {ioo} J the set of CUSP8 of r is finite. Sometimes
we speak of the cusp a E Q U {ioo} of r; this means the orbit of Q under
r. For example) r(l) has one cusp ioo, r, has two cusps: ioo and -1, and
ro(lc) for prime k has two cusps: ioo and O.
Let' be as above and consider a cusp 0 of r. Let B = (:) E r(l)
be such that B(ioo) = cr. Then /o(r) := (CT + d)- I(Br) is a modular
form of weight k and some multiplier system XO for B-1 r B. We say I
is meromorphic, holomorphic, or Ads zero or pole oj order r at Q if /0 is
meromorphic t holomorphiC t or has zero or pole of order r at ioo. We 88Y
that I is R-singular if orders of poles of I at all cusps are < R.
A moduJar form of weigbt k and muJtipJier system X for r is calJed a
meromorphic modular form, 8 holomorpAic modular form, or a cusp fonn if
it is meromorphic, holomorphic, or vanishes at all cusps of r, respectively.

258
Affin Alge6ra" TAetti hnc'ionl, and Modular Fonn.
Chi 13
A holomorphic modular form of weight 0 is a constant. This aiiowl one
to identify modular Corms. In what follows, f wiliitand, for e 2 . 'r , as usual.
Using various specializations of classical theta functions, we can con-
struct modular (orJ118. In (act. given a holomorphic function F on Y and
o
0, (J e ., we define a holomorphic function Fa," (r) on 1{, by
(13.6.1) Fa'(T):= (FI(a'lo»(T,O,O) = F(T,-O+TP,-!(PI-o+TP».
For example,
(13.6.2)
e'(T) = ewii(al) L: e21rik{ah)qihl'/2.
-
.,EM+'-.>'-
(13.6.3)
Furthermore, it is clear by (13.4.3) that
(FI(AJ»a'(T) = r1aH",ea+cf"(T)I(AJ)1
where ((:  ) Ii) E Mp(2,R).
A special case of this is
(13.6.4)
(FI(SJ»a'(T) = F-la(T)I{sJ)'
Now it is easy to prove the following
PROPOSITION 13.6. Given positive integers, and m, put
Frn" = {ea'(T) lee Thrn, '0 e M, ,p EM}.
Then every function from Fm" is a holomorphic modular form ot wei,he
it tor r(mn)nr(,) (resp. r(mn)nr,nr(,» it the lattice M is even (resp.
odd).
Proof. Let A = (::) e r(mn) (resp. f(mn) n f,). Then by Theorem
13.5 and (13.6.3) we have
v(A)8°"(r) = 8C1Q+'tco+cI ( or + 6 ) ( cr + d ) -f
 A cr + d '
where v(A) E C, Iv(A)1 = 1. On the other hand, by (13.6.2) we have
e'(T) = :i:eaH,ea+4(T) if (: ) e f(6).

Ch.13
Affine Algebra", Theta Function" tJnd Modular Forms
259
Thus, every function from Fm,. satisfies the required transformation prop-
erties.
Furthermore, it is clear from (13.6.2), that .Tm..I(Te,j) = Fm.. for I: = 1
or 2, according to whether M is even or odd. Using this, as well 88 (13.6.4)
and (13.4.1), we conclude that the linear span of Fm.I is invariant under
M P2(Z) if M is even. Since all functions from Fm,. are holomorphic at
the cusp ioo, we deduce, by Lemma 13.12 in 513.12, tbat all or them are
holomorphic modular forms, provided that M is even. The general case is
reduced to this one by the change of variables r -+ 2r.
o
COROLLARY 13.6. Let M be a Z-lattice a/rank l and let (.1.) be a positive-
definite Q-valued bilinear form on M. Let f : M -. C be constant on Cagets
of some sub/attice of finite jndex and Jet a E Q 0z M. Then
leT) = L lh)qh+ G I2
.,eM
is a holomorphic modular (orm of weight it (or some r(N) and some
multiplier system.
Proof. Replacing M by a sublattice of finite index. we can assume that
((1) = 1. Moreover, since replacing (.,.) by l(.'.) corresponds to replacing
r(N) by r(kN)t we can assume that M is an integral lattice. But then
I(r) = e,;G.
o
(13.6.5)
The most popular examples of holomorphic modular forms are these:
In,m(r) = E qrn U +n / 2m)2, 9n,m( r) = L( -lY qrn(j+n / 2rn)2,
iEI jEI
where m, n E 4Z, m > O. Since fn,m(T) = 8n,m(r, 0, 0), we obtain from
(13.5.6), provided that n E Z:
( . ) 1/2
fn,rn(-r- 1 ) = ;: E t 1dn / m h,rn(r).
iEZmod 2mJ
(In fact (13.5.6) implies (13.6.5) only for integral m; to get half-integral
711, one needs to take (ala) = 1 in Example 13.3.) Since 9n,m = fn,4m -
f(n+2m).4mt we derive from (13.6.5), provided that m, n E ! + z:
1 ( _ir ) 1/2  (2k+l)n
(13.6.6) gn,rn(-r-) = 2m LJ (2(08 2m )gi+t,rn(r).
'EZ
°Si<m-I

260
Affine Alge6ns., TlaetJ Func'ion" tJnd Modular Fornal
Ch. 13
A special case or this is
(13.6.7)
'!,t(-r- 1 ) = (-ir)1 / 2't.t(r).
Now we consider some examples of modular forms defined by an infinite
product, and among them the Dedekind "..function
,,( r) = t w it-/12 II (1 - t 2trin1' ), r E 'H..
nl
. Our starting point is the identity (12.7.4) for the affine algebra Ail).
Using (12.1.5) and the inrormation from Examplel12.1 and 13.3, it can be
written 88 follows (we put 0 = Ql):
( 13.6.8)
(9 1 ,2 - 8-1.2)(T,Z,U) .
= e 2 . i ( 11'- t(.10)-2") II (1 - q" )(1 - q"-I e 2 ..("o»( 1 - q" e- 2 ..(.,o».
n1
Note that we have Cor m, r E Z, m > 0:
Im/2-r.m/2(r) = (81,2 - e_l,2)o.ro(mr).
Hence, we obtain from (13.6.8) (recall that (oIQ) = 2):
(13.6.9)
'm/2-r.m/2(r) = q(2r-m)2/ 1m IT (1 - qm J )(1 _ qmJ-(m-r»(l _ qmj-r)
J1
A special case of this for m = 3, r = 1 is Euler's identity:
( 13.6.10)
,,(r) = L(-1y q t<J+i)2 (= 'i.t(r».
JEI
Comparing this with {13.6.7}, we obtain the classical functional equation
for the f7-function:
(13.6.11 )
,,(_!) = (-ir)t,,(r).
'r
Using (13.4.1) we deduce that ,,(r) is a hoJomorphic cusp-form of weight
1 Cor r(l} and a multiplier system X" such that X4 = 1. Note that:
X,,(S) = e- fti / 4 , X,,(T) = efti/12.

Ch.13
Affine Alge6nJi, TAe'. F"nctionl, Grad Mod.lor Form.
261
513.1. We shall apply the theory or theta functions to affine algebras.
Let g(A) be an affine algebra of type X) and let M be the lattice intro-
duced in 16.5.
o
Since, by definition, M = l(W · 8 V ), and (,VIIY) = 2a;1 (see 56.4), we
see that the lattice M is alway. interal; moreover, it is even if and only
if the affine algebra is not of type A . Using the information about finite
root systems given in 56.7, one easily computes (or this lattice the constant
n :: n(M) introduced in 513.5; it iI found in the following table.
-y(A) l n = n(M)
ACI) odd 2( t + 1)
t
even l+ 1
B(I) D(I) A(2) odd 8
l. t t , U-l
211t 4
41t 2
d J ) E(I) D(2) 4
t. t 7 , t+ 1
E(l) d l ) D(3) 3
e , 2' 4
F(J) A(2) E(2) 2
4 , 2l t 6
E1 1 ) 1
Fix a positive integer k. Using (6.1.1), we can write (cf. SlO.l and 112.4):
pit = { E . I level() = i, (,Q) E Z for Q E QY}.
Recall also the definition (rom S 13.2:
Pi = { E ". I leve]() =}, (IQ) E l for Q eM}.
Using (6.5.8) aod (6.5.9) we obtain:
( 13.7.1 ) p" :> pi,
(13.7.2) p. = p' if r = 1 or 00 = 2.
LEMMA 13.7. Let e, ,\ E Pi, be a clASSical theta function associated witb
the lattice M. If A E r(kn), then
9AI(AJ) = v(A,j; i)9A,
where v(A.j;k} E C, Iv(A,j;k)( = 1.

262
Affine Alge6rol, TAtfa Func'ioR&, anti Modular Form
Ch. 13
Proof Since n = 2 is the only case when M is odd, and since r, c r(2k),
the lemma follows from Theorem 13.5.
o
Define the space of anti-invarian' classical theta functions of degree k
by:
o
(13.7.3) Th; = {F E Th. I F(w(v» = l(w)F(v) for w e W, v e }.
Using Proposition 6.5 and (6.5.10), we see that the space Th; consists of
holomorphic functions F on Y, satisfying the following four properties:
(Tl-)
(T2- )
(T3-)
(T4-)
F(w(» = f(w)F() for w E Wi
F(>' + 2iriQ) = F() for Q E M;
F(>' + 06) = e to F() (or a E C;
DF = O.
Given  E p., we introduce the anti-invariant classical theta function of
degree  and characteristic I:
(13.7.4)
A.\ = E (w)81C/(.\).
.
WEW
By (13.7.3), Proposition 6.5 and (6.5.10), we have:
( 13.7.5)
A" = e-1Wl' E (w)elC/('\).
wEW
Let P i + = P+ n Pi, P,,++ = P++ n Pi.
PROPOSITION 13.7. a) The set {AA I A e P i ++ mod C6} is a C-basis of
Th;.
b) 1£), e P++, then:
A.\I(sJ) = (-i)tIIM e /kMI-i
x E L (w)e-(IC/(i)Iii)Aj
PEP,++mod c, .
UlEW
-
A( T + 1, %,1.1) = elril,\l/i A,\(T, %,1.1).

Ch.13
Affine Alge6ral, Theta Function" and Modular Form,
263
Proof By Proposition 13.3, the set {A r ,\ e Pi} spans Th i . Thanks to
Lemma 12.4 and Proposition 6.5 we have:
(13.7.6) Pm mod (mM + C6) = U w(Pi+ mod C6) (disjoint union).
.
wEW
But we clearly have:
(13.7.7)
A" = 0 if ri(IJ) = I' (or some i.
Hence, by (13.7.6), the set {A,\ I ,\ E Pi++ mod C6} spans Th;. These
A are linearly independent by Proposition 13.3 and (13.7.6), proving a).
By Theorem 13.5,
A" l(sJ) = (-i)IIIM./kMI-I
x E ( E t'(w)e- 1fi (w(A)IP) )e,..
pep,mod (iM+CI) W .
wE
This together with (13.7.6 and 7) proves b).
o
513.8. We turn now to the proof of the (ollowing crucial result:
LEMMA 13.8. Let g(A) be an af1ine algebra o(ype Xl) or A). Then
a) Th; = 0 jf k < h V and Th;v = CAp.
.
b) Apl(sJ) = (-i) it+I+1 Ap.
Proof. a) follows (rom (13.7.2) and Proposition 13.7a). By a) and Propo-
sition 13.7b), we have:
Apl(s,j) = (_i)IIIM./h v MI-I cAp where c = E (w)e- ty(w(p)lp).
.
wEW
By Corollary 10.5, fel = IM./h v Mlt. On the other hand. identity (10.4.4).
applied to g, gives
c = q (-2iSin wT!Q» ) .
aE4+
.
It follows that iJA+1c is a positive real number. completing the proof.
o

264
Affine Alge6ra" Theta F.nction" anti Modular Form,
Ch, 13
The proof of Lemma 13.8 gives us the Collowing useful identity
(13.8.1)
II 21m 1r!O) = 1M- /h v Mp/2,
.
aE4+
.
Since,
(13.8.2)
(pIQ)  (piS) < h v.
Recall the Cormula Cor the normalized character (see (12.7.11» for A e
P i.
+.
XA(r.z,u):= qmAch£(A) = AA+p/A,.
Note that a special case of (12.2.2) Cor 9 = 0 is
(13.8.3) XA( T, Z, u) = e 2 l'iiu trL(A) e 2r i.r qL:- e(i) if g(A) = leg).
Proposition 13.7b) together with (13.7.2) and Lemma 13.8.b) imply im-
mediately the following important transformation law for normalized char-
acters:
THEOREM 13.8. Let g(A) be an alfine algebra of type X?> or A>, and
let A E p,:. Then
a) XA( -t,, tc -1Wl> = E SA.A'XA'(T,Z, u), where
A/EPmod CI
SA,A' = i'A+IIMe/(k+hv)Mf-i E £(w)e- h'IA1:+A'+ i
.
tuEW
XA(r+ Itz.u) = e 2 .. im :\XA(r,z.u).
(13.8.4)
b) The linear span of the {XA}AEP mod C6 is invariant under the following
action of SL 2 (Z).'
( 0 " ) ar + 6 Z C(IJI)
C d '!(T,Z,U)=!( CT+d ' CT+d ,u- 2 (CT+d) '
o
The matrix
S() := (SA,A' )A,A'EP mod C6

Ch.13
Affine Alfe6,.., TAeta F.nc,ionl, Grad Modular Form,
265
has a number or remarkable properties. First, we obviously have
(13.8.5 )
, 5(1:) = S(i)'
Furthermore, Corollary 13.5 gives us
(13.8.6)
S(i) is a unitary matrix,
and we have from (13.8.5) and (13.8.6):
(13.8.7) Sit = S(l) (complex conjugate).
Given A e P. we let
, ,-
A = iAo + A + (A,d)6,
where eA is the highest weight of the i-module contragredient to L(X).
Then
( 13.8.8)
S[k) = (bA,t..., )A,MEP:modC6'
This is immediate from 8 2 ./(TtZ,U) = I(r,-z,u). Finally, letting for
,\ep+
x,\(e') = tr. e', JJ E ii,
L(A)
and using the Weyt character formula, we can rewrite the formula for SA.A'
as follows:
(13.8.9)
I -
a 2i - (A
SA.A' = a(A}XX,(e- ... ),
where
(13.8.10) a(A) (= SA,tAo) = 1M- /(k + hV)MI-t II 28in w(; + :!Q) .
. +
QE+
Remark 13.8. Due to (13.8.2), o(A) is a positive number since (AIQ) S i
o
if Q e +.
Incidentally, (13.8.6) gives the following generalization of (13.8.1):
(13.8.11)
E a(A)2 = 1.
AEP mod C,
Example 19.8. If g(A) is of type Al), then
( 2 ) 1/2. w(j + l)(j' + 1)
S(-j)Ao+jA..(-jl)Ao+i'A. = k + 2 8m k + 2 ·

266
Affine Alge6ra" TlaetG Function" anti Modular Form,
Ch. 13
S13.9. Unfortunately P,  p. in the rest of tbe cases, i.e., when 00 = 1
and r = 2 or 3. As a result. Lerruna 13.8 and Theorem 13.8 fail in these
cases. We shall indicate now how to handle them.
For A of type XJ;) = Al' Dl' E2), and D3) we let A' be of type
XJ;:> = Dl' Al' E2), and D3), respectively. Let X; = (aj/oj)"K J
(resp. = (Ol+l-J/ar+l_j)AI.+1-J) if A = Al or Dl (resp. = E1 2 ) or
I.
D3». Let Ai =  +a'Ao (i = 1,...,/), A = Ao, p' = EM, k' = k,
i::O
t 0
P' = E lA, + C6 t M' = QV, W' = W. Then P' is the set of integral
.=0
weights for g(A'), etc., and the analogue of (13.7.2) is
(13.9.1)
Pi = p,i
We denote by e. A. X the corresponding functions for g(A'). Then the
formula (13.5.1) and Proposition 13.7b) can be rewritten as follows:
(13.9.2)
6>.(-!,, 1.1 - (:1:» = (-ir)I.!2IM* /kMI-t
T T 2r
x L e- 2fi(II IA) e ( !:, !:, 1.1),
r r
pEP'.mod 11M'
(13.9.3)
A( _.!., ,1.I - (:1:» = (-ir)'12/M- /kMI-!
r T 2r
x L E £(w)e-(w(:r)lii)Ap(.!:,,1.I).
r r
pEP'...mod CI WEW
In the same way as Lemma 13.8b). we deduce
(13.9.5)
(13.9.4) Ap( -, ;,1.1 - (I:) = (_ir)t/2( -i)ll+IIM'/ M,- tA/(;' ;, u),
and the following analogue of (13.8.1):
n 2sin 'Ir!Q) = 1M- /h v M'/1/2 if aor = 2 or 3.
.
OEA.
From (13.9.3) and (13.9.4) we deduce the following analogue of Theorem
13.8.

Ch .13
Affine Alge6,.a, TAet. F.raclioft6, Grad Motl.ltJr Form,
261
.
THEOREM 13.9. Let SeA) be an aIline algebra of type X};) with r > 1,
(10 = 1, and Jet A E P:. Then:
a ) '\I ,\( _1!. u - !!1!} )
A  , T ' 2,-
where
-
-
LA'EP'. mod C, SA,A'XA't (  , ,u),
+
· 2 ' -  
SA,A' = i'A+IIM./(.t + hV)MI-1IM'/Mli L t(w)e- H'v (UI(A+1)IA +p);
.
tuEW
( + 1 ) - 2..im- ( )
XAT ,Z,u -e AXAT.%,U.
b) The linear span of the {XA} AEP.mod CI is invariant with respect to r o( r)
+
under the action (13.8.4).
Proof. Since'T = ST- 1 S-l, using (13.4.2) we see that a) implies b).
o
In what follows, in order to state all cases in a uniform fashion, in the
cases r = 1 or 00 = 2 we let A' = A, Ai = A, P = pi, M = M', p = pi,
etc., and let r' = I.
SI3.10. Now we can prove a transformation law for the string functions
for general affine algebras, and branching functions in the nontwisted case.
,..
If 9 is semisimple, we let for £(;):
S A A ' = n S A A , ·
, ( j ) . (j)
"> 1
J_
If 9 is a reductive subalgebra of g satisfying conditions or Theorem 13.10b)
below. we let
s ,= I M. / k M l -if!-2.i(  (o)I .\ (o),/'o
A(o) ,A(O) 0 0 0 ,
S). A' = TI S). ( ' ) ' ·
t , t (j)
.>0
J_
THEOREM 13.10. a) Let g(A) be an affine algebra or rank l + 1. and Jet
- t
A E P+. Then
c(-) =: IM./kMI- 1 / 2 (-iT)-t/2
T
L
S e 21ri(  I ). ')/. c ,A' ( T )
A A' \, - ·
JAr
A'EP'mod CI
'ep"mod (iM'+CI)

268
Affine Algebra" TlaettJ F.raclion" Gntl Modular Form,
Ch. 13
b) Lei 8 be a llenU.imple finite-dimensional Lie algebra with a Cartan
subalgebra ij and the coroot lattice (jv C ij", and let 8 be a reductive
Bubalgebr8 of g such that its center 8(0) i9 spanned over C by the lattice
Mo := Cr n 8(0). Let A E P:, where i = (i l . i2, . . .), i. > O. Then:
1  -r A' .
(13.10.1) bN-;i g) = L.J SA,,\'SA,A.bA.(Ti g).
A'EP mod C,
A'Ei': mod (C'+oMo)
Proof. We shall give a proof of b). The proof of a) is the same (in the
nontwisted case, a) is a special case of b».
Consider the column vectors
.
x(r, z, u) = (XA)AEP mod c" X(T, z, u) = (XhEP mod (C'+ioMo)I
.
where we let XA = XA(I)(e(o/'1(T)lo) (see (12.12.8», and consider the
matrix of branching functions:
B ( r ) - ( 6 A ( r. it »
- A ,., AEP: mod C,
AEP mod (ci+ioMo)
Then equation (12.12.8) can be written in a matrix form:
(13.10.2)
.
x(rt l , u) = B(r)x(r,z, u),
.
z E ij.
Apply the transformation S to both sides of (13.10.2)t using Theorem
13.8a) and (13.5.1):
S(t)x(r,z,u) = B(-)S(.)(r,z,u).
Plugging the expression for X given by (13.10.2) in this equation, we get:
..:.. 1 · ..:..
S(t)B(T) X (T, z, u) = B( -;)S(t) X (T, z, u).
Since the (unction {xAhEP mod (C'+ioMo) are linearly independent (see
Propositions 13.7a) and 13.3), we deduce: S(i)B(r) = B(-)S(i)' which t
due to (13.8.5) is equivalent to
1 -r
(13.10.3) B( -;) = S(A:)B(T)S(t),
proving b).
o

Ch.13
Affine Alge6ra., TAe'" Ftanclio"., anti Modular Form.
269
Remark 13.10. ConditioDs on g ::) 9 in Theorem 13.10b). simply mean
that 9 is a Lie algebra of a closed reductive subgroup of the semisimple Lie
group corresponding to g.
Since obviously
C(T + 1) = e21rimA.lc(T)i
b(r + 1; i) = e21fi(mA-mA)b(T; g),
Theorem 13.10 implies
COROLLARY 13.10. a) If g(A) is an aIline algebra and i E Z+, chen ehe
linear spaD of all the Btring (unctions ct with A E p is invariant under
the action ofro(r') uDder the (projective) action
(13.10.4)
(13.10.5)
(: :) .j(r)=(cr+d)l/2j( :;: ).
b) Under the assumption of Theorem l3.l0b), given k E Z+, the linear
span of all the branching (unctions b(g) with A E P: mod C6 is 5£2(Z)-
invariant under the action
(13.10.6)
( 0 6 ) . j(r) = j( or + 6 ).
c d cr + d
Proof. a) (or r/ = 1 and b) follow (rom Theorem 13.10. (13.10.4)t and
(13.4.1). a) for r' > 1 is proved by the same argument as that used in the
proof of Theorem 13.9.
o
SI3.11. In this section g(A) is an affine algebra of type X) and rank
t + 1. We give here the proof of the "very strange" formula. Given 0, fJ e
o
., we define the associated specialization 1°.lJ or A, as follows. Put:
f1 a . fJ = {'Y E f11 hlAo + (J) = 0 and hla) E Z}.
It is easy to see that dat is a finite reduced root system and that
A 0 0
f1:'" := {'Y E f1 afJ I f E f1+ U lf1+}
is a set of positive roots. RecaJJing the definition (13.6.1), we put
( ) Q.fJ
ja.fJ(r):= Ap II (1- e-'Y)-I (r).
-rE 6+'.

270
Affine Alge6nJ', TJaelG F.nction6, Grad Modular Form,
Ch. 13
(13.11.1)
l
LEMMA 13.11.1. Let a,p e E QQj. Then jQ,P(T) is a modular (orm of
i=l
weight l(t + I:'I) for fen), some n.
Proof. This is similar to that of PropOIitioD 13.6. using the formula
Q, _ r A,( r, -a + rfJ, l(fJla - rfJ»
I (r) - Q,IQ n 2i( -')'1(0 - 0') + r(fJ -Il'»'
p' - "YE A:
where a'. {J' are such that  a' ./J' = e.
o
We need one more (act.
LEMMA 13.11.2. Let 6 1 ,6 2 ,... be a periodic sequence or integers with
m
period m, such that 6j = bm-J (or j = 1,..., m - 1. Set b = E b j . For
J=:1
c e C put
(13.11.2)
00
le(r) = qe II(1- ).i.
j=l
Then feeT) is a modular form ({or r(n), some n) jf and only if:
b 1 m-l
C =  - - E j(m - j)6 j .
244m J=1
Proof. Recall that the functions gn.r(T) given by (13.6.9) are modular
forms. It is easy to see that if c is given by formula (13.11.2), then Ic(r) can
be represented as a finite product of functions of the form 9m.,.(T) (1 S
r  m - 1) and a power of'1(mT); hence, /c(r) is a modular form. Con-
verselYt if /c/(r) is a modular form, then qe-e' is also a modular form; it
follows that c = c'.
o
Now we are in a position to prove the u very strange" formula (12.3.7).
Proof 01 (12.9. 7). Note that /0.." is nothing other than the specialization
of type 8 of A, n (1 - e- 7 )-1. Hence setting e 2 . i , = q,;/r t we
,.e 4+
(.,JAo+"Y,)=o
obtain (using (12.3.1»:
(13.11.3)
/0,.." (r) = q t 1v Iii-A y ....1 2 II (l _ q{ y'j(,;r).
· > 1
J_

Ch .13
Affine Alge6rtJ, Tlaeta Function" and Modular Forms
271
But /0;.,. is a modular (orm by Lenuna 13.11.1. Using (13.11.3), we apply
Lemma 13.11.2 to complete the proof.
o
We bave the following important corollary of the "very strange" formula:
COROLLARY 13.11. a) For aD arbitrary alIine algebra o(type xj;> one
has:
(13.11.4)
'pJ2 dim g(XN,)
2h v = 240 0
b) fEr> 1 and 00 = 1, then
(13.11.5)
IpV 1 2 _ 11, 2 _ dim g(XN)
2h v 2h v 24r
Proof The classical "strange" formula (13.11.4) for r = J is a special case
of the "very strange" formula for , = 0, since 2 = Ip12. The formula
(13.11.5) for Ip v l 2 /2h v follows from this since p for A is the same as pV
for' A and h V (or XJ;J is independent of r. The remaining formulae are
checked, case after case, by making use of the "very strange" formula for
s = o.
o
The following theorem also may be deduced from the "very strange"
formula, but we shall give another proof, which is simpler. In order to
state the theorem, we need to introduce the following subset of the set
{Ot 1, . . . , l} :
J == {j J OJ (resp. oj) = 1 if r = 1 (resp. r > I)}
( ::: (Aut S(A» · 0).
'I'JIEOREt 13.11. Let A E P. Then
( 13.11.6)
2k(Alp) > h v (AlA),
and the equality holds jf and only jf A = kA j mod CD with j E J.
Proof. Inequality (13.11.6) is equivalent to
I p A 1 2 I P 1 2
h V - k  hV I

272
Affine Alge6ra" TIat(J F"Rction., tlnd Modular Form,
Ch. 13
which follows from
(13.11.7)
IL - :12 < 11-1 2 if: E Car.
h V - h V
To prove this inequality note that
o
o  (QI %) S 1 if: E Cal for all n e +.
Hence. in the case r = 1 we have, using Corollary 8.7:
o  E «01:)2  (01:» = hV(zl:) - 2( p lz),
.
oE6+
which is an equivaJent (orm of (13.11.7). We have an equality if and only
. -
if (olz) = 0 or 1 for all Q e 4+, which means % = A, with a, = 1, proving
the theorem in the case r = 1.
In the case 00 = 1 and r > 1 we have, using Corollary 8.7 (note that
;(Zt z ) = 2r E (QI%)2 + 2 E (QI:)2):
. .
oE6.. (lEA.,
o  r E «al:)2 - (al:» + E «al:)2 - r(alz»
.
oE4+.
= rhV(zlz) - 2r(plz),
.
oE4.f
which is (13.11.7). We have an equality if and only if (:.OV) = 0 or 1
o _
for all Q E d+, which means z = A, with G = 1 t proving the theorem
in this case as weJJ. In the remaining case, A), one checks directly that
(13.11.7) holds with equality only fOf Z = 0 (or uses a similaf argument
o
with n replaced by D\ {Ot}).
o
PROPOSITION 13.11. Let A e P':',  > 0, and let ,\ e peA). Then
m > IpI 2 i
At - - 2h v i + h v
with equality if and only if A = kAi mod C6 and Oi = 1 (resp. or = 1) in
the case r = 1 (resp. r > I), and A = w(A), w E W.
Proof. By definition (12.7.6) of mA,A. (13.11.8) means
(13.11.8)
IA + pl2 _ 'p12 ,,2
 k + h V  T'

Ch .13
Affine AI,e6rtJ, TAet. Fwflction., Gntl Modular Form.
273
By Proposition 11.4. it suffices to prove this for  = A, in which case
this inequality is equivalent to (13.11.6). The cases of equality £ollow from
Proposition 11.4 and Theorem 13.11.
o
Note tbat the inequality (13.11.8) in the nontwisted case is a special case
of Proposition 12.12b).
513.12. We shall now find Jevels and upper bounds of orders or poles
at all cusps (or the string and branthing functions. For this we need the
followins
LEMMA 13.12. a) Le& V be a space of modular forms of weight i which is
invariant with respect &0 the (projective) action (13.10.5) with t = -2k.
Suppose that all (unctions {rom V have poles of order < R at ioo. Then
orders of poles of these {unctions at all cusps are S R (i.e., all (unceions
from V are R-singular).
b) Let I(r) be a modular (orm of integral (resp. half-integral) weight (or
some principal congruence subgroup and some multiplier system. Suppose
that I(r + n) = j(r) for some n E Z, n > O. Then f(r) is a modular form
(or r(n) with a trivial multiplier system (resp. with a multiplier syseem
"ith values in :I: 1).
Proof. a) is clear since 5L 2 (Z) acts transitively on the set of cusps. For b)
see t e.g. t Schoenberg (1973].
THEO REM 13.12 . a) Let g( A) be an a/line algebra of type X) and rank
l + 1 and let A E P:, i > o. Let
, = Icm{ i, It v , i + h v , n}.
Then all string functiOlJ8 ct(T) are modular forrm of weight -it and a
multiplier system with valueIJ in (:!:1)' for the ,roup r(.). Furthermore,
all of them are R-singular, where
R = dim I(X,) It .
2400 i + h v
b) Under ihe assumptions and notation of Tbeorem 13.10, let
v "y · ·
· = tern{ Iti + hi ,ni, It, + hi (i  1), i o , no := n(M o ), fti (i  1),24}.

274
Affine Alge6t1J', TAe'. F.nclion., anti Mo4..ltJr Form,
Ch. 13
Then all branchiD8 functions 6t( Tj 8) are modular (orms 0( weight 0 with
a trivial multiplier sys&em (or &be &roup r(.). Furthermore. aU of them are
(c(t) - c(k»/24-sin,ular.
Proof. To prove a), we use identity (12.7.13). Due to Lemma 13.7. this
identity implie8 that the function F := E c( T)9A satisfies
AEP: mod ('M+CI)
the transformation property F(A.(r. z, u» = C(A)F(,., z, u), where C(A) E
C and IC(A)I = 1, for A e r(nh Y ) n r(n(i + h V ». Since the summation
in F is taken over a subset of p" mod (M + CcS) (see 13.7.1). applying
Proposition 13.3 and a.gaiD Lemma 13.7. we lee that ct are modular forms
of weight -it for the sroup r(nh V ) n r(n(i + h V » n r(n), and apply
Lemma 13.12a).
To complete the proof or a), note that mA.l  -R, by Corollary 13.11
and Proposition 13.11, hence the order of a pole at ioo for all c is < R.
If r = 1 or 00 = 2, we derive that all ct are R-8ingular, due to Corollary
13.10a) and Lemma 13.12b). In tbe rest of the Casel the proof is similar
using Theorem 13.10a). Corollary 13.11 b), and the fact that fo(2) and
fo(3) have only two cusps, iQO and o.
The same argument as above, using (12.12.8) in place 0£(12.7.13), proves
that b(T; g) are modular forms or weight 0 for r(.) with a trivial multiplier
system. All of them are (c(i) - c(k»/24-singular, since due to (12.8.12),
Proposition 12.12b) implies that
mA - mA  -(c(i) - C(k».
o
Note that Theorem 13.12 tells us that there exists an effective integer
6, depending only on 8, g, and , such that the first 6 coefficients of a
branching function determine the whole function.
513.13. In this section we study the asymptotic behavior of characters
as p := -if' ! 0, or an affine algebra of type X;> and rank t + 1. We shaH
write 1(/1)  ,(13) if lim/(fJ)/g(fJ) = 1.
JO
First of all, we have from the definitions:
(13.13.1)
e A ( _(ifJ)-l, %, -iJ1(zlz)/2)  1 or 0
depending on whether .\ E Ao + kM + C6 or not, where k:: le\'el(.\);
(13.13.2)
'1( -l/ifJ)-1  e ft '121J.

Ch.13
..
Affine Algt6rtJ.f, TAela F"ncl;on.f, and Modular Fonns
215
Now t 5- 1 (ijJ, - ipz, 0) = (-( ifJ)-l ,z, -i{J( zlz )/2), hence, applying (5, ".1/2)
to both sides of (13.5.1), we obtain:
(13.13.3)
e(iPt-ip%,O) = I M ./ k MI- 1 / 2 p-t/2
- .i -
x L e-r(A/ii)e,,(-(ip)-l.z. -iP(zlz)/2).
pEP.mod (iM+C6)
Hence, by (13.13.1) we have:
(13.13.4) e,.(ip, -ifJz. 0)  1M. /I:MI-1/2p-L/2.
Similarly, from (13.6.11) and (13.13.2) we deduce:
(13.13.5) '1(ifJ)-1 '" pt/2e,"/12/J.
o 0
Let R+ = A+ if r = 1 or 00 = 2, and R+ L\ otherwise. The
asymptotic! of the AA is given by the following
PROPOSITION 13.13. Let"\ E Pi++. Then
It. v dim .( X N )
(13.13.6) AA(il3, -ifJzJ 0)  b(A, z)fJ-l/2e- 1:l6.r ,
where
(13.13.7)
6(-\,z) = IM./kMI- 1 /'6(p,z) IT 2Sin1r (X k o) ,
aER+
( 13.13.8)
b(p, z) = 1M' /MI- 1 / 2 II 2sin ]I"(zlo).
oER+
Proof. We have from (13.13.3):
AA(i/J, -i{Jz, 0) = 1M. /I:MJ-I/2/J-1/2
x L E(W) L e- 2r '(UI(A)IPJ/te,,( _(iP)-I, %, -ifj(zlz)/2).
WEW PEP.mod (tM+C'>
p resul ar
o
We may take p to be regular (Le., such that W p :: 1), since nonregular JJ
give a zero contribution (cr. (13.7.7». Plugging in the explicit expression
(13.2.5) for eA, we get:
(13.13.9) A>. (iP, -i(Jz,O) = 1M. /kMI-l/2p-l/2eJtifJ(zlz)
x E E( w) E L e- ap(UI(i)Iji)- "'Y17 +2..k(')lz).
WEW pEP.mod (iM+CI) "YEM+ f
" regu lar

276
Affine Alge6ra, Theta Function" and Modular Fonn
Ch. 13
In order to complete the proof we need the following:
LEMMA 13.13. Let JJ E M be regular, and let ( = p ifr = 1 or ao = 2,
and { = pV otherwise. Then we have:
o
11J1 2  1  12 with equality if and only if IJ = t1(p),tT E W.
Proof. We consider the case when r = 1 or 00 = 2; then  = p . (In the
o
rest of the cases  = pV and the proof is similar.) In this case M = QV
I - 0 1-
(see (6.5.8», hence M. = E ZA i . Let tT E W be such that (T- (I') E P++.
i=1
t I.
Then u- 1 (JJ) = E he Ai = E 6iQrt where 6.t 6 i > 0, b i E Z. Hence we
.=1 i=l
have:
Ipl2 - Ipl2 = 1(1-1(p}1 2 - fpl2 = (1-1(p) _ pl(1-1(p) + p)
l
= L)b i - 1)( Ad (1-1(p) + p).
.=1
This completes the proof since tr- 1 (JJ) + P is a linear combination of the
0'( (i = 1,. . . t t) with positive coefficients.
o
End of the proof of PropoJition 13.13. By Lemma 13.13, it is clear from
(13.13.9) that the asymptotic behavior of A). (ijJ, -ipz, 0) as (J 1 0 is deter.
_ _ 0
mined by the terms with -, = tT(f.)/A:, Jl = u(), (T E W. Thus, we have as
P ! 0:
A  ( i 11. - iIJ z , 0)  1M. / k M ,- 1/2 (J - t /2 e - .1(1 1 / i IJ
x L !(w) L e- 2f1 (l1w(A)I()e- 2 I'i(I1(()I%)
. .
tuEW tlEW
= 1M. /kMJ-l/2{J-e/2e--tfI2/ill
X (  t(w)e- ap(W(()I,\») (  E(W)ehi(CJ(()I%»).
-EW tlEW
Hence, from the Weyl denominator identity and tbe "strange formulas"
(13.11.4) and (13.11.5) we obtain asymptotics (13.13.6). Formula (13.13.8)
is equivalent to identity (13.8.1) in the cases r = 1 or 00 = 2. In the
remaining cases it is equivalent to the identity (13.9.5).
o

Ch.13
Affine Algebra!, Thefa F.ncion', tJnd Modular Form,
271
An immediate corollary of Proposition 13.13 is the asymptotic8 or char-
acters. For A E P: one has:
(13.13.10)
XA(ip, -i/3Zt 0)  a(A)e"e(i)/J2",
where (cf. (13.8.10»
.
(13.13.11) a(A)=IM.J(k+h Y )M'I-1/2 n 2sinll' (:) t
aER+
 dim g(XN)
(13.13.12) c(k) = k + h Y ·
o
Rfmark 13.13. Let z E . and consider the "Hamiltonian" H = -d - z.
Then
(13.13.13)
chL(A)(ip,-i{jz,O) = trL(A)e- 2 11'PH,
hence (13.13.10) has the following interpretation:
(13.13.14)
trL(A) e- 2PH ,..., a{A)eW'c(i)/12rP as {3! O.
In analogy with statistical mechanics, {J = t, where T is the tempera-
ture. Thus (13.13.13) is the Upartition function" and (13.13.14) is its "high
temperature limit." It is remarkable that this limit is independent of z.
S13.14. We are now in a position to study the asymptotic behavior of
string and branching functions.
Consider the set Ri of vacuum pairs of level t (see 512.11). If (A;'\) E
R" then clearly (A + a6;;\ + 06 + k o1 ) E R, for any a E C and 7 E Mo.
Such pairs are called equivalent. Denote the set of all eq\livalence classes
--..
by Ri. This is a finite set. Here and further, in the case of branching
functions we keep the assumptions and notation of Theorem 13.10b); in
the case of string (unctions we have: io = k and Mo = M.
,..,
In general, the set RII is quite complicated, but in the case of string
functions the answer is very simple. Namely, recalling the definition of J
in iI3.11, due to Proposition 13.11 and (12.4.5) we have:
(13.14.1)
RI: = {(kA j i kAj) I j E J} in the case of string (unctions.
THEOREM 13.14. a) Under the assumptions of Theorem 13.10a) ODe has:
c{i,8)  IJII P/ MI- 1 IM. /MI 1 / 2 a(A)(p/,,=)l/2 e "C(i)/12r/J.

278
Affine Alge6ral, TlaettJ Function" Gnd Mod.lar Form,
ChI 13
b) Under the assumptions of Theorem 13.10b), let
d(A,.\) = E SA,A'S",>" multA,(.\'j g).
-
(A'.A')E R.
Then one of the two possibilities (i) or (ii). holds:
(i) d(A,.\) = 0 and 8"NiP; g)/e tr (e(J:)-c(Ir»/12 11 = 0,
(ii) d(A,) is a positive real number and
6Z( ipj g) ,.." dCA, .\)e tr (e(t)-c(k»/12 JJ .
Proof. We first prove b). By the definition or the branching function in
S12.12 and since it is a convergent Belies for 1m T > 0, we have:
( 13.14.2)
":(-j g)..... e- 2tr (m A ,-m>.,)/11 mult A ,(.\'; g)
if multA'('\'; g) =F 0, but multA,(A' + n6; 9) = 0 for n > o. Replacing T
by -t in (13.10.1), and using Proposition 12.12b), we see that the leading
term of b( r; g) 88 T -+ 0, is given by the 8ummands corresponding to the
-
pairs (A'; ') from R.. Due to (12.8.12) we thus obtain:
Yrn 6N iP; g)/ etr( e(t)-c(k»/1211 = d(A,.\).
Since the left-hand side is a positive real number as soon as fJ is a positive
real number, this completes the proof of b).
We could prove a) in a similar way by explicitly evaluating d(A, ).
We shall give, however, another. more illuminating proof. Note that if
 e P(A), then, by Proposition 12.5, there exists p E Z+ 8uch that A - 1'6 E
[W()], where [ ] stands for convex hull. Then A - (n + p)6 E [W('\ - neS)]
for all n E Z+, and since also  - n6 E [W(A - neS)] for all n E Z+, we
obtain by Proposition 12.5:
multJ\(A - n6)  multA( - n6)  multA(A - (n + p)6)
for all n e Z+. This implies for (J > 0:
(13.14.3)
c(iP) ..... c(iP) if.\ E peA).
Considering the theta function identity (12.7.13) asymptotically as P 1 0
we obtain, using (13.14.3), (13.13.4), and (13.13.10):
(13.14.4) a(A)e tre (t)/12r" ..... 6 A IM- /kMI-1/2p-l/2cZ(iP),

Ch.13
Affine Alge6ral, Theta Fanclion" and Modular Form$
279
where b A = #{ E Pf mod (kM + C6) I  = A mod Q}. The proof of a)
now follows from (13.14.4) and the following simple formula:
(13.14.5)
000
6 A = aolQv. /ItMI/IQv. /QI.
o
It is clear that
d(kAo. kAo) = L a(A')a(') multA'p'; g) > 0
-
(A' ,')ER.
. . ...., .
since (kAo. kAo) E Rt. On the other hand no example when L('\) C L(A),
but d(A.) = 0, is known. It is Datural to conjecture that no such example
exists. In fact, I would like to propose .an even stronger conjecture:
CONJECTURE 13.14. If (A'; I) E RJ: and multA(; g)  0, then
-
.
SA,h'S>".).' > o.
Let (A;"\) be a vacuum pair; then we have by (13.14.2):
b (- i ; g) ,... e;r(e(I:)-e(i»f 12 P muJtA{; g).
We deduce from this and Theorem 13.14 as fJ ! 0, the following result.
PROPOSITION 13.14. The vector (multA(; g)}(A;A)ER. is an eigenvector
""'T
with eigenvalue 1 of the matrix (SA,A,8A,A' )(A;A),(A';A')ER. .
o
Note that if Conjecture 13.14 is true, then, by the Frobenius-Perron
theory op08ition 13.14 determines uniquely the vector, since
multkAo(kAo; g) = 1.
\Vc conclude this chapter with a very special case of the developed theory,
\vhich is important to the st.ring theory. IJet g J g be as in 'rheoren113.10b).
One checks directly that c(k) - c(k) is a strictly increasing function of k
.
when kEN. Hence, due to {->roposition 12.12c), the equality c(k) = c(k) is
I>ossibJe onJy jf k = 1. I n this case g is calJed a conformal subalgebra of g.
-
It follo\\1s froln (13.13.10) that a £(O)-ulodule L(A), A E p, k > 0, viewed
-
as a .c(g)-nlodule, decomposes into a finit( sum:
(13.14.6)
L(A) = EB muJt"p; g)ip)
). E pia
+

280
Affine Alge6rtJl, TJaeC4 Func,ion" and Mod.lor Form,
Ch. 13
if and only if c(i) = c(k) and it is semisimple (then necessarily k = 1).
Looking asymptotically at characters 88 {J ! 0 in (13.14.6), we deduce from
(13.13.10):
(13.14.7)
a(A) = E a() muItA(; 8).
EP'
+
Thus, the (generally irrational) number o(A) plays a role or & dimension.
It is called the uymptotic dimenajon of L(A).
FjnaJJy, notice that, due to Proposition 11.12b), the coset Vir-module is
trivial if c(k) = c(k), hence
(13.14.8)
multA(; g)  0 => (A;) E R., if c() = c(k),
and we may use Proposition 13.14 to compute these multiplicities.
513.15. Exercise..
13.1. Show that 9(i)(T) = E erin'" (where n 2 = E nl) is a holomorphic
nEI'
modular form of weight it for r, and a multiplier system Vi such that
vi(T 2 ) = I, VieS) = e-rliI4. Using Gauss identity (Exercise 12.4) show
that 8(1)(T) = "a(',/+IN21 .
13.t. Let
I.(T) = q-I/48 n (1 + qi+1/2) = e-ri/24 ,,<l(T + 1» = ,,(T)2
jEZ+ fJ( r) '1( r/2)'1(2r)'
12(T) = q-l/48 n (1- qi+1/2) = "(T I 2) ,
iEI+ '1(T)
h(T) = 9 1/24 II (1 + qi+1) = ,,(2T) .
iEI+ ,,(T)
Show that these are modular forms of weight 0 for r(48) with a trivial
multiplier system. Check that ft(-l/T) = It(T), h(-l/T) = V2h(T),
and deduce that the linear span or /1, 12, and /3 is SL2(l)-invariant under
the action (13.10.6).
13.3. Let d 1 , d 2 ,... be an m-periodic sequence or integers. Put
(13.15.1)
n .=EdiP (  ) (k=lt2,.,.),
;1' 1

Ch. 13
Affine Alge6ral, TAeta F.nctionl, anti Modular Form,
281
where,.. is the CI8S8ical Mobius function (p( 1) = 1, p( n) = (-1)' if n is a
product of . distinct primes, pen) = 0 otherwise). Suppose that
(13.15.2)
(t, m) = (., m) implies d, = d,.
Show that then n. = 0 unless ilm. Prove a converse statement.
[If there exists a prime p such tbat  = p' it t P does not divide it, p' does
not divide m, then
n. = 1: :i:(d"/I - d,.-J/I) == 0.)
.Ii l
13.4. Let 8 be a simple finite-dimensional Lie algebra of rank i,  the
Killing form on g,  a Cartan 8ubalgebra. A the root system, A+ a subset of
positive roots, and p their half-sum, n = {al,... Ql} the set of simple roots,
I.
-00 = E a.ai the highest root, W the Weyl group. Let M be the lattice
i=l
spanned by long roots. Let. = (o, 81, . . . ,6t) be a sequence of nonnegative
t
integers; put m = .0 + E 0iai. Define. e . by +(A"cri) = ../2m
i=1
(i = 1,....1). Let A,+ = A+ nZ{cri 13i = 0 (i = O,...,l)}. let W, be
the subgroup of W generated by reflections in 0 E A.+, and let p, be the
half-sum of the elements from d,+. Put D,() = n (IQ)/(p,IQ) for
oE4.+
 E .. Let  j , Ij (j = 1,..., I) be the Chevalley generators of g. Define
the automorphism IT, of 9 by
u.(ej) = e2fti'jlm J u.(/j) == e-hi'jlm Jj (j = 1,.. ., I),
let 9 = ED Ij be the corresponding l/ml-gradation and put d j (8) = dim 9J.
j
Show that (cr. (12.3.6»:
(13.15.3) q",_1.1:a II (1 _ qi)dj
"> 1
1_
L l(W) L D.(w(p) + o)qm 1 tu(p)+a--1.u:a
tUEW.\W QE1M
and prove the "very strange" formula:
-
-
( 13.15.4)
1 1 m-l
lip - ,,,2 = 24 dim, - 4 2 L j(m - j)d j (6).
m j=l
Here and further IIAII2 stands for -t(, ,\).

282
Affine Alg6ra', The'. F."ctionl, Gnll Mod.'ar Form,
Ch. 13
13.5. In the notation of Exercises 13.3 and 13.4, show tllat
det./IJ(1 - e 4rip ) = dh vl .
where d is tbe determinant of the C81tan matrix of g and h V = 110'011- 2 .
19.6. In the notations or Exercise 13.4, &88ume that the sequence dJ =
dj mod m(') satisfies condition (13.15.2), i.e., that the automorphism 6, iI
quasirational (see Exercise 8.12). Define nt by (13.15.1). Deduce from
Exercises 13.3 and 13.4 that
II'1(kr)". = right-hand side of (13.15.3).
.1'"
Show that
mllp - .1I2 =  E nt.
'1m
19. 7. Let   be a quasirational automorphism of g and let ni be defined
by (13.15.1). Show that
det.(l - fa') = II (1 - qi )n"/i .
Jim
Deduce (rom that another type of "..runction identities, using Exercise 12.10
and the "strange" formula.
In Exercises 13.8-13.12, 9 is an affine algebra with a symmetric Cartan
matrix, 80 that g is of type At, D" or E,.
19.B. Let 9 = E& Ij( I ; 1) be the gradation corresponding to the aute>
JEI/"1
morphism of type (1 ; 1). let dj = dim gj( I ; 1) and let n, be defined by
(13.15.1). Show that "1 = i + 1; n" = -1; Ri = 0 if i does not divide h;
n. = -nj = %1 or 0 if ij = hand i 'F 1.
[Use Lemma 14.2d).]
13.9. Show that the application of the specialization I in Exercise12.9
gives the foJlowing identity:
E qllA'Y+plfl- I/;Jf' = (qA)t II tp(" )".n .
-yE Q JIA
13.10. Using (14.2.8), deduce from (0.10.1) that
F. (e( -Ao)L(Ao» = dim, L(Ao) = tp( q)t II tp(qi)-ni.
jJh

Ch.13
Affine Algebra" Tlaeta Function" Gnd Modular Form,
283
J 9.11. Apply the principal specialization to the formula given by Lemma
12.1 to deduce that .
F. (e( -Ao)L(Ao» = (  mult(Ao - j6)t/ )  qllh 7 +P1I'-IIP1I'.
..,EQ
19.1 t. Deduce from Exerciaes 13.8-13.11 the formula:
E multL(Ao)(A o - j6)t/ = V'(q)-t.
.>0
J_
Using the same method, give an alternative proof of (12.13.4).
19.13. Consider an affine algebra of type XJ;). Let
G( ) = 1rirdimg(X N ,) n<l - 2winr ) multnl
l' exp 12(0 + 1) > e ,
"_1
where a = h jf r = 1 and a = h v if r > 1. Check the following table:
g(A) G(.,. )
X(l) r A(2) '1( T )t
i 0 2l
A(2) '1( r )l-l '1(2r)
2l-1
D(2) '1( T )'1(2r )l-l
l+1
E2) '1( T )2'1( 2r)2
D(3) ,,( r)'1(3r)
4
J3.J. Let A E P+ be such that (A, d) = o. Then the eigenspace decom-
position of L(A) with respect to oold is of the form
(13.15.5) L(A) = $ L(A)_j,
jEI+
where L(A)_; is the eigenspace with the eigenvalue - j. Show that
dim L(A)_j < 00 and that (13.15.5) is the basic gradation of L(A), i.e.,
the gradation of type (1 to. . . . . 0). Provided that A is a symme tric affine
matrix, show that
q-lI24)dimL(Ao)_j)t/ = q(r)-l L qlhl'.
jO "EM
In particular, in the case 9 = Es, the right-hand side is (qj(T»!, wherej(T)
is the celebrated modular invariant, the generator of the field of modular
forms of weight 0 for 5£2(Z).

284
Affine Algebra" TAt'a Fvnction" Gnd Modular Form,
Ch. 13
13.15. Generalize Exercise 13.14 to the case or twisted affine algebras.
13.16. Show that for BP>, aU A E P+ mod C6 of level 1 are A o . AI, and
At. Show that. up to W-equivalence, all the string (unctions of level} are:
C Ao - C As C Aa _ - C As and C At
Aa - All AI Ao' A,.
Show that (cf. Exercise 13.2):
(13.15.6)
(13.15.7)
( 13.15.8)
c: = 'l(rr'-1'l(2r) = 'l(r)-l h(r),
c - c = 'l(r)-I-l'l<ir) = '1(r)-'I2(r),
c: + c: = '1<lr)-1'l(r)2-',,(2r)-1 = '1(r)-IIt(r).
[A(r) := 'l(r)t+l'1(2r)-lc: is holomorphic at cusps ioo and 0 of ro(2) and
hence is a constant. This implies (13.15.6). Formula (13.15.7) is deduced
from (13.15.6) by replacing r by -t and using Theorem 13.10a). Formula
(13.15.8) follows from (13.15.7) by replacing T by r + 1.]
19.17. Show that for tbe hyperbolic Kac-Moody algebra g(A), where A is
from Exercise 3.8, one haa:
muJt( kQ l + kQ2 + (3) = p(k).
[Use Exercise 11.7 and Exercise 13.12 for 9 of type Al).]
13.18. Let  e p+. Using Proposition 13.2, show that
A  (')
eAo = L.J A.,.A"t
liE p.+lmod C.
,. ++
where
cp(r) = E t(w) E qi.(HI>h+(k+l>-IUI(,.)-.-aI1t.
· -rE M
tuEW
13.19. Deduce from Exercise 13.18 that for the affine algebra of type XJ;>,
X = A, D, or E, one haa:
XA X Ao = L (cplt(r)/G(r») XI"
" EP.+'mod C,
++

Ch.13
Affine Alge6raa, Theta Functioh, Grad Modular Form.
285
where G(T) is defined in Exercise 13.13, 80 that we have the CoJlowing
formula (or the branching (unctions for the tensor product decomposition
of the g(X»-module L(A) 0 L(Ao):
6:. Ao (T) = V'(T)/G(T).
13.20. Let 8 be a simple finite-dimensional Lie algebra and let j C 8 :=
-
beg be the diagonal imbedding. Con8ider the£(g)-module L(A/)L(A"),
-
where L(A') and L(A") are integrabJe £(i)-modules of )evel k' and k " ,
respectively. Show that the coset Vir-module has conformal anomaJy
,,, .' ( i' 1;" Ie' + " )
c(k ,k ) == (dun g) i' + hV + k" + hV - i' + k" + h V I
-
where h v is the dual Coxeter number or .l(g), and that
L'ti = ! ( A'IA' + 2p) (A"IA" + 2p) _ {} )
o 2 i' + h v + k" + h v k'+ kIt + h v ,
......
where (1 is the Casimir operator for leg).
IS.tl. Deduce the following decomposition as
g' (Al») $ Vir-modules (j, k E Z+, j < k) :
L«c - j)Ao + jA 1 ) 0 L(Ao)
= E L«k + 1- .)Ao + 'Ad 0L(c(k), hl,'+1)
O<'<+l
' = jmod 2
where
(i) _ _ 6
c -1 (k+2)(k+3)'
hell) = «k + 3)r - (k + 2).)2 - 1
r,. 4(k+2)(k+3) ,
and L(ee'), hh is a positive energy Vir-module such that c = eCA:) I the low-
est eigenvalue of Lo is equal to h} and has multiplicity 1,
d t do-e(') /24 .(.) h
an rL(e(').Ia}) q = f/Jr,. J were
(i) _ 1 (
({Jr,. - '1 Ir(""3)-.(I:+2),('+2)(1:+3) - '''(1:+3)+.(1:+2).(1:+2)(1:+3» ,
13. ft. Deduce that the Vir-module L(c(j:),h!l> with i E Z+ and 1 , S
r  i + 1 are unitarizable, and that, letting
X} = tr L ( c(.) h(.) ,40-e(") /24,
. ".'

286
Affine Alge6ro'J TAetCl F.nc'iora._ Grad Modular Form,
Ch. 13
one hu for these modules the following coefficient-wise inequality of
.
f-eXpanSJOD.:
(13.15.9)
x}  }.
13.13. Show that (see Exercise 13.2):
11 = l + fP1, 12 = l- fPlt 13 = .
13.1./. Let 6 = 0 or 1/2 and let CI, denote the associative algebra on
generator. .pJ, j E Z + 6, and the following defining relatione:
1/J.1PJ + .pJ 1/J. = 6iJ.
Let  denote the irreducible CI,-moduJe which admits a nonzero vector
to) such that
,,;10) = 0 for j > O.
Let
Ln = -  L j'tPjtPn-j for n ::I: 0,
jE6+Z
Lo = ! ( ! - 6 ) + L j1/J-j1/Jj.
8 2 . 6 Z
JE + +
Show that [tPm, Ln] = (m + n/2)tPm+n and deduce that the map dn  L"
defines a representation or Vir in V, with c = i.
13.RS. Show that the elements
"'J' · · · tPj21Pj 110) with 0  jl > i2 > ... > i,
form a .basis oC 11,. Define an Hermitian Corm on V, by dedaring these
elements to be orthogonal and to have square length 1 (resp. 2), if jl < 0
(resp. i1 = 0). Show that ,pi and ,p-i are adjoint operators and deduce
tbat V, is a unitarizable Vir..modu]e. Show that
tr q L o-l/48 = 2/3.
V o
tr 9Lo-l/48 = 11-
V'/2
13.£6. Let Ir,+ (resp. V;-) denote the linear span of elements oC even (resp.
odd) degree. Taking (or granted that (or h -F 0, !I or Is, the Vir.module
L(!,h) is not unitarizable (see I(ac-Raina [1987] Cor a proof), show that
Vi  L<4,O), Vi  L(i, j), V o +  V o -  L(i.h) (cr. Exercise 13.22).
Deduce that xl = <p!1 for 1 < 8 < r < 2. (It follows from Feigin-Fuchs
(1984 AI H] that this is true for all Vir-modules from Exercise 13.22.)

Ch.13
Affine Algel,.., Theta Function., find Mod.lar Form
287
13.!7. Deduce from Theorem 13.10b) the following transformation formula
(1 $ II 5 a  k + 1):
cpi'l(-!) = (8/(i+2)(i+3»1
· T
x " ( _l ) (aH)(r+,) sin 1I"ar sin 1I"bs U') () ( r ) .
L-i k + 2 k + 3 T ',I
1<.<r<i+l
- - -
13.t8. Deduce the following decomposition of the g(El»-moduJe (viewed
as a g(El»EB Vir-module):
1 1 1 1 1
L(Ao)@L(Ao) = L(2Ao)@L( 2 ,O)+L(A7hL( 2 '2)+L(Ad@L(2' 16 ).
13.H9. Show that a«(T. A) = o(A) for any (f E AutS(A). Deduce that for
A of type X) with X = At D, or E one has (see 113.14):
a(A) == IJ/- t if A E P.
13.30. Let 9 be a simple simply-laced finite-dimensional Lie algebra of
rank l and let g be its 8ubalgebra generated by elements Ei' Fi' i E
""
{O.l,... ,t}\{j} (see S8.2). Show that a level! £(g)-module L(A) viewed
-
88 a l(g)-module decomposes into a direct sum of OJ irreducible summands
as follows:
L(A) = EI1 t().
EPnp(A)
13.31. Recall that we have a surjective map 1/1 : M- - AutS(X?». Show
that
2.i( Al o) S S
e ",M = A.eJt( a )(M).
Deduce that the matrix 8(1) in the simply-laced case is the character table
of the group M-/M. divided by 'JI 1 / 2 .
13.92. Show that for the diagonal embedding 9 c 9 := 9 $ 9 one has
-
Ri/.ill = {k'Aj,k"Aj;(k' + 1:")Aj , j E J},
and therefore Conjecture 13.14 holds in this case. Deduce, using Exercise
13.31, that
b ' ,A /I UP) _ / M. /Qla( A')a( A" )a( A)e 11'( C( 1:')- i(")- i( 1:' + 1:" »/12fJ .

288
Affine Alge6,.., The'. F.nc'ioft., anti Mod./ar Form.
Ch. 13
19.39. Show that for 0 < z < lone has:
II (1 - e- 2 ...,,-2d')(1 - e- 2 1'(n-l)P+bi) ..., _i e l'i'e- ft (,#-,+1/6)/".
nl
Deduce that for % E Int Cal and r = lone baa:
· v v 
A,(i/l,'-z,O)  p-l/2(_i)IA+le-." 1.-1/" I/.
Deduce that for At peP': one haa:
"" ( i {J - 11 +., 0 ) ..., S. e af.1 .\ _I:t.
A A ' i + h v ' A,"
13.3. Let 1 be a finite eet with a distinguished element 0 and let V be a
finite-dimensional vector apace with a buis {XA}, indexed by A e 1. Let S
be an invertible operator OD V 8uch that all the entries or the O-th row of
its matrix (SA,,,) are nonzero. Show that there exists unique operators ;
and t/J t indexed by ,\ E 1 J on V lIatisfying the following three properties:
(i) t/>'>. are diagonal in the basia {XA},
(ii) 4P(XO) = X.\,
(iii) fJ'J. = S'>. 5- 1 .
Show that (both for' and "): 4P>.4Pp = E" N Ap 4P" (the numbers NAp are
called fusion coefficients), where
(13.15.10) S>., Sp, = E N: p 5", .
SOf Sac SOt
"
Equivalently:
N il _  8>-,5", (5- 1 )",
A" - L...J I
So,
,
13.35. Let g be a simple finite-dimensional Lie algebra, let ij" be its Cartan
-  -
8ubaJgebra, Jet  C  be the root system, + a subset of positive roots, p
their half-sum, 8 the highest root, h V the dual Coxeter number, let QV be
the coroot lattice, M = II(Qv) for the normalized invariant form (.1.), let
-   -
p+ C  be the set oCdominant weights,let P+ = {.,\ E P+ 1 ("\IH V )  k}.
Given .,\ E 15 +, either ro("\ + p) = .\ + P mod (k + hV)M for some Q E X
and we let t("\) = 0, o! else there exists a unique W E Wand unique X E 
such that w(+p) == +p mod (i+hV)M and we let e('\) = e(w). Now let
in Exercise 13.34, 1 :=  and SA" := SA+Uo,,,+u o ' Show that the fusion
coefficients, as defined by (1.3.15.10), are given by the following formula:
N A " = E t(r) mult>.@p II,
1"EP.
f="
where mult>.@p II stands for the multiplicity of £(11) in I(.\)  L(IJ). Show
that NAp" := N is symmetric in all three indices.

Ch .13
Affine Aigebrd6, TAet" F."clio"8, "ntl Modular Forml
289
13.96. Sbow that for G = ,i 2 one haa (i,j" e l+ = P +):
N. = { I if Ii - j IS.  i + j, i + j + . e 2Z and  2'.
IJ 0 otherwise.
13.37. (Open problem). Show that for the Kac-Moody algebra from Ex-
ercise 3.8 one has
mult Q' S p(l - (ola),
where (010) is defined by (o;laj) = ta;j and pen) is the c)888ical partition
function (cf. Table H3 in S11.15).
13.38. (Open problem). If .\ E P+ n P(A)I., then 6Ng)  o.
13.39. (Open problem). If £(1\) is arl irreducible highest-weight module
over an affine algebra, then for z E Int( Caf) one has:
tr L(A) e 2 W'1J(4+')  DptIJ e. c /121J
for some positive real numbers D and G (called the asymptotic dimension
and growth of L(A), respectively) and a nonnegative real number w (called
the weight of L(A).
S13.16. Bibliographical Dotes and co mm ents.
The theory of theta functions is an extensive subject which has its of.igin in
the works of Jacobi and Riemann. The treatment of the part of the theory
presented in 5513.1-13.5 is fairly nonstandard; it is based on the ideas of
Kac-Peterson [1984 A). A presentation of other topics of theta function
theory may be found in Mumford [1983], [1984].
The exposition of 1513.7-13.14 mainly follows Kac-Peterson [1984 A]
and Kac-Wakimoto (1988 A]. Some of the results of these sections have
been previously known. Thus, Lemma 13.8b (excluding the .case A» is
due to Looijenga [1976], who used it to give a theta-function proof of the
Macdonald identities. An independent theta-function proof of the Mac-
donald identities was also found by Bernstein-Schvartzman [1978]. Van
Asch [1976] was the first to use properties of modular functions to prove
the Macdonald specialized identities.
Modular invariance of characters and branching functions, especially
the matrices S(i) (computed by Kat-Peterson [1984 AJ), have been play-
ing a fundamental role in recent developments of conformal field theory,
see Cardy [1986J, Gepner-Witten [1986], Capelli-Itzykson-Zuber [1987J,
Verlinde (1988), and many others. A special case of the "very strange"
formula, reproduced in Exercise 13.6, is proved in Kac {1978 AJ by the same

290
Affine Alge6ral, TlaetG Fwncl;onl, Gnd Modular Form,
Chi 13
method 88 (12.3.6). Important easel of tbe "very strange" formula were
found earlier by Macdonald (1972]. Exercises 13.8-13.12 are due to Kac
[1978 A]. For Al) Exercise 13.12 was previously obtained by Feingold-
Lepowsky (1978]. Exercise 13.16 is taken (rom Kac-Peterson (1984 AJ.
Exercise 13.17 is proved by Feingold-Frenkel (1983] by a more complicated
method.
Exercise 13.14 is taken from Kac [1980 B]. It is intimately related to
the work of Conway-Norton [1979]. who suggested that there must exist
a "natural" graded module over the Monster group, such that the corre-
sponding generating (unction is qj( r). In Kac (1980 E) such a module was
constructed for the centralizer of an involution of the Monster t by analogy
with the Frenkel-Kac [1980] construction. Frenkel-Lepowsky-Meurman
[1984] and {1989] (see also Borcherds (1986]) managed to extend a modifi-
cation of this construction to the whole MODster t recovering the construc-
tion of Griess (1982] of the action or the Monster on & 196883-dimensional
commutative nonU80ciative algebra. It would be rair to say that though
the relation of the representation theory of affine algebras to the theory
of modular forms is quite clear, the similar relation for the Monster group
remains mysterious.
An explicit expression for all string functions is known only for the sim-
plest affine Lie algebra, or type A 1), due to Kac-Peter80n [1980], [1984 A].
After a lengthy calculation. they turn out to be certain peculiar "indefinite"
modular (orms discovered by Heeke around 1925. Another expression for
these string functions was found by Distler-Qiu [1989] and Nemescbansky
(1989 A] by using the free field construction described by Exercise 9.18. The
open problem posed in Exercise 13.19 or previous editions haa been lO)ved
by Kac-Wakimoto [1988 A]. Exercise 13.18 is taken from Kac-Wakimoto
(1988 B]. Exercises 13.21 and 13.22 were found independently by Goddard-
Kent-Qlive [1986J and Kac-Wakimoto (1986]. The inequality (13.15.9)
is actually an equality, which follows from Feigin-Fuchs [1984 A, BJ. Due
to Friedan-Qiu-Shenker [1985] the Vir-module L(c, h) with c < 1 that
is not listed in Exercise 13.22 is not unitarizable. The unitarizable Vir-
modules L(c, h) with c > 1. are those with h  0 (Kac [1982 B)). Exer-
cises 13.24-13.26 are weB-known. Exercise 13.27 is taken from CappeUi-
Itzykson-Zuber [1987 A), and Exercise 13.28 from Kac-Wakimoto [1986].
Exercises 13.29-13.32 are taken from Kac-Wakimoto [1988 A] and Exer-
cise 13.33 from Kac-Wakimoto [1989 AJ. Exercise 13.34 is a formalization
of the work of Verlinde (1988].
As shown by Kac:-Wakimoto [1988 B], [1989] the class or modules
L() with modular invariance properties is much larger than the cl888 of

Ch .13
Affine Algebra!, The'a Function!, Gnd Modular Forms
291
integrable modules. They probably correspond to some non-unitary CFT,
and are related to the quantum gravity theory of Knizhnik-Polyakov-
Zamolodchikov {1988]. Highest weights A of all these modules lie in the "pos-
itive" half-space {,\ E. I Re('\,K) > _h V } (see the list itl Kac-Wakimoto
l1989 AJ), and on the critical hyperplane {A E . I (A, K) = -h V }.
Several authors studied the L(A) corresponding to  on the critical hy-
perplane: Wakimoto (1986), Hayashi (1988J, Ku {1988 B], Malikov {1989J,
Feigin-Frenkel [1989 A] (these modules are modular invariant if and only
if (A + p, a V)  N for all a E l1i-e. Finally, at least a conjectural con-
nection between the modules from the "negative" half-space of an affine
Kac-Moody algebra and the modules over the corresponding quantized fi-
nite type Kac-Moody algebra at q a root of unity was found by Lusztig
(1989 C).

Chapter 14. The Principal and Homogeneous
Vertex Operator Constructions
of the Basic Representation.
Boson-Fermion Correspondence.
Application to the Soliton Equations.
14.0. The highest-weight module L(Ao) over an affine algebra g(A) is
called the 641;C repre,entatioft of g(A). In this chapter we construct the
basic representation explicitly in terms of certain (infinite-order) differential
operators in infinitely many indeterminates, called the vertex operators. In
a similar fashion, we construct representations of affine aJgebras of infinite
rank. In the case of A = Aoo this construction is essentially the boson-
fermion correspondence in the 2-dimensional quantum field theory.
These realizations are applied to construct the so-called soliton solutions
of hierarchies of partial differential equations, the celebrated KdV and KP
equations among them.
514.1. Let SeA) be an affine algebra of type XJ;) and rank l + 1 (from
Table Aff r). Recall that g(A) = a'(A) + Cd, where g'(A), the derived
subalgebra, is generated by the Cheval ley generators ei, ,. (i = 0,. . . , t),
and d is the scaling element. Recall that the center of g'(A) coincides
with that of g(A) and is spanned by the canonical central element K (see
Chapter 6). Let 'J(A) = g'(A)/CK I so that we have the exact sequence
(14.1.1)
..
o -+ CK -+ g'(A) -+ }(A) -+ O.
Recall that relations deg ei = - deg Ii = 1 (i = O. . . . t t) define the principal
gradation g'(A) = ED gj( I); it induces the principal gradation I(A) =
jEI
EB Jj ( I ) so that dim J j ( I ) == dim 8j ( 1 ) for j  O.
jEI
The element
I
1= E.-(ed E J1(1)
1=0
is called the cyclic elemen' of J(A). Let i = {z e 11 [z,l] = O} be the
centralizer of e in SeA). It is dear that i is graded with respect to the
292

Ch. 14
Vertez Opena'or Con',",ciora6 Gntl Soliton Ellualion,
293
principal gradation of g(A):
· = ED'i'
jEZ
The subalgebra . = ..-1(1) is caJJed the princ;ptJl su6algebra of ,,/(A)
(or g(A». It is graded with respect to the principal gradation of g'(A):
· = €I).j.
iEI
Note that
(14.1.2)
(14.1.3)
dim'J = dim 1'j for j :F 0,
80 = CK.
The last relation is clear by Proposition 1.6.
The nonzero integers of the set which contains j with multiplicity dim .j
are called ezponenu of the affine algebra g(A). We will compute the expo-
nents beJow.
514.2. We study tbe principal subalgebra _ by making use of an explicit
construction of g(A), discussed in Chapter 8.
Let g be a simple finite-dimensional Lie algebra of type XN and let p
be a diagram automorphism of 9 of order r (= I, 2, or 3).
Let Ei' Fi' Hi (i = 0,... ,t) be the elements of 9 introduced in 58.2.
Using the results of 56.2, we see that
(14.2.1)
l
L oj Hi = O.
;=0
Here a are the labels of the diagram of the tranSpate of the affine matrix or
type xt> in Tables AII'. Recall that the elements E i (i = 0,... ,t) generate
the Lie algebra B.
Let 0i be the labels of the diagram of the affine matrix xt>. The integer
I
h("> = r L Oi is called the r-Ih Corder Rum6er of g. By Theorem 8.6 a),
i=O
the relations
deg E i=-degF i =l, degHi=O (i=O,...,t),

294
Ver1ez Opena'Dr Con,'nac'ion, Gratl Soli'ofl EtutJ'ionl
Ch, 14
define a Z/hCr>Z-gradation
( 14.2.2)
D = EB Dj ( . ; r),
J
called the r.,"nei,G' INdG'ion of 8.
The element
I.
E= EEi
1=0
is called the r.c,clic elemen' or g. Denote by s(r) the centralizer of E in
g. It is graded with respect to the r-principal gradation:
s(r) = EB SY).
JEI/1a(r)1
PROPOSITION 14.2. a} dim Di( . i r} = 1 + dimSY) (j E Z/h(r)z).
b) s(r) is a Cartan Bubalgebra ot 9.
c) The subspaces sir) and SY) are orthogonal (resp. non degenerately
paired) with respect to a nondegenerate invariant bilinear form on g jf
i + j  0 mod hCr) (resp. i + j == 0 mod her».
Proof. Using automorphisms of 9 of the form E i ....... AiEi, F i t-+ Ail Fat
the r-cycHc e)ement E is conjugate to a multipJe of an arbitrary element
I.
of the fOfm E Ci E. t where a]) CI  O. Therefore. it is sufficient to prove
4=0
the lemma for one of the eJementB or this form, say E ' . We take E' =
t.
E vafEi. Put F' = -waCE'}, where Wa is the compact involution of g;
1=0
l
then F' = E .;o:r F, t and we have
i=O
I
[E',F') = Ear H, = 0
4=0
by (14.2.1). Hence ad E' commutes with its adjoint operator ad F', with
respect to the (p08itivdefinite) Hermitian form (.f.)o (defined in 5 2 .7).
Therefore, E is a semisimple element, the ortbogonal complement B =
ED Bj to s(r) in 8 is ad E-invariant, and the restriction or ad E to B is
jEZ/Ia(r)1
invertible. Since [E, Bj] C Bj+l' we conclude that dim BJ = 1 (= dim Bo)
for all j e Z/h(r)z, proving a).

Ch. 14
Verlez Operator Con,',..c'ionl Gnd Soliton Equation!
295
From a) we deduce
(14.2.3) dims = dimS(r) + h(r)t.
But one easily verifies by inspection that
(14.2.4) dims - h(r)t = N.
Comparing (14.2.3 and 4) gives
(14.2.5) dimS(r) = N = rank g.
Since s(r) is the centralizer of the semisimple element E, (14.2.5) proves
b).
The first part of c) follows from Lemma 8.1; the second part now follows
from the fact that tbe restriction of a nondegenerate invariant bilinear form
to any Cartan subalgebra is nondegenerate.
o
The nondecreasing sequence of integers mr) < mr) < ... from the
interval (1, her) - 1}, in which j appears with multiplicity dim sy), is called
the set of r-exponents of g. They }lave the following properties.
LEMMA 14.2. a) The number o(r-exponents is equal to N = rank 9, and
we have:
1 = mr) < m)  · · · < m) = her) - 1.
b) m (r) + m (r) - h (,.)
j N-j+l - ,
c) lli and j have the same greatest common divisor with her), then i and
j have the same multiplicity among the exponents.
d) dim gj( I; r) = l+ (multiplicity of j among r-exponents).
e) Let r = 1, so that N = t. Denote by Cj the number of roots of 9 of
height j. Then
(14.2.6) dim Dj( I ; 1) = Cj + CIa(l)_j (or 1  j  h(1) - 1;
(14.2.7) Cj + cla( 1 )+1-j = l (or 1 5 j  h(l);
(14.2.8) dim Bj(2, 1,.. .,1; 1) = I for aJJ j.
f) Let 9 be of type A2t. Then
(14.2.9) dim 9j(2, 1,.. ., 1; 2) = l (or all j.
Proof Proposition 14.2 b) implies that the number of exponents equals N.
It is clear that 1 appears among the exponents with multiplicity 1. Now
a) and b) follow by Proposition 14.2 c). Part d) follows from Proposition
14.2 a) and the definitions.
The proof of the rest of the statements uses the following:

296
Ver1ez Operator Con,'nction, tJnd Soliton Equation,
Chi 14
SUBLEMMA 14.2. Automorph;sms (T o£type (1, 1,...,1; r) or (2, 1,...,1; 1)
are rational, i.e., tT' is conjugate to iT jf the order of tr and , are rela..
tively prime. In particular, they are quasiratjonal (see Exercise 8.12 for
the definition).
Proof. Let (1 be of type (1,. . . , 1; r). Then the order of t1 is h(") and the
dimension of its fixed point set is t. The element (I', where, and h(r) are
relatively prime, has the same properties. But Theorem 8.6 shows that
there is a unique. up to conjugation, automorphism with these properties.
The proof for (I of type (2,1, ... ,I; 1) is similar, using (12.4.3).
o
The statement c) of Lemma 14.2 follows from d) and the sublemma. In
order to prove e) consider the automorphism t1h(l) = 0'1,1 (resp. O'h{I)+l =
U(2,1,...,1;1) of 9 of order h(l) (resp. = h(l) + 1). We have for 1n = h(l) (resp.
= h(l) + 1):
(14.2.10) (1m(Ej) = (ex p 2i )Ei'
This implies (14.2.6) and
(14.2.11) dim gj(2, 1,...,1; 1) = Cj + CA(l)+I_j (or 1 :S j :S h(1).
If h(l) + 1 is a prime number, then, by the sublemma, all the eigenspa-
ces of (11&(1)+1 have the same dimension t; this together with (14.2.11)
proves (14.2.7 and 8) (or an exceptional Lie algebra g, since for 9 of type
G 2 , F4, ESt E7, Es, respectively, h(l) + 1 = 7,13, 13, 19,31 is a prime num-
ber. If 9 is of classical type At, Btt CI., or DI.' one cbecks (14.2.7) directly;
using (14.2.11) this gives (14.2.8) u well, proving e). Finally, f) is checked
directly.
( 211"i )
(1rra(Fj)= eXP-m Fj.
o
Using Lemma 14.2 one easily computes the r-exponents. For instance, if
j and h(") are relatively prime, then j is a r-exponent of multiplicity 1 by
Lemma 14.2 a) and c) (this takes care of all exceptional Lie algebras except
E 6 , r = I, in which case one should check that 2 is not an exponent). A
complete list of r-exponenta is given in Table Eo-
Rtmtlrk /-1.1. The numbers ml) (j = 1,.. .1) are the ordinary exponents,
and h(l) is the ordinary Coxeter number or a simple Lie algebra 9 of type
X,. Indeed, comparing Lemma 14.2 d) with (14.2.6) gives
Cj + CA(I)-J = i + (multiplicity of j among exponents),

Ch. 14
Verlez Operator COft6'ructioR6 Gnd Soliton EqutJtion&
291
which is one or the definitions of the exponents and the Coxeter number
of g. Note also that aU r-exponents have simple multiplicity with one
exception: 9 = Dl, where t is even, and r = 1; in such case t - 1 has
multiplicity 2.
TABLE Eo
9
r h(r)
(r) (r) (r)
m 1 ,m 2 ,...,mN
Al 1
Bt 1
Ci 1
Dt 1
E& 1
E7 1
E8 1
F4 1
G 2 1
Au 2
A2l-1 2
Dt+l 2
D4 3
Ee 2
t+l
2t
2l
2/- 2
12
18
30
12
6
4l+2
4/- 2
2/+2
12
18
1,2,3,...,t
1,3,5,...,21- 1
1,3,5,...,21- 1
l,a,5,...,2l-3,1- 1
1,4,5,7,8,11
1,5,7,9,11,13,17
1,7,11,13,17,19,23,29
1,5,7,11
1,5
1 , 3, 5, . . . t 2i - 1, 2l + 3, . . . , 4l + 1
1, 3, 5, . . . . 4t - 3
1 t 3, 5, . . . , 2t + 1
1.5,7,11
1,5,7,11,13,17
514.3. Now we return to the affine algebra g'(A) of type X). By
Theorem 8.7, we have
(14.3.1) g(A)  €a(t i  gj mod A(t')( I i r»,
jEI
where the isomorphism is defined by
(14.3.2) "'( ei) ..... t  Ei, ",(Ii) ..... t- 1  Fi (i = 0, . .. It).
Moreover, (14.3.1) is the principal gradation of J(A). Hence
(14.3.3) l = t 0 E is the cyclic element or )(A);
(14.3.4) Ij = t i  sJr)mod Aft') (j e l).
Finally, notice that
hCr) = rh,
where h is the Coxeter number of g/(A).
The (ollowing proposition now follow8 Crom Proposition 14.2.

298
Ver1e Operalor Con,'ruc'ionl anti Soli'on Equation,
Ch, 14
PROPOSITION 14.3. a) dim gj( I) = t + dim'J (or j e Z\{O}.
b) i is a commutative ,ubalgebra otJ(A).
c) The set o( exponent, of g'(A) i,
{mr) + jh(r), where, = 1,..., N;j e Z}.
o
From Proposition 14.3 a) and Proposition 10.8 we deduce:
COROLLARY 14.3. Dual affine algebras have the same exponents.
o
Using Table Eo one immediately gets the list of exponents of all affine
algebras as the set of all integers satisfying the conditions given by the
following table:
TABLE E
A(1)
t.
B(I)
I.
d 1 )
l
DCI)
E!l)
1)
El)
FCl)
4
1)
A(2)
21.
A(2)
21.-1
D(2)
t+l
D(3)
4
E1 2 )
i _ 0 mod (l + 1)
i == 1 mod 2
i == 1 mod 2
i == 1 mod 2, i == t - 1 mod (2i - 2)
i == :I: 1, :J:4,:l:5 mod 12
i == :1:1, :1:5,:1:7 t 9 mod 18
i == ::i:l,:i::7,:!:11,:J:13 mod 30
i == :i: 1 mod 6
i == :i: 1 mod 6
i == 1 mod 2 such that i _ 0 mod (2l + 1)
i == 1 mod 2
i == 1 mod 2
i E :i: 1 mod 6
i == :i: 1 mod 6
Note that the multiplicities of exponents are 1, except for the case A =
Dl), I even, when the multiplicity of (1- 1)(2. + 1)(. E Z) is 2.
FinallYt recall that by Theorem 8.7, g/(A) can be constructed as the
central extension of J(A) by CK:
(14.3.5)
I'(A) = EB(t J @ Ij mod h(p)( I i r» (9 CK.
iEI

Ch. 14
Verlez Operator Contruc'iora' and Soliton E9utJ'ion
299
with the bracket
( 14.3.6)
1 ( dPl(t) )
[PI (t)0g1 I P2(t)0g2) = P 1 (t)P 2 (t)[gl , 92]EDj; Res dt P 2 (t) (gdg2)K,
where (.1.) is the normalized invariant bilinear form on g and h is the
Coxeter number of g(A).
514.4. Recalling tbe definition of the infinite-dimensional Heisenberg
algebra (59.13), note the following simple lemma:
LEMMA 14.4. The princjpal Bubalgebra _ of g'(A) is jsomorphic to the
inlinite Heisenberg algebra.
Proof Proposition 14.3 b) implies [."'j] C CK for all i,j E Z. By Propo-
8ition 14.2 c) and formula (14.3.6), we have [.;,'j] = 0 for i 'I -j and
(a, .1_.] = CK for every a e 'f, a  0, proving the proposition.
o
Now we are in a position to make the first step in the "principal con-
struction tt of the hasic representation.
PROPOSITION 14.4. Let g'(A) be an affine algebra, where A is of type
xt), with X = A, D, or E, and let. be the principal subalgebra of g'(A).
Then the basic g'(A)-module L(A o ), considered as an .-module, remains
irreducible.
Proof Let L(Ao) = EB L(Ao)j be the principal gradation of L(Ao) and let
.>0
J_
dim, L(Ao) = E dim L(Ao)jqJ be the q-dimension of L(Ao) (see 510.10).
'>0
J_
Then by Proposition 10.10 we have
(14.4.1)
dim. L(Ao) = II (1 - vi )dim' 'j(2.1.",l)-dim ,/( I ).
.> 1
J_
But the algebras g(' A) under consideration are precisely the algebras from
Table Aff 1 or A). Hence, thanks to (14.2.8 and 9), we have:
( 14.4.2)
dim' gj (2, 1, . . . , 1) = l for all j #- o.
Furthermore, by Proposition 14.2 and formula (14.3.4), we have
(14.4.3)
dimgj(l) =l+dimaj.

300
Verlez OpenJ'or COR,truc'ion. Gnd Soliton EquGtion,
Ch, 14
Substituting (14.4.2 and 3) into (14.4.1), we obtain
(14.4.4) dim q L(Ao) = IT (1 - qi)- dim'j.
'>1
J_
On the other hand, the Z-gradation , = ED IJ induces a Z-gradation
iEf
U(.) = ED U(')J; set UJ = U(')-J(vo), where Va e L(Ao)Ao is a nonzero
jEZ
vector. Then
(14.4.5) UJ C L(Ao)J.
But by Lemma 9.13 a), the .-module U = ED Uj is isomorphic to the
.>0
J_
canonical commutation reJations representation; hence we have
(14.4.6) L(dimUj)qi = II(I-f/)-dim' j .
JO jl
Comparing (14.4.4) with (14.4.6) gives equality in (14.4.5) for all j. Hence
the .-module L(Ao) = U is irreducible.
o
S14.5. Proposition 14.4 allows us to identify the restriction of the basic
representation to the principal 8ubalgebra with the canonical commutation
relations representation. In order to extend this to the whole Lie algebra
g(A) we need a lemma about differential operators.
Let R = C{Zl, %2, · · .] be the algebra of polynomials in infinitely many

indeterminates Zi and let R = C[[Zl, %2, . . . J] be the algebra of formal power
series (i.e., algebra of all linear combinations of (finite) monomials in the
Zi or, in other words, the Cormal completion or R (see 51.5). A differential
operator on R is a (generally infinite) sum or the form
L L f},...i r 8 8, ... 8 8 . I where Pi,...i r E fl.
>0 1 < . < < . %'1 Z' r
r _ _'1_ '.._1,
""
A differentia) operator defines a linear map D : R -+ R (in fact, every such
linear map can be realized uniquely as a differential operator). A particular
case is the multiplication by PER. Another example is the operator T).
defined by
(T/)(Zl. %2,' · .) = 1(%1 + 1 J %2 + 2, . ..).
Note that
(14.5.1)
IJ
T.\ = exp E Ai 8
i 1 %i
by Taylor's formula.

Ch. 14
Verlez Operator COftlt",ctionl dnd Soliton Equation,. 301
.,.,.
LEMMA 14.5. Let D : R .... R be a linear map.
a) 1£ (z"D] = >.,D, i = 1,2,.." then D = D(I)exp(- >'ik).
.
b) If l 8:i ' DJ = /liD, ; = 1,2,..., then D(l) = cexp LJJ.z., c e C.
I
c) 1£ [Zit D) = >'iD and [ ":i ' D) = PaD, i = 1,2".., then D =
cexP(JA'Zj) eXP(->,j 8:. ) (orsomecEC,
Proof. a) We replace D by DT,,; then our statement is equivalent to the
following: if [Zi, D] = 0, i = 1,2,..., then D = D(l). But D(/) = D(l)/
for all / E R by an easy induction on the total degree of I.
b) We rep!ace ? by exp- LJAji)Dj then our statement is equivalen,t o
the followmg: If [/;;, DJ = 0, 1 = 1,2,.", then D(I) = const. ThIS 18
obvious: [ a:. ,D] = 0 implies ,,:. (D(I» = 0, i = 1,2,.." and hence
D( 1) = const.
Part c) follows from a) and b).
o
The operators o( the (orm (ex p  >'jZj ) (ex p -  Pj :j ) are caUed
vertex operntors. These operators were discovered by physicists in the string
theory.
S 14.6. In this section we shall construct. in terms of differential oper-
ators in infinitely many indeterminates, the basic representation L(Ao) of
an affine Lie algebra g/(A) with the Cartan matrix A o( type XJ;), where
either r = 1 and A is symmetric or r > 1. This construction is calJed the
principal reG/ization of the basic representation.
Let 9 be a simple finite-dimensional Lie algebra of type XN, let 9 =
e gj( I ; r) be its r-principal gradation. E E gr( 1, r) the r--cyclic
iEZ/A(r)1
element, s(r) = $.sjr) the centralizer of E in 9 (see S14.2). According
J
t(J Proposition 14.2 b), s(r) is a Cart an subalgebra of 9 (graded by the
r-principal gradation). Let l:1 c s(r).. be the set of roots of 9 with respect
to s(r). For {3 E  pick a nonzero root vector Ap E 9 and decompose it
\vitI} respect to the principal gradation:
A" = EA",i (summation over j E Z/h(r)z).
j

302
Vertez Opera'or Con,'ruc'ionl Gntl Soli'on Egu4,ion,
Ch. 14
Since s(r) n 9 0 (1; r) = 0, the subspace go(l; r) ( = t C H j ) is the
J==1
linear span of the Ao,o, (J E . Therefore, one can choose t. root vectors
ApI' · · · , Atlt with respect to s(r), corresponding to some roots fJl t . · . t {3t E
(s(r»*, such that their projections on 90(1; T) form a basis of this space.
Since ad E is a nondegenerate operator on g/ s<r) t which shifts the in-
duced gradation by It we obtain that the images of the elements AtJttj in
of s(r) are linearly independent. For all {J and j we have
[E, A..,J] = (Il, E)AJ+T with (P, E) -I: O.
Let 1j (j = 1,..., N) be a buis of .sir) such that Tj E s(rlr) and
m.
J
(11ITN + l-J) = h6iJ for all i, j = 1,.. . , N (see Proposition 14.2 c)). We set
(14.6.1)
).  j = (fJ, 1j ) .
Then t clearly. we have
(14.6.2)
[T i , Ap,;] = ;\PiApIHmr)
(here j + mr>ia viewed u an element of l/h(r>z).
Note that by (14.2.4), all the elements Ai.j and T. form a basis of g,
since they are linearly independent. Note also that (ApjIT) = O.
We use the c'principal" realization of the Lie algebra g'(A) defined by
(14.3.5 and 6). The map
(14.6.3)
e. ..... t 0 Elf Ii  ,-1 0 ii, or...... Hi + (ai/aj h)K
provides an isomorphism with this realization.
Let E+ = {hi, b 2 , .. .} be the sequence of positive exponents of g(A)
arranged in nondecreasing order. We introduce the following basis of the
principal subalgebra .:
K . 10. 'T' 6 _ 1 , _. rr
, Pi = , i '0' .l i' , q, = i · 0 .j N + 1-.' .
where i = 1, 2, · · .. Here and further on. i' is defined to be the element of
{l,...,N} congruent to i mod N. By (14.3.6) we have
(14.6.4)
[Pi t Vj J = 6 ij K for All ;, j = 1. 2. . . . .
The degrees of these elements in the principal gradation are
(14.6.5)
deg Pi = b i = - deg q, .

Ch. 14
Vertez Opera'or Con.truc'ion, tJntl Soli'on Equation,
303
Assume now that g(A) is an affine algebra of one of the types listed at
the beginning of the section and let L(Ao) be its basic module. Since L(Ao)
has level 1, the element K is represented by the identity operator. Due to
Proposition 14.4 and Corollary 9.13 we can identify L(Ao) with the space
R = C(XI. X2, 00 0] so that K operates as an identity, Pi as ai ' and qi as
multiplication by Xi (i = 1,2).. .). From (14.6.5) we see that the relations
( 14.6.6)
deg Zi = hi (i = 1, 2, . . . ),
(together with deg PQ = deg P + deg Q) define the principal gradation of
L(Ao).
In order to extend tbe realization of the basic representation from I to
the whole Lie algebra ,,'(A), we extend, in an obvious way, the identification
A  A
of L(Ao) with R to that of L(Ao) with R, where L(Ao) denotes the formal
completion of L(Ao) by its principal gradation (i.e., t(A o ) = n L(Ao)j).
j
Let n ga be the formal completion of g(A). This space is not a Lie algebra;
Q
however, the adjoint action of g( A) can be extended to it in an obvious
way. We introduce the following elements of this completion, depending on
a parameter z E CX:
X t3 (z) = L z-; (t j @ A t3 ,j) , /3 E !1.
jEZ
Then X(z) maps L(Ao) into i(Ao}.
A.
LEMMA 14.6. The operator X"(z) : R  R acts &9 the following vertex
operator:
(14.6.7)
rtJ(z) == (Ao, AJ'.o) ( ex p t AIlj'zl;Zj )
J==1
(  )
-1 _6, lJ
X exp - L AII.N+I-j l b j % J  .
i=1 v
Proof. Using (14.6.2) and (14.3.6)>> we write:
(p" XII(z)] = [tl. 0 T,I, L z-j (t i 0 A IIJ )]
jEI
= L z-it iH . 0 (T,I, AII,j]
jEI
= AIJ,,%6. L z-j-l.tiH. 0 ApjH.
JEI
= p"Z.. X(Z).

304
Verltz Operalor Con,truction, Grad So/ilon Equation,
Ch. 14
Similarly.
[q..X(z)] = ,N+l_,/6;1 z-&, X"{z).
Now the lemma follows from Lemm 14.5.
o
We expand the vertex operator r(z) defined by (14.6.7) in powers of z:
r"(z) = E r"JzJ.
Jez
Th..en r IJJ are infinite-order differential operators, which map R = L(Ao)
into itself. Since all the element. A#.J and T, form a basis of g, we can
reformulate Lemma 14.6 as follows.
THEOREM 14.6. Let g'(A) be an affine algebra such that either A is sym-
metric «(rom Table Air 1) or A is (rom Tables Air 2 or 3. Set R =
C [Zt,Z2,.. .]. Then the identity operator, the operators Zit 8:' (j =
J
1,2,...), and riJ (i = l,...,l;j e Z) form a basis of a Lie subalge-
bra of the algebra of differential operators preserving R. This Bubalgebra
is isomorphic to g'(A), and the representation of it on R is equivalent to
the basic representation of g'(A).
The realization of the basic representation given by Theorem 11.6 is
called the pnnciptJlveriez operator con""'ction.
SI4.7. We write down the explicit formulas for the principal vertex
operator construction of the affine algebra g'(A) of type All (n  2).
In this case. 9 = sln(C). and (zly) = trzJI. Let E.j (i,; = 1,...,n)
denote the n x n matrix which is 1 in the i,j-entry and 0 everywhere else.
We take:
Eo = En!,
Ei=Ei,i+! (i=I,....n-l),
Fi = Ei + t ,i (i = 1. . . . , n - 1).
H. = Eii - Ei+l,i+l (i = 1,..., n -1).
Fo = E ln ,
Ho = Enn - Ell,
The I-principal Z/nZ-gradation of 9 is given by setting deg Eij = j - i for
i '# j and deg D = 0 for a traceless diagonal matrix D. We set
o 1 0 ... 0
o 0 1 ... 0
n-1
E = E Ei :::
1=0
. . . . . . . . . . . . . . . .
o 0 0 ... 1
1 0 0 ... 0

Ch. 14
Venez OpertJlor COft,'",ction, Gntl Soliton Equa'ion,
305
and let S be the centralizer or E in ,. Then S is a Cartan 8ubalgebra of
g, with basis
7J = EJ (j = 1 t . . . , n - 1).
We have deg 7J = j mod n, and
(7iITn-j) = n6iJ for i,; = 1,..., n - 1.
For two distinct n-th roots or unity, £ and '1, define n x n matrices
A ( · -j ) n
(I,.,) = £" iJ=l.
Then A(t.,,) E " are root vectors with respect to S:
[7i, A(t,,,)] = (£' - 'Ii) ACI,,,) Cor i = 1,. .. ,n - 1.
Let A(t,,,),J be the homogeneou8 components oCthe ACt,,,). The Ti together
with the homogeneous components of the ACt,l) (orm a basis of g. In the
notation of (14.6.1), we have
(t .,,).J = ei - rI for j = 1,.. . , n - 1.
We use the principal realization of g'(A):
8 / (A) = EB (t i 8j mod n( I ; 1» E9 CK,
iEI
with the bracket given by
(a(t) E9 >'K, bet) E9 pK] = a(t)6(t) - 6(t)a(t) E9 ; (Res tr dt) b(t») K.
We also have h = nand E+ = {j E Z+ Ij __ 0 mod n}. For m = 1,2,...;
m _ 0 mod n, we set
e m"., -l,-m'J"
Pm = .lm', 9m = m I "'n-m'.
We can use this simpler indexing than in tbe general case above, because
no exponent baa multiplicity greater than 1. We have
[P',lm) = 6'm K
for all tt m > 0 not divisible by n.

306
Ver1ez Opena'or COft,'",ction, Gnti Soli'on Equation,
Ch, 14
We identify the space of the basic representation with the space of poly.
nomials
R = C[zm;m e Z+,m -. Omod n].
Then 1 is a highest-weight vector, and putting degxm = m defines the
principal gradation of R. For i = 1 t . . . , n - 1, we set
X(I,IJ)(Z) = L ,-i,i A(I,IJ)J"
lEI
Then K, Pm t qm (m E Z+,..., m  0 mod n) and the homogeneous com-
ponents t j A(.1),j of X(l:,l)(z) form a basis of g'(A). By Lemma 14.6 and
Theorem 14.6, the basic representation (7 of g'(A) on R is given by:
t1(K) = 1.
lJ
u(Pm) = 8 I U(qm)=zm (mEZ+lmOmodn),
Zm
D'(t J ACI,fJ).j) = the coefficient of z-J in
(1 00 ) ( 00 -m 8 )
£ exp E Zm (Em - "m)Zm exp E Z (,,-m - £-m) 8 .
£ - " m m
=1 ",=1
Indeed, to compute (Ao, A(I,fJ),o) in (14.6.7), we observe that
cr = (Enn-El1)+kK. Qr = (Eii- E .+l.i+l)+k K (i = l,....n-I).
n-I 
Hence A(I,IJ),O = .E «£/,,) + (£/,,)2 + ... + (£/,,'1)Q'1 - K. Since
J=1
(A o , o) = 6 0 ,1 and (Ao, K) = 1, we obtain (A o , A(I.,,).o) = .
514.8. We turn now to the homogeneous vertex operator construction.
This construction is based on considering the homogeneous Heisenberg sub-
algebra t, see 58.4 (in place of the homogeneou8 Heisenberg 8ubalgebra ,
in the principal vertex operator construction). It is not true that L(Ao)
is irreducible 88 a t-module. One can still prove an analogue of Proposi-
tion 14.4 (see Exercise 14.6) and proceed along the lines of the proof in
the principal case. This way allow8 one to "discover" the construction. A
shorter way to prove that the construction "works" is to apply eorne ver-
tex operator calculus (which is more familiar to physicists). Here we shall
use the latter method, referring the reader to Exercises 14.5-14.7 for the
former. For the sake of simplicity, we shall not consider the twisted cue.

Ch. 14
Vertez Operator Con5'",c'ion, Inti Soliton Equa'ion,
307
Let 9 =  ED ( EB CEo) be a simple finite-dimensional Lie algebra of type
oEA
Ai , Dt, or Ell with commutation relations defined by (7.8.5). Let (.1.) be
the normalized invariant form on 9 (defined by (7.8.6». Let
£(g) = Crt, ,-1) @c 9 + CK + Cd
be the associated affine algebra of type AI), DI), and Ell), respectively.
Consider the complex commutative associative algebra
( 14 .8.1 )
v = S«(B(t J @ » @c: C[Q],
;<0
where S stands (or the symmetric algebra and C[Q) stands Cor the group
algebra of the root lattice Q C  of g(X t ). Let Q ....... eO denote the inclusion
Q c C(Q]. As before, "en) will stand for t n  u (n E I, u e g). For n > 0,
u E . denote by u( -n) the operator on V of multiplication by u(-n). For
n  OJ u E , denote by u(n) the derivation of the algebra V defined by
the formula
(14.8.2) u(n)( v(-rta) @ eel!) = ncSn,-rn(ulv) @ eO + cSn,o(alu)v(-rta) @ eel!.
Choosing dual bases Ui and u' or , define the operator Do on V by the
formula
( 14.8.3)
Do = t(Ui(O)U'(O) + E Ui(-n)U'(n»).
i:l n > l
Furthermore, for Q e Q, define the sign operator co:
(14.8.4)
c o (/0 e 6 ) = (QJ 13)/ @ e,".
Finally, for Q E A C Q introduce the tJenez operator
(14.8.5)
( zi ) ( Z-i )
fel!(z) = exp E -;-a(-j) exp- E -;-a(j) eel!zel!(O)c o .
jl J i1 J
Here z is viewed as an indeterminate. Expanding in powers of z:
r eI!(z) = E r)z-J-l J
jEI
we obtain a sequence of operators r) on V. Now we can state the result.

308
VeMez Oprator Con,t",ctiofl' tJnd Solil01l Equation,
Ch, 14
THEOREM 14.8. The map IT : £(g) -+ End V given by
K ..... 1,
u(n) ..... u( n),
""n) ..... r(n)
,c,o Q ,
d ...... - Do
for u e , n E Z t
for 1 e A, neZ,
,..
defines the basic representation otthe affine algebra l(g) on V.
Proof. First, we check that this map is a representation. Since Eo(z) H
r a(z), according to (7.7.3) and (7.7.4) we need to check the following rela-
tions (all other relations trivially hold):
(14,8.6)
(14.8.7)
( 14.8.8)
[Do, r 0(:)] = (zt + l)r o(z),
[r 0(%1), r -0(%2)] = -0(zl)6(Z1 - %2) + 61 (%1 - %2),
[r 0(11). r(Z2)] = £(a,p)r 0+(%1)6(Zl - %2) if a + {J E A.
Relation (14.8.6) follows from relations
( 14.8.9)
[Do, u(n)] = -nu(n),
1
[ Do , e "f) = - 2 (-y h )e "f ,
which are obvious. In order to check (14.8.7 and 8), we let for "y e :
r(z) = exp L 1(%!) z=fJ I
Jl =FJ
r( z) = e"f z "f(0) C"f'
Then we have:
( z ) (QIP)
(14.8.10) r;;(Zl)r:(Z2) = r:(Z2)r;;(Zl) 1 - ;; I
(14.8.11) r(Zl)r(Z2) = eO+ P zr(O) z:<O) z1o IP )coCpE(Q, t1).
Here and further on, by (1 - %2/Z1)"', m E Zt we meaD its power series
expansion in %2/'1' The second of these formulas is trivial and the first
one, viewed as aD identity of formal power aeries in zt 1 and zt 1 t follows
from the following two facts:
( 14.8.12)
(14.8.13)
eAe B = eBeAe[A.B] for two operators A and B
such that [A, B] commutes with ..4 and B;
exp ( - L: zi Ii) = 1 - z.
Jl

Ch. 14
Vertez Opera' or Con,truc'ion, and Soliton Equation,
309
The folJowing formula is immediate by (14.8.10 and 11):
(14.8.14)
( Z ) (ol) ( I)
r o(ll)r(12) = 1 - ;; zt £(o,{J)
x ( ex p L (Q(-j)z{ +,8(-j)» ) ( exp- L: (Q(j)zij +P(j)Z;i» )
°>1 J j>1 J
J_ _
x eO+z(O)z(O)coc.
Using (14.8.14) we obtain (or a E A:
(14.8.15) [r Q(%I), r -0(%2)] = 61 (%1 - %2)
x ( ex p L: Q(-:-j) (z{ _oz{» ) ( exp- L or(!) (zi j _ I;j» ) ( !!. ) O<O).
. > 1 J .> 1 J %2
J_ J_
Here we have used that
(14.8.16)
6(.11 - %2) = .1;1 ( 1 _ !!. ) -1 + 1.1 ( 1 _ !! ) -1 I
%2 '-- %1
and therefore
(14.8.17)
6I(Zl - %2) = %;2 (1- *) -2 _ %.2 (1- *) -2.
Applying (7.7.5) to (14.8.15) proves (14.8.7). The proof of (14.8.8) is sim-
ilar. Thus, (1 is a representation of £(g) on V.
Furthermore t it follows from (14.8.9) that
try qDo = E q("(h)/2 / fP(q)l..
"YEQ
Comparing this with (12.13.6), we lee that (f is the basic representation
with the highest-weight vector 1  1.
o

310
Vertez Operator Con.'ruclion, Gnd Soli'on Equation,
Ch. 14
S14.9. The basic representation or the infinite rank affine algebra of type
Aoo (and Boo) can be constructed using one of the two approaches discussed
in 5514.6 and 14.8. In this section and the next, we shall discuss yet anothr
approach, which is of fundamental importance for mathematical physics.
This is usually referred to 88 the boson-fermion correspondence.
An infinite expression of the form
it "i 2 " b " · · · ,
where iI, i2, . .. are integers such that
i 1 > i 2 > i3 > ... I and in = in-I - 1 for n::> 0,
is called a semi-infinite monomial. Let F be the complex vector space with
a basis consisting of all mi...infinjte monomials, and let H(. ,.) denote the
Hermitian form on F for which this basis is orthonormal.
Define the chtJrge decomposition
F= E9 F(m)
mEZ
by letting
1m) = m A m - 1 " m - 2 /\ .. ·
denote the vacuum vector of charge m and F(m) denote the linear span of
all semi-infinite monomials of charge m, that is, those which differ from
1m) only at a finite number of places.
Given a semi-infinite monomial '{J = 11 1\ 12 A... of charge m, we asso-
ciate with it a partition  = {1  2  · ..  O} by letting 1 = i1 - m
2 = ;2 - (m - I), .. .. This is clearly a bijective correspondence between
the set of all semi-infinite monomials of given charge m and the set of all
partitions Par (Le., the set of all finite nonincreasing sequences of non-
negative integers). We define the energf, of I{J to be equal to the size of
the partition Ip, Ilpl := i' Let F j m ) denote the linear span of all
.
semi-infinite monomials or charge m and energy j.
We have the energy decomposition:
F(m) = E F}m).
lEI
It follows from the above discussion that
(14.9.1)
dim lj(m) = p(j).

Ch. 14
Verlez Operdtor CDn"c"O'" and SDliton Efua'ionl
311
hence we have
dim, pm) := E dim Fjm)«I = l/rp(q).
JEI
For j E Z, introduce the wedging and conlNc'ing operators .pi and !/Jj
on F by the following formulas:
(14.9.2)
{ 0 if j = i, for some .,
1Pj (it A i 2 A.. · ) = ( 1) " '.. " r ..'
- 11 A.. -I. " l/\ 1.+ 1 " · .. I I, > J > 1_+1.
... { 0 if j f. i, Cor aJ I "
tPj(!11\!21\...)= ( 1) ' + '.' · · A . c . ·
- !1 A 12 " · · ·  _ J ,,+ 1 · -. 1 J:: I,.
It is straightforward to check that the operators tPj and .pj are adjoint with
respect to the Hermitian form H(., .), and that the following relations hold:
(14.9.3) t/Ji"; + t/Jjt/Ji = 6ij. t/Jit/Jj + t/Jjt/Ji = O. t/Jit/J; + t/Jjt/Ji = O.
Thus, the operators tPJ and .pj generate a Clifford algebra CI. It is cJear
that the Cl-module F is irreducible and that
1PjfO) = 0 for j $ 0,
1/1;10) = 0 for j > o.
Remark J. 9. RecaJl that given a vector space V with a symmetric bilinear
form (.,.), the 8880ciated Clifford GIgt6ra CI V is defined 88 Collows:
CI V = r(V)/ J,
where T(V) is the tensor algebra over V and J is its 2-sided ideal generated
by elements of the form %@1/- (zly) (Z,1I E V). Given a maximal isotropic
subs pace U of V, the algebra CI V has a unique irreducible module Fu,
called the "p;n module, which admits a nonzero vector 10) such that U '0) =
o. The algebra Cl is the Clifford algebra associated with the space V =
}; CtPi + "I; Ct/J; with the symmetric: bilinear form (tPiltPi) = 6jj. aU other
. ,
= 0, and F is its spin module associated with the 8ubspace U = E C,pi +
iSO
E C.p;.
i>O
All our further calculations are based on the following commutation
reJations, which follow from (14.9.3):
(14.9.4)
[1Pi.pj, tPi] = 6ltj 1Pi;
[1/JitPj, ",] = -6 1c .1/Jj.

312
Verlez Opern'or Con,t",c'iora6 Gntl Soliton Equation,
Ch. 14
The embedding r : gloo -+ CI defined by
r{E ij ) = tPi1/Jj
defines a representation r or gleo on F, producing thereby a representation
r m of gloo on F(m) (or each m E Z. Note that the representation r is
unitarizable for the antilinear involution (J t-+ -'a of gloo, in the sense that
r(' a ) and r(a) are adjoint operators with respect to the Hermitian form H
(or all (J E gloo.
It is easy to check that gl; acts by "derivations" on F, i.e., for (I =
(Oij) E gloo one has:
(14.9.0 )
r( a ) (il A b " · · · ) = a · i 1 A 1.2 " · · · + 11 " (J · 12 " · · · + · .. ,
where a. j = EOij!. Here (and further on) we use the usual rules of the
- i
exterior algebra to express the right-hand side of (14.9.5) in terms of semi-
infinite monomials. Thus F can be viewed 88 an infinite generalization of
the usual exterior algebra. For this reason r is usually called the infinite
wedge reprelenttJtion.
9100 is the Lie algebra of the group
OLeo = {o = (Oij)iJEZ I (J is invertible and all
but a finite number of Gij - 6 i j are OJ.
The corresponding action R of GLoo on F is given by
(14.9.6)
R(g )(11 A i 2 A.. · ) = g · i 1 A 9 · b " · · · ,
that is (14.9.5 and 6) are related by
(14.9.7)
exp r(o) = R(ex» a), G e 9100.
Using the standard exterior algebra calculus. we get the following formula
for the representation Rm of 9 e OLeo on F(m):
(14.9.8)
Rm(g)(il"""') = L (detg;::::::) il "b"'" ,
1,>J2>."
where g;1:',:'.: denotes the matrix located at the intersection of the rows
il,j2,... and columna il. i2,... of g.

Ch. 14
Verlez OprtJ'or COftlf,..ctionl Gntl Soliton E9uation,
313
It is clear that the representations r m are irreducible. Moreover, we
have:
r m ( EiJ ) Jm) = 0 if i < j, or i = j > m,
rm(EJj}lm) = 1m) if j  m.
Thus, as an sloo-module. F(m) is isomorphic to L(Am), where Am is the
fundamental weight, defined as usual by (Am,Qj) = 6mj (see i7.11 for
notation concerning 9 1 00, 0 00 , etc.).
Tbe representation .ro of 9100 on F(O) is called the 6asic repsen'(Jtion.
The glco-module F(m) does not extend to Goo since for example
ro(diag(..., i, i+l,.. . »10) = (o + A-1 + ... )10),
which is generally divergent. In order to avoid this anomaJy, we modify the
representation r by letting
f ( E.. ) _ { t/J.1/Jj if i -F j or i + j > 0,
I} - ./.. .1. · r . · < 0
'Y J ¥'i 1 · + J _ ·
It is easy to see that this extends by linearity to a representation r of the
completed infinite rank affine algebra a oo on F with K == 1 (see 7.11 for the
definitions). We obtain thereby for each m E Z an irreducible representation
Trn of aoo on F(m).
As in S14.1, we call
 = 1: w(ei) E oo
i
the c,clic elemtnt of %00' and denote by i its centralizer in ZOO. Here
1r : Zoo -+ Zoo is the canonical homomorphism. The 8ubalgebra _ = ",-1 (i)
is called the principal .a6tJIge6ra or Zoo.
In the case of 0 00 , we have
1 = L Ei,i+1'
iel
hence the eJements eJ = E Ei.ifj (j E Z) form a basis of i. One irnmedi..
tEl
ately finds using (7.12.1):
¥,(r",l") = m6m,-n.
Hence - = E e'm + CK, where W(lm} = em and [., K]  0, with com-
mEI
mutation relations
(14.9.9) [8m, .n] = m6m,_nK,
i.e., - is the oscillator algebra. Comparing (14.9.2) with (14.9.9), we obtain,
in the same way as in 514.4, the following:

314
Vertez Operator Con,'",ctioft, Gnd Soliton EqutJtion,
Ch, 14
PROPOSITION 14.9. Viewed &9 an .-module, the ooo-modulu 1'<"') are
irreducible.
114.10. The operators .pj and tPj are calld free fennion,. The bOlOniza-
tion consists of introducing free 6o,on.t:
Qn = r('n), n e z.
Explicitly:
On = E .pj1/Jj+n if n E Z\{O},
iEI
00 = E.pJ.pj - E 1/Jj.pJ ·
J>o JSo
By (14.9.9) we have:
(14.10.1)
[Om, On] = m6m.-n.
Note also that
(14.10.2)
ooIF("') = m1t
and that Om and Q-m are adjoint operators.
Physicists would call F the fermionic Fock space. We introduce now tbe
bosonic Fock space
B = C(Zl, 2,.,. ;9. ,-I],
which is a polynomial algebra on indeterminates ZI. %2, . .. and 9,9- 1 . De-
fine a representation r B of the oscillator algebra. on B as follows:
B ( (J
r 'm) = 8 I
Zm
B lJ
r ('0) = 98f'
rB(._m) = mZm
if m > 0;
rB(K) = 1.
Then by Proposition 14.9 and the uniqueness of the canonical commu-
tation relations (Corollary 9.13), there exists a (unique) isomorphism of
s-modules
t1:F::'B,
such that O'Om» = 9 m . Note that 0'( F(m» = B(m) := q"'C[%I, %2, . .. J.
The map tT transports the Hermitian form H(. t.) on F to a Hermitian
form HB(., .) on B, and the energy decomposition (14.9.1) or F to that of
B. Explicitly they are given by the following proposition.

Ch. 14
Veriez Opera'or Con,'",ction, Grad Soliton EtJua'ion.
315
PROPOSITION 14.10.
a) HB(qm P(z), q"Q(z» = 6mn P (, iJ!;,... ) Q(z)'=o'
b) Energy of qm z{1 2 . . . = ;1 + 2j2 + · · · .
Proof. a) follows from (9.13.4). b) follows from the simple observation that
the operator 1Pi1/Jj increases energy by i - j (thus the energy decomposition
is nothing else but the principal gradation).
o
Using iT, we can also transport the operator of multiplication by 9 from
B to F, obtaining the (unique) operator on F. which we again denote by
q, such that
qlm) = 1m + 1), Q1Pi = -Pi+19 (m, i E Z).
(The second equation follows from the first one and the equation qQi = Oi9,
i e z \ (O}.) Note that q and q-l are adjoint operators.
The fermionization consists of recoDstructing fermions .pi and 1/J; in
terms of bosons Qi. In order to do that introduce (as before) the gen-
erating series (fermionic fields):
1/I(z) = Ezj-Pj,
jEI
1/I-(z) = Ez-j.pj.
leI
Introduce also the following operators:
z-n
r+(z) = exp E -;;-cr nl
"1
zn
r _(z) = exp E -;;-cr_ n .
nJ
As before, we view z as a formal parameter, 80 that r :j:(z) are viewed as
generating series of operators f; on F:
r%(z) = E r;z?".
nEI+
Since ri = I, we may consider the generating series r %(z)-I. Note that
r + ( z) and r - ( z) are adjoint operators in the sense that r:- and r: n are
adjoint operators on F. Note also that in the bosonic picture we have:
(14.10.3)
-n 8
r+(z)=expE- 8 '
n1 n Zn
r _(z) = exp E zn zn .
">1
-

316
Verlez Operwtor COfl,t,..ction. Grad Soliton Efuation,
Ch. 14
Finally, with a partition  e Par t we associate the Scbur pDlynomial
S(z) aa usual. First. define the elementary Schur polynomials S".(z) by
the generating series
(14.10.4)
E Sm(z)zm = exp L z"z"
mEI n1
(= II e'..).
n > l
We have:
Sn&(X) = 0 for m < 0, So(x) ::: 1, and
Xkl Xk2
Sm(x) = L k\ k\ '" for m > O.
k 1 +2k2+...==m 1. 2.
Given  = {1  2  .. · } e Par, let
SA(Z) = det(Si+j-;(%»l;JS'I'
Here and further on, z stands for (%1, Z2,. ..).
Now we are in a position to state the boson-fermion correspondence:
THEOREM 14.10. a) One has:
.p(z) = zOOqr _(.t)r +(z)-I,
.,-(z) = q-lz- a or _(z)-Ir+(z).
b) If 'P E F(rn) is a semi-infinite monomial, then
(T() = qms_(z).
Proof. In order to prove a) we directly check the following relations of
operators on F:
[aj. tP(z)] = ,i ,p(z).
[ Q j . 1/1. ( z )] = - zi 1/J. ( z ) .
The first of these two equatiQDS transports to B as follows:
(, (T1jJ( Z )0-- 1 J = zj (O'1P(z )cr- 1 ).
I
-J
[zi 1(1"'( i)(1-1] = «(1tP( Z )(1-1).
J
Hence, by Lemma 14.5, the operator
(1'1/1(z)cr- 1 : B(m) -+ B(m+l)

Ch. 14
Vertez Operator Con,'",ction, and Soli'on Equation,
317
is of the form
D'.p(Z)(T-l = Cm(z)qf(z), where
f(z) = ( ex p EziZj ) ( exp- E  ) .
il J1 J J
But the coefficient of 1m + 1) in 1j1(z)(m) is ,m+l, hence Cm(z) = zm+l.
This proves the first of the two formulas in a). The proof of the second
formula is similar.
In order to prove b) we let g(lI) = exp E Jljl j , where 1'1,112,... are some
.>1
J_
complex numbers, and compute
(14.10.5)
D'(Rm(g(y»)i 1 1\ i 2 " ... ) = R(g(y»P(Z)J
where P(z) = 0'(i 1 1\ 12 1\ · · .). We shall obtain the result by comparing
the coefficient of the vacuum 1m) (= vacuum expectation value), which we
denote by F(y), on both sides of (14.10.5).
We must first settle a minor technical problem since g(JI) does not lie in
GLoo. It does lie, however, in the group
-
OL oo = {o = (Oij )i.jEI I a is invertible and all but a finite
number of OiJ - 6ij with i  j are OJ.
The corresponding Lie algebra is
-
gloo := {(a'j)iJEZ I all but a finite number of Oij with i  j are OJ.
-
!he representations R and r extend from GLoo (resp. gloo) to GLoo (resp.
gloo), and formulas (14.9.7) and (14.9.8) still hold.
Since R:'(g(y» = exp E IIj 8:. ' we obtain:
-> 1 J
J_
F(y) = ( ex p E IIi 8' ) p(z)I=o = P(z + y)lr=o.
.>1 J
J_
Thus, from the calculation in the bosonic picture, we have
(14.10.6)
F(y) = Pc,,).
On the other hand, by definition:
,(y) = exp EliYj = L: Sm(y) e m .
Jl mO

318
Ver1e% Optralor Cona'raclions anti Soliton E9utJlion.
Ch. 14
The latter expression is a matrix Q with matrix entries 4"," = S"-",(y).
Now we can read off the coefficient or 1m) in the expansion of
CT m (R(o)(i1 "b " .. · » from (14.9.8), whic::h gives det a::':''i;;;'-2..... This
is easily seen to be Sia- m ..,-(m-l),...(I/). Comparing this with (14.10.6)
completes the proof.
o
As & consequence of (14.9.8) and Theorem 14.10 b), we obtain the Col-
lowing formula for the action of GLoo on B(m):
(14.10.7)
RB ( o ) S =  ( deta.\l+m t A :a+ m -l.... ) S .
m A L.J ",+m."1+m-I...."
peP.,.
Another corollary or Theorem 14.10 is a vertex operator construction of
the g/oo" and Goo-module L(Am):
COROLLARY 14.10. Let r m (resp. r m ) denote the representation of
gloo (resp. 0 00 ) on L(Am). Then L(Am) can be identified with the space
C[ZI,Z21'.'] such that
( 14.10.8)
(14.10.9)
where
. '"
LiJEZ z1z;Jrm(EiJ) = la6:'J 61 r(Zl,Z2),
Li.JEI z1z;J rm (Ei i ) = 1-6/61 «Zd Z 2)mr(Zl,Z2) -I),
(14.10.10) r(%I. %2) = (ex p EJ1 (zf -z{)Zj ) (ex p - Ejl ajj,;j :j )'
Proof. Note that we have on F(m):
(14.10.11)
E zi z;J rm (E{j) = .p(Zz)tP-(Z2)'
i .j E I
Note also that, by (14.8.12 and 13):
(14.10.12)
r +(Zl)-lr _(%2)-1 = r _(Z2)-1r +(Z2)-I(1 - %2/ Z 1)-1,
Applying Theorem 14.10 b) to (14.10.11) and using (14.10.12) and (14.10.2
and 3) gives (14.10.8).
In the case of r m we observe {rom (7.12.1) that we must simply subtract
i(%I/Z2)i = (1- %2/%1)-1 from the right-hand side of (14.10.8) to get the
right-hand side of (14.10.9).
o

Ch. 14
Vertez Operator Con"",ciora' Gratl Solilon EfutJI;on,
319
Remark 1./.10. It (ollows from the proof of Theorem 10.12 b) that the
boson-fermion correspondence tT : F -+ B can be constructed explicitly as
follows. Let H(z) = E %jOj and let V'",(a) denote the coefficient of 1m)
jEN
in a E F in the semi-infinite monomial basis. Then
«r(V) = L CPm(eH()v)qm.
mEI
514.11. We proceed to explain how one uses the representation theory
developed in previou8 sections to construct solutions of soliton equations.
The starting point is the following trivial observation:
Remark J./ .11. Consider a representation of a group G on a vector space
V and a vector Va E V. Let OJ be an operator on V commuting witb the
action of G and suppose that Vo satisfies the equation
(14.11.1)
01 V = v,  e C.
Then any element of the orbit G · Vo satisfies this equation.
In this section we shall appJy this observation to the group G Loo acting
diagonally on F  F, Vo = 10) 0 10) and
0 1 = E ¥'j @ ¥'; ·
;EI
It is clear that 01 V O = 0; using (14.9.4) one checks immediately that Ot
commutes with gloo and hence with GLoo. Thus, due to Remark 14.11,
any element T or the orbit G Loo 10) satisfies the equation
(14.11.2)
L "j(T) @ "j(T) ::: O.
jEI
LEMMA 14.11. The orbit G Loo 10) is the set of all nonzero solutions T e
F(O) of equation (14.11.2).
N
Proof. Write T = TO + L CjTj as a linear combination of semi-infinite
;=1
monomials 'i, among which TO has the greatest energy. If among the Ti
witl1 i > 1 there exists one, say T2, of the form
(14.11.3)
ro(Eij)TO with i < j,

320
Vertez Opernlor Con8ruc'ion' tJntl Soli'on Equation,
Ch, 14
we can kill off the term C2r2 by replacing r with (exp-c2E;j)r. Repeat-
ing this procedure a finite number of times we arrive at an element of the
form ro + VJ, where none or the semi..infinite monomials appearing in cp is
equal to TO or is of the form (14.1.3). Since TO + cp satisfies (14.11.2), it
follows that cp = O. Since TO E GLooJO), we obtain that T E CLeoIO).
o
Using the boson-fermion correspondence iT : F -+ B, we may view
(14.11.2) 88 an equation on T E B(O) = C[Zl, 2:2, . . .]. We shall rewrite it in
a more explicit form u8ing Theorem 14.10 a). By definition, (14.11.2) can
be rewritten as
(14.11.4)
,O-term of .p(z)r @ ".(z)r = O.
The isomorphism a: F(O) =. B(O) = C[ZI,2,"'] extends to an isomor-
phism
t'O\ · F (O)  F (D) ,.., C[ " · II II ]
(f '0' tr. \01 -+ ZI'Z2"" '%1 '%2'." .
We can transform (14.11.4) to the bosonic picture using Theorem 14.10 a)
and (14.10.3), obtaining
( 14.11.5 )
Res,=o (exp E zJ(zj - zj')\ (exp - E zJ. ( 8 8 - 8 8'1 )) r(z/)T(Z"):O.
 J >1 ')  .>1 J z, z,
- J_
Introducing new variables
(14.11.6)
1 ( ' " )
Zj = 2 ZJ + Zj ,
1 ( I " )
IIj = 2 ZJ - Zj ,
80 that
a lJ I)
-=-+-
8z J o /Jz'. lJz'.' t
J ,
the latter equation becomes:
8 8 {J
8Yj = 8zj - 8z'J '
Res,=o ( ex P 2 E zJ yJ ) ( exp - E z:J  ) r(z + )r(z - y) = o.
jl jl J 8y}
Arter expanding exponentiala with the help of the generating series (14.10.4)
Cor elementary Schur polynomials and taking the z-l_ term, this equation
becomes:
(14.11.7)
E Sj(2y)Sj+l( -a,)T(Z + 1/)T(Z - 1/) = 0,
J?O

Ch. 14
Verlez Operator COfl6'MlC';On8 cratl Soliton Ef.Gt;on,
321
where Dr stands for (, i 8:2 ' i S:, ' · · · ).
Given a polynomial P(Zl, 2, . . .) depending on a finite number of the
%j t and two COO-functions f(z) and g(z), we denpte by P(Dl, D2,... )/. 9
the expression
p ( 8: 1 ' 82 ' .. .) l(zl + Ul, Z2 + u2, ... ),(ZI - U., Z2 - u2,. .. )Iu=o.
The equation P( D)f · 9 = 0 is called a Hirota 6ilineGr equation. For exam-
ple. if P = z t then from Leibniz' formula we obtain:
n ( ) 8i/ 8n-i
Di I., = E(-I). : lJzII lJ n- .
i=O 1 %1
Note that P / ./ = 0 if and only if P( z) = - P( -%). This is called a trivial
lJirota bilinear equation.
Now we rewrite equation (14.11.7) in a Hirota bilinear form by using the
following trick based on Taylor '8 formula:
P(8v)r(z + y)r(z - y) = P(Du)r(z + 1/ + u)r(z -1/- u)I.,=o
(14.11.8) = P(8u) ( ex p L YJ Q. ) r(z + u)r(z - u)lu=o'
'>1 J
J_
We obtain:
(14.11.9)
ESj(2 g )SJ+1(-v)(ex p Ey,D.)r. r = O.
JO ,1
- 1 1
Here D stands for (D 1 '"2 D2t 3 Da, . . .).
If we expand (14.11.9) to a multiple Taylor series in the variables )lJ,
1/2,. · · 1 then each coefficient of the series must vanish, giving U8 thereby
a nonlinear partial differential equation in a Hirota bilinear form. This
system of equations is called the KP laierarcA,.
For example, the coefficient or ,1,. is the following Hirota biJinear equation
(r  1):
(14.11.10),.
(2S r + 1 (-D) - DIDr)T' T = O.
Using the explicit formula (or Sm(Z) given in 114.10, we see that the Hirota
biJinear equation (14.11.l0)r is trivial for r = 1 and 2, whereas (14.11.10)3
becomes, after dropping odd monomials and multiplying by 12:
(14.11.11)
(D: + 3Df - 4D1D3)r. r = O.

322
Verlez Opera'or COft,'",c'ion. Ind Soliton E,"t1'iofl.
Ch. 14
Thia is the Kadomtlev-Petviuhvili (KP) equation in the form of BirotL
Namely, putting
8 2
(14.11.12) U = 2 8=2 101 r; 1 = Z, Z2 = lit 3 = t,
:1: 1
the equation (14.11.11) takes it. cluaieal Corm:
38 2 u 8 ( au 3 8u llJ3U )
(14.11.13) i 8,2 = a; 8i - 2 u a; - i 8%3 ·
In order to construct polynomial solutions of the KP hierarchy, Dote
that any semi-infinite monomial of charge 0 liee in the orbit GLooIO). and
hence satisfies (14.11.2). Using Theorem 14.10 b), we obtain the followins
remarkable result:
PROPOSITION 14.11.1. All Schut polynomials SA(Z) are solutions o(the
KP hierarchy.
o
Another type or solution, the so-called soliton solutions, are constructed
by using the vertex operator r(Zlt%2) (see (14.10.10». Let UI.... 1 UN.
Vl, · · · J tiN be some indeterminatee. In what follows. by the expreuioDI
(Uj - Ui)k, (Uj - Vi)k, etc., with i < j, we mean the Cornlal power series
expansion u1(1 - uiJUj)k, u1(1 - viJUj)k, etc.
Let 1"(ZI,%2,"') be a formal power series. Then using (14.8.13) we
derive by induction on N the following formula:
(14.11.14)
r(UN,VN).. .r(Ul,vl)r(zl,z2,.")
- II (UJ-Ui)(VJ-Vi) ( "' ( ' ' » )
- ( )( ) exp L.., L.., Uj - vJ Zi
li<JSN uJ - Vi Uj - Ui . j=l
)( r ( .. .,%, - t i(Uj' - vi'),.. . ) .
J=1
In particular, we have: feu, v)2 r (z) = 0, 80 that
(14.11.15) expar(t., v) = 1 + ar(u,v), a E C.
Given complex numbers 01, . · · . aN; Ul. . . . , UN; and VI, . . . , tJN 8uch that
Ui  tJj if i 'F j. we let
r _. . . ... _ N jtt I ... .. U 11 ; . ....." N ( z) = (1 + (IN r ( UN, tJ N » · . . (1 + CJ 1 r ( u 1 . tJ 1 » · 1.
Since 1 is a solution of the KP hierarchy and r(u, v) lies in a completion
of ,/00 by Corollary 14.10, we obtain:

Cb. 14
Vertez Openalor Con,'rwc,iora, Gratl Soli'on Ef utJ 'ion6
323
PROPOSITION 14.11.2. The (unction
T ClS,...,O N;U i ,...,U" ;" 1,.. .,. N (% )
-
-
r (Uj., - Uj,. )( Vj., - Vj,. )
E II OJ,, II (Uj.. - Vj,. )(Vj" - Uj,.)
O$rSN .,=1 1"<pSr
lj, <J4j < ...jrN
r
X exp E E(u:" - vt)Zt.
iI.,=l
is the solution of the KP hierarchy.
o
The T-Cunction T 01 ,....ClN;..'.....UH;tI',...,VN(Z) is called an N -oliton &olution
of the KP hierarchy. or curse, (rom the point of view of the single equation
(14.12.4), the indeterminates Z4,ZS,... are arbitrary parameters; an N-
soliton solution T, or rather u(z) = 2 '!°f! , describes the interaction of N
uSa
waves during time %3, in shallow water in plane coordinates %1 and Z2.
For example, the I-soliton solution
u(z, II, t) = (2 log Tl;a;'(Z, II, t»
of the classical KP equation (14.11.13) takes the following form:
1 1
u(x, y, t) = 2 (a - b)2[cosh 2 «a - b)x + (a 2 - b 2 )y + (a 3 - b 3 )t + const)r 2 .
SI4.12. We shall examine here what Remark 14.11 gives us when ap-
plied to an arbitrary principal vertex operator construction. We start
with some general remarks concerning an arbitrary symmetrizable Kac-
Moody algebra g(A). Let L(A) be an integrable highest-weight module
over g(A) and let VA be a highest-weight vector. Recall that L(A) car-
ries a (unique) positive-definite Hermitian eontravariant form H such that
H(VA,VA) = 1 (see Theorem 11.7 b». This form determines a positive-
definite Hermitian form on the tensor product L(A) 0 L(A) by letting
H(u @ v, u'  Vi) = H(u, u')H(tJ, v'). Then we have an orthogonal direct
sum of modules
(14.12.1)
L(A) 0 L(A) = Lhigh ED L 1aw ,
where Lhi8h, the laighe' component, is the g(A)-submodule generated by
the vector VA 0 VA, and Llow is ita orthocomplement. Notice that Lhilh is
isomorphic to L(2A) by the complete reducibility theorem (Corollary 10.7).

324
Verlez Opera'or Con,'",c'ioft, anti Soli'on E9utJlion.
Ch, 14
Let G be the group of automorphisma of L(A) generated by I-parameter
groups exp te, and exptJ., i = 1,2,..., nJ where i, Ii are the Chevalley
generators. Denote by VA the G-orbit of VA and by VA its formal completion
(see S 1.5).
Finally, let (} be the generalized Casimir operator <S2.5), and let O 2
be the operator on L(A) 0 L(A) introduced in 52.8, which commutes with
g(A). We can now state the result.
PROPOSITION 14.12. a) lIv E VA, then
(14.12.2)
(14.12.3)
(14.12.4)
v 0 v e Lhilh;
O(v 0 v) = 4(A + piA}" @ Vi
C1 2 (v @ v) = (AIA)v0 v.
b) The conditions (14.12.2)-(14.12.4) on v e L(A) are equivalent. (In fact,
any nonzero vector tI E L(A) satisfying (14.12.3) lies in VA; see Peterson-
Kac [1983J.).
Proof, If v = VA, then (14.12.2 and 3) are obvious, and (14.12.4) is clear if
we choose
(14.12.5)
02 = L Ee)0e,
aEAU{O} i
where e) are bases of 80' Q E 4 U to} such that (e)lel) = 6ij. This
implies a).
It is straightforward to check that equations (14.12.3 and 4) on II e L(A)
Are equivalent. The equivalence oC (14.12.2 and 3) follows Crom the following
lemma (since LJow is a direct lum or the L(A') with A' < 2A):
LEMMA 14.12. Let A,A' e P+. It A > A', then
(A + 2p1A) - (A' + 2p1A') > o.
Proof. IA + pl2 -lA' + pI 2 = (A + A' + 2plA - A') > O.
o
Now let L(Ao) = C[Zj;j E E+J be the principal vertex operator con-
struction of the basic: representation or an affine algebra. Then L( Ao) 0
L(Ao) can be thought of as the apace of polynomials on two sets or vari.
abies: C(zj,zj;j E E+J. We introduc:e new variables zJ and 71J, j E E+, by

Ch. 14
Venez Operator Con,t,..ctionl and Soliton Equalion,
325
(14.11.6),80 that L(Ao) @ L(Ao) = C[Zj, IIj;j E E+]. Then the principal
subalgebra _ acts on L(Ao)  L(Ao) 88 follows:
lJ
pj  , qj  2% j ,
vz.
J
K ....... 2.
Note that by (9.13.4) and Proposition 14.4, we have the Collowing for-
mula Cor the contravariant Hermitian Corm on L(Ao):
(14.12.6)
- 1 8
H(P,Q) = P(...,-: Jr,...)Q(...,Zj, ...)(0), j E E+.
J VZj
Now we are in a position to rewrite equation (14.12.2) in terms of the
Hirota bilinear equations.
LEMMA 14.12. r @ T e Lhish if and only jf
1
P( · · · , 2J Dj , · · · ) T · T = 0
(or all P(y) e C(Yj;j E E+] n Llow C L(A o ) @ L(A o ).
Proof Applying Lemma 9.13 b) to the .-module Llow, we have
( 14.12.7)
Llow = (Llow n C[YJjj E E+]) @ C[Zj;j e E+].
Hence, T @ T e Lhilh if and only if
(14.12.8)
H (Qi(Z)Pi(Y), r(Z')T(ZIl») = 0,
.
where Qi E C[Zj;j E E+] are arbitrary and Pi E C[Yj;j E E+]nLlow'
Applying (14.12.6), condition (14.12.8) is equivalent to:
E ( 818 ) ( 8 18 )
Qi -;-'- 2 .- {j ,... Pi - () '- 2 .- 8 ,... T(%+J/)T(-y) l _ o 0 =0,
i UZI J Zj 111 J IIj ,- ,-=
where P and Q are the same as above. Since this holds for arbitrary Q,.
and since the 8ubspace C[II] n Llow is invariant, under complex conjugation,
we get the result.
o
The elements from the space
Hir := Cry;;; e E+] n Llow

326
Vertez OperaCor Con,'",c,ion, Gnd Soliton Equation,
Ch, 14
are called HirotG ,ol,nomiGII, and the element. (rom the Cormal completion
...-.
VAo in C[[Zj;j E E+]] oCthe orbit of 1 are called r-junc'ion,. Thus, in order
to derive applications of Proposition 14.12 to PDE, we have to be able to
construct explicitly two things: a) even Hirota polynomials; b) r-func:tioDs.
Denote by H. c C('j i j E E+] the space of Hirota polynomials of prin-
cipaJ degree k, 80 that H ir = EB Hi (recall that the principal degree is
.
defined by deg Yj = j). By formulas (14.12.7) and (14.12.1), we have
(14.12.9)
00
dim, Hir:= L(dim H)q
i=O
= «dim, L(A o »2 - dim, L( 2A o» II (1- II').
lEE.
(Recall that dim. L(A) can be computed by Proposition 10.10.)
Let us apply formula (14.12.9) to the infinite rank affine algebra of type
Aool Formula (10.10.1) can be written in this case in the following nice
form:
(14.12.10) dim, L(A'1 + · · · + A,.) = IT (1 - q'i-'jH-i)/Ip(q)n,
l < i<JSn
where 'I ?= '2  ...  Ira are arbitrary integers. In particular, we have:
dim, L(Ao) = cp(q)-l,
dim, L( 2A o) = (1 _ q)<p(q)-2.
Hence we obtain: dim. Hir = q<p(q)-l, i.e.,
(14.12.11)
dim Hit = p(k - 1).
Denote by P>'I.>.....(D) the coefficitmt of the monomial 11: 1 11: 2 ... in
(14.11.9). It is easy to see that
(14.12.12)
PA 1 .A,....(II) E Hir1+).2+...+1.
On the other hand, note that
02 = Oint.
Hence equation (14.11.2) implies (14.12.4). Conversely, if v satisfies
(14.12.4), then H(OiOl(VV),(vv» = 0, hence H«(}l(VV),Ol(VV» =
0, hence t.I satisfies (14.11.2). It follows that the polynomials PAI.A 2 .... (y)
span Hir, by Proposition 14.12 b). Comparing (14.12.11) and (14.12.12),
we arrive at the following result.

Ch. 14
Vtrlez Operator Con,truction, tJntl Soliton E9uatioR$
327
THEOREM 14.12. The Hirota bilinear equations PAIIA2,...(D)r.T = 0 form
a basis of the KP hierarchy of Hirota bilinear equations.
o
Note that for j $ 3 (reap. j = 4) the number oCtrivial Hirota equations is
p(j - 1) (resp. p(3) - 1). Hence (14.11.11) is the nontrivial Hirota equation
of lowest principa! degree.
514.13. The most celebrated example, the Korteweg-de Vries (KdV)
equation, occurs in the context of the principal vertex operator construction
of the basic representation of the affine algebra 9 of type A\l). Recall the
basic realization of this algebra:
,,= .t 2 (C[t,,-l]) + CK + Cd,
do
[a(t), 6(t)J = a(t)6(t) - 6(t)a(t) + (Res,=o dt 6)K,
da(t)
[8, K) = 0, [d, a(t») = t tit ·
Recall the principal vertex operator construction of the g-module L(Ao)
(cr. 514.6 and Exercise 14.12):
L(Ao) = C{Zl t Z3, %5, . . .];
lJ
Hj ...... 0 _, H_j t-+ jZjt j E N odd ;
z.
J
K ......1 . 2d- !A o ...... -  J 'z.
2 L", J {Jz. '
jEN... J
,,_ 1
A(z) := L...J z J Aj ...... -(r(z) - 1),
JEZ 2
where
- ( 0
H 2 j+l = t J t
and
)I
. ( -1 0 )
A 2 j = t J 0 I I
. ( 0
A 2 j+l = t J
-t
),
r(z) = (ex P 2 E zJ Zj ) (ex P -2 L zJ 8. ) '
iEN a 44 jENoct4 J J
We choose the following dual bases of g:
{-l2H2i+1' AJ' K, d} and
{H-2j-l' A-i' d, K}, j E ZI

328
Ver1ez Opera'or Con,t",ction, Grad Soliton EfUGtion,
Ch. 14
and take the corresponding operator 02 on L(Ao)  L(A o ). Then calcula-
tions similar to thoee in 114.11 show that equation (14.12.4) is equivalent
to the following hierarchy of Hirota bilinear equations:
(14.13.1)
(E 5 n (4Yl. 0. 4Y3. 0..., )5n( -r D !, O,-iD3' 0". .) - 8 E JYJ DJ )
EN JEN...
X (ex p E YJ DJ) r · r = O.
J EN...
On the other hand, formula (14.12.9) gives in this case (cr. Exercise
14.3):
diffit Hir = II (1- 9 2J - 1 )-1 - II (1- 94i-2)-1 .
Jl jl
It is clear that all Hirota equations of odd principal degree are trivial.
Also, by (14.13.2), we have dim H2 = 0 and dim H4 = 1. Looking at
the coefficient of 1/11/2 in (14.13.1), we see that the unique Hirota bilinear
equation of (the lowest) principal degree 4 is
( 14.13.2)
(14.13.3)
(Dt - 4DID3)r. r = o.
The same change of (unctions and variables in the corresponding Hirota
bilinear equation 88 in (14.11.2) (except that thre is no %2), gives the
Korteweg-de Vries equation:
(14.13.4)
8u 3 {Ju IlJ3u
8t = 2 u a; + 4' 8z 3 '
The N-soliton solutions for the KtlV hierarch, (14.13.1) are constructed
in the same way &8 those (or the KP hierarchy. The answer is as follows:
TCS1.....ClN;Ul.....UN(Zl, %3,. ..)
r
E II OJ.. II
o<,.<N ,,::1 1<"<1'<"
ljl <j,,<"'jrS;N --
( U. - U. ) 2 r
1 1.   ,
( u. u ) 2 exp2 L.J  Uj"Zi.
1., + J. i> 1 .,=1
i od d
The function 2 ::3 log rOl.....ClNiUl.....UN (z) describes the interaction of N
I
waves during time %3 in a narrow channel in coordinate %1.
In particular, the I-soliton solution u(z,t) = 2(logrl;G(z,t))z describes
a solitary wave:
u(z,t) = 2a[cosh(o% + 0 3 , + const)]-2.

Ch. 14
Verlez Operator COflt",ct;on. anti Soli'on Equation6
329
Actually, all these result. for the KdV hierarchy can be deduced from
the corresponding result. for the KP hierarchy using the so-called reduc-
tion procedure. This amounts to putting Vi = -U,. The representation-
theoretical meaning of this procedure is explained in Exercises 14.11 and
14.12.
Another celebrated example, tbe nonlinear SchrOdioger (NLS) equation,
is related to tbe homogeneous vertex operator construction of the same
basic representation of the same affine algebra g of type Al). We let
Q=( 1)'
e=( ),
/ = ( )
and choose the following dual bases of g:
{'"a, ,n e , ,n /, K, d}, and {it-nOt ,-n " ,-n e , d, K}.
We have: Q = Za, (ala) = 2, E(mcr, nQ) = (_l)mn, 80 that cQ(/ @
e"Q) = (-1)". Letting 9 = e Ol , we identify C[Q] with C[q,q-l]. Thus, the
homogeneous vertex operator construction can be described as follows:
L(Ao) = C[ZII %2, · · .; 9, 9- 1 J;
Q(n) .- 2 and
8z n
K .-. 1,
lJ
o( -n) ...... nZn fOl n > 0, Q(O) ...... 2n-'
 lJq ,
( lJ ) 2 lJ
d.... - q- - LnZn-i
8q nl (Jzn
E(z) := L E(n)z-n-l t-+ r + (.z),
nEI
F(z) := L F(n)z-n-l .- r _(z),
nEI
where
r:!:(z) = ( exP:l::Ezj ) ( eXP=F2L %J  ) q*lz-J:2fItC:l:a
.>1 .>1 1 lJzJ
1_ J_
(note that z2fl;(qn) = z2nqn).
Taking the operator O 2 on L(Ao) 0 L(Ao) corresponding to the above
choice of dual b88e8 and performing ealculatioDs similal to those in 114.11,
we find that equation (14.12.4) is equivalent to the following hierarchy of

330
Verfez O,trafor Con,'",c,ioft. tlntl Sol,'ora E9"4';on.
Ch, 14
Hirota bilinear equations on T = E T,(X)q' (m, n E Z):
.eZ
(14.13.5)
( m - n)2 + 2 EjJlJDJ) (ex p EJI,D, )rm · r n
J1 .1
+ (_I)",-n ESJ(2,)SJ+2 m -2n-2(-2D)(exp EfI,D,)rrn-1' TnH
J1 ,1
+ (_l)rn-n ESJ(-2 f1 )SJ-2m+2n-2(2D)(exp EfI,D,)rrn+1 · T n -l = O.
J1 '1
Note that this hierarchy iI invariant with respect to the translation m ....
m + r, n ...... n + r. r e Z. We look at the coefficient of IIf Uld restrict.
oureelvea to the caee m - n = 0 t -1, or 1. For. = 0 or 1 we then get trivial
equations, whereas (or , = 2 we obtain the following equations respectively
on functions Tn(Z), T n -l(Z). and 1"n+l(Z):
DlTn · r n + 2Tn-l Tn+l = 0,
(14.13.6) (Dl + D2)r" · '-n+l = 0,
(Dl + D 2 )r n -l · r n = O.
Letting  = ZI, , = Z2, 9(%,') = rl/ro, 9.(,t) = r-I/ro, u(z,') = 108ft),
we transform equations (14.13.6) into the following equations respectively:
u.... = -qq.,
-9, + 9. + 2qu.. = 0,
,; + q; + 2q.u... = O.
Excluding u. we arrive at the NLS sY8tem of PDE'. on functions 9 and 9*:
{ q, = 9.. - 29 2 9- ,
(14.13.7)
q: = -q; + 2'19*2.
Note that imposing the constraint
,*(z, it) = :l:q(z, it) := g(z, t)t
we get the classical nonlinear SchrOdinger equation on the function g:
(14.13.8) ig, = -g :i: 21g12,.
For any sequence of ligns, say, + + - · · " and numbers (11, (12. . . . e C,
%1, %2, · · · E CX such that 1%11 < 1-'21 < ... we can construet a solution of
the NLS laieNJrc.A, (14.13.5):
++-... ( )
Tal ,42"";'1 .'2.... Z I q
= .. .(1 + Gar -(%3»(1 + 02r +(%2»(1 + air +(zl»1 @ 1.
The proof that this is a solution is the lame as in 514.11 for the KP hier...
archy.

Ch. 14
Vertez Operotor Con&tructiOR! tJnd Soliton Equation!
331
514.14. Exercise..
Jl.l. Show that Proposition 14.4, Lemma 14.6, and Theorem 14.6 hold if
Ao is replaced by A e P.
1 . £. Let 9 be a simple finite-dimensional Lie algebra of rank l, let h be its
Coxeter number, and Eo be the set of exponents. Given a positive integer
j, let Cj denote the number of roots of 9 of height j and let Mj denote the
number of times j appears in Eo. Check their following properties:
Cj = Cj-l - Mj-l t
Mj = MIa-j.
Deduce by induction on j the following formulas (cf. Lemma 14.2 e»):
Cj + CIa-J = t + Mj,
Cj + C1a+l-j = i,
Cj +Ch+2-j =l-M j - t .
1 f. 9. Let A be an affine type matrix such that 'A = X  1 ) and let 9 be a
simple finite-dimensional Lie algebra of type X,. Let L(A) be an integrable
g(A)-module or level 1. Deduce from Exercise 14.2 the following formulas
for q-dimensions (cr. (14.4.4»:
dim, L(A) = II II (1 - qi+nh)-l t
jEE o nEZ+
dim, L(2A) = II II (1 - qi+n")-l(1_ qi+1+ n ("+2»-1.
jEEo nEI+
Using these formulas, rewrite (14.12.9) in a more explicit form:
dim, Hi,. = II IT (1 - qJ+",,)-l - II IT (1 - qi+1+ n (h+2»-1.
jEE o nEZ. jeEo nEI+
14.. Show that the principal vertex operator construction of the basic
g'(A)-module L(Ao) given by Theorem 14.6 extends to the semidirect prod-
uct g/(A) )4 Vir by lettins
d 2....  ... 1  ' ( h(r) · d 1  .
o  hC")  o_JoJ + 4h<")2 L-) - ), n  2h(") £....J °nMr)-jOJt
jEB+ jeEo jEE
where, for j E E+, we let OJ = k, Q_j = jZj. The conformal central
charge is then equal to N. 1
In Exercises 14.5-14.7 we discus8 the homogeneous vertex operator con-
struction of the basic representation tr.

332
Verle,- Operator Con,tructiOft8 anti Soliton Equation,
Ch. 14
. 1.5. Let Ttl be the group introduced in Exercise 12.19 (acting on the space
L(Ao}}, and let ( : QY x QY - {::H} be aD asymmetry function. Show
o
that Ttl .::. QV x {:l:l} with multiplieation:
°v
(0,0)(11,6) = I:(Q,{1)(o + (1,(6), o,{JeQ t d,6e{:!:1}.
14.6. Let g'(A) be an affine algebra, where either r = 1 and A il symmetric t
or r = 2 or 3. Let i = CK E9 (ED gjf). Using Proposition 12.13, show that
lEI
there are no nontriviallubspaces in L(Ao), invariant with respect to all

oper atols from D'( t) and TiT.
o
14.7. Assume now that A is a symmetric affine matrix, 80 that A is of type
. 0 0
All Dl. or El. Then QV can be identified with the root lattice Q of g(A) via
o
identifying their basel or ..... 0i, preserving the bilinear form (.1.); let A C
o 0 0 0 0
Q be the root system of g(A). Recall that sa := g(A) = $( ED CEo) with
.
crEA
o
commutation relations (7.8.5). Choose t; E T" with image "1 e Qr and such
o 0
that t'; = l(,8,-y)t+, {J,1 E Q. Recall that ,,'(A) = C[t,t-1Jc 9E9CK
.. 0 0
and t = CK$( ED (t J @». Put  = ED (I*J @)t 80 that t = t_ eCK (Dt+
lEI j>O
is the homogeneous Heisenberg subalgebra. Using Exercise 14.6, show that
..
L(Ao) considered as (T I t)-module t can be identified with the space V :=
o _
C[Q] c S( t_) with the following action of T" and t:
t(e-Y 0 P) = l(P, 'Y)e"+  p.
tr(t i  h) = h(k), treK) = I.
Check that
{I .  t
t 0 h, E? ( % ) ) = (1, h) z E-y ( z) , h E  t t E Z;
t:ct · E..,(z) = z-(a,..,) i..,(z), Q E QY.
Deduce the homogeneous vertex operator construction of the basic repre-
sentation of g'(A). Let {Ui} and {u i } be dual bases of t Introduce the
fo))owing operator. on the space V: Do by formula (14.8.3), and
1 l
(14.14.1) Dm = 2' EE Ui(-j)U,(j + m) for m e Z\{O}.
lEI i=1
Show that the map d. ..... Dil C ..... 1I extends the g'(A)-module V to a
g'(A) )4 Vir-module.

Ch. 14
Verft% Operator COft,'",ction, anti Soliton E9utJ,ion,
333
ll.8. Let Q be an arbitrary even lattice and let  be the complex span
of Q. Introduce the space V by formula (14.8.1) and operators u(n), for
u E  and n E Z by (14.8.2). Let Dn be operators on V defined by
formulas (14.8.3) and (14.14.1). Show that they form a representation of
the Virasoro algebra in V with conformal central charge c = dim. Let
E : Q x Q --+ {::I: I} be an asymmetry function and let C-y, 'Y e Q, be the
corresponding sign operators on V. For "1 E Q define the vertex operator
r,,(z) by formula (14.8.5). Show that this is a primary field of conformal
weight ! ( -r1"Y)' Prove the following formula:
(14.14.2)
f'YN(ZN)...f'Yz(Zl)(l@l)= II t(1;,1J)(z;-Zj)('Yihi)
l<i< .<N
_ J_
N · N
x ( II exp L 1i( -j) ) exp L "(.,
-=1 jEN J i:l
,
where by (Zi - Z j)m we mean the power series expansion of (- Z j)m X
(l-z./Zj)m.
1.9. In the proof of Theorem 14.8 deduce that IT is a basic representation
from (14.14.2) (avoiding thereby the use of the character formula).
1.10. Equation (14.12.4) is equivalent to the system of equations of the
form P = 0. where P e S2(L.(A»; show that these P generate the ideal
of functions from S(L.(A» vanishing on VA.
[Let CM denote the eigenvalue of the Casimir operator {1 on L(M). Show
that (0 - C,A)V' = ls(, - 1)«0 - C2A)V 2 )V.- 2 . Now use Lemma 14.12.]
1  .11. Fix n > o. Let VI. . . . , V n be the standard basis of en. We identify
Crt, ,-1] 0 en with Coo = ED CVj by setting V n ,+. = ,-t  Ui. This
iel
gives us an embedding gln(C[f,t- 1 ]) -+ 0 00 . Show that the 8ubalgebra
thus obtained consists of aU matrices (Cij)iJEI E 0 00 such that cHnJ+n =
Cij. Show that the restriction of the central extension 000 -+ 0 00 to this
.....
subalgebra is isomorphic to (,(g/n) := gl,,(C[tt ,-1]) ED CK with bracket
[A(t), B(t)] = A(t)B(t) - B(t)A(t) ED (Res tr dt) B(t») K.
Show that the fundamental Goo-module L(A,) remains irredueible when
-
restricted to £(g/n).
14.1 t. Consider the vertex operator construction of the fundamental 000.
module L(A,), 8 = 0,..., n -1, on the space C[j;j = 1,2,.. .]. Show that

334
Venez OperGtor Con,Cnaciioft' Grad Soliton E9uatioft,
Ch. 14
the subspace C[x j; j  0 mod n} is invariant with respect to the affine
- - (1)
algebra £(st n ) := {A(t) + CK I A(t) E £(gln), tr A(t) = O} of type An-It
viewed as a luballebra of Goo. Show that we obtain the principal vertex
operator conltruction of the l(.ln)-module L(A.), given by tbe lollowing
formulas:
( n-I ) J 8
1) (0) .
1 + L E,,'+l ..... - {) (reap. - JZ_j)
.=1 zJ
if j > 0 (resp. < O),j  0 mod n;'
n-l ( n t )
  z-i-nm  £-J Em) +  e- j Em+l)
   J-iJ  J-i+nJ
.=0 mEI i=i+l j=1
(,+1 l
t-+ 1 r( (Z , z) - 1 '
f- l-
where £ runs over all nth roots of 1 different from 1.
Deduce that a r..function Cor the KP hierarchy is a r-function for the
All-hierarchy if and only if T is independent of Zj with j E 0 mod n.
Deduce that the set of all solutions of equation (14.2.4) in L(A,) is VAoU{O}.
1.13. Consider the Clifford algebra CIs on generators i, i E Z, with
defining relations
;iJ + jtPi = (-1)'6.,-J'
and consider it 8 irreducible module V witb the vacuum vector 10) subject
to conditions
4>iI O ) = 0 for i < o.
Introduce the neJj'ral/ermionic fitld
(z) = E .z',
lEI
and the operator q on V defined by
910) = o (0),
q;i = tPiq.
We have: 9 2 = i. Introduce the nutral 60,0,., An, n e Zodd, by
\ -!  ( I V+l. .
"n - 2  - J Y'JY"-J-n,
Jl
and show that
1
(AmI An] = 2m6m,-n.

Cb. 14
Vertez OperGtor Con,'",ct;onl IntI Soli'on EfutJtionl
335
Let tr ::: Us : V  C[Zl t %3, Zs, . · . ; q]/(9 2 -1) be an isomorphism of vector
spaces such that
-1 8
O'mO' = 8 I
Zm
(110) = 1, O'oIO} = q;
-1 1
O'.\-mO' = 2mzm
for m E N odd .
Show that
(1(z)u-J = , ( exp E z jzi ) ( exP-2 E z.j 8' ) .
JEN.d. jENodd J J
The map iT is <;a)led the hOlon-fermion cOJTe&pondence 01 type B. Explicitly:
O"(v) = cpo(eHs(e)v) + CPt (eHs(z)v)q,
where HB(Z) = L z,..\,. and 'Po (resp. 'PJ) (0) is the coefficient of 10)
nENodd
(resp. qIO}) in o.
1./.1./. Lel Fij = (-l)iEij - (-l)iE_i,_i' i,j E Z, (i,j):# (0,0), be the
standard basis of 8° 00 . Show that the map
p( Iii) = t/Jit/J-i
defines an irreducible looo-module on the space V o (resp. VJ) of even (resp.
odd) elements of V, which is isomorphic to L(A o ). We have: tT = (7'0 ED tTJ :
V o Ea VI  C[ZI, Z3t.. .] $ qC[ZI, %3,. ..], where (f is the boson-fermion
correspondence of type B. Show that the map
{ A, At. · if i -l J . or i = J . > 0 J
A ( F... ) _ Y'.Y'-J r
P IJ -  . .A _J . - _ 2 1 1 . C . · 0
V'Y' 1 I=J< ,
p{K) = 1
extends by linearity to the boo-module L(A o ). Show that the map tTo :
t'o .::-. Cfz J I Z3, · · .] gives us an equivalence of the representation p and the
following vertex operator construction of 6 00 :
 i -j 1 1 - Z2/Z1
 Zl Z 2 F;j.... _ 2 1 / (rS(Zl'%2) - I),
i ° E l + %2 %1
.J
where rB(Zl,Z2)::: ( ex p E (z{ +)Zj ) ( exP-2 E 6;-;-:6;; k ) .
jENo4. jeNodd J J

336
Verlez Opera'or Con,Inc'ion. Gnd Soli'on EfutJ,ion.
Ch. 14
1./ .15. Show that the operator
of = L(-l)J;J @;-}
JEI
on V @ V commute. with 6 00 , Show that the equation
Of(T0r)=o(T)0;o(T)t ,-eVo,
transferred from V o to C[Xl,X3,".] via 0"0 gives the so-called BKP hierarchy
of Hirota bilinear equations:
E S}(2Yl ,0, 2Y3," · )SJ( -Dl' 0, -D3"") (ex p E Y,D,) r · r = O.
j1 .EN...
Show that the equation or the lowes.t principal degree of this hierarchy is:
(Dr - 5DfD a - 5D: + 9D1Da)T." = o.
Construct soliton solutions of the BKP hierarchy.
1./.16. Given n > 0, show that the lubalgebra {(Cij)iJEI  CK e 6 00 8uch
that Ce+nJ+n = e'J} is isomorphic to an affine algebra g'(A), where A
is of type A) (resp. Dl) if n = 2i + 1 (resp. n = 21). Consider the
realization of the basic representation of 6 00 on the space C[Zj; j positive
and odd]. Show that the lubspace C[Zj;j ;. 0 mod n] is invariant with
respect to g'(A). giving the principal vertex operator construction of its
basic representation. Show that the corresponding vertex operators are (€
is an nth root of unity. l  1):
( exp E (l-i)z}ZJ ) ( ex P -2 E (I_C}) zJ 8' ) .
JEN... jEN... J J
14.17." Show that the energy decomposition (principal gradation) of the
moduJe F(') is: deg(i l Ai 2 A...) = (sum of the j > . that occur in the set
{iI, i 2 ,. · · }) - (sum of the j S . that do not occur in this let). Equating
q-dimensions, deduce Euler'. identity:
(,)-1 = 1 + E ,1 2 /(1 - ,)2 ... (1 -: ,.)2.
i1
Let H be the energy operator on F, i.e. HtF") = jI, and let 00 be the
J
charge operator (see 114.10). Computing in b0800ic and (ermionic pictures
trF ,H zGO derive once more the Jacobi triple product identity:
II (1 + zt")(1 + z-I,"-I) = E zl qiU+1)/2/'{)(t).
"EN jEI

Ch. 14
Vene% Operator Con6'",c,iora, dntl Soliton E,..'ionJ
337
1.18. Using the boson-fermion correspondence, show that
S" (8) s,..(z)lz=o = 6",...
1.19. Let %00' where z = 6, C, or d, be the completed infinite rank affine
algebra of type Xoo (see 57.11). We call
l':= L (ei) E oo
i
the cyclic e/emtnt of Zoo J and let i be the centralizer of l in Zoo. Then i is
graded with respect to the principal gradation: . = ED ij. The subalsebra
. jEI
. = -1(1') is called the principtd nbGIgdru of ZOO' Show that dime} =
dim Ij = 1 for j odd, = 0 for j nonzero even and '0 = CK. Show that
l' = L: Ei.i+! for 100 or oo; l' = Eo.2 - E-l.l + r: E.,.+! - r: E..i+l
jEI i1 is-l
-. -
for d oo ; and that {e1 }jEleclcl form a basis of i in the cases 6 00 and Coo;
{P}JENO.. U {('lY}JENO.. form a basis of I in the case d oo . For 600t Coo.
or doc put r = i. 1, or i. respectively. Introduce the following elements of
oo:
Pj = eI t
Ij = ej (j E d).
rJ
Then we have the following commutation relations in zoo:
(p ] 6 K (, ' t) ' e N odd ).
f ,fj = iJ
Show that in the cases boo and doc t
dim f L(Ao) = II (1 - qi)-l.
iE""...
and deduce that L(Ao) remains irreducible when restricted to ..
LI.tO. Show that for the fundamental coo-module L(A,) (. = Ot It .. .) one
has
dim, L(A,) = (1 - 9 2 .+ 2 )/"'(9).
Denote by p, : F<-,-2) - F(-I) (. e l+) the map of the exterior
multiplication by E - j I\ j + 1 . Show that p, is an injective homomorphi.m
.>0
J_
or coo-modules. Deduce that the coo-modules L(A,) and F<-')/p,(FC-,-2»
are isomorphic.

338
V"'e% O,era'or Con,truction. Clntl Soliton E,"tltion,
Ch. 14
J.tl. Define a lubgroup P or GLoe by:
P = {(OiJ )iJEI I Oil = 0 (or j  0 and i > O}.
This is the normalizer of the weight space of L(Ao)Ao. The group Eoo of
all finite permutations of Z has a natura) embedding in OLeo (by tT( flj) =
vU))' Denote by E_ (resp. E+) the subgroup ofEoo ofpermutatioDS which
fix all j :5 0 (resp. j > 0). Let U+ (reap. U_) denote the subgroup of OLeo
of all upper- (reap. lower-) triangular matricel with ones on the diagonal.
For 8 right coset w E Eoo/E+ X L- we put p = U-J: n (n uew uU_u- 1 );
P; is a finite-dimensional subgroup of U +. Show that G = U PwP is a
w
disjoint union, where w runs over Eoo/E+ x E_, and that presentation on
the right is unique.
14.". Deduce from Exercise 14.21 that we have a disjoint union:
SLn(C(t, ,-1]) = U SL n (C(t%l]) diag(t.t.,.. . I t.t" )SLn(C(t]),
where k 1 S ,..  n are integers and E k; = O. (These kind. of de-
compositions are called BruAG' and BirkAoJ! tlcompoitioft' for + and -.
respectively. )
----
1.e3. Consider the upper-triangular matrix g(z) e GLeo introduced in
514.10, and Jet C E GL oo , w E Eoo. Show that, in notation of 514.10, for
v = CwIO) we have:
1'. ( z ) = det ( g ( z ) C ) i o ......'" .
" O....,m
Deduce that in the principal realization o( the basic Goo-module in the
space C[ZI. %2,.,,], the orbit OLeo · 1 together with 0 contains all polyno-
mials of the Corm:
(14.14.3) det(g(z)C):::",
where C e GLoo. Show that, moreover, every polynomial from GLoo · 1
can be uniquely represented in the (orm (14.14.3) such that io < ... < i m
and C E PWt where w 10) = :1:10 1\.. .Aim A 1- m -1) and m is lufliciently
large. Thus, we obtain all polynomial80lutions of the KP hierarchy. U.inS
the principal IU balgebra, show that if P( ZI. 2:2, . . .) lies in G Leo · 1, then
its "translate" P(Zl + Cl, %2 + C2, . , .), where Ci e C. also lie. in GLeo · 1.
ll.IS. Show that one hu a natural bijection between the set Eoo/E_ x E+
and the set of all partitions. Namely, given w e Eoo, let m be the greatest
integer, such that w( m) = mj arrange all w(j) with j S m in deereuins
order: jrn > jrn+1 > "', and put 60 = jrn - m, 6 1 = irn-l - (m - I), . . . .
Then 6 0  6 1  · .. is the &88ociated partition.

Ch. 14
Vtrlez Operator Con,Iruction, Gnd Soliton Equation,
339
1.e6. Given the two partitions" and 6', define the skew Schur polynomial
S6'6'(Z) = S6,(8)S6(Z)i in particular S6'{O}(Z) = S6(Z) is a Schur poly-
nomial. Show that S"6'(Z) is, up to a sign, the coefficient of w'(IO) in
g(z)w(IQ» where b and 6' are the partitions associated with wand w'.
1.e7. Let V be the n-dimensional vector space (n = 1,2,... ,00) over C
and Jet AV = ED A 1 V be the Grassmann a1gebra over V. Denote by
iEI+
Vi the set of all decomposable elements of AV (i.e., elements of the form
vIA.../\ t1t where Vi e V). Denote by Pit : V.\{O} -+ Grn.i the canonical
map onto the Gr88smannian GrnJ of k-dimensional 8ubspaces of V defined
by Pi(V)1\. · .I\ v i) = ECvj. An element v e V defines a wedging operator
j
on A V:
v( VI A V2 /\ · · · ) = v A VI 1\ V2 1\ · .. ,
a.nd an element lEV. defines a contracting operator:
f( VI 1\ V2 A.. · ) = {It fJl)V2 A v3 1\ · · · - (I, V2)Vl " v3 " · · · + · .. .
Show that the wedging (resp. contracting) operators anticommute and that
vI + Iv == (I, v). Deduce that the wedging and contracting operators
generate a Clifford algebra on V EB V. with the bilinear form
(v $I'v' e I') = (v, I') + (/, v').
Choose bases {f.} of V and {e;} of V. such that (e; J ej) = 6ij, and let
n
0 1 = E fj  ej be an operator on A V  A V. Show that 0 1 commutes
;=1
with the action of GL(V) on AV and r E AtV is a decomposable element
if and only if
(14.14.4) (}l(T @ r) = o.
Show that written in the basis ei s A... 1\ ei. (i 1 < ;2 < ... < j) of
Ai V t equation (14.14.4) gives the classical Plucker relations. Deduce from
Exercise 14.10 that Plucker relations generate the ideal of relations of V.
Show that the equation
(14.14.5) 01(T r') = 0, T e A'V, r' E A"V, k  r,
is equivalent to pA:(r) ::) /3r(r'). Show that equations given by (14.14.5)
generate the ideal or relations of the variety (called the affine /lag va.riety)
V := {(rl " " , rn) E V 1 X..  x V" I pA:(ri) :J Pr(rr) for It > r}.
1. .
14.28. Put Zj = 1(t1 +... + r N ). Show that SA:(Zl, Z2,...) is equal to the
trace of the matrix diag(f1,"', fN) in the GLN(C)-module Si(CN) (and
hence, by Schur, 5.(z) is the trace of this matrix in the GLN(C)-module
corresponding to the partition 6; see, e.g., Macdonald [1979]).

340
Vertez Operator Con,truction, and Soliton Equation,
Ch. 14
14.R9. We identify Coo with e[t, t- 1 ]  en as in Exercise 14.11. This gives
us an embedding of tbe group T := {diag(t'l J . . . . t''') I k i e Z, E k. = O}
into the group of all permutations of Z. For gET there exists a per..
mutation w E Eoo such that 9 = wg', where 9' is a permutation of Z
which leaves invariant the set of nonp08itive integers. Let 6 be the par-
tition associated with wand put I, = S.; show that, up to a sign, the
polynomial 8, is independent of the choice of w. Let 9 be an affine al-
gebra of type A121; we identify the basic g-module with the subspace
C(zJ;j  0 mod n] OfC[Zl,Z2," .]. Let G be the associated group and let
V = G · 1 be the orbit of the highest-weight vector. Show that " E V
(9 E T), i.e., ., is a polynomial solution of the A121-hierarchy. Let
PeG be the preimage of SLn(C[t]) under the canonical homomorphism
d -+ SLn(C[t, ,-1]). Using the Bruhat decomposition (see Exercise 14.21),
show that
v = UP'" (a disjoint union).
lET
14.30. We keep the notation of Exercise 14.29. Let n = 2; for k E Z
we denote by 8, the polynomial 8, with 9 = diag(t',t-). Show that
Sir = S{2i-l p ...l} if k > 0 and Ii = S{-2i.-2.-1.....1} if k  O. Show that
dim(,+{si) + C8) = i + 1 = dimP · 8,. Deduce that all polynomial
solutions of the KdV hierarchy are of the form
(14.14.6)
COS{.,i-l.....1}(ZI + CI. %2 + C2... .), where Ci e C.
[The projectivisation of the set. P · 8. is isomorphic to C'. We have an
injective map I: C t -. C defined by f«Cl,...,Ci» = 8i(ZI +Ct.Z2 +
C2,.' .) and such that dim /(C.) = k. Deduce that I is an isomorphism.]
J.31. Show that the polynomial S{2,l} = lzf - %3 is a solution of the
A121-hierarchy for n  4, but that not all polynomials of the set p. SU , 1}
are of the form (14.14.6).
J.92. Define the ;nfinitt GnJlsmannian Gr 88 follows. Consider the infi-
nite-dimensional space Coo = E9 CVj and its 8ubspaces Cr = E9 CVj.
. jEI jSk
Denote by Gr (resp. Gr n ) the set of all 9ubspaces U of Coo such that
U ::> Cr (resp. U :> C r and dim U ler = n - k) for k < O. Then
Gr = U Gr n (disjoint union) and GLoo acts transitively on each Gr n . Let
nEZ
V n = {r e F(n) I nl(r r) = O} (= OLeo In) U {O})

Ch. 14
Ve"'% Operator Cora,'ruc'iora, Grad Soliton E,.tJI;on,
341
(see S14.11), and denote by Pn : V n \{O} -+ Gr n the map defined by
Pn(!1 " h " · · .) = E CVij t which is GLoo-equivariant, 8urjective, with
j
fibers CX. Prove infinite-dimensional analogues of all facts stated in Exer-
cise 14.27.
1-1.33. Show that the equation
- 01(T@r')=O, rEF(i+n), T'eF(i>, nO.
when transported to B, is equivalent to the following hierarchy of Hirota
bilinear equations (called the n-th MKP hierarchy):
(14.14.7)
LSj(211)SJ+R+1(-D> (ex p LYrDr)T' T' = O.
jO rl
Deduce Crom Exercise 14.32 that the set of all polynomial solutions consid-
ered up to a constant factor of the KP hierarchy is naturally parametrized
by Gr; and that oC the union of all nth MKP hierarchies, by tbe infinite
flag variety
:F = {. · · ::> U 1 :> U 0 ::> U -1 ::> · · · I Uj E G r j }.
1 . 3. Denote by P).I ,).2,... ;n (D) the coefficient of the monomial yf 1 11: ' . . .
in (14.14.7). Let L7lh denote the highest component of the ,Ico-module
B(i+n)  B(). L(n) its orthocom p lement. Hir(n) = L(n) n C ( fl J and
'0' 'Ijw' 'low II
Hir(n) = $HirJn , the principaJ gradation (cC. !14.12). Show that the
1
polynomials PA11,....;n(y) form a basis of the space H,<:rn+t' 80 that
dim Ht> = p(j - n - 1). Show that 2PO;1(X) = x - X2, and hence
(Dr - 2D 2 )r' . T = 0 is the equation of lowest degree of the first MKP.
Show that using the change (14.11.12) of variables and functions together
with v(x, y, t) = lag(r'IT), we obtain:
U = v r - v: - tls-s-,
called the Miura transformation.
If 35. Show that a Hirota polynomial P(1I) is odd if and only if P(Y) e
A2(L(Ao), and that any odd polynomial is a Hirota polynomial.
[Use the fact that 21/j = zj - zj'.]

342
Verle% Operutor Con,t",ction, Gnd Soliton Equation,
Ch. 14
14.36. Show that (or the operators Ln introduced in Exercise 9.11 one haa
(cf. Exercise 7.24):
[Ln,tJ1..] = (-n/2 - i + in).pt-n'
1.37. Show that for the KP hierarchYt the formal completion of the space
H ir is invariant witb respect to the vertex operator
Z(u, v) = (ex p (ul: - Vl:)y.) (ex p -  (u-II - v-I:) lJJ: )
+ (ex p - (ui - Vi)YII) (ex p   I(u- k - v-II) 8:" )'
1.98. Put O-J: = 2-. kYIl and OJ: = 2t 'D:. for k > 0, and let L niO denote
the operators on tbe space V = C[Yl, Y2, . . .] introduced in Exercise 9.17
with ,\ = }J = O. Let
Z(u) = lim(l- !)-2(Z(U,v) - 2),
v...u u
where Z(U,l1) is the vertex operator introduced in Exercise 14.37. Show
that Z(u) = E u- j Ljt O , and therefore, the 8ubspace Hir of V of Hirota
j
polynomials (or the KP hierarchy and its orthocompJement HirJ. are in..
variant with respect to the Virasoro operators Lj;o,j E Z.
1.39. Using the boson-fermion correspondence, transport the action of
the operators L" o defined in Exercise 14.38 to F(C), and show that the
elements of this 8pace killed by all Li:jo with Ie > 0 are linear combinations
oCthe elements nA n-1 A..."!" I-n), where n = 0,1,2,.... For
n = 0,1, 2,. · ., let Qn(Y) denote the Schur polynomial S{n.....n}(y)' where
n is repeated n times. Deduce that if a polynomial P(y) is killed by all LJ:;o
with Ie > 0, then P(y) is a linear combination of the Qn{Y), n = 0, 1,2,....
J."O. Deduce from Exercise 14.39 that the polynomia]s Qn(Y), n = I,
2, · · ., are Hirota polynomials for the KP hierarchy. (For n = 2 we thus
recover the polynomial in (14.11.11).)
J..jJ. Show that Hirl. is irreducible under Vir and that Hir decomposes
into a direct sum or irreducible Vir-modules generated by (highest-weight)
vectors Qn(Y), n == 1,2,. .. .
J.£. Deduce from Exercise 14.39 that the highest-weight vectors of the
aoo-module L(Ao)  L(Ao) are scalar multiples of the polynomials Qn(Y):
n = 0, 1,2, . . . .

Ch. 14
Verlez Operator Contruction' dnd Soliton Equation,
343
Jl.3. Let L(A) be an integrable highest-weight module over 0 00 (A =
E i. Ai, where k i are nonnegative integers and all but a finite number of
i
them are zero). Let  = {'\i}iEI be such that k, = '\i - i-l and i = 0
for sufficiently big i, and let hi be the hook lengths of tbe (infinite) Young
djagram of . Show that dim, L(A) = n(l - qla i )-I.
i
In Exercises 14.44-14.51 we discuss a pseudodifFerential operator ap-
proach to the KP and KdV hierarchies.
J 4 .l./. A (formal) pseudodifferential operator is an expression of the form
P(, 8) = L Qj(z)8i,
jel
where OJ(z) are formal power series in z and OJ(z) = 0 for j :> o. The
multiplication of pseudodifferential operators is defined by letting
· " ( j ) dlt6 ._,
l)I · biz) = L- k dz t lJ1 ·
i > O
Show that the pseudodifferential operators (orm thereby an associative al-
gebra \It. Show that 'It carries an anti-involution · determined by conditions
b(z). = b(z),
lJ- = -8.
14.5. Denote by U+ (resp. U_) the space of expressions of the form (formal
oscillating functions):
EOj(z):je U
iel
(resp. E OJ (z)zi e- U ),
jel
where aj(x) are fornlal po\ver series in x and aj(x) = 0 for j » O. Show
that one has a representation of the algebra 'it on the space U + (resp. U _ )
determined by the condition (b(x)8 j )e ZX = b(X)ZieZX (similarly for U_).
For P = E a j 8 j E \Ii let
jEZ
p+ = LOj8i t
°>0
J_
P- = LOjcY.
j<O
Show that if P(z,8),Q(z,o) E  are such that
, I II
Res=o(P(z tz)e zQ(z"._z)e- Z "} = 0,
then (PQ-)_ = o.

344
VeMez OperalDr Construction, anti Solilon E9utJ'ion,
Ch. 14
1-1../6. Consider now pseudodifferential operators of the (orm P(Z, 8),
where z = (ZI, z,....), 8 = 8:. ' and Z2, Z3,... are some parameters. and
denote z · z = E zi ZJ. Let r(z, z) = e6.r +(z)-I, r.(z, z) = e-'.r +(z)
J1
(see (14.10.3». Given a nonzero Cormal power aeries r(z). introduce the
associated wave function
w(z, z) = r(z, z)r(z)/r(z) := w(z, z)e'.,
and the adjoint wave (unction
w.(z, z) = r.(z, z)r(z)/T(Z) := w.(z, z)e-",
where w(z. z) = 1 + Wl(Z)z-l + W2(Z)z-2 +'.., w.(z, z) = 1 + wi(z)z-l +
w;(z)z-2 + .... We may write:
W(Zt z) = Pe,.e,
w.(z, z) = Qe-'.,
where P = 1 + wl(z)8- 1 +.", Q = 1 - wi a-I + w;a- 2 - .... Note that
equation (14.11.5) of the KP hierarchy can be written 88 follows:
( 14.14.8)
Rea,:o w(z', Z)W.(Z",Z) = o.
Deduce from Exercise 14.45 tbat if r is a solution of the KP bierarchy, then
Q = (p*)-I,
1./.7. Show that r(z) is determined up to a constant function by its wave
function, namely:
J:II {} log T = - Res.: o z" ( 1: z-j-l a {J - - O {J ) log w(z, z).
uZn Jt Zj Z
14.48. Let L:= P8p-l == 8 + Ul(X)a- 1 +..., and let Bn = (Ln)+_ Show
that: Bo = I. B 1 = 8, B 2 = 8 2 + 2U1. B J = 8 3 + 3U18 + 3U2 + 3U1rll and
that Ul = (logr)zJ%l.
Prove that equation (14.14.8) imp]jes
IJL
(14.14.9) - lJ = [Bn, L], n = 1,2,. ...
z"
[Apply :, -.B" to (14.14.8) and use that ( 8:.. - B,,)w(z,z) = ( I:' +
(L")_P)et. to conclude (rom Exercise 14.45 that
(14.14.10) a lJP = _(plJp-l)_p (Sato equation).
Zn
Differentiating LP = P8 by z" and using (14.14.10). obtain (14.14.9).]

Ch. 14
Venez OpertJlor Con,truction, and Soli'on EqutJlionl
345
1.9. Show that (14.14.9) implies
{14.14.11)m."
[  - Bn, 0 - Bm ] = 0 (Zakharov-Shabat equation).
OZn oX m
Sho\v tllat (14.14.11)2,3 is the classical KP equation (14.11.13) on the func-
tion u = 2Ul.
1./.50. Fix an integer n > 2. Show that the following conditions are
equivalent:
a) r(z) is independent of Zjn, j e N,
b ) Jh!L = zRj W
1fi""::: J
,.
e) P is independent or Zjn, j E N,
d) Ln is a differential operator, i.e., (L")_ = O. Deduce that if a solution
T of the KP hierarchy is independent of Zn, then it is independent of all
Zjn,jEN.
14.51. Let n = 2. 'l'hen condition d) of Exercise 14.50 becomes L2 _
a 2 + u(x) := S, where x = (Xl, X3t. . .), so that (14.14.9) reduces to
85 1/2
(14.14.12)j = [(Si+ J / 2 )+, 8 1 / 2 ].
8z 2 j+l
Show that (14.14.12)1 is the classical !(dV equation (14.13.4).
14. St. Consider the following subsystem of the NLS hierarchy (cf. (14.13.6»:
Dfr n · Tn + 2Tn-l Tn+l = 0, n E Z.
Show that after letting Un (z) = (log ";:. )( Z, C2, C3, . . .), this system be-
comes the classical Toda lattice:
(u ) - e U ,,- U .- 1 .U_+l-U"
n II - -  .
14.53. Derive the following explicit form of the solution
r+..'+ ( z q) - "1-: ( z )q "
Gll...tGN;I....,6N ' - L-J R
of the NLS hierarchy (see 514.13):
Tn = E I;......;.(z) if 0 < n  N,
lSi J <...<i.SN
= 1 if n = 0 and = 0 otherwise, where
Ii, .....;. (z) = II (Zi, - Zit )20;, ... Oi. Zit ... Zi.
lS'<fSn
x exp E Zi(zt 1 + .. · + zt.).
'>1
-

346
Verlez Openator COftl'",ctioftl anti Soliton EqutJtiofil
Ch. 14
1 . 5. Derive the following formula for the nth derivative of the 6..function:
67) = n[z;"-I(1 - %1/ Z 2)-n-l + (-l)Rzln-l(l - Z2/Z1)-n-l].
In Exercises 14.55-14.63 we shall outline a more invariant approach
to the affine algebra of type Ace" the corresponding group and the infi-
nite Grassmannian, and an application of this approach to the algebraic
geometry of curves.
14.55. Let V be a vector space. Two subspaces A and B of V are called
commen!una61e, write A  B, ifdimA/(AnB) < 00 and dimB/(AnB) <
00. Show that this is an equivalence relation. Define the /tJt;fJe dimen,iDIJ
of A and B by:
dim A - dim B = dimA/{A n B) - dim Bj(A n B).
Show that this is well-defined.
1./.56. Let End(V,A) = {g E End V I gA + A ,..., A}. Show that tbia
is a 8ubalgebra of the (associative) algebra End V. Let gl(V, A) be the
associated Lie algebra and GL(V, A) = End(V, A»)( the associated group.
Show that
GL(V, A) = {g e (End V)X I gA  A).
Show that the map deg: GL(V,A) -.. Z defined by deg(g) = dimA-dimgA
is a homomorphism onto the additive group of Z. Let GL(V,A)O denote
the kernel of deg. Define the Gr8Ssmannian
Gr(V, A) = {U C V r u  A},
and let Gr n = {U e Gr(V, A) I dimU -dmA = n}. Show that GL(V,A)O
acts transitively on Or". Show that taking V = COO and A = ego. we
recover the Gr88smannian considered in Exercise 14.32, and that 0 00 is a
subalgebra of the Lie algebra gl{COO J ego).
1.51. Let Endftn A = {g E End A f dimgA < oo}, let glftnA be the associ-
ated Lie algebra and let G LAnA = {I + 9 , 9 E Endfln A, 1 + 9 is invertible}
be the associated group. Choose a projection r : V --+ A and let
--
End(V,A) = {(g,/) e End(V,A)EBEndA l7rg1r -I E EndftnA}.
Show that this is an algebra independent of the choice of 1f'. Show that
(14.14.13)
o - End"n A  E;;d(V, A) .!.!... End(V, A) _ O.

Ch. 14
Verlezo OptrtJtor Con,t,.."ctionl Gntl Soliton Equation4
347
where ;2 is the inclusion in the second summand and PI is the projection on
the first summand, is an exact sequence of associative aJgebras. Considering
..--...
the section End(V, A) ...... End(V, A) defined by g  (g, 1rg1r), show that the
associated Lie algebra ii( A) is isomorphic to the vector space gl(V, A) $
glflnA with the bracket:
[(g,/),(9',/')] = ([g,g/J,l/,/'] + A(g,g'»,
where A (g, g') = [rg1r t 'Wg'1I'] - 1r[g, g']r. Pushing out the 8ubspace of
traceless elements, we obtain from (14.14.13) the centra.l extension of the
Lie algebra gl(V, A):
(14.14.14)
0- C - 91(V, A) -+ g/(V,A) -+ 0
with the 2-cocycJe tP(g,g') = trv([r,gJg'). Show that (14.14.14) gives the
central extensions discussed in 57.12 and Exercise 7.28. Consider the group
---- - -
GL(V t A) = End(V, A»)( and the embedding GLftn(A) ....... GL(V, A) given
by /....... (1, I). Show that we obtain thereby an exact sequence of groups:
..---....
1 -+ GLftn(A) -+ GL(V, A)  GL(V, A) -+ 1.
Pushing out the gubgroup of determinant 1 elements gives a central exten-
.
Slon
----
(14.14.15) 1 -+ C X -+ GL(V, A) -+ GL(V, A) -+ 1.
Let GL(V.A)O = ((g,l) E GL(V.A) I 9 E GL(V,A)O}.
1./.58. Let Aoo = {a = (Oij)i.;EI I a is invertible and Qij - 6 ij = 0 for
Ii - jl :> O} c GL(COO J ego) be the group corresponding to the Lie 81-
-:-0 -
gebra a oo , and let Aoo = Aoo nGL(COO, ego)o. Formulas (14.9.4) extend to
a representation o£ooo (resp. Aoo) on the Clifford algebra Cl by derivations
(resp. automorphisms), which we denote by p (resp. P). Show that the
formula
r(g,/)zIO) = p(g)zIO) + Z,.(g -/)10)
defines a pre8entation of the Lie aJgebra i1{COO, ego) on F, which pushes
down to gl( COO , ego) ::> a oo , producing thereby the representation r con-
structed in  14.9. Show that the formula
Ro(g,J)zIO) = (P(g)z)(gl-l)IO)
defines a representation of the group G L( COO , ego)O on F(O), which pushes
down to a projective representation of the gr ou p :.r on F(O) so that
--:"'0 00 t
Po(AoofO» = GrOt .

348
Verlez Oprn'or COR,tructioR' and Soliton Equation!
Ch. 14
1..59. Let V be a vector space and let K be a 8ubspace of V. Given a
subspace D of V, let HO(D) = D n K and Hl(D) = }J  K . Assuming
that the latter two spaces are finite-dimensional, let X(D) = dim HO(D) -
dim Hl(D). For two commensurable 8ub8paCe8 D 1 and D2 of V prove the
"abstract Riemann-Roch theorem":
X(D 1 ) - X(D 2 ) = dim D 1 - dim D 2 .
14. 60. Let X be a smooth algebraic curve. For a point p eX, let A,
denote the formal completion of the local ring at p and let Kp be its field
of fractions. Let Vx = n' K, be the space of allele! on X (all but a finitely
"ex
many components I, of aD adele are in A p ). Given a divisor D = L kpp,
,ex
let A(D) = {(I,), e Vx I v,,(/,)  -k,}. Show that deg D = dim A(D) -
dim A(0).
1.61. Let K, be the field of rational functions on X and let K = {(I), I
/ E K} C Vx be the 8ubspace of principal ade/e,. Show that
Hi(X,Ox(D» = Hi(A(D», i = 0,1.
Deduce from Exercise 14.59 the "easy part" or the classical Riemann-Roch
theorem.
1.6e. Show that for commuting g,g' E gl(V,A), 1/JA{g,g') = tPA(9,g')
is independent of the choice of 1r, that 1PA(g,g') = 1PA'(9,9') if A  A',
and that 1/J A (g, g') = 0 if dim A < 00 or dim V / A < 00. Show that if
g, g' e gl(V, A)ngl(V, B) and gg' = g'g, then one has the "abstract residue
theorem n :
1/J A (g, g') + t/J s(g, g') = t/J A+B (g, 'g') + 1/J An 8(g, g').
1.63. Considering / and 9 E K, as operators of multiplication in Vx, show
that tPAp(/. g) = Res p gdf. Deduce from Exercise 14.62 the classical residue
theorem:
E Res p gdJ = 0, g, I E K-.
'EX
S14.15. Bibliograpbical Dotes and co mm ents.
The exceptional role or the basic representation in the theory or affine al.
gebras was pointed out in Kac [1978 A). Quite a few explicit constructions

Ch. 14
Verlez Operator COR!tructionl Gnd Soliton Equation!
349
of this representation are known. First, it is the principal vertex opera-
tor construction, obtained by Kac-Kazhdan-Lepowsky-Wilson {19B!]. In
the simplest case of AP> this construction had been previously found by
Lepowsky-Wilson (1978J. The second is the homogeneous vertex opera-
tor construction, obtained by Frenkel-Kac [1980J and, in a different form,
by Segal (1981). The third is the spin realization found independently by
Frenkel [1981J and Kac-Peterson (1981). The fourth is the infinite wedge
representation found by Kac-Peterson [1981J. Kac-Peter80D [1985 C} found
a general approach. which attaches to every element w of the finite Weyl
o 0
group W (or, more generalJy, of Aut Q) a vertex operator construction of
the basic representation so that Cor w = 1 (resp. w = Coxeter element)
one recovers the homogeneou8 (resp. principal) construction; for w = -1
one recovers the construction of Frenkel-Lepowsky-Meurman [1984}. An
exposition of this approach based on the vertex operator calculus was later
given by Lepowsky [1985]. Further constructions of the basic representa-
tion are the path space realization by Date-Jimbo-Kuniba-Miwa-okado
(1989 A] given in the framework of statistical lattice models, and the stan-
dard monomial basis construction by Lakshmibai-Sheshadri {1989J done in
the framework of geometric theory of infinite-dimensional flag varieties.
The construction of the integrable modules of higher levels in the frame-
work of the vertex operator calculus is discussed by Lepowsky-Wiison
(1984], Lepowsky-Primc [1985), Meurman-Primc (1987J, Misra [1989 A],
Frenkel-Lepowsky-Meurman [1989). There are at least three explicit con-
structions of aU integrable modules over AI), the vertex operator calculu8
construction by Prime [1989J, tbe path space construction by Date-Jimbo-
Kuniba-Miwa-okado [1989 BJ, and the standard monomial basis construc-
tion by Lakshmibai-Sheshadri [1989J and Lakshmibai [1989).
The vertex operator construction of basic representations of affine alge-
bras of type different from A-D-E were studied by Goddard-Nahm-Olive-
Schwimmer (1986), Bernard-Thierry-Mieg [1987 BJ, Misra [1989 A], and
others.
The exposition of UI4.1, 14.3-14.7 closely foIJow8 Kac-Kazhdan-
Lepowsky-Wilson [1981}. 514.2 is taken from Kae [1978 A]. The resuJts
of S14.2 in the case r = 1 are due to K08tant [1959J. His proof is longer,
but does not use the case-by-case inspection of (14.2.4). Theorem 14.8
is due to Frenkel-Kac [1980], but the exposition in !14.8 is different and
doser in spirit to the physicists' approach. Vertex operators r Q(z) and
the Virasoro operators Dm play a crucial role in the dual string theory
(see surveys by Mandelstam (1974], Schwartz [1973J, [1982]). The work
Frenkel-Kae [1980] linked this theory to the representation theory of affine

350
Verle% Operator Con,'ruclion, and Soliton E9UtJtioft'
Ch. 14
algebras. Physicists were able to use this in the revival of the string the-
ory started in 1984 (see Gross-Harvey-Martinec-Rohm (1985 B] and the
book by Green-Schwartz-Witten [1987}). Note t]lat using the vertex opera-
tors r Q(z), Frenkel (1985J and Goddard-Olive [1985 A] constructed certain
(reducible) representations of (nonaffine) Kac-Moody algebras.
Sinte the pioneering work of Skyrme [1971] (see also Mandelstam [1975]),
the boson-fermion correspondence baa been playing an increasingly impor-
tant role in 2-dimensional quantum field theory. More recentJy, it has
become an important ingredient in the work of the Kyoto school on the
KP hierarchy. The remarkable link of the theory of soliton equations to
the infinite-dimensional Grassmannian was discovered by Sato [1981] and
developed in the framework of the representation theory of affine algebras
by Date-Jimbo-Kashiwara-Miwa {1981]. [1982 A, BJ, {1983 A, B].
The infinite wedge representation was constructed in a more invariant
form by Kac-Peterson [1981] and effectively used in the representation
theory of the Virasoro algebra by Feigin-Fuchs [1982]. Theorem 14.10
is due to Date-Jimbo-Kashiwara-Miw&t wbo use the spinol formalism.
Exposition of SS14.9 and 14.10 folJows Kac-Peter80D [1986J and Kac:-Raina
[1987]. The approach developed in 1514.11-14.13 is due to Kac-Wakimoto
[1989J. The equation (14.11.9) of the KP hierarchy first appeared in Date-
Jimbo-Kashiwara-Miwa (1981]; Proposition 14.11.2 is also due to them.
Explicit formulas for the Hirota equations of the KP hierarchy were found
by Lu [1989 AJ. The fact that all Schur polynomials are 8olutions of the
KP hierarchy (Proposition 14.11.1) was discovered by Sato [1981] and has
become the starting point of this line of research.
The first part of Exercise 14.2 is due to Kostant [1959]; and its last part
and Exercise 14.3 are due to Kac-Wakimoto [1989 AJ. Exercises 14.5-14.8
are taken from Frenkel-Kac {19801.
Exercise 14.10 is taken from Kac-Peterson [1983]; the finite-dimensional
case is due to Kostant. In the case of the GLn(C)-moduJe A 1 C", equa-
tion (14.13.3) is no other than the classical Plucker relations (see Exercise
14.27). This equation plays a key role in the theory of infinite-dimensional
groups developed in Peterson-Kac [1983]. The main result of this paper i8
that the set of solutions of (14.13.3) actually coincides with VA U {OJ (cr.
Lemma 14.11). This result has many important applications to the .true-
ture theory of Kac-Moody algebrasj for instance, the conjugacy theorem
of Cartan 8ubalgebraa is an easy consequence of it.
Exercises 14.13-14.16 are from Date-Jimbo-Kashiwara-Miwa (1982 A,D).
The general reduction procedure is discussed in their paper [1982 B)j and
the BKP hierarchy, in their paper [1982 A]. The approach taken here to

Ch. 14
Vertez Operator Constructionl and Soliton EqutJtionl
351
the BKP hierarchy follows that of Kac-Peterson [1986]; it was developed
further by You [1989].
Note that the formula in tbe introduction of Date-Jimbo-Kashiwara-
Miwa [19S1] gives only the "big" cell of the "Birkhoff decomposition," and
not the set of all polynomial solutions. The set of all solutions together with
its Bruhat decomposition is described in Exercise 14.23. The background
on the infinite flag varieties and Bruhat and Birkhoff decompositions (see
Exercises 14.21 and 14.22) may be found in Garland-Raghunathan [1975],
Pressley [1980J, Lusztig [1983J, Tits [1981J, (1982], Kac-Peterson [1983],
[1984 B), {1985 AJ, (1987J, Peterson-Kac (1983], Atiyah-Pressley [1983J.
Goodman-Wallach (1984 AJ, Freed [1985], Pressley-Segal [1986], Kazhdan-
Lusztig [1988), K08tant-Kumar [1986J, [1987], Kumar [1984], (1985),
[1987 A], [1988J, Lakshmibaj [1989J, Lakshmibai-Seshadri (1989], Matbieu
[1986 C), [1987], [1988], Arabia [1986], and others.
Exercises 14.30-14.34 are taken from Kac-Peterson (1986]. I also have
benefited from lectures by and discussions with M. Kashiwara. AdJer-
Moser [1978] have studied some polynomials related to tbe KdV equation,
found their degrees and a recurrent formula, but failed to find an explicit
formula.
Exercise 14.38 is due to Kac s Vena, and Yamada (see Yamada [1985J; the
fact that Vir acts on Hir for KP was conjectured by Wakimoto-Yamada
[1983]). The description of singular vectors of the Virasoro algebra given
by Exercise 14.39 was conjectured by Goldstone and proved by Segal [19811
by a more complicated method. Exercise 14.42 seems to be new.
Given a representation of a group G on a vector space V and vectors
v e V, v. E; V., the (unction Iv,tI. (g) = (g · v, v.) on G is called a matriz
coefficient It is we)) known (see, e.g., ViJenkin [1965J) that many speciaJ
functions may be viewed as matrix coefficients restricted to some subset of
G. and most of the properties of these functions may be derived from tbis
fact. Exercise 14.26 shows that the skew Schur functions are special (unc-
tions associated with the basic representation of GLoo. As a result, many
properties of these (unctions can be given a group-theoretical interpreta-
tion. A systematic study of matrix coefficients of the groups associated
with Kac-Moody algebras W&5 st.arted by Kac-Peterson [1983].
Segal-Wilson [1985] gave & group-theortical interpretation to the 80-
called qUa8i-periodic solutions of the KdV..type equations via the wedge
representation.
All the representations constructed in this chapter are, in & certain sense,
related to the projective line. A representation related to an elliptic curve is
studied in Date-Jimbo-Kashiwara-Miwa {1983 BJ. Representations related

352
Vertez Operator Con,truction, Gnd Soliton Equation,
Ch. 14
to algebraic curves are also considered by Cherednik {1983 A, Bl.
There arc a number of papers on tile applications of the theory of affine
algebras to completely integrable Halniltonian systems. Here are some of
t}lern: Adler-van lvloerbeke [1980 A, B], (1982), Leznov-Saveliev-Smirnov
[1981], Leznov-Saveliev [1983}, Mikhailov-Olshanetsky-Perelomov [1981],
Reiman-Semenov-Tjan-Shanskii {1979], [1981], Ueno-Takasaki (1984),
Goodman-Wallach (1984 BJ.
DrinCeld-Sokolov [1981], (1984J, were probably the first to notice a link
between affine Lie algebras and the KdV-type equations. With each vertex
of the Dynkin diagram of an affine Lie algebra they associate a KdV -type
hierarchy of PDE, giving a uniform explanation for a large variety of scat-
tered results in the area. Wilson [1984] gave a general approach to the
construction of the solutions of these hierarchies of equations.
Exercises 14.44-14.51 are taken mainly (rom Date-Jimbo-Kasbiwara-
Miwa. Exercise 14.52 is taken from ten Kroode-Bergvelt [1986]; and
Exercise 14.53, from Kac-Wakimoto [1989]. Exercises 14.55-14.58 use
materia) from Kac-Peterson [1981], Arbarello-De Concini-Kat [1989],
Pressley-Segal [1986]. Exercise 14.61 goes back to Cheval ley (1951]. Exer-
cises 14.62 and 14.63 are due to Tate [1968]. A similar approach was used
by Arbarello-De Concini-Kac [1989] to prove an "abstract reciprocity law"
and to derive from it the classical Weil reciprocity law.

Index of Notations and Defini tions
Chapter 1.
St.l.
(,n,nV)
generalized Cartan matrix
realization of A
indecomposable matrix
root basis
coroot basis
simple roots
simple coroots
root lattice
n
n v
Q 1, 0'2, · .. ,Qn
V V V
Q 1 , 0'2 ) · .. 'On
S 1.2.
Q
Q+
hta
>
-
g(A)
R_ and n+
height of Q E Q
partial ordering on .
-
t
S 1.3.
g(A)
A
Lie algebra associated to A
Cartan matrix of g(A)
rank of g(A)
quadruple associated to A
Kac-Moody algebra
Cartan subalgebra
Chevalley generators
derived algebra of g(A)
n
(g(A),, D, n V )
g(A)

ei, Ii (i = 1, . .. ) n)
g'(A)
' =  n g/(A)
g(A) = ED 8a
GEQ
root space decomposition
go
mult Q
4
4+ and _
root space attached to Q E Q
multiplicity of a e Q
set of roots
sets of positive and negative roots
353

354 Index oj NO(Jtion tJnd Definilion.
g(A) = n_ e  E9 n+ triangular decomposition
w Chevalley involution
51.5. V = E9 V o M -graded vector space
oEM
U = fi)U n Va graded 8ubspace
tJ e Va homogeneous element
formal topology on V
formal completion of V
g(A) = e gj(') gradation of type 6
jEI
g(A) = E9 gj(l) principal gradation
Jel
g'(A) (direct definition)
S 1.6. ( center of g(A) or g'(A)
11.8. g-l  go ED 81 local Lie algebra
Chapter 2.
52.1.
B
symmetrizable matrix
symmetrization of A
symmetrizable algebra g(A)
S2.2.
52.3.
12.5.
v :  -+ .
(.1.)
(.1.)
invariant symmetric bilinear form
standard invariant form
restricted g(A)- or g'(A)-module
generalized Casimir operator
52.6.
o
.0 0
P
U(g(A) = E9 UfJ
6EQ
U(g'(A) = e U '
EQ 
.
t(A)
S2.7.
S2.8.
S2.9.
"'0
(zlY)o = -(wo(z)ly)
O 2
compact form of g(A)
compact involution
Heisenberg Lie algebra of order n

lntlez of Notation, and Definition,
355
Chapter 3.
53.1. g(' A) dual Kac-Moody algebra
QV dual root lattice
v dual root system
53.4. locally nilpotent element
locaJly finite element
S3.6. V - or '-diagonalizable module
V weight space
 weight
multv  multiplicity of ,\
integrable g(A)- or ,'(A)-module
53.7. r; (j = 1, . .. , n) fundamental reflections
W Weyl group
S3.8. G" group IL880ciated to an integrable
S(A)-module 1r
--
W.
D. .
53.11. W = Tj1 . . . Ti, reduced expression
t(w) length of w
53.12. C fundamental chamber
w(C) chamber
X = Uw(C) Tits cone
C V dual fundamental chamber
Xv dual Tits cone
Coxeter group
53.14. integrable Lie algebra
Chapter 4.
14.3.
54.7.
54.8.
A
S(A)
o( finite, affine or indefinite type
Dynkin diagram of A
label. of S(A) in Table Air
54.9.
00, · · · ,(It
6 =' (aD, (11 t · .. , at)
9
highest root

356
Index of Notation, Gnd Dtfinition,
Chapter 5.
S5.!. re
A r ,
+
QV
S5.2.
15.3.
55.6.
S5. 7.
S5.8.
15.9.
15.10.
55.11.
15.13.
Chapter 6.
56.0.
56.1.
16.2.
56.3.
56.4.
r(l
set or real roots
set of positive real roots
dual real root
reflection at Q E A".
long and short rea) roots
set of imaginary roots
set of positive imaginary roots
support of Q E Q
im
&im
+
8UPP Q
l.
6 = E O;Q;
;=0
z
null root
imaginary cone
a root basis
of hyperbolic type
Lorentzian lattice
strictly imaginary roots
A
An
'im
affine algebra
v Y
00 , · · · ,at
. hand h Y
Coxeter number and dual Coxeter number
A belongs to Table Aff r
canonial central element
scaling element
normalized invariant form on
affine algebra
r
K
d
(.1.)
Ao
o

X or S-
o
"
o
d
o
Q
o
projection of  e . or S C . on .
d"" tire
.' I
9 = 6 - GOOo
seta of short and long real roots

Indtr of Notatioft6 and DefinitioR8
357
(.,.)
0
16.5. W
'0
M
T
S6.6.
J6.8
normalized invariant form
on finite-dimensional algebra
Wal
Gal
;(ZtY)
group of translations
special vertex of S(A)
affine Weyl group
fundamental alcove
KiJJing form
integral lattice
Chapter 1.
S7.1. (, = C[t.t- I ]
Res P(t)
57.2.
algebra of Laurent polynomials
residue of P(t) E L, at t = 0
nontwisted affine algebra
loop algebra
o
2-cocycle on £:(9)
central extension of £(9)
realization of a non-twisted affine algebra
Lie algebra of regular vector fields on CS"
£(g)
tP
leg)
£(g)
57.3. b = ID Cd j
jel
Vir
- 0
Vir  £(g)
57.4
7.7
17.8.
i7.9.
57.11.
17.12.
Virasoro algebra
semidirect product of Vir
and affine aJgebra
Eo and Fa
zen)
z(z)
6(Zl - Z2)
,"0z
current
formal 6-function
primary field
conformaJ weight
asymmetry function
diagram automorphism
infinite affine matrices
£(atfJ)
II
Aoo, A+ oo ,Boo ,Coo ,Doo
,/00
(foe. boo, coo, d oo

358
57.13.
Indez 01 Notation, IInJ Dtfinition6
4 00 t 6 00 t Coo, doo
o
G(l)
T(z)
Chapter 8.
58.2. .c(g, 0', m) = ED (t i 0 gj modm)
jEI
58.3
18.4.
58.5.
58.6.
58.7.
S8.8.
Chapter 9.
59.1.
59.2.
59.3.
S9.4.
l(g, 11, m)
£(g, (1, m)
6 0 and 9°
t
a -1-. {J
(1 'i.
VJ,
P(V)
II e V, v  0
D() = {p E  .IJJ  A}
o
VA
M(A)
M'(A)
U(b) U(.) V
L(A)
L.(A)
B
(v)
completed infinite rank affine algebras
o
loop group associated to G
energy-momentum tensor
realization of & twisted affine algebra
homogeneous Heisenberg subalgebra
automorphisms of type (8; k)
covering homomorphism
realization or type. of
an affine algebra
quasirationaJ automorphism
rationaJ automorphism
let of weights of a g(A)-module V
weight vector of weight 
category of (certain) g(A)-modules
highest weight module
highest weight
highest weight vector .
Verma module with highest weight A
unique proper maximal submodule of M(A)
induced module
irreducible module with highest weight A
primitive vector
primitive weight
irreducibJe module with lowest weight-A
contravariant bilinear fOfm on L(A)
expectation value or v

19.6. (V : L(p)]
59.7. e(A)
£
chV
S9.10.
(A,ar)
-
S9.11. M()
59.13. .
Rd
S9 .14.
Indtz oj No'tJCionl and Dtfini'ioRI
359
multiplicity of L(JI) in V
Cormal exponential
formal character
highest weight module over g'(A)
label of the weight A
a
trv qA
Vir = Vir _ Ef) Viro E9 Vir +
M(c, h) and L(c, h)
infinite-dimensional Heisenberg algebra
canonical commutation relations representation
vacuum vector
oscillator algebra
triangular decomposition of Vir
c
tr v qd o
conformal central charge
formal character of a Vir-module V
Chapter 10.
SIO.I.
S 10.2.
S J 0.5.
S 10.6.
S 10.8.
J 1 0.9.
p
p+
p++
peA)
-
E
R= n (l_e(_Q»muha
oEA+
l( W )==( -1 )t(w)
K
e
weight lattice
set of dominan.t integral weights
set of regular dominant integral weights
set of weights of the g(A)-module L(A)
(generalized) partition function
chv character of a g(A)-module V
Y(V) region of convergence of chv
Xc cOmplexified Tits cone
Y = {h E I L (multQ)le-(o,h)I < ooJ
aE+
g(A) = EB9j( I)
F
FI
principal gradation of g(A)
specjaJization of type 8
principal specialization

360
Indez of Notation, and Definition,
SI0.10. V = e (.)
jEI+
v= E9 (I)
jEI+
dim. V = L dim (I )qi
.>0
J_
S 10.11. Veal
Chapter 11.
S 11.5 · H ( ., .)
S 11.6.
S 11.9.
511.11.
511.13.
0 1
g(a)
D() and D,()
n re and n im
SA
Chapter 12.
o
S12.1. x
512.2. F
o
d()
00
cp(q) = n (1 - qn)
"=1
S 12.4.
'1
(At K)
Ai
pit t P:
S 12.6.
! 12.7.
max( A)
00
o = E multL(A)( - n6)e- n '
n=O
e
A
mA
X A
mA.

gradation of type, or V
principal gradation of V
q..dimeosion
contravariant Hermitian form on
a g(A)-module
unitarizable g(A)-module
generalized Kac-Moody algebra
basic specialization
Euler's product function
Dedekind's rrfunction
level of A E P+ or of module L(A)
fundamental weights
maximal weight
set of maximal weights of L(A)
classical theta function
modular anomaly of A
normalized character
string function

512.8 ·
J 12.9.
S 12.10.
512.11.
!12.12.
512.14.
Intl% of Notation! tJnd Definition!
361
Uc(g')
Tn
c(k)
h A
A(.)
k(i)
L'
n
.
J, ,ra
L.;i
n
multA(; g)
b(g)
U(A,'\)
T"
Chapter 13.
SI3.1. N
Nz
-- --
S13.2. Th = e Th m
mO
P, = {,\ E '(16) = k,I EM.}
e
'H
D
Th = EB Th m .
rnO
S13.3.
513.4.
r(n)
ro(n)
r,
S = (°1 -0 1 )
T = ( )
Mp2(R)
G = M P2(R)  N
theta function identity
restricted completion of U(g')
Sugawara operators
conformal anomaly
vacuum anomaly
i- th level
Virasoro operators
Dynkin indices
coset Virasoro operators
.
multiplicity of occurrence of L()
in L(A).
.
vacuum palr
branching function
basic gradation
Heisenberg group
algebra of theta functions
classical theta function
upper half-plane
Laplace operator
space of classical theta functions
principal congruence subgroup
metaplectic group

362
S 13.5
S 13.6.
S 13.7.
513.8.
S 13.9.
513.11.
S13.13.
Inder of Notat;ofl$ and Definition,
n = n(M)
Fcrt( T)
q = e 2w .1-
In,m( T)
9 n t m (1')
Th-
m
A
P:
p+
S(t) = (SAtA' )A,A'EP
'A
a(A)
A'
A'.
J
pi
pi
pi
i
M'
W'
9 A
A
X
cA
IjllJ(r) = (Aut S(A» · 0
fJ = -iT
!(fJ)  g(fJ)
least positive integer such that nM. C M
and n(,,11) e 2Z Cor all-, e M.
modular (orm of weight k
and multiplier system X
cusp
meromorphic modular form
holomorphic (orm
R-singular modular form
cusp form
anti-invariant classical theta
(unctions of degree m
mod c,

!13.14.
513.15.
Chapter 14.
514.0.
S14.1.
SI4.2.
514.4.
514.5.
514.6.
Indez 01 No'(Jtiofl tJnd Definitionl
363
R+
a(A)
c(k)
-
Ri
asymptotic dimension
let of equivalence cJasseJJ
of vacuum pairs
d( A, ,\)
conformal 8ubalgebra
11,/2, /3
p(n)
j(r)
c(i)
h(Il)
r..
(I-) (I)
Xr.. and VJ,...
CI,
V,
Nr.,.
classical Mobius function
modular invariant
Clifford algebra
Cl,-module
fusion coefficients
L( Ao )
basic representation of an affine
algebra g(A)
cyc)jc e]ement of I(A)
principal 8ubaJgebra of S(A)
exponent8 of g(A)
r-th Coxeter number of a simple
finite-dimensional Lie algebra"
r-principal gradation of 9
r-cyclic element of 9
r-exponents of ,
vacuum vector
l
.
dim aj
her)
g = ffigj(l;r)
j
E
(r) (r)
m) S. · ·  m N
'+ and ._
differential operator
vertex operator
principal vertex operator
construction
principal realization of g(A)

364
S 14.8.
S 14.9.
514.10.
514.11.
SI4.12.
Indt-z of No'a'ion tJnd Dt-finitionl
E+
r(z)
sequence of positive exponents of g(A)
vertex operators for g(A)
homogeneous vertex
operator construction
vertex operatr
semi.infinite monomial
fermionic rock space
charge
vacuum vector of charge m
set of all partitions
energy
wedging operator
contracting operator
Clifford algebra
spin module
infinite wedge representation
of gloo (resp. GLoe or 0 (0 )
cyclic eJement
principal subalgebra
free bosons
bosonic Fock space
Schur polynomial
boson.fermion correspondence
r a(z)
i 1 /\ 12 " · · ·
F
1m)
Par
.pj
1/Jj
Cl v
,. (resp. R or f)
l E tJ oo
, C 000
Qn
B
S(z)
a
r +(z) and r _(z)
r(ZI.%2)
0 1 = E tPj fl} 1/;j
jEI
P(D)/. 9
vertex operator for 4 00
Hirota bilinear equation
KP hierarchy
Kadomtsev-Petviashvili equation
N801ition solution of KP hierarchy
highest component of L(A) @ L(A)
orthocomplement of Lhilh
group generated by the exp tei, exp t/i
G..orbit of a highest weight vector VA
u", = (u, - tuu - uzu)z
Lhilh
Llow
G
VA

]nde% of Notation. Gntl Dfifti'ionl
365
....... formal completion of VA
VA
Hir = EBHj space of Hirota polynomiaJs
j
T-function
514.13. u, = UU + iu% Korteweg-de Vries equation
KdV -hierarchy
ig, = -gz:c ::!: 21g129 non-linear Schrodinger equation
NLS hierarchy
514.14. Dm Virasoro operators
fJi neutral fermions
Cis
(18 b0800- fermion correspondence
of type B
BKP hierarchy
Eoo group of finite permutations of Z
Bruhat and Birkhoff decompositions
8 6 / 6 ,(%) skew Schur polynomial
Grnt i Grassmanian
Plucker relations
Gr. Gri infinite Grassmanians
MKP hierarchy
:F infinite ftag variety
Miura transformation
Z(u,v)
t- algebra of pseudodifferential
operators
w( Z J z) wave (unction
L
Sato equation
Zakharov-Shabat equation
Toda lattice
A"'8 commensurable subspaces
dim A - dimB relative dimension
End (V t A)
gl(V, A)

366
Indez of Notation, Gntl Definition.
GL(V, A) and GL(V, A)O
End ftn A
-
End(V, A)
GLftn(A)
Gl(V, A) and GL(V, A)O

00
HO(D) and HI (D)
Vx
adeles of an algebraic curve X

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