Infinite-Dimensional Lie Algebras

§1.9.

Bibliographical notes and comments.

The study of Kac-Moody algebras was started independently by Kac [1967], [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) Chevalley introduces a general algebraic 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 [1968] is a simple adaptation of these. The material of §§1.5 and 1.6 is taken from Kac [1968 B]. Exercises 1.4, 1.8 and 1.14 are taken from Kac [1968 B]. A somewhat different approach to the construction of graded Lie algebras was 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 from Kac-Peterson [1987]. The rest of the material of Chapter 1 is fairly standard. The problem of isomorphism of Kac-Moody algebras has been settled quite recently. Namely, as shown by Peterson-Kac [1983], any two maximal ad-diagonalizable subalgebras of a Kac-Moody algebra are conjugate, and hence two Kac-Moody algebras are isomorphic if and only if their Cartan matrices can be obtained from each other by reordering the index set. The question for arbitrary &(A) remains open.

Chapter 2. The Invariant Bilinear Form and the Generalized Casimir Operator

§2.0. In this chapter we introduce two important tools of our theory, the invariant bilinear form and the generalized Casimir operator ft. The operator Q is a "second order" operator which, in contrast to the finitedimensional theory, does not lie in the universal enveloping algebra of g(A) and is not defined for all representations of &(A). However, Q is defined in the so-called restricted representations, and commutes with the action of &(A). Remarkably, one can manage to prove a number of results (including some classical ones) using only Q. §2.1. Note that rescaling the Chevalley generators: e,- H-* e,-, /,- i-+ e,/,(i = l , . . . , n ) where e,- are nonzero numbers, we get a^ »—• e,a^, which extends to an isomorphism f) i—• fj (nonunique, if det A = 0). This extends to an isomorphism: &(A) —• g(DA), where D = diag(ei,..., e n ). An n x n matrix A = (<*ij) is called symmetrizable if there exists an invertible diagonal matrix D = diag(ti,... ,e n ) and a symmetric matrix B = (6 tJ ), such that (2.1.1)

A = DB.

The matrix B is then called a symmetrization of A and Q(A) is called a symmetrizable Lie algebra. Let A be a symmetrizable matrix with a fixed decomposition (2.1.1) and let (f),n,II v ) be a realization of A. Fix a complementary subspace f)" to t)' = X^O*)' in f), and define a symmetric bilinear C-valued form (.|.) on I) by the following two equations: (2.1.2)

(a?\h) = (a,-, h)€i for A € M = 1 , . . . . n;

(2.1.3)

(h'\h") = 0foih',h"€l)".

Since a]', . . . , a£ are linearly independent and since (by (2.1.1 and 2)) we have (2.1.4)

(ay\aj) = bijuej (i,j = 1,..., n),

there is no ambiguity in the definition of (.|.). 16

Ch. 2

The Invariant Bilinear Form

17

LEMMA 2.1. a) The kernel of the restriction of the bilinear form (.|.) to I)' coincides with c. b) The bilinear form (.|.) is nondegenerate on f).

Proof, a) follows from Proposition 1.6. If now for all h G f), we have 0

n

n

c a y

c e a

= (Z) • 1 * « IM = (Z) » 1 « « «»^>)

n

then

c c a !C i l « * t = 0 and hence ct- = 0,

i = 1,.. . , n , proving b).

• Since the bilinear form (.|.) is nondegenerate, we have an isomorphism v : f) —• V defined by

and the induced bilinear form (.|.) on f)*. It is clear from (2.1.2) that (2.1.5)

!/(*}') = *.•«.•>

« = l,...,n.

Hence from (2.1.4) we deduce: (2.1.6)

§2.2.

(<*t|<*j) = &ij =

flijAn

i , i = 1 , . . . , n.

Our next basic result is the following theorem.

THEOREM 2.2. Let g(A) be a symmetrizable Lie algebra. Fix a decomposition (2.1.1) of A. Then there exists a nondegenerate symmetric bilinear C-valued form (,\.) on Q(A) such that:

a) ( . | . ) i s i n v a r i a n t , i . e . ( [ x , y ] \ z ) = ( z | [ y , z ] ) for all x , y y z G b ) ( . |. )k

IS

defined by (2.1.2 and 3) and is nondegenerate.

d) (.lOlfl^+g.* is nondegenerate for a ^ 0, and hence QQ and 0_ a are nondegenerately paired by (. | . ) . e

) [*>y] = (^ly)^" 1 ^)

for

«

€ g a , y € g - a , a G A.

/. Consider the principal Z-gradation (see §1.5)

18

The Invariant Bilinear Form

Ch. 2

N

and set &(N) =

® fy for N = 0,1,

Define a bilinear symmetric

form (. | . ) on 0(0) = f) by (2.1.2 and 3) and extend it to 0(1) by

Then the form (. | . ) on 0(1) satisfies condition a) as long as both [x, y] and [j/,z] lie in 0(1). Indeed, it is sufficient to check that

or, equivalently,

which is true, due to (2.1.2). Now we extend ( . | . ) to a bilinear form on the space &(N) by induction on N > 1 so that (0,|0j) = 0 if | i | , | j | < N and i -h J # 0, and also condition a) is satisfied as long as both [x, y] and [y> z] lie in &(N). Suppose that this is already defined on g(N — 1); then we have only to define (x\y) for x E Q±N,V € Q^N- We can write y = X^[iii,v»], where Ui and V{ i

are homogeneous elements of nonzero degree which lie in &(N — 1). Then [x, U{] € &(N — 1) and we set (2.2.2)

(z|y)

In order to show that this is well defined, we prove that if i,j,s,t EZ are such that |i + i | = |s + *| = W, i + j + « + * = 0, |t|,|i|,|«|,|<| < ^ and Xi e 0,-, Xj G 0 j , x , 6 0,, x, G 0t, then we have (on $(N - 1))

(2.2.3)

(tt*<.*iL*JI*i) = (*.-|[*i,[^, *«]])•

Indeed, using the invariance of ( , | . ) on Q(N — 1) and the Lie algebra axioms, we have

= (Xi \[X,, [Xj, Xt]] + [[Xj, X,), Xt))

Ch. 2

The Invariant Bilinear Form

If now x = X > - X L » have

then b

19

y t h e definition (2.2.2) and by (2.2.3) we

Hence this is independent of the choice of the expressions for x and y. It is clear from the definition that a) holds on &(N) whenever [x,y] and [y, z] lie in &(N). Hence we have constructed a bilinear form (. | . ) on 0 such that a) and b) hold. Its restriction to t) is nondegenerate by Lemma 2.1 b). The form (. | . ) satisfies c) since for h Gfy,x G 0a and y G 0/? we have, by the invariance property: 0 = ([/», x]\y) + (x\[h, y]) = ((a, h) + (/?, h))(x\y). The verification of e) is standard. For x G 0 a j V € 0-a> where a G A, and fc G f), we have ([*,»] - (x\y)u-\a)\h)

= (*|[»,&]) - (*|y)
Now e) follows from b). It follows from b), c) and e) that the bilinear form (. | . ) is symmetric. If d) fails, then, by c), the form (. | . ) is degenerate. Let i = Ker(. | . ) . It is an ideal and by b), we have iPI t) = 0, which contradicts the definition

of 9(A).

a §2.3. Suppose that A = (a,j) is a symmetrizable generalized Cartan matrix. Fix a decomposition (2.3.1)

A=

di*g(e1,...,cn)(bii)?Jsl

where ef- are positive rational numbers and (6,-j) 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 e,"1 and 6^:

By definition, it has a solution over C. Hence, it has a solution over Q. We can assume that A is indecomposable. Then for any 1 < j < n there exists

20

The Invariant Bilinear Form

Ch. 2

a sequence 1 = t'i < %i < • • • < ik-i < i* = ,;' such that ciit,tt+1 < 0. We have: (2.3.2)

a W s + 1 6 i s + 1 = ai.+lti.eim(8

= 1 , . . . , k - 1)

Hence CJ€\ > 0 for all j , completing the proof. From (2.3.2) we also deduce Remark 2.3. If A is indecomposable, then the matrix diag ( e i , . . . , e n ) is uniquely determined by (2.3.1) up to a constant factor. Fix a nondegenerate bilinear symmetric form (. | . ) associated to the decomposition (2.3.1) as defined in §2.1 From (2.1.6) we deduce: (2.3.3)

(a,|a.) > Ofor t = l , . . . , n ;

(2.3.4)

(oti\otj) < 0 for i ^ j ;

(2.3.5)

v

2

_u

,

Hence we obtain the usual expression for the generalized Cartan matrix:

A= We extend the bilinear form (. | . ) from f) to an invariant symmetric bilinear form (. | . ) on the entire Kac-Moody algebra &(A). By Theorem 2.2 such a form exists and satisfies all the properties stated there. It is easy to show that such a form is also unique (see Exercise 2.2). The bilinear form (. | . ) on the Kac-Moody algebra g(A) provided by Theorem 2.2 and satisfying (2.3.3) is called a standard invariant form. §2.4. Let A be a symmetrizable matrix, let &(A) be the associated Lie algebra and let (. | . ) be a bilinear form on &(A) provided by Theorem 2.2. Given a root a, by Theorem 2.2 d) we can choose dual bases {eL* } and {e^}Q} of 0 a and 0_ a , i.e., such bases that (e£'|e~£) = 6ij (t, j = 1 , . . . , mult a). Then for x G 0 a and y € 0-a we have (2-4.1)

(x|y)

«

Ch. 2

The Invariant Bilinear Form

21

The following lemma is crucial for many computations: LEMMA

2.4. If a, /? G A and z G 0/?_<*, then we have in &(A) ® g(A)

(2.4.2)

£ «« 9 [z,e«] =

/ We define the bilinear form (. | . ) on g(A) ® g(i4) by: (x y|xi ® j/i) = (x|xi)(y|j/i). Pick e G 0 a and / G 0-/?. It suffices to check that pairing both sides of (2.4.2) with e ® / gives the same result. We have:

by Theorem 2.2 a) and (2.4.1). Similarly,

Applying again Theorem 2.2 a) gives the result.

• COROLLARY

2.4. In the notation of Lemma 2.4, we have

(2.4.3)

£[e ( _tM a ')]] = - Q M - J ] - ^ ] in 9(A),

(2.4.4)

£ e<_'i[z, eW] = 8

Proof. Apply to (2.4.2) the linear maps from g(A) ® g(v4) to g(^4) and to U(&(A))t defined by x ® y i-+ [x, y] and x ® y i-* xy, respectively. D

22

The Invariant Bilinear Form

Ch. 2

§2.5. Let Q(A) be a Lie algebra associated to a matrix A, f) the Cartan subalgebra, 0 = 0 0 a the root space decomposition with respect to f). A a

$(.4)-module (resp. g'(j4)-module) V is called restricted if for every v £V, we have $Q(v) = 0 for all but a finite number of positive roots a. It is clear that every submodule or quotient of a restricted module is restricted, and that the direct sum or tensor product of a finite number of restricted modules 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 (. | . ) is a bilinear form provided by Theorem 2.2. Given a restricted 0(^4)-module V, we introduce a linear operator Q on the vector space V, called the (generalized) Casimir operator, as follows. First, introduce a linear function p £ f)* by equations

If detA = 0, this does not define p uniquely, and we pick any solution. It follows from (2.1.5 and 6) that

(2.5.1)

( p K ) = i(a.-K-)

(i = l , . . . , n ) .

Further, for each positive root a we choose a basis {e a } of the space 0 a , and let { e _ a } be the dual basis of g_ a . We define an operator fio on V by

One could easily check that this is independent of the choice of bases (see Exercise 2.7). Since for each v G V, only a finite number of summands ei^eL (v) are nonzero, fl0 is well defined on V. Let i/ 1 ,ix 2 ,... and ti1, ti 2 ,... be dual bases of f). The generalized Casimir operator is defined by

We record the following simple formula: (2.5.2) which is clear from (2.5.3)

A =

Ch. 2

The Invariant Bilinear Form

23

We make one more simple computation. For x G 0 a one has u

i>

x

\

=

I

Hence, we have I ^ « ' « , - , x I = x((a\a) + 2i/" 1 (a)) for x G 0 a .

(2.5.4)

§2.6. to fj:

Consider the root space decomposition of U(&(A)) with respect

?, where r] = (/?, A)i for all h G f)}. Put £/^ =

U(Q'(A))

fl C/^, so that U(gf(A)) = (&UL

Now we prove the

following important theorem: THEOREM

2.6. Let &(A) be a symmetrizable Lie algebra.

a) IfV is a restricted &'(A)-module and u G U'Q, then: (2.6.1)

[Qo, ti] = -ti (2(p\a) + (a\a) + 2i/" 1 (a)) .

b) If V is a restricted fl(i4)-moduie; then 12 commutes with the action of 0(A) on V. Proof, b) follows immediately from a) and (2.5.4). If a) holds for u G U'a and tii G £/£, then it holds for uu\ G C^4+^[Qo, uui] = [Qo, ti]tii + tipo, tii = -ti(2(p|a) + (a|a) a) + (a|a) (a

24

The Invariant Bilinear Form

Ch. 2

Hence, since eai1 e-ai (i = 1 , . . . , n) generate Q'(A), it suffices to check (2.6.1) for u = eai or e_ a i . Applying (2.4.4) to z = eai and using Lemma 1.3, we have:

= 2[e.ai>eai)ea,

+2

Thanks to (2.5.1), this is (2.6.1) for u = eQi. Similarly, [QOie^ai] = 2e-.cr.[ecrt, e_ai] = 2e_ o r .i/- 1 (a t ), which, by (2.5.1), is (2.6.1) for u = e_ai. D COROLLARY 2.6. If, under the hypotheses of Theorem 2.6 b), there exists v G V such that e,(v) = 0 for all i = 1 , . . . , n, and h(v) = (A, A)t; for some AGl)* and ail ft G f), (2.6.2)

n(«) = (A + 2p\A)v.

If, furthermore, U{${A))v = V, then (2.6.3)

n = (A

Proo/. Formula (2.6.2) follows from the definition of Q and formula (2.5.2). Formula (2.6.3) follows from (2.6.2) and Theorem 2.6.

• Remark 2.6. One can define the Q-gradation U(g'(A))

= $ K

and the

map i/" 1 : Q —• fy; without using the Lie algebra 0(^4). (Here, as in Remark 1.5, the symbol Q denotes the free abelian group on generators ori, . . . , an.) Indeed, the Q-gradation of U($'{A)) is induced by the Qgradation of Qf(A) defined in Remark 1.5. If A is symmetrizable, we fix a decomposition (2.1.1) and define i/" 1 as a homomorphism of abelian

groups such that i/~ 1 (a i ) = e~la( for infinite n as well.

(i = 1 , . . . ,n). These definitions work

Ch. 2

The Invariant Bilinear Form

25

§2.7. Let A be an n x n-matrix over R. Let (()«, II, IIV) be a realization of the matrix A over R, i.e., f)n is a vector space of dimension 2n — I over R, so that (f) := C ®R J)R, n, IIV) is the realization of A over C. We define the compact form l(A) of g(A) as follows. Denote by u0 the antilinear automorphism of g(A) determined by:

o;0(/i) = -h for A G 1)R. We call u>0 the compact involution of 0(^4). The existence of u>0 is proved by the same argument as that of a; in §1.3. Then l(A) is defined as the fixed point set of o;0; this is a real Lie algebra whose complexification is &(A). Note that this definition of the compact form coincides with the usual one in the finite-dimensional case. Now let A be a symmetrizable matrix over R and let (. | . ) be a standard form on &(A). Define a Hermitian form on g(A) by:

(x\y)0 := -(uo(x)\y). Theorem 2.2 implies the following properties of this Hermitian form. The restriction of (. |. )o to Qa is nondegenerate for all a G AU{0}; (0a|0/?)o = 0 if a ^ /?; the operators ad u and — adu;o(ti) for u G &(A) are adjoint to each other, i.e., ([u,z]|j/) 0 = -(x|[o;o(w), y])0 for all x, y G &(A); in particular, the restriction of (. |. )o to t(A) is a nondegenerate invariant R-bilinear form. We will return to the study of the Hermitian form (. |. )o after developing some representation theory. §2.8. The following variation of the above results is very useful for applications. PROPOSITION 2.8. Let g be a Lie algebra with an invariant non-degenerate bilinear form ( . | . ) , let {#,•} and {j/,} be dual bases (i.e. (x,- \yj) = 6ij), and let V be a ^-module such that for every pair of elements u,v G Vf Xi(u) = 0 or yi(v) = 0 for all but a finite number of i. Then the operator

is defined on V ®V and commutes with the action of g. Proof.

(2.8.1)

W e h a v e t o s h o w t h a t for e v e r y : G 0 o n e h a s :

]£([z>x«] ® » + Xi ® t z '^) = °'

26

The Invariant Bilinear Form

Ch. 2

Write: [z, x,-] = ]T) a t ;Xj, [2, y,] = YlPjiVj 5 taking inner product of the first i i equation with j/y and the second with Xj> we obtain: *«•; = ([*>**]!«*).

Pv = ([*.VI-]|*J)-

Using in variance of (. | . ) , we deduce:

It follows that otij = —Ajj proving (2.8.1). D

§2.9. Here we consider the most degenerate example, the Lie algebra 0(0) associated to the n x n zero matrix (including n = oo). In this case [*i,ei] = 0, [/,-,/;•] = 0, [ajj] = Sirf, (i,j = l , . . . f n ) , so t h a t

n

The center of 0(0) is c = J2 C a v . Furthermore, dim f) = 2n and one can choose elements di, . . . , dn £ f) such that

and One defines a nondegenerate symmetric invariant bilinear form on the basis of 0(0) by:

(efl/.) = 1, (a?|*) = l> M

t h e others

= °-

Note that p = 0 and the Casimir operator is

Set Ci = ^C(a t ^ - aj) C c. Then the Lie algebra 0;(O)/ci is a Heisenberg Lie algebra of order n, i.e., it has a basis e,-, /,- (i = 1 , . . . ,n), z, such that [eiyfj] = iij2r (i,,; = 1 , . . . ,n), and all the other brackets are zero.

Ch. 2 §2.10.

The Invariant Bilinear Form

27

Exercises.

2.1. The matrix A = (afj) is symmetrizable if and only if aij = 0 implies aji = 0 and a

«i»2a«2«3 • • • a«k«i

=

a

i 3 si a i3«3 • • • a«i«j. f ° r

a

^ l"i» • • •»**•

#.#. Show that the bilinear form (. | . ) is uniquely defined by properties a) and b) of Theorem 2.2. 2.3. Let ( . | . ) be a nondegenerate symmetric invariant bilinear form on %{A). Show that the matrix A is symmetrizable, that (. |. )lb ls defined by (2.1.2 and 3) for some choice offy",and that (. | . ) satisfies all the properties of Theorem 2.2. 2.4- Let 0 = 0 0j be a Z-graded Lie algebra, which is generated by 0_i -f t

0o + 0i. Show that an invariant symmetric bilinear form on the subspace 0-1 + 00 + 01 such that (0« |0j) = 0 whenever i + j / 0, can be (uniquely) extended to such a form on the whole 0. (Here the property of invariance is understood to hold whenever it makes sense.) 2.5. Let (. |. )i and (. |. )2 be two nondegenerate symmetric invariant bilinear forms on 0(A) and assume that A is indecomposable. Show that there exists an automorphism <j> of g(A) which leaves 0;(-4) pointwise fixed and preserves I), such that the bilinear forms (x|y)i and ((x)\{y))2 are proportional. Show that any two invariant bilinear forms on Qf(A) are proportional. 2.6. Show that the adjoint representation of 0(A) is restricted if and only if dim g(A) < oo. 2.7. Let {xi} and {y,} be bases of n+ and n_, dual with respect to an invariant bilinear form on 0. Show that fio = 25^y,x,- is independent of the choice of these bases. Prove an analogous statement for f)22.8. Let 0 = 0(J4) be a simple finite-dimensional Lie algebra. Choose a basis v i , . . . , Vd of 0 and the dual basis v 1 , . . . , vd with respect to the form (. | . ) . Set Q = ^2 ViV%. Show that Q coincides with the Casimir operator defined in §2.5. Show that p is the half-sum of positive roots of 0. 2.9. Show that the 0(>l)-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 0(A)-module.

28

The Invariant Bilinear Form

Ch. 2

2.10. Let 0o be a finite-dimensional Lie algebra with the bracket [, ]o and a nondegenerate invariant symmetric bilinear form ( . | . ) . Let d be a derivation of the Lie algebra 0o such that (d(x)\y) = — (x\d(y)) for x,y € 0o- Set 0 = 0o 0 Cc 0 Crf, where c and d are some symbols, and define a bracket [,] on 0 b y : [x,y] = [z,y] 0 + (<*(*)|y)c for x,y G 0o, [c, x] = 0 for x e 0; [d, x] = d(x). Check that this is a Lie algebra operation. Extend the bilinear form ( . | . ) from 0o to 0 by setting (d\d) = (c\c) = 0, (c\d) = 1, (c| 0 o) = 0, (d|flo) = 0. Show that this is a nondegenerate symmetric invariant bilinear form on 0. 2.11. Let 0 be a solvable n-dimensional Lie algebra with a nondegenerate invariant symmetric bilinear form. Show that either 0 is an orthogonal direct sum of (n — l)-dimensional and 1-dimensional Lie algebras, or 0 can be constructed as in Exercise 2.10 from an (n — 2)-dimensional Lie algebra 00-

[Let i be a 1-dimensional ideal in 0. Show that i lies in the center (using the fact that coadjoint orbits are even-dimensional). If i is isotropic, consider i 1 and the Lie algebra 0O = i 1 /*-]

§2.11.

Bibliographical notes and comments.

Theorem 2.2 is due to Kac [1968 B] and Moody [1968]. For finite-dimensional 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 Q was introduced by Kac [1974]. The idea of its definition is borrowed from physics. We take the usual definition of the Casimir operator (see Exercise 2.8):

(

a>0

)

t

i

we rewrite it by using commutation relations:

ft = £ „-!(«) + 2 £ £ e« «W + £ Uiu\ or>0

cr>0

t

t

Ch. 2

The Invariant Bilinear Form

29

and then replace the first summand, which makes no sense, by a "finite" quantity 2t/~1(p). The proof of Theorem 2.6 is along the lines 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" Casimir operators was constructed by Kac [1984], using ideas of Feigin-Fuchs [1983A].

Chapter 3. Integrable Representations of Kac-Moody Algebras and the Weyl Group

§3.0. In this chapter we begin a systematic study of the Kac-Moody algebras. Recall that this is the Lie algebra g(A) associated 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 classical Weyl group in the finite-dimensional theory. However, in contrast to the finitedimensional case, W is infinite and the union of the W^-translates of the fundamental chamber is a convex cone, which does not coincide with the whole "real" Cartan subalgebra ^R. §3.1. Let us first make some remarks on the duality of Kac-Moody algebras. Let A be a generalized Cartan matrix; then its transpose i A is again a generalized Cartan matrix. Let (f),II,II v ) be a realization of A; then (f)*,II v ,II) is clearly a realization of xA. So, if (g(A), f),II,n v ) is the quadruple associated to A, then (0(*A), f)*,IIv,II) is the quadruple associated to %A. The Kac-Moody algebras Q(A) and 0(*J4) are called dual to each other. Note that the dual root lattice of

is the root lattice of Q^A). Furthermore, denote by A v C Qv (C f)' C fy) the root system A(*A) of 0( t A). This is called the dual root system of Q(A). In contrast to the finite-dimensional case, there is no natural bijection between A and A v . §3.2. Recall some well-known results about representations of the Lie algebra sli (C). Let

be the standard basis of 5^2 (C). Then

[e,f] = h, [h,e) = 2e, [h,f] = -2f. 30

Ch. 3

Integrable Representations and the Weyl Group

31

By an easy induction on Jfc we deduce the following relations in the universal enveloping algebra of (3.2.1) (3.2.2) LEMMA

[*,/*] = - 2 * / * , [h,ek] = [e, /*] = - * ( * - I)/*" 1 4- it:/*"1/*. 3.2. a) Let V be an st2(C)-module and let v eV be such that h(v) = At; for some A G C.

Set Vj = (i!)- 1 /'"^). Then: (3.2.3)

h(vj) =

(\-2j)vj.

If, in addition, e(v) = 0, then: (3.2.4)

e(vj) = (\-j

+

l)vj-1.

b) For each integer Jfe > 0 there exists a unique, up to isomorphism, irreducible (k + l)-dimensional ^(C)-module. In some C-basis {vj}!j=0 of the space of this module the action ofs£2(C) looks as follows: h(Vj) = (* - 2j)Vj; f(Vj) = (j + l) W i + 1 ;

e(Vj) = (k + 1 - ; > , _ ! .

Here j = 0 , . . . , k and we assume that Vk+i = 0 = v_i. Proof. Formulas (3.2.3 and 4) follow from (3.2.1 and 2). Let now V be an irreducible (fc-fl)-dimensional s^2(C)-module. Let u G V be an eigenvector of h with eigenvalue /i. It follows from (3.2.1) that if es(u) / 0, then it is again an eigenvector for h with eigenvalue \i + 2s. Since V is finite dimensional, there exists v = es(u) ^ 0, such that e(v) = 0 and h(v) = Xv. We set Vj = (j!)""1/-7^). As V is finite dimensional, we deduce from (3.2.4) that A is a nonnegative integer, say m, that {VJ} are linearly independent for j = 0 , . . . , m and that vm +i = 0. Hence m = Jfc and b) follows.

•

32

Integrable Representations and the Weyl Group

Ch. 3

§3.3. Let g(A) be a Kac-Moody algebra and let e,-, /,• (i = 1 , . . . , n) be its Chevalley generators. Set 0(t) = Ce f -+Ca v + C/i; then 0(,) is isomorphic to «^2(C), with standard basis {e,-,a^,/,}. We can deduce now the following relations between the Chevalley generators:

(3.3.1)

(ad et)l-**ej = 0; (ad ftf-ufi

= 0, if i # j .

We prove the second relation; the first one follows by making use of the Chevalley involution u> (see (1.3.4)). Denote v = /,-, 0,;- = (ad/j) 1""^'/^. Consider 0(>1) as a 0(,)-module by restricting the adjoint representation. We have: a

Hv)

= - a t j v l e «( v ) = 0 if i ^ j .

Hence Lemma 3.2 a) together with the properties (Cl) and (C2) of A imply «ij - (1 - *«) + l ) ( a d / 0 " f l y / i = 0 if i ± j . It is also clear that e* commutes with 0,j if k £ i, k ^ j (by relations (1.2.1)), and also if ib = ,;' but at;- ^ 0. Finally, if Ar = j and atJ- = 0 we have:

o (by (cs)). So, [cjbjfltj-] = 0 for all Ar and we apply Lemma 1.5.

§3.4. Now we need a general fact about a module V over a Lie algebra g. One says that an element x G 0 is locally nilpotent on V if for any v EV there exists a positive integer N such that xN(v) = 0. LEMMA 3.4. a) Let y\, j/2> • • • ^ a system of generators of a Lie algebra 0 and iet x 6 0 be such that (adx)Niyi = 0 for some positive integers N{, 1 = 1,2, — Then ad x is locally nilpotent on Q.

b) Let vi, i>2, • • • be a system of generators of a Q-module V, and let x E 0 be such that ad x is locally nilpotent on 0 and xNi(vi) = 0 for some positive integers Ni, i = 1,2, Then x is locally nilpotent on V. Proof. Since ad x is a derivation of 0, one has the Leibnitz formula:

(ad.)*!**] = E ())[(ad«)'y,(ad«)*-'*], •=0 ^ ^

Ch. 3

Integrable Representations and the Weyl Group

33

proving a) by induction on the length of commutators in the y,-. b) follows from the following formula (for A = \i = 0):

(3.4.1) (x-\-v)

a= > [ )((?idx-\)sa)(x-LLr~s,

k > 0, A,/xGC,

which holds in any associative algebra. In order to prove (3.4.1), note that adx = Lx — RXi where Lx and Rx are the operators of left and right multiplication by x, and that Lx and Rx commute (by associativity). Now we apply the binomial formula to Lx — A — /i = (ad x — A) + (Rx — fi).

a Applying the binomial formula to adx = Lx — i^, we obtain another useful formula (in any associative algebra):

(3.4.2)

§3.5.

(adx)*a =

LEMMA 3.5. adet- and ad/,- are locally nilpotent on

Proof. By relations (1.2.1) and (3.3.1), we have: (ade t -) |fliil+1 x = 0 = (ad/<) |a « l+1 a?, if x is e;- or ft. Also, (adc,-)2& = 0 = (ad/,-)2& if A € t Now we can apply Lemma 3.4a).

• §3.6. A g(^)-module V is called fy-diagonalizable if V" = 0 A 6 b * VA, where VA = {v G V||A(t;) = (A,A)t; for A G f)}. As usual, VA is called a weight space, A G I)* is called a weight if KA ^ 0, and dimV^ is called the multiplicity of A and is denoted by multv A. Similarly, one defines an f)'-diagonalizable ^'(^-module, its weights, etc. An f)-(resp. fj'-Jdiagonalizable module over a Kac-Moody algebra g(A) (resp. $'(A)) is called integrable if all et- and /; (i = 1 , . . . ,n) are locally nilpotent on V. Note that the underlying module of the adjoint representation of a KacMoody algebra is an integrable module by Lemma 3.5. The following important proposition justifies the term "integrable."

34

Integrable Representations and the Weyl Group

PROPOSITION

Ch. 3

3.6. Let V be am integrable %(A)-module.

a) As a g^y module, V decomposes into a direct sum of finite dimensional irreducible ^-invariant modules {hence the action o/0(«) on V can be "integrated" to the action of the group SX2(C)). b) Let A G fy* be a weight ofV and let at- be a simple root of g(A). Denote by M the set of all t G Z such that A + t<*i is a weight of V, and let mt = multv(A -f ta,-). Then (i) M is the closed interval of integers [—p,q], where p and q are both either nonnegative integers or oo andp — q = (A,a^) when both p and q are finite; i/multv A < oo, then p and q are finite; (ii) a : V\+t0Ci -+ Vx+(t+i)Qi «'* an injection fort G [-p,-%(*,otl[)); in particular, the function t >-+ mt increases on this interval; (iii) the function t\-+mt is symmetric with respect to t = —^ (iv) if both A and A + t*i are weights, then ei(V\) ^ 0. Proof By (3.2.2) we have: (3.6.1)

eif?(v) = Ar(l - Ar-h (A, a^))/*" 1 W + / * * (

We deduce that the subspace

is (0(,-) + ^-invariant. As et- and /,- are locally nilpotent on V, dimU < oo. By the Weyl complete reducibility theorem applied to the 0(t)-module Uy the latter decomposes into a direct sum of finite dimensional ^-invariant irreducible 0(,)-modules (cf. Exercise 3.11). So, each v G V lies in a direct sum of finite dimensional ^-invariant irreducible 0(,)-modules, and a) follows. For the proof of b) we use a) and Lemma 3.2 b). Set U = J2 V\+kai\ this is a (0(t) + f))-module, which is a direct sum of finite dimensional irreducible ($(,•) -f ^-modules. Let p = — inf M, q = supM. Both p and q are nonnegative as 0 G M. Now all the statements of b) follow from Lemma 3.2 b), as (A + ta,-,^) = 0 for t = - | ( A , a / ) .

• The following corollary of Proposition 3.6 b) (i) and (iii) is very useful. COROLLARY 3.6. a) If A is a weight of an integrable Q(A)-module V and A + a,- (resp. A — &i) is not a weight, then (A, a^) > 0 (resp. (A, a^) < 0).

Ch. 3

Integrable Representations and the Weyl Group

b) If A is a weight ofV, multiplicity.

35

then A — (A,a^)af- is also a weight of the same

Remark 3.6. Let V be an integrable g'(A)-module. Then, clearly, Proposition 3.6 and Corollary 3.6, with f) replaced by f)', still hold. Furthermore, the local nilpotency of e,- and /, on V guarantees that V is /i,-diagonalizable, and hence Jj'-diagonalizable provided that n < oo. This follows from a general fact which will be proved in §3.8. §3.7. Now we introduce the important notion of the Weyl group of a Kac-Moody algebra &(A). For each i = 1 , . . . , n we define the fundamental reflection rt of the space f)* by rt.(A) = A - ( A , a ) > < > A G Ij*. It is clear that r, is a reflection since its fixed point set is TJ = {A G f)1(A,c*y) = O}, and r,(a t ) = -<*<. The subgroup W of GL(f)*) generated by all fundamental reflections is called the Weyl group of &(A). We will write W(A) when necessary to emphasize the dependence on A. The action of r, on fy* induces the dual fundamental reflection r/ on I) (for the dual algebra 0(M)). 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. 3.7. a) Let V be an integrable module over a Kac-Moody algebra &(A). Then multv A = multv w(X) for every A G fy* and w G W. In particularf the set of weights ofVis W-invariant. PROPOSITION

b) The root system A of $(A) is W-invariant, and mult a = mult w(a) for every a G A, w G W. The following fact will be needed later. 3.7. If a G A + and r t (a) < 0, then a = a,-. In other words, A+\{a t -} is r,--invariant. LEMMA

Proof follows from Lemma 1.3.

36

Integrable Representations and the Weyl Group

Ch. 3

§3.8. In this section we outline a somewhat different approach to the Weyl group. Let a be a locally nilpotent operator on a vector space V. Then we can define the exponential 1

1

exp a := Iv + yya + —a2 + . . . , which has the usual properties, in particular, exp ka = (exp a)*

(Jfc G Z).

Let 6 be another operator on V, such that (&da)Nb = 0 for some N. Then one knows the following formula: (3.8.1)

(exp a)6(exp - a ) = (exp(ad a))(6).

This easily follows by using formula (3.4.2). LEMMA 3.8. Let IT be an integrable representation of space V. For i = 1 , . . . , n set

Q(A)

on a vector

rf = (exp/,)(exp-ef)(exP/«)Then a) r?(Vx) = Vri(x) b)rfeAutQ(A)

Proof. Let v G Vx. Then h(r?(v)) = (\,h)rf(v) if (a,-,h) = 0. Hence for a) and c) we have only to check that <*?(r*(v)) = —(\,aY)r*(v). This follows from

(r i T 1 aJ"'* = -«*V-

(3.8.2)

By (3.8.1) it is sufficient to check (3.8.2) only in the adjoint representation of s^(C); using (3.8.1) again, one has to check (3.8.2) in the 2-dimensional natural representation of ^ ( C ) . But in this representation we have

,= (J J),exp(-eO=(J ~ 1 1),rf=(J ~ and (3.8.2) is clear. b) follows from (3.8.1) applied to the adjoint representation.

•

Ch. 3

Integrable Representations and the Weyl Group

37

Remark 3.8. Let (K, TT) be an integrable g(A)-module whose kernel lies in f). By Proposition 3.6, the action of the subalgebra 0(,) (i = l , . . . , n ) on V can be integrated to a representation 7rt- : SX2(C) —> GL(V). The groups 7r,(SX2(C)) generate a subgroup GT in GL(V). The group GT can be viewed as an "infinite dimensional" group associated to the Lie algebra Q(A). The elements r* = TT,-( J "^) (i = 1 , . . . , n) generate a subgroup WT C G*. The group W* contains an abelian normal subgroup Dr generated by {rj)2 (i = 1 , . . . ,n) such that W(A) ~ W*/D*. We conclude this subsection by the proof of a general fact a special case of which was mentioned in §3.6. One says that an element x of a Lie algebra 0 is locally finite on a 0-module V if any v G V lies in a x-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)* b is finite-dimensional. 3.8. Let g be a Lie algebra and let V be a g-module such that 0 is generated by the set Fy of all ad-iocaily finite elements which act locally finitely on V. Then PROPOSITION

a) 0 is spanned over C by Fy. In particular, if g is generated by the set F of all ad-iocaiiy finite elements, then g is spanned by F. b) If dim 0 < oo, then V is ^-locally finite (i.e. any element ofV lies in a finite-dimensional Q-submodule of V). Proof. Let &y denote the C-span of Fy and let a G Fy. Using (3.8.1) we deduce that Qy is invariant with respect to automorphisms exp<(ada). Since lim((exp*ada)6 - b)/t = [a, 6], it follows that [a, &y] C 0v, proving a), b) follows from a) and the Poincare-Birkhoff-Witt theorem.

• §3.9. Let A be a symmetrizable generalized Cartan matrix and let (.|.) be a standard invariant bilinear form on PROPOSITION

3.9. The restriction of the bilinear form (.|.) to f)* is W-

invariant. Proof. As |r,(e*,)|2 = |-c*i| 2 = |a,| 2 ^ 0, it suffices to check that (A|a t ) = 0 implies (r,(A)|a t ) = 0. But r,(A) = A by (2.3.5). D For a converse statement see Exercise 3.3.

38

Integrable Representations and the Weyl Group

Ch. 3

§3.10. We return to the study of the Weyl group W of a Kac-Moody algebra &(A). Let us start with the following technical lemma. LEMMA s

(1 <

s

(3.10.1)

3.10. If on is a. simple root and rtl . . . ^(c*,) < 0 then there exists < 0 such that r tl . ..ri§ . ..ritri

= rh .. .r.-.^r,-.^ . . . r lV

/. Set /?* = r,-fc+1 . ..rt< (c*t) for Jfc < 2 and /?* = a,-. Then, by the hypothesis, /?o < 0; on the other hand, pt > 0. Hence, for some s we have: &_i < 0, p9 > 0. But /?,_i = r tf (/?,); hence by Lemma 3.7, /?, = a,,, and we obtain: (3.10.2)

ai$ = w(ai), where w = r,-,+1 . . . rt-f.

But (3.10.3)

u;(a.) = aj (weW)=>

ti;(aV) = a / .

Indeed, M; = ti)|^ for some ti) from the subgroup in Aut g generated by rf*1 (i = l , . . . , n ) (see §3.8). Applying w to both sides of the equality [0<*i>0-<*J = Ca,y, we obtain (by Lemma 3.8 b)) Ciy(a^) = CaJ. Since w(ai)(w(aY)) = (a.-,ttty> = 2, we get ^ ( a y ) = aj. Now we can conclude from (3.10.2 and 3) that rit = tor,-*!;"1. Multiplying both sides of this by r tl . . . r,-f_l on the left and by r «.+i • • • r»«r» o n ^ n e r ^h^ completes the proof.

• §3.11. The expression w = r tl ...r t -, 6 W is called reduced if s is minimal possible among all representations of w G W as a product of the r,-. Then s is called the length of u; and is denoted by £(w). Note that detj,* rt- = — 1 and hence (3.11.1)

detj,. w = (-I)'**) for ti; 6 W.

The following lemma is an important corollary of Lemma 3.10.

Ch. 3

Integrable Representations and the Weyl Group

39

3.11. Let w = r tl . . . r,-f G W be a reduced expression and let <*,• be a simple root. Then we have

LEMMA

a) £(wri) < £(w) if and only ifwfai)

< 0.

b) ti;(of
c) (Exchange condition) If £(wr{) < £(w), then there exists s, such that 1 < s < t and r

«. r ».+i • • • r*<

=

r

«.+i • • •

r

itri'

Proof. By Lemma 3.10 (applied to it;), u;(a t ) < 0 implies that £{wri) < £(w). If now w(a{) > 0, then iuri(<*t) < 0 and hence £(w) = £(wrf) < £(wri), proving a), b) follows immediately from a). Finally, if £(wri) < £(w), then a) implies iu(a t ) < 0 and applying Lemma 3.10 to w we deduce the exchange condition from (3.10.1), multiplying it by (r»i • • n.-i)"" 1 on the left and by rt- on the right.

• §3.12. Now we are in a position to study the geometric properties of the action of the Weyl group. Recall the definition of f)u C f) from §2.7. Note that f)n is stable under W since Q v C f)R. The set C={he

i)R|(a,-, h) > 0 for i = 1 , . . . , n}

is called the fundamental chamber. The sets w(C), w 6 W, re called chambers, and their union

= |J w(C) is called the Tits cone. We clearly have the corresponding dual notions of C v a n d X v in f)J. 3.12. a) For heC, the group Wh = {w e W\w(h) = h} is generated by the fundamental reflections which it contains. PROPOSITION

b) The fundamental chamber C is a fundamental domain for the action of W on X, i.e., any orbit W • h of h £ X intersects C in exactly one point. In particular, W operates simply transitively on chambers. c) X = {h £ ^ R | ( « , h) < 0 only for a finite number of a € A+}. In particular, X is a convex cone. d) C = {h G f)g| for every w G W, h — w(h) = JZ c * a ^ where c,- > 0}.

40

Integrable Representations and the Weyl Group

Ch. 3

e) The following conditions are equivalent:

(i) \W\ < oo; (ii) * = *)*;

(iii) |A| < oo; (iv) |A v | 0 and therefore (w(oti9)9h') > 0. But by Lemma 3.11 b), w(otit) < 0, and hence

(w(aig)}hf) < 0. So, (w(ai9),h') = 0 and (ait,h) = 0. Hence rit(h) = h and both a) and b) follow by induction on £(w). Set X' = {h G f)R|(o,^) < 0 only for a finite number of a G A+}. It is clear that C C X' and it follows from §3.7 that X' is r,-invariant. Hence Xf D X. In order to prove the reverse inclusion, let h G X' and set Mh = {a G A+|(a,/i) < 0}. By definition, |M*| is finite. If Mh ^ 0, then at- G M^ for some t. But then it follows from Lemma 3.7, that |Mr.(/i)| < |M/»|. Induction on |M^| completes the proof of c). The inclusion D of d) is obvious. We prove the reverse inclusion by induction on s = £(w). For £(w) = 1, d) is the definition of C. If £{w) > 1, et w = rai . . . r fV We have /i - w(h) = (h - r tl . . . r.-^^A)) + r %i • • • n,_i(A — r»,(A)) and we apply the inductive assumption to the first summand and Lemma 3.11b) for A v to the second summand, completing the proof. Now we prove e). (i) =*> (ii) since an element h! of W • h with maximal height of (hf — h) lies in C. In order to show (ii) => (iii) take h in the interior of C. Then (a, —h) < 0 for all a G A + and hence |A| < oo by c). (iii) =£* (i) because of (3.12.1)

{w(a) = a for w G W and all a G A} => w = 1.

To prove (3.12.1), note that if a reduced expression w = r tl . . . rt< is nontrivial, then Lemma 3.11 b) implies that w(ctit) < 0, contradiction. The fact that (iv) is equivalent to (i) follows by using the dual root system. Finally, to prove f) we may assume that h G C. Then f) follows from a) by applying the equivalence of e)i and e)ii to Wh operating on f)/C/i.

• Note that, by Proposition 3.12 b), the Weyl group W operates simply transitively on chambers, the stabilizer of a point from the interior of a chamber being trivial.

Ch. 3 §3.13. relations

Integrable Representations and the Weyl Group

41

Recall that the group with generators r i , . . . , r n and defining

r? = 1 (i = 1 , . . . ,n); (ri^)"1" = 1 (i,j = 1 , . . . ,n)

(3.13.1)

is called a Coxeter group. Here m,j are positive integers or oo (we use the convention x°° = 1 for any x). 3.13. The group W is a Coxeter group, where the m,j are given by the following table: PROPOSITION

0

1

2

3

>4

2

3

4

6

oo

Proof. First we check relations (3.13.1). The relation r? = 1 is obvious. Furthermore, the subspace t := Rat- +Rc*j is invariant with respect to both r,- and rj, and the matrices of rt- and ry in the basis a», Qfj of t are l~Q ~^tJ J and I _ a .. _l J. Hence the matrix of r,^- in this basis is (3.13.2)

f-l V

+ *Mi -«i<

a,A -1/

and we obtain (3.13.3)

det tinrj - XI) = A2 + (2 - ayait-)A + 1.

Hence r^j has an infinite order if a^aji > 4 and the order is given by the table in the rest of the cases (f)* = t © {A 6 f)*|(A,a}') = (A, a)) = 0} if 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.

• §3.14.

Exercises.

3.1. Let A be a symmetrizable matrix and let A = DB be a decomposition of the form (2.3.1). Show that xA is also symmetrizable; more explicitly % A = DVBV, where Dv = Z?"1, £ v = DBD. Show that the corresponding standard forms on I) and f)* (defined in §2.1) induce each other.

42

Integrable Representations and the Weyl Group

Ch. 3

3.2. Let e, / , h be the standard basis of s/2(C). Show that one has in the enveloping algebra: T m

fnl

min(m,n) min^m,r»;

_ -, .j /n fn

/ _ ^ _

R

_

Deduce that if / is locally nilpotent on a s/2(C)-module V, then Ve is A-diagonalizable. [Prove this for m = 1 by induction on n, then use induction on m]. 3.3. There exists a nondegenerate symmetric W-invariant bilinear form on I) if and only if A is symmetrizable. Any such form can be extended from f) to an invariant nondegenerate symmetric bilinear form on the entire Lie algebra &(A). This form satisfies all the conclusions of Theorem 2.2. 3.4- Let V be an integrable g(^4)-module. Show that

3.5. The set A+ is uniquely defined by the following properties: (i) n C A+ C Q+; 2ai $ A+ for i = 1 , . . . , n; (ii) if a E A + , a z/z a,-, then the set {a -f koti,k G Z} fl A+ is a "string" {a - poti,..., a + ga t }, where p, q G Z+ and p - q = (a, a / ) ; (iii) if a 6 A+\II then a — a, G A + for some i. 3.6. Prove that ^(w) = \{a G A + |u;(a) < 0}|. 3.7. Let J4 = f b ~£ J be a generalized Cartan matrix of rank 2, and assume that ab > 4. Show that the Weyl group W^(^4) is an infinite dihedral group. Let ab > 4; then (xa\ + ya2\xa\ + yc*2) = bx2 — abxy + ay2 is a Winvariant quadratic form on f£, and XVU-XW = {A G bil(A|A) < 0}U{0}. / 2 -2

0 \

3.#. Let i4 = ( -2 2 - l ), and let W = W(A) be the associated Weyl \ o -l

2 /

group. Show that the map rx t-+ ( J - i ) ' r 2 ^ induces an isomorphism 7r : W —• PGL2(Z).

("Q1 j ) »r3 ^

(j j)

[Use the fact that PSL,2(JL) is generated by elementary matrices and 7r(ri)7r(r3)]. .S'.P. Let A = (dij) be the matrix of Exercise 3.8, f) the Cartan subalgebra of g(A),ai,<22>a3 simple roots. Define a standard bilinear form on f) by (ai\aj) = dij. Set 71 = - a i - a 2 , 72 = ^ 1 , 73 = - a i - a 2 - a 3 . Define

Ch. 3

Integrable Representations and the Weyl Group

43

a map \i : f)* —• S^C) (symmetric 2 x 2 matrices) by fi(aj\ + 672 + C73) = (ba/2 6 f ) , and define the action of PGL2(1) on 5 2 (C) by g(S) = gS^g). Check that /x is a W-equivariant map, and that (<*]<*) = —2det/i(a) for a E l ) * . Using Proposition 3.12 b), (5.10.2) and Exercise 3.8, deduce that a quadratic form ax2 + bxy + cy2 such that a,6,c G Z, 4ac > b2 and a > 0 can be transformed by GL,2(Z) to a unique quadratic form such that a > c > b > 0. 3.10. In this exercise we outline a geometric proof of Proposition 3.13. Let W be the Coxeter group on generators r,- (i = l , . . . , n ) and relations (3.13.1), and let TT : W —• W be the canonical homomorphism. We construct 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,x) ~ (w2)y) O {x = y and w^lW2 lies in the subgroup of W generated by those r,- which fix x}. Define an action of W on U by: w(wi,x) = (wwi,x). This is obviously well defined. Show that there exists a unique continuous W'-equivariant map : U —• X such that <£(l,x) = x for x G C (W operates on X via TT). Let Y = {x\x is fixed by at least three reflections from W} and set X1 = X \ y , f / ; = U\4>~l(Y). Show that ' :U' -^ X1 is a covering map. Deduce that <j>' is a homeomorphism and hence TT is an isomorphism. 3.11. Show that for an irreducible l)-diagonalizable module over s£2(C) all weight spaces are 1-dimensional. Classify these modules. Classify the ones which are integrable. Prove that every finite-dimensional s/2(C)-module is completely reducible. [Use the Casimir operator and Exercise 3.2]. 3.12. Let w = r tl .. .r tf be a reduced expression of w £ W. Let /? G A+ be such that w~l((5) < 0; show that the sequence /?, r tl (/?), r^r,-^/?), • •. contains a unique simple root, say <Xj(p)- Let A Gfy*;show that A-u;(A)=

[Use the identity A — w\r,-(A) = (A — wi(A)) 4- tx;i(A — r,(A)) and induction ont.] 3.13. Show that the action of r ^ on 0(^4) preserves every invariant bilinear form and commutes with the Chevalley involution.

44

Integrable Representations and the Weyl Group

Ch. 3

3.14. Show that 0 -* c -> g'(A) -> fl;(i4)/c -> 0 is the universal central extension of g'(j4)/c, where A is a generalized Cartan matrix. [Let

gf(A)/c be an epimorphism with a finite-dimensional kernel. Then ip~l(ei + /* + (I)7C)) is isomorphic to g* 0 (), where & ~ s^2(C) and fj is commutative, and is completely reducible on g. Let e», aY, fi € g^ be the preimages of ei,ot( ,f%\ show that they satisfy relations (1.2.1) and hence generate a quotient $' of Qf(A) by a central ideal. Show that g is a direct sum of g' and a central ideal from £)]. 3.i«5. Show that InteriorX = {A G I)RK<*> A) < 0 only for a finite number of a G A+}. 3.16. An element x of a Lie algebra g is called locally finite if ad x is locally finite on g. Put 0fin = linear span of {x G g|x is locally finite }. For a 0-module V, put Vfin = {v G F|dim J2 Cz*(v) < 00 for every locally finite element x of g}. Show that 0fin is a subalgebra of g and Vfin is a 0-submodule of V. 3.17. Let 0 be a Lie algebra such that g = 0fin; then g is called integrable. Let V be a g-module such that V = Vfin; then V is called integrable. Let G* be a free group on generators 5 := {x G g|x is locally finite }. Given an integrable representation TT of g on V, define a representation /(TT) of G* on V by I(*)(x) = J ] ^77r(x)n> * € 5. Let AT* be the intersection of kernels n>0

n

*

of all /(TT), where TT runs over all integrable g-modules. Put G = G*/N*; denote by exp x the image of x G S under the canonical homomorphism G* —• G. The group G is called the group associated to the integrable Lie algebra g. Show that if g is a simple finite-dimensional Lie algebra, then G is the associated connected simply connected Lie group, and x »-• expx 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 §3.6.) 3.18. Show that any Kac-Moody algebra g(^4) is integrable. (As shown in Peterson-Kac [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 g be an integrable Lie algebra and G the associated group. Then G acts on the space of every integrable representation V of g, so that exp x is locally finite on V for every x G S (defined in Exercise 3.17). Open problem: let V be a G-module such that every exptx is locally finite on V and is a differentiate function in t on every finite-dimensional x-invariant subspace. Show that V can be "differentiated" to an integrable g-module.

Ch. 3

Integrable Representations and the Weyl Group

45

3.20. Let A be the extended Cartan matrix of a complex connected simply 0

connected algebraic group G. Show that the group G associated to the 0

Kac-Moody algebra 0(J4) is a central extension of the group

G{C\t,t~1]).

n

3.21.

Let 0 be the Lie algebra of polynomial vector fields £ Pii&~

sucn

«=i

that YJ fir** =

cons

t - Show that 0 is an integrable Lie algebra and that the

i

associated group G is a central extension of a group of biregular automorphisms of C n . Show that 0 is generated by all locally finite elements of the Lie algebra of all polynomial vector fields. 3.22. Show that the group G* described in Remark 3.8 depends, up to isomorphism, only on the Z-span of the set of weights of x. Show that if the 0(j4)-module V is irreducible then the kernel of the adjoint action of W* on Q(A) lies in { ± M 3.23. Let q be a complex number ^ 0, ± 1 . Let Uq($t2) denote an associative algebra on generators e,/, a and a"1 and the following defining relations: aa~l

= a~la = 1,

aea~l — q2e,

afa~l — q~2f,

Let k E Z+ and let V be the (Jfc + l)-dimensional vector space with a basis t>o» • • •, v*. F ° r i € N let \j] = (gJ — q~~*)/(q —tf"1)-Show that formulas (we assume v_i = 0, vk+i = 0) : e(v,-) = ± [ t - t + l]wf-_i, /(v,-) = [i + l]v,+i, a(v,-) = ±qk~2%Vi define a representation of the algebra Uq(st2) on V. Show that this representation is irreducible if and only if q is not an n-th root of unity, where n < Jfc, and that, under this assumption, these are the only two irreducible (Jfc + l)-dimensional (7y(s^2)-niodules. Show that as q —• 1, we get the (Jfc + l)-dimensional irreducible representation of s^(C) (or rather of U(s£2(C))) described by Lemma 3.2b). 3.24. Let for s G Z and j £ N : [a; s] = (aq° - a-lq-*)/(q

- q~l) and

L^ J Show that we have the following identity in Uq(s(,2) (which is a "^-analogue" of that of Exercise 3.2):

[

^

/I

V~^

/

fa;—m —n+2,;l

^

[m)V [n)\\ j ^ [n - ; ] ! I > J [m - j]\ 3.25. (Open problem.) Classify all simple Z-graded (by finite-dimensional subspaces) integrable Lie algebras.

46

§3.15.

Integrable Representations and the Weyl Group

Ch. 3

Bibliographical notes and comments.

The Weyl group of a Kac-Moody algebra was introduced by Kac [1968 B] and Moody [1968]. The exposition of §§3.6-3.9 follows mainly Kac [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 "symmetrizable" case. In the framework of general groups generated by reflections it was introduced by Vinberg [1971] (see also Looijenga [1980]). The exposition of §§3.10-3.13 follows mainly Kac-Peterson [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 (cf. 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 [1975], and Tits [1981], [1982]). Various aspects of the theory of Kac-Moody groups may be found in Arabia [1986], Kumar [1985], [1987A], [1988], KostantKumar [1986], [1987], Mathieu [1986C], [1988], [1989A], Slodowy [1985B], Tits [1985], [1987], Bausch-Rousseau [1989], and others. The algebra Uq(si2) constructed in Exercise 3.23 is the simplest example, first considered by Kulish-Reshetikhin [1983], of a "quantized Kac-Moody algebra" 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

§4.0. In order to develop the theory of root systems of Kac-Moody algebras 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 nxn matrix A = (a,j) which satisfies the following three properties: (ml) A is indecomposable; (m2) aij < 0 for i ^ j \ (m3) aij = 0 implies a,-,- = 0. 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 column vector u = *(ui, W2,...) we write u > 0 if all U\ > 0, and u > 0 if all Ui > 0. §4.1. Recall the following fundamental fact from the theory of linear inequalities: A system of real homogeneous linear inequalities A,- > 0, i = 1, . . . , m, has a solution if and only if there is no nontrivial linear dependence with nonnegative coefficients among the A,-. We shall use a slightly different form of this statement. LEMMA 4.1. If A = (a^) is an arbitrary real mx s matrix for which there is no u>0, u ^ 0, such that (*J4)II > 0, then there exists v > 0 such that Av <0.

Proof Set A,- = Y2<*ijXj, where the Xj form the standard basis of the dual of R*. Then the lemma is a consequence of the "fundamental fact" for the system of inequalities: r-At->0

(i=l,...,m)

• 47

48 §4.2.

A Classification of Generalized Carian Matrices

Ch. 4

We need one more lemma.

LEMMA 4.2. If A satisfies (ml)-(m3), then Au > 0, u > 0 imply that either u > 0 or u = 0.

Proof. Let j 4 u > 0 , t i > 0 , t i ^ 0 . We reorder the indices so that u, = 0 for i < s and tit- > 0 for i > s. Then by (m2) and (m3), Au > 0 implies that a,j = cy,- = 0 for i < s and j > s, in contradiction with (ml).

• §4.3.

Now we can prove the central result of the chapter.

4.3. Let A be a real n x n matrix satisfying (ml), (m2), and (m3). Then one and only one of the following three possibilities holds for both A and i A: THEOREM

(Fin) det A ^ 0; there exists u > 0 such that Au > 0; Av > 0 implies v > 0 or v = 0; (Aff) coranki4 = 1; there exists u > 0 such that Au = 0; Av > 0 implies Av = 0; (Ind) there exists u > 0 such that ylti < 0; At; > 0, v > 0 imp/y v = 0. Proof Replacing t; by — t; we obtain that in cases (Fin) and (AfF) there is no v > 0 such that Av < 0 and Av ^ 0. Therefore each of (Fin) and (AfF) is not compatible with (Ind). Also (Fin) and (AfF) exclude each other because rank A differs. Now we shall show that each A together with %A is of type (Fin), (Aff), or (Ind). Consider the convex cone KA = {u | Au > 0}. By Lemma 4.2, the cone KA can cross the boundary of the cone {u \ u > 0} only at the origin; hence we have KA n {u | u > 0} C {u | u > 0} U {0}. Therefore, the property (4.3.1)

KAn{u\u>0}t{0}

is possible only in the following two cases: 1) KA C

{U

I u > 0} U {0}, or

2) KA = {u | Au = 0} is a 1-dimensional subspace.

Ch. 4

A Classification of Generalized Carian Matrices

49

Now, 1) is equivalent to (Fin); indeed, det A^ 0 because KA does not contain a 1-dimensional subspace. Clearly, 2) is equivalent to (AfF). We also proved that (4.3.1) implies that there is no u > 0 such that Au < 0, An ^ 0. By Lemma 4.1 it follows that if (4.3.1) holds, then both A and % A are of type (Fin) or (Aff). If (4.3.1) does not hold, then both A and %A are of type (Ind), again by Lemma 4.1.

• Referring to cases (Fin), (AfF), or (Ind), we shall say that A is of finite, affine, or indefinite type, respectively. 4.3. Let A be a matrix satisfying (ml)-(m3). Then A is of finite (resp. affine or indefinite) type if and only if there exists a > 0 such that Aot > 0 (resp. = 0 or < 0). COROLLARY

§4.4. We proceed to investigate the properties of the matrices of finite and affine types. Recall that a matrix of the form (aij)ijes, where 5 C { l , . . . , n } , is called a principal submatrix of A = (a,j); we shall denote it by As- The determinant of a principal submatrix is called a principal minor. LEMMA 4.4. If 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 5 C { 1 , . . . ,n} and let As be the corresponding principal submatrix. For a vector u, define us similarly. Now, if there exists u > 0 such that Au > 0, then Asus > 0 and = 0 only if a,j = 0 for i G S, j
• §4.5. LEMMA 4.5. A symmetric matrix A satisfying (ml)-(m3) is of finite (resp. affine) type if and only if A is positive-definite (resp. positivesemidefinite 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 < 0. The cases (Fin) and (AfF) are distinguished by the rank. Now let A be of finite or affine type. Then there exists u > 0 such that Au > 0. Therefore, for A > 0 one has: (A + XI)u > 0, hence A + XI is of finite type by Theorem 4.3. Hence det(^4 + XI) ^ 0 for all X > 0 and all the eigenvalues of A are nonnegative.

•

50

A Classification of Generalized Carian Matrices

Ch. 4

§4.6. LEMMA 4.6. Let A = (ay) be a matrix of finite or affine type such that an = 2 and a^aji = 0 or > 1. Then A is symmetrizable. Moreover, if (4.6.1)

a,-lt-3a,-ajs.. .aj,j, ^ 0 for some distinct fi, ...,*', with s > 3,

then A is of the form

/

2

-til

2 0 0

\ ~

0 0

0

2 ,-i u n-l

2

)

where ui, ..., un are some positive numbers such that u\... un = 1. Proof. It is clear that the second statement implies the first one (cf. Exercise 2.1). Suppose now that (4.6.1) holds. Taking the smallest possible s in (4.6.1), we see that there exists a principal submatrix B of A of the form: 2

-

0 0

2

)

By Lemma 4.4, B is of finite or affine type and therefore by Theorem 4.3 there exists u > 0 such that Bu > 0. Replacing B by the matrix (diagu)~ 1 5(diagii), we may assume that *ti = ( l , . . . , l ) . But then Bu > 0 implies that the sum of the entries of B is nonnegative:

(4.6.2)

Since 6,-6J- > 1, we have 6t- + b\ > 2; hence, by (4.6.2) we obtain that 6t- + &(• = 2 and therefore 6,- = 6{ = 1 for all i. As in this case det B = 0, Lemmas 4.4 and 4.5 imply that B = A.

D

Ch. 4

A Classification of Generalized Cartan Matrices

51

§4.7. We proceed to classify all generalized Cartan matrices of finite and affine type. For this it is convenient to introduce the so-called Dynkin diagrams. Let A = (a«j)Jj= i be a generalized Cartan matrix. We associate with A a graph S(A), called the Dynkin diagram of A as follows. If a ij<*ji < 4 and |a,j| > |a Jf |, the vertices i and j are connected by |a t; | lines, and these lines are equipped with an arrow pointing toward i if |a,j| > 1. If a a ij ji > 4, the vertices i and j are connected by a bold-faced line equipped with an ordered pair of integers |a t ;|, \aji\. It is clear that A is indecomposable if and only if 5(^4) 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 of finite type if and only if all its principal minors are positive. b) A is of affine type if and only if all its proper principal minors are positive and det A = 0. c) If A is of finite or affine typef then any proper subdiagram of S(A) is a union of (connected) Dynkin diagrams of finite type. d) If A is of finite type7 then S(A) contains no cycles. If A is of affine type and S(A) contains a cycle, then S(A) is the cycle Ay from Table Aff 1. e) A is of affine type if and only if there exists S > 0 such that AS = 0; such a 6 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 §2.3). Now a) and b) follow from Lemma 4.5. c) follows from Lemma 4.4, d) follows from Lemma 4.6, e) follows from Theorem 4.3.

• §4.8. type.

Now we can list all generalized Cartan matrices of finite and affine

4.8. a) The Dynkin diagrams of all generalized Cartan matrices of finite type are listed in Table Fin. THEOREM

52

A Classification of Generalized Cartan Matrices

Ch. 4

b) The Dynkin diagrams of all generalized Cartan matrices of affine type are listed in Tables Aff 1-3 (all of them have £ + 1 vertices). c) The numerical labels in Tables Aff 1-3 are the coordinates of the unique vector 6 = t (ao,fli,..., at) such that AS = 0 and the a,- are positive relatively prime integers. Proof. First, we prove c). Note that AS = 0 means that 2at- = Y^mjaj

f° r

j

all i, where the summation is taken over the j's which are connected with i, and rrij = 1 unless the number of lines connecting i and j is s > 1 and the arrow points toward i; then ra;- = 5. Now c) is easily checked case by case. It follows from c) and Proposition 4.7 e) that all diagrams in Tables Aff 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 remains to show that if A is of finite (resp. affine) type, then S(A) appears in Table Fin (resp. Aff). This is an easy exercise. We do it by induction on n. First, det A > 0 immediately gives (4.8.1) A2i C2 and G2 are the only finite type diagrams of rank 2; Ay' and A2 are the only affine type diagrams of rank 2. (4.8.2) ^ 3 , B3I C3 are the only finite type diagrams of rank 3; A2\ G(21}, D^ A^\ D^ are the only affine type diagrams of rank 3.

C2 ,

Furthermore, by Proposition 4.7 d) we have (4.8.3) S(A) has a cycle, then S(A) = A^\ By Proposition 4.7 and inductive assumption we have (4.8.4) Any proper connected subdiagram of S(A) appears in Table Fin. Now let 5(^4) be of finite type. Then 5(^4) does not appear in Tables Aff 1-3, has no cycles 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 D/, Ee, E*?, Eg. Similarly we show that if S(A) is not simply-laced (i.e., has multiple edges), then it is Bi, Ct, F4, or G2. A simply-laced diagram with no cycles and branch vertices is At.

Ch. 4

A Classification of Generalized Carian Matrices

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.

• We fix once and for all an enumeration of the vertices of the diagrams of generalized Cartan matrices A in tables below as follows. In Table Fin vertices are enumerated by symbols a i , . . . , a/. Each diagram Xj1' of Table AfF 1 is obtained from the diagram Xt by adding one vertex, which we enumerate by the symbol ao, keeping the enumeration of the rest of

TABLE Fin At

Bt

o —o

—o — o Of/—i at

o —o

— O =£> O

Ofi

Ct

Dt

Of/-i

aa

(2)

at

O — o ofi aa

— O <= O or/-i at

(2)

o —o

— O — O

(4)

ai

aa s

*

O — O — o -- o — o

E7

o —o

4

Ofi

aa

O?3

F^

O — 0

=>o

a4

O?3

(3)

a5

T

G2

- o — o — o

o — o — o -- O — 0 — O — 0

0f3

o ^ c>

a* — O O?4

as

a«

(2)

(1) (1) (1)

54

A Classification of Generalized Carian Matrices

Ch. 4

TABLE Aff 1

>2) "~

1 1

3) 2)

1 1

o —o— O — 1 2 2

1

2

2

2

2

1

°1

4)

o—o—o— 1 2 2

O —O

2

1

O — O ^

1

2

O —O —

1

2

3

3

• O—O

4

2

Pi o—o—o—o—o 1 2 3 2 1

O — O — O — O—O—O—O

1

2

3

4

3

2

1

O3 O — O — O — O—O—O—O—O

1

2

3

4

5

6

4

2

the vertices as in Table Fin. Vertices of the diagrams of Tables Aff 2 and Aff 3 are enumerated by the symbols oro,...,»/. The numbers in parentheses in Table Fin are det A. The numerical labels in Tables Aff are coefficients of a linear dependence between the columns of A.

Ch. 4

A Classification of Generalized Cartan Matrices

55

TABLE Aff 2 a0

oro

o r < _ , or.

<*s

0'<=O oi a0

1 2

O =*O a/_i a^

3

2

1

O - O - O < = O - O

TABLE AflF 3 or0

§4.9. We conclude the chapter with the following characterization of Kac-Moody algebras associated with generalized Cartan matrices of finite type (cf. Proposition 3.12 e)). 4.9. Let A be an indecomposable generalized Cartan matrix. Then the following conditions are equivalent:

PROPOSITION

(i) A is a generalized Cartan matrix of finite type; (ii) A is symmetrizable and the bilinear form (. |. )j, B is positive-definite; (iii) \W\ < oo; (iv) |A| < oo; (v) &(A) is a simple finite-dimensional Lie algebra; (vi) there exists a G A+ such that a + at- £ A for all i = 1, . . . , n. Proof. By Lemma 4.6 and Proposition 4.7, the bilinear form (.|.)&« is positive-definite if A is of finite type; thus, (i) =^ (ii). If (ii) holds, then by Proposition 3.9, the group W lies in the orthogonal group O((. | . ) ) and hence is compact. Since W preserves the lattice Q, it is discrete. Hence \W\ < oo, which proves (ii) => (iii). The implication (iii) => (iv) follows

56

A Classification of Generalized Carian Matrices

Ch. 4

from Proposition 3.12 e); (iv) =»• (v) is clear by Proposition 1.7 a), and for (v) => (vi) we can take a to be a root of maximal height. Finally, let a + ot{[ ^ A for all i. Corollary 3.6 a) implies that (a, a?) > 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 (a,a^) = 0 for all i. But in this case, by Lemma 1.5, a — at- £ A+ for some i. Hence, if A is of affine type, Proposition 3.6 b) implies that <* + <*,- G A+, a contradiction. This proves the last implication (vi) =» (i).

• Remark 4-9. A root of a finite root system A which satisfies condition (vi) of Proposition 4.9 is called a highest root. It is easy to see that there exists a unique such root (cf. Proposition 6.4 in Chapter 6); it is given by the formula t 0 =

where a,- are the labels of the extended Dynkin diagram from Table Aff 1. §4.10.

Exercises.

4-1. An indecomposable generalized Cartan matrix A is said to be of strictly hyperbolic type (resp. hyperbolic type) if it is of indefinite type and any connected proper subdiagram of S(A) is of finite (resp. finite or affine) type. Show that a matrix ( *h ~° j , such that a and 6 are positive integers, is of finite (resp. affine or strictly hyperbolic) type if and only if ab < 3 (resp. ab = 4 or ab > 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.) 4.2. Let A be of type TVA%r (p > q > r), i.e., let its Dynkin diagram be of the form

Set c = i + i + i . Then A is of finite (resp. affine or indefinite) type if and only if c > 1 (resp. c = 1, c < 1). Show that for c < 1, the signature of A

Ch. 4

A Classification of Generalized Carian Matrices

57

is (+ H r —)- 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 det A = pq + pr + qr- pqr. [To prove the statement about the signature, delete the branch point.] 4*3. Introduce the following diagrams with n vertices (n > 5): En{— Tn -3,3,2)

o—o—• • • —o—o—o—o

AEn

0—0

JO£jn

O ^ = O — • • • —O — O — O — O

CEn

o =£• o — •• • — o — o — o—o

DEn

o — o—

0—0—0

?

o — o —o—o

Show that #10, BE\o, C#io, and DE\o are all hyperbolic matrices of rank 10. Show that a hyperbolic matrix of rank 7, 8, or 9 is one of the following list: r 4 , 3 ,3, T5>4,2, AEn, BEn, CEn, DEn (n = 7,8, or 9). 4-4- Classify all (strictly) hyperbolic matrices. 4-5. Show that E\Q is the only symmetric hyperbolic matrix with determinant — 1. Show that d i a g ( l / 2 , 1 , . . . , l)(BE\o) is an integral symmetric matrix with determinant —1. 4-6. Show that if A is symmetrizable and hyperbolic, then the corresponding "symmetrized" matrix has signature (4- H h —). 4- 7. Let A be a strictly hyperbolic matrix. Show that the group Aut X of all linear automorphisms of the Tits cone X acts with a compact fundamental domain. 4-8. 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 -2 2 -1 - 1 - 1 2 provides an example of a cone in R3 such that Aut X has a compact fundamental domain, but does not act transitively on the interior of X.

58

A Classification of Generalized Carian Matrices

Ch. 4

4.10. Using Exercise 1.4, show that if A is an indecomposable generalized Cartan matrix and A is not from Table Aff, then the Lie algebra $!(A)/t is simple. 4.11. Let A = (ciij) be a symmetric matrix. Set t* = 5^a«*c*> where Ck are nonzero numbers. Check that

Assume that atJ- < 0 for i ^ j . Deduce that if there exist positive numbers ci, . . . , c n such that ** = 0 (Jfc = 1 , . . . , n), then the quadratic form J2 aikti€k is positive-semidefinite. Use this to prove Proposition 4.7 e). Dei,k

duce also that this quadratic form with an indecomposable A is positivedefinite if and only if there is no c\ > 0, . . . , cn > 0 such that tk < 0 (*=l,...,n). 4A2. Let 5 ' and S" be affine Dynkin diagrams. Let A be a generalized Cartan matrix with the Dynkin diagram

S(A)=\ Show that det A = 0. 4-13. Show that an indecomposable generalized Cartan matrix is affine if and only if it is degenerate and all its principal minors are nonnegative. 4.14- Show that the following is a complete list of connected Dynkin diagrams of generalized Cartan matrices of infinite order such that any principal minor of finite order is positive: AOQ

o—o—o— •••

A+oo

O-O-o

BOQ

O ^ = O —O — O— • • •

CQO

O = » O —O — O — • • •

£>oo

§4.11.

O — O — O— O

Bibliographical notes and comments.

Theorem 4.3 is due to Vinberg [1971], as well as most of the results of §§4.44.7. The list of affine diagrams appears in Kac [1967] and Moody [1967]. The notation appears naturally in the context of the theory of finite order automorphisms (Kac [1969 A]), as we shall see in Chapter 8. Exercises 4.8 and 4.9 are taken from Vinberg-Kac [1967]. Exercise 4.11 is taken from Bourbaki [1968]. The rest of the exercises are fairly standard.

Chapter 5. Real and Imaginary Roots

§5.0. In this chapter we give an explicit description of the root system A of a Kac-Moody algebra &(A). Our 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. §5.1. A root a G A is called real if there exists w G W such that w(a) is a simple root. Denote by Are and A+c the sets of all real and positive real roots respectively. Let a 6 A r c ; then a = u;(a,-) for some a,- G II, w 6 W. Define the dual (real) root a v G A V r c by a v = w(ot{). This is independent of the choice of the presentation a = w(cti). Indeed, we have to show that the equality ti(c*,-) = otj implies tx(afy) = aj for u G W. But this is (3.10.3). Thus we have a canonical W-equivariant bijection A r c —• A V r c . By an easy induction on ht a, one shows, using Proposition 5.1 e) below, that a > 0 if and only if a v > 0. We define a reflection ra with respect to a G A r c by

Since (a, a v ) = 2, this is a reflection, and since ra = wriw"1 if a = tu(at ), it lies in W. Note that r a . is the fundamental reflection r,. The following proposition shows that real roots have all the "classical" properties. PROPOSITION

5.1. Let a be a real root of a Kac-Moody algebra &(A).

Then a) mult a = 1. b) Jba is a root if and only if k = ± 1 . c) If (3 G A then there exist nonnegative integers p and q related by the equation

such that 0 + ka € A U {0} if and only if -p < k < q, k € Z. 59

60

Real and Imaginary Roots

Ch. 5

d) Suppose that A is symmetrizable and let (.|.) be a standard invariant bilinear form on 0(^4). Then (i)

(a|a)>0; if a = £ Jb,at, then ibf(af|a,) G (a|a)Z.

(iii)

e) Provided that ±<x £ II, there exists t such that |htr t (a)|< |hta|. Proof. All the statements a)-d) are clear if a 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) follow from Proposition 3.9. Statement d) (iii) follows from the fact that a v G ^2 Za^ and the following formula: (5.1.1) Finally, suppose the contrary to e); we may assume that a > 0. But then —a G C v and, by Proposition 3.12 d) for the dual root system, —a + w(a) > 0 for any w G W. Taking w such that w(a) £ II, we arrive at a contradiction.

• The following lemma will be needed in the sequel. 5.1. Suppose that A is symmetrizable. Then the set of all a = »<*« eQ such that

LEMMA

(5.1.2)

ki(ai\ai) £ (a\a)l

for all i

is W-invariant. Proof. It suffices to check that r,(a) again satisfies (5.1.2), i.e., that (k{ — (a\oii))(ai\ai) G (a|a)Z. This is equivalent to (5.1.3)

2(a|a,.) G (a|a)Z,

which follows from (5.1.2): 2(a|aO =

Ch. 5

Real and Imaginary Roots

61

Let A be a symmetrizable generalized Car tan matrix, and let (.|.) be a standard invariant bilinear form (see §2.3). Then, given a real root a we have (ck|a) = (a,|a,) for some simple root a,-. We call a a short (resp. long) real root if (a\a) = min,-(a,-|at-) (resp. = max t (a,|a f )). These are independent of the choice of a standard form. Note that if 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 ra-f 1 different square lengths since an arrow in S(A) points to a shorter simple root. It follows that if A is a nonsymmetric matrix from Table Fin, then every root is either short or long. Furthermore, if A is a nonsymmetric matrix from Table Aff, and A is not of type A^ with / > 1, then every real root is either short or long; for the type A^t w ^ n ' > * there are real roots of three different lengths. In this chapter, we shall normalize (.|.) such that the (a,*|at-) are relatively prime positive integers for each connected component of S(A). For example, if A is symmetric, then (a,|a,) = 1 for all i, and all real roots are short (= long). §5.2. A root a which is not real is called an imaginary root Denote by A*m and A+ 1 the sets of imaginary and positive imaginary roots, respectively. By definition, A = Arc U Afm

(disjoint union).

It is also clear that A i m = A1^1 U (-A1^71). The following properties of imaginary roots are useful. PROPOSITION

5.2. a) The set A^ 1 is W-invariant.

b) For a G A 1 ^ there exists a unique root (3 G - C v (i.e., (/?,a^> < 0 for all i) which is W-equivalent to a. c) If A is symmetrizable and (.|.) is a standard invariant bilinear formt then a root a is imaginary if and only if(a\a) < 0. Proof. As A+" C A+\II and the set A + \ { a t } is r,-invariant (by Lemma 1.3), it follows that A^71 is r,-in variant for all i and hence W-in variant, proving a). Let a 6 A+" and let /? be an element of minimal height in W - a C A + . Then -/? G C v . Such a /? is unique in the orbit W • a by Proposition 3.12 b), proving b). Let a be an imaginary root; we may assume, by b), that - a G C v (since (.|.) is W-invariant). Let a =

62

Real and Imaginary Roots

Ch. 5

kt > 0; then (a|a) = £ * , ( a | a , ) = £ iKfJMa.aV) < 0 by (2.3.3 and 5). t

t

The converse holds by Proposition 5.1 d).

§5.3.

For a = 5^*«a* € Q

we

^ e fi ne the support of a (written supp a)

to be the subdiagram of S(A) which consists of the vertices i such that jfet- ^ 0, and of all the edges joining these vertices. By Lemma 1.6, supp a is connected for every root a. Set: K = {a G Q+\{0} | (a, a)') < 0 for all t and supp a is connected}. LEMMA 5.3. In the above notation, K C A$?. Proof. Let a = £ i,at- 6 /C. Set i

« a = { T e A + 17 < a } . The set fta is finite, and it is nonempty because the simple roots, which appear in the decomposition of a, lie in f2a . Let ft = J^ mi&i be an element i

of maximal height in Q a . It follows from Corollary 3.6 a) that (5.3.1)

supp/? = supp a.

First, we prove that a £ A + . Suppose the contrary; then a ^ j3. By definition: (5.3.2)

P + otit A+ if *,- > mi.

Let A\ be the principal submatrix of A corresponding to supp a. If A\ is of finite type, then {a G Q+ | («,Qf^) < 0 for all t} = {0} and there is nothing to prove. If Ai is not of finite type, then, by Proposition 4.9, we have (5.3.3)

P := {j € supp a | Jb;- = m ; } ^ 0.

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)

( / ? , ^ ) > 0 it

ieR.

Set /?' = ^2 mtof,-. Since supp a is connected, (5.3.1) and (5.3.4) imply (5.3.5)

(/?,a,?) > 0 if i e R;

((3', <*/) > 0 for some j £ R.

Therefore, by Theorem 4.3, the diagram R is of finite type.

Ch. 5

Real and Imaginary Roots

63

On the other hand, set

Since suppa' is a connected component of supp(a — /?), we obtain that (5.3.6)

(<*',<**} = (« - P><*i) for i G # .

But (a, o%) < 0 since a 6 AT, hence by (5.3.4 and 6):

This contradicts the fact that R is of finite type. Thus, we have proved that a G A + . But 2a also satisfies all the hypotheses of the lemma; hence 2a G A+ and, by Proposition 5.1 b), a G Ay 1 . D

§5.4. roots.

Lemma 5.3 yields the following description of the set of imaginary

THEOREM 5.4.

A^1

=

\J

w(K).

Proof. Lemma 5.3 and Proposition 5.2 a) prove the inclusion D. The reverse inclusion holds by Proposition 5.2 b) and by the fact that suppa is connected for every root a (by Lemma 1.6).

• §5.5. The following proposition shows that the properties of imaginary roots differ drastically from those of real roots. 5.5. Ifa£ A1™ and r is a nonzero (rational) number such that ra G Q, then ra G A i m . In particular, na G A i m ifn G Z\{0}.

PROPOSITION

Proof. By Proposition 5.2 b) we can assume that a G —C v fl Q+; since a G A, it follows that supp a is connected and hence a G K. Hence ra G K for r > 0 and therefore, by Lemma 5.3, ra G A i m .

•

64

Real and Imaginary Roots

§5.6.

Ch. 5

Now we prove the existence theorem for imaginary roots.

THEOREM

5.6. Let A be an indecomposable generalized Cartan matrix.

a) If A is of finite type, then the set A%m is empty. b) If A is of affine type, then

t

where 6 = ^ <*,•«,-, and the at- are the labels ofS(A) in Table Aff. t=0

c) If A is of indefinite type, then there exists a positive imaginary root a = J2 *«a« suc^ that k > 0 and (a, a^) < 0 for all i = 1, . . . , n. i

Proof Recall (see Chapter 4) that the set {a G Q+ | (a,<*,Y) < 0, i = 1 , . . . , n} is zero for A of finite type, is equal to 26 for A of affine type, and there exists a = E *«a« s u c n that *« > 0 and (a, a?) < 0 for all i if ^4 is of t

indefinite type. The theorem now follows from Theorem 5.4.

• §5.7. It follows from Proposition 4.7 that if a is a null-root, i.e., a G A(A) is such that a\^' = 0, then suppa is a diagram of affine type which is a connected component of S(A) and a = kS, k € Z. We proceed to describe the isotropic roots. PROPOSITION 5.7. Let A be symmetrizable. A root a is isotropic (i.e., (a\a) = 0) if and only if it is W-equivalent to an imaginary root (5 such that supp/? is a subdiagram of affine type of S(A) (then (3 = k6).

Proof Let a be an isotropic root and let (.|.) be a standard bilinear form. We can assume that a > 0. Then a G A^ 1 by Proposition 5.1 d), and a is W-equivalent to an imaginary root /? G K such that (/?,«^) < 0 for all i, by Proposition 5.2 b). Let /? = ^ fc,at- and P = supp/?. Then i€P

(P\P) = E *<(fl*<) = 0, where i, > 0 and (fla,) = §|asf (/?,<) < 0 for t G P. Therefore (^,afV) = 0 for all i G P, and P is a diagram of affine type. Conversely, if /? = k6 is an imaginary root for a diagram of affine type, then (p\p) = k2(6\6) = k2j^ai(6\ai) = 0, since (6,aV) = 0 for all i. D

Ch. 5 §5.8. roots.

Real and Imaginary Roots

65

Now we give a description of the Tits cone X in terms of imaginary

PROPOSITION

5.8. a) If A is of finite type, then X = f)R.

b) If A is ofafBne type, then X = {ft G t)B | (6, ft) > 0}

c) If A is of indefinite type, then (5.8.1)

X = {ft G hi | (a, ft) > 0 for ali a G A ^ } ,

where X denotes the closure of X in the metric topology oft)*. Proof, a) holds by Proposition 3.12 e). If A is of affine type, then A r c +k6 = A r c for some k (see Proposition 6.3 d) from Chapter 6). Using Proposition 3.12 c), we deduce immediately that {h€ttm\

(6,h) > 0} C X and {ft G f)R | (6,h) < 0} CiX = 0.

If (6,ft) = 0 and ft £ R"" 1 ^)* then (a,-, ft) < 0 for some i, completing the proof of b). In order to prove c), denote by X1 the right-hand side of (5.8.1). By Proposition 5.2 a), X1 is W-invariant; also it is obvious that X1 D C. Hence

X'DX. To prove the converse inclusion it is sufficient to show that for ft G X1 such that (a,-, ft) € Z (i = 1, . . . , n) there exists only a finite number of positive real roots 7 such that (7, ft) < 0 (see Proposition 3.12 c)). Recall that by Theorem 5.6 c) there exists /? € Alf71 such that (/?, 0%) < 0 for all t. If 7 € A; c , then r7(/?) = /3+sy G Ay 1 , where s = - ( / ? , 7 V ) > h t 7 v . As ft G X', we have (/?+$7,ft) > 0. Hence there is only a finite number of real roots 7 such that (7, ft) < —1, which is the same as (7, ft) < 0. D Proposition 5.8 has a nice geometric interpretation. Define the imaginary cone Z as the convex hull in fyg of the set A+" U {0}. Then the cones Z and X are dual to each other: X = {ft G f)n I (a, ft) > 0 for all a G 'Z}. In particular, Z is a convex cone (cf. Proposition 3.12 c)). Note also that Z C —X . Exercises 5.10 e) and 5.12 give another description of the cone

66

Real and Imaginary Roots

Ch. 5

In the next subsection we will need LEMMA 5.8. The limit rays (in metric topology) for the set of rays {R+a | a e A r + e } liein~Z.

Proof, We can assume that the Cartan matrix A is indecomposable. In the finite type case there is nothing to prove since |A| < oo by Proposition 4.9. In the affine case the result follows from the description of A in Chapter 6. In the indefinite case we choose 0 G A ^ such that (/?,aty) < 0 for all i (see Theorem 5.6 c)). Then (/?,a v ) < - h t a v for a € A^f and ra(/3) = P -f kot 6 A^7*, where Jfc > 1, proving the lemma in this case also.

•

§5.9. A linearly independent set of roots II' = {a[, a^,... } is called a root basis of A if each root a can be written in the form a = ±5ZJfc|C*J, where Jfc,- € Z+. 5.9. Let A be an indecomposable generalized Cartan matrix. Then any root basis II' of A is W-conjugate to II or to —II.

PROPOSITION

Proof. Set Q'+ = ^ Z + a £ . By Theorem 5.4, the set of rays through a e i

A1™ is dense in Z, which is convex. It follows that A ^ lies in Q'+ or — Q'+, and changing the sign if necessary we can assume that A1™ C Q'+. It follows by Lemma 5.8 that the set A+ fl (—Q^_) is finite. If this set is nonempty, it contains a simple root a*. But then |A+ 0 (-ri(<2' + ))| < |A+ Pi (—Q+)\. After a finite number of such steps we get A + C w(Q+) for some w e W and hence II = it; (II').

• Remark 5.9. II is W-conjugate to —II if and only if A is of finite type. §5.10. A generalized Cartan matrix A is called a matrix of hyperbolic type if it is indecomposable symmetrizable of indefinite type, and if every proper connected subdiagram of S(A) is of finite or affine type. Note that if A is symmetrizable, then a standard invariant bilinear form (.|.) can be normalized such that (ai\otj) are integers. Hence a = min lal 2 exists and is a positive number. a€Q||2>0'

5.10. Let A be a generalized Cartan matrix of finite, affine or hyperbolic type, and let a = Yl^jaj € QLEMMA

Ch. 5

Real and Imaginary Roots

67

a) If \a\2 < a, then either a or —a lies in Q+. b) If a satisfies (5.1.2), then either a or —a lies in Q+. Proof. Suppose the contrary to a); then a = f3 — 7, where /?, 7 G Q+\{0}, and the supports Pi and P2 of 0 and 7 have no common vertices. Then

a > M 2 = |/?| 2 +M 2+2(-/?|7).

(5.10.1)

There are two possibilities: (i) both P\ and Pi are of finite type; (ii) P\ is of finite type, P2 is of affine type, and they are joined by an edge in S(A). In case (i) we have |/?|2 > a, I7I2 > a, and (—0\j) > 0, which contradicts (5.10.1). In case (ii) we have |/?|2 > a, \j\2 > 0, and (-/?bO > 0, which again contradicts (5.10.1). The proof of b) is exactly the same using the observation that |/?|2,-|7|2 G Z|a| 2 , since:

PROPOSITION 5.10. Let A be a generalized Cartan matrix offinitey affine, or hyperbolic type. Then

a) The set of all short real roots is {a e Q I \a\ = a = min |a»| }. b) The set of all real roots is

{<* = £ * i a i € Q I H 2 > 0 and *j|c*i|2/M2 G Zforaiij}. j

c) The set of all imaginary roots is

{«eQ\{0}|H2<0}. d) If A is affine, then there exist roots of intermediate squared length m if and only if A = A%/, £ > 2. The set of such roots coincides with { a G Q | | a | 2 = m}. Proof. Let a G Q be such that |a| 2 = a. Then \w(a)\2 = a for w G W and hence, by Lemma 5.10 a), w(a) G ±Q+ for every w eW. Without loss of

68

Real and Imaginary Roots

Ch. 5

generality we may assume that a E Q+. Let /? be an element of minimal height among ( W - a ) n Q + . As (/?|/?) = a > 0, we have (/?|a,*) > 0 for some i. If now ft ^ a,, then rt(/?) E Q+ and ht(rf(/?)) < ht(/?), a contradiction with the choice of /?. This shows that a is a real root, proving a). The proof of d) is similar. By Proposition 5.1 b) (i) and (iii), A r c lies in the set given by b). The proof of the reverse inclusion is the same as that of a), using Lemmas 5.1 and 5.10 b). Let now a E Q\{0} and |a| 2 < 0. By Lemma 5.10 we may assume that a E Q+. The same argument as above shows that W a C Q+. As above, we choose an element (3 E W • a of minimal height. Then (/?,a^) < 0 for all i. Furthermore, supp/? is connected, since otherwise /? = 71 + 72, where supp7i and supp72 are disjoint unions of finite type diagrams which are disjoint, and, moreover, are not connected by an edge in S(A); then |/?|2 = | 7 l | 2 + |7 2 | 2 > 0, a contradiction. So, /? E K and therefore 0 E A i m by Lemma 5.3; hence a E A*m. Now c) follows by Proposition 5.2 c).

•

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)

l u - X = {h E b« I (h\h) < 0}.

We obtain also the following corollary of Propositions 5.9 and 5.10. Note that an automorphism cr of the Dynkin diagram S(A)y induces an automorphism of the root lattice Q by
b) If A is of a symmetric matrix of finite, a/Sue, or hyperbolic type, then the group of all automorphisms ofQ preserving (.|.) is ± Aut(A) K W. Proof. If a E AutQ and
•

Real and Imaginary Roots

Ch. 5

69

Remark 5JO. a) Suppose that in Corollary 5.10, A is not symmetric but the square length of one of the simple roots, say e*i, is 1 and that of all a,- / oc\ is 2. Then

A rc = {a e Q I (a|a) = 1 or 2}. Indeed, by Proposition 5.10 b) the only additional condition occurs if (a|a) = 2, and in this case Jbi £ 2Z; but (a\a) = k\ mod 2, hence (a\a) = 2 implies that k\ G 2Z. Therefore, by Corollary 5.10 a), the conclusion of Corollary 5.10 b) still holds. b) By Remark 5.9, if A is of finite type, one can drop ± from the conclusion of Corollary 5.10 b). §5.11. In this section we apply the results of §5.10 to the study of the standard Lorentzian lattice An (n > 3) with the bilinear form (.|.) which in some Z-basis vo, v i , . . . , v n - i is given by (5.11.1)

_

=

a r

t=0

(This is the only odd unimodular integral n-dimensional Lorentzian lattice.) First, we choose another basis of A n : a

= V\ — V2,. . . , <* n -2 = t>n-2 — U n - 1 ,

n - l = Vn_i,

an = — vo — v\ — V2 — vs (resp. = — vo — v\ — v%)

if n > 4 (resp. n = 3). This shows that A n is the root lattice corresponding to the generalized Cartan matrix (2(a,|c*j)/(<*,-|e*,)) with the following Dynkin diagram BEn: o—o—6

o

O —O=>O<=O

o

if

n > 5,

and Of3

Note that in the basis {v,}, the fundamental reflection r,- for i < n — 2 is the transposition of v,- and v,-+i, r n _i is the sign change of v n _i, and r n is represented by the following matrix: /

2 1 1 1 - 1 0 - 1 - 1 -1 - 1 0 - 1 -1 -1 -1 0

\ if n > 4, /n-4/

and

70

Real and Imaginary Roots

Ch. 5

0

Denoting by Wn the group of all permutations of v i , . . . , v n _i with simultaneous arbitrary sign changes, we have thus obtained that the Weyl group 0

Wn of type BEn is generated by Wn and R^. The fundamental chamber C% C Rn = R 0 z An of the group Wn is given by the following inequalities: *i > *2 > • • • > *n-i > 0,

Jk0 > *i + k2 + k3

(k3 = 0 if n = 3).

n-l

Note that C* C L+ := {a = £ *•'«< I *o > 0,-sg + x? + •• • + x2n^ > 0}, i=0

the upper half of the light cone. Thus, the dual Tits cone X% lies in L j . Note that Wn C O\n := {g € GL n (R) | (/ • L+ = L+, (/ • An = A n } . Now, BEn is hyperbolic if and only if n < 10. It follows from §5.10 that, for 3 < n < 10 we have: (5.11.2)

Aim = {a e An | (a\a) < 0}\{0},

(5.11.3)

A r e = {a G An | (a|a) = 1 or 2}.

(If n > 4, (5.11.3) follows from Remark 5.10. For n = 3, due to Proposition 5.10 b), all a G A 3 with (ot\a) = 1 are roots, but the a = kioti + Jb2c*2 + *3 a 3 € A3 with (a|a) = 2 should satisfy the additional condition £2,^3 G 2Z; however, this easily follows from (c*|a) = 2*i(*i - * 2 ) + (*2 - *a) 2 = 2.) Thus: (5.11.4)

A = {a G An | (a\a) < 2}\{0}.

It follows from Corollary 5.10 a) that (5.11.5) COROLLARY

Wn = OXn

for

3
5.11. Let 3 < n < 10. Then

a) Aii integral solutions of the equation

-*g + x? + • • • + x\ = 1 (resp. = 2) form a Wn-orbit of the vector v\ if n > 3 and a union ofWn-orbits of v\ and vQ + vi -h v2 if n = 3 (resp. form a union of Wn-orbits of vx -f v2 and t>o + vi + v2 + V3 if n > 3 and a Wn-orbit of vi + v2 if n = 3). b) All integral solutions of the equation

Ch. 5

Real and Imaginary Roots

form a ±Wn-orbit of VQ + v\ if n < 9 and a union of two ±Wn-orbits VQ + vi and 3v0 + v\ + v2 H h vg if n = 10.

71 of

•

We conclude this section with the following general construction of hy0

perbolic lattices. Let Q be the root lattice corresponding to a generalized Cartan matrix of finite type Xi> Let a i , . . . , a/ be simple roots and 0 be the highest root (see Remark 4.9). We normalize the standard invariant form on 0

Q by the condition (6\0) = 2. Let i/ 2 be the 2-dimensional hyperbolic lattice with basis u\,ui and bilinear form (iii|ii2) = 1, (ui|ui) = (^2^2) = 0. o

Let Q be the orthogonal direct sum of lattices Q and H2- Then Q is the root lattice corresponding to the Dynkin diagram Xg obtained from X^' by adding an addition vertex a_i joined with the vertex ao by a simple edge. This follows from the following choice of basis of Q: The lattice Q (resp. diagram Xf)

is called the canonical hyperbolic exien-

0

sion of the lattice Q (resp. diagram Xt). Note that

(5.11.6)

f

Note that if X = A, By C, or Dy then Xf* = XEt+2 (cf. Exercise 4.3), and that JS?f = T4,3,3, Ef = T5f4f2, Eg = J5?10 = T7,3>2 (cf. Exercise 4.2). The most interesting example is Eg = #10. The corresponding root lattice Q is the (only) even integral unimodular Lorentzian 10-dimensional lattice. Since E\Q is of hyperbolic type, we can apply the results of §5.10. Thus, the Weyl group of E\$ coincides with the group of automorphisms of the root lattice Q preserving the upper half of the light cone, A =

{a e Q I (a\a) < 2}\{0}, etc. §5.12. In conclusion, let us make one useful observation. Recall that g(A) = g(A)/t, where r is the maximal ideal intersecting f) trivially. However all the proofs in Chapters 3, 4, and 5 used only the fact that (5.12.1)

(adc-y^e,- = 0 = ( a d / O ^ / j for all i ^ j and some Ny.

In other words, we have the following: 5.12. Let g be a quotient algebra of the Lie algebra by a nontrivial Q-graded ideal such that (5.12.1) holds. Then all the statements of Chapters 3, 4, and 5 for Q(A) hold for the Lie algebra 0 as well. PROPOSITION

72

Real and Imaginary Roots

Ch. 5

We deduce the following: COROLLARY

5.12. Let

Q

be as in Proposition 5.12. Then

a) The root system of $ is the same as that of &(A), the multiplicities of real roots being equal to 1. b) If A is of finite type, then 0 = 0(^4). Proof, a) follows from the proofs of Proposition 5.1 a) and Theorem 5.4, while b) follows from a).

• Remark 5.12. We shall see in Chapter 9 that Corollary 5.12 b) holds for an arbitrary symmetrizable generalized Cartan matrix. §5.13.

Exercises.

5.1. Show that for a G Are(A) one has:

Show that if A is a nonsymmetrizable 3 x 3 matrix and a = ot\ + oti + <*3, then a G Aim(A) and d i m ^ , 0_ a ] > 1 (cf. Theorem 2.2 e)). 5.2. Show that dim[0 a ,0_ a ] = 1 for all a G A(^4) if and only if A is symmetrizable. 5.3. If dim &(A) = oo, then |A r c | = oo. 5.4- The set A + ( J 4 ) is uniquely defined by the properties (i) and (ii) of Exercise 3.5 and the following property (iii); if a G A+, then suppa is connected. [Let A;+ satisfy (i), (ii), and (iii)'. Then A' + \{a f } is rj-invariant, hence A y C A'+. If now a G A'+\Ay, then W(a) C A;+ and /? of minimal height from W(a) lies in K] 5.5. Let A = (ciij) be a symmetric generalized Cartan matrix. Show that {a € Q\{0} | (o|o) < 1} D A. Show that the converse inclusion holds if and only if A is of finite, affine, or hyperbolic type.

Ch. 5

Real and Imaginary Roots

73

5.6. Let A = (aij) be a finite, affine, or hyperbolic matrix, let B = (6 t; ) be the corresponding "symmetrized" matrix, and let B{x) = Yl^*jxixj be the associated quadratic form. Show that all the integral solutions of the equation B(x) = 0 are of the form sw(6), where s G Z, w G W^-A), and 6 is the indivisible imaginary root of A(A')> where A' is a principal affine submatrix 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 J3 <*ijXiXj = 2 are of the form w(a t ), where w G W, a,- G II. 5.7. Show that the sublattice of the lattice A n consisting of vectors with even sum of coordinates is isomorphic to the root lattice of type D^^5.8. Show that the Weyl group W is a subgroup of index 3 in the group of automorphisms preserving (,|.) of the root lattice of type C4. 5.9. Let a G A1^1 be such that -a G C v and (a,a-') ^ 0 for some i G suppa. Then the subdiagram {i G suppa | (a,a^) = 0} C 5(^4) is a union of diagrams of finite type. [Denote by T a connected component of this subdiagram and let a = J2ki<*i'> set /? = X) Jb,at-. Then (/?,a,^> > 0 for all i and > 0 for some • G T.] In Exercises 5.10-5.15 we assume A to be indecomposable. 5.10. An imaginary root a is called strictly imaginary if for every 7 G A r c either a + 7 o r a — 7 i s a root. Denote by A* im the set of strictly imaginary roots. a) If a G A+ and (<*,(*?) < 0 (i = 1 , . . . ,n), then a G A " m . Deduce that if a G A' m , ry(a) ^ a for all 7 G A r e , then a G A* tm . b) If a G A^'m and (a,afV) < 0 for i = 1 , . . . ,n, then a + /3 G A + for every c) If a G A y m , /3 G A*+m, then a + /? G Af+m. d) A " m is a semigroup. e) Z = R + A ^ m . 5.11. Let \y denote the linear function on QK := R ®z Q defined by (A,v, a,) = ^ (j = 1,.. •, n). Then Z = {a e QR n - X R | (ii;(Asv), a) > 0 for all w G VF and those s = 1 , . . . ,n for which the principal submatrix (dij)ij^s is of indefinite type}. 5.12. Z is the convex hull of the set of limit points for R+A!jf.

74

Real and Imaginary Roots

Ch. 5

5.13. If A is a matrix of finite or affine type, /? G A, a G A r c , then the string {/? + fca,Jb G 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 /? G A and a G A r e , the string {/?+ Jba} contains at most four real roots. 5.15. Let J4 be symmetrizable of indefinite type. Then the following conditions are equivalent: (i) A is of hyperbolic type;

(ii) X u - * = {*€ fa |(A|A)<0}; (iii) Z = - X V ; (iv) Aim = {a € Q\{0} | (a|o) < 0}. 5.16. Let A be symmetrizable and let (.|.) be a standard form. Let a, ^ G A f . Then (a|/?) < 0. [One can assume that —a G C v .] 5.17. Under the hypotheses of Exercise 5.16 assume that (a\(3) < 0. Then [Since the cone Xw is convex, we can assume that —(a + /?) G C v . But suppa and supp/? are connected, and since (c*|/?) < 0, supp(a + /?) is connected and we can apply Lemma 5.3.] 5.18. Under the hypotheses of Exercise 5.16, assume that <* + /? G A!^1 and that a and /? are not proportional isotropic roots. Then (<*|/?) < 0. [Use Exercise 5.9.] 5.19. Let a, /? G A^f be such that (c*,/?v) > 0. Show that (/?,a v ) > 0. Furthermore, show that (Z+c* -f T.+P) H A = {a, /?, a -f /?} n A!Jf. [To show that (/?,a v ) > 0, one can asssume that a is a simple root; then (a,/? v ) > 0 (resp. = 0) if and only if rp(a) < 0 (resp. = a), which is equivalent to rp(av) < 0 (resp. = a v ) . Furthermore, if ma + n/3 G A for some m^n G Z+, then (ma + n/3>(3v) > 2n, hence ma G A and m < 1; similarly, n < 1; in particular, 2(a + /?) is not a root.] 5.20. Let a = ]£*,<*,• € # and let /? = X^mia,- G Q+ be such that *

t

a - P G Q+. Using the identity (a show that (a — /?|/?)< 0 unless (ala) = 0 and /? is proportional to a.

Ch. 5

Real and Imaginary Roots

75

The rest of the exercises deal with Kac-Moody algebras &(A) of rank 2, i.e., A= (

) , where a, 6 are positive integers, and a >b.

We associate to Q(A) the field F = <}(y/ab(ab - 4)); when ab ^ 4 we denote by A H-* A' the unique nontrivial involution of F. Fix the following symmetrization of A: B=

[-a

2a/b)

and the corresponding standard form (.|.) on f)*, so that (c*i|ai) = 2, (<*2|<*2) = 2a/6, (ai|a2) = —a. Introduce the following numbers: -ab + Jab(ab - 4) ; \ b

1=

Co = e if a / 6 and

e

= -6^-1'

= ry if a = 6.

5.21. Assume that ab / 4. Show that in the basis c*i, a 2 of f)J, one has C v = {(x,y) | 2x > ay, 2y > bx}. Show that if ab > 4, then T

= {(x,y)\-r,'y<x<-r,y}.

5.22. Show that c and e' are eigenvalues of riT2, and that fk-\

_

fik-\

'

*

[Use the fact that for a 2 x 2 matrix a with eigenvalues Ai and A2, one has ^ a Ai — A2 Ai — A2 5.23. Assume that ab ^ 4. Let xyf + x'y be the trace form of the field F. Show that the map a

=

is an isometry of the lattices Q and Zfr] C F, so that the fundamental reflections r\ and r*i of Q induce automorphisms of the lattice Z[ry]: n(A) = - V ,

r2(A) = -eX,

76

Real and Imaginary Roots

Ch. 5

and the group from Corollary 5.10 a) maps isomorphically onto the group generated by multiplication by the unit e0, the involution ' and — 1. Showthat <j>(Q+) = {z + yyjab{ab - 4) G Z[iy] | y > 0 and x > 0 if y = 0}, ) = {cwi|, e—»f - e - w i / f - c w + 1 , where n > 0},

5.24- Show that in the case when a = b = m, m ^ 2, we have F = Q(\/ra 2 - 4), 7] = e0, e = e^; if in addition m 2 - 4 is square free except possibly for a factor of 4 when m is even, then Z + | ( 1 + \/rn 2 — 4)Z is the ring of integers of the field F. Deduce that under the above hypotheses on m, the number \(m + y/rri2 — 4) is a fundamental unit of the ring of integers of the field Q(>/ra2 — 4), i.e., eo together with —1 generate the group of all integers of norm 1. 5.25. Show that + dj+ia2 and c,-+ia?i + dja2 (j G Z + )}, where the sequences Cj and dj are defined by the following recurrent formulas for j > 0: Cj+2 + Cj = adj+i dj+2 + dj = 6CJ+I and Co = d0 = 0, ci = 0. Show that more solutions exist only if the right-hand side is a and a/b = r 2 , where r is an integer > 1, and they are (rx, ry), where 6x2 — abxy + ay2 = 6. 5.27. Show that 2 a +(- 2 ^ ) = 0 ' i + 0' + l)tt2 and

/ 2 A

+ V -1

-4 2

for even j G Z + ; j a i + ^(j + l)a 2 and 2(.; + l)ai + ja2 for odd j G Z + } .

Ch. 5

Real and Imaginary Roots

77

5.28. Show that A

/ 2 + (_3

—3 \ 2 7

^

a i +

^i+2<*2

and <^2j>2«i + 2jOL2

(j € Z+)},

where j is the j t h Fibonacci number: to = 0,

§5.14.

<^i = 1,

<^-+2 = <£j+i + j for j € Z+.

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 §§5.1-5.7 and §5.10 is taken from 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 §5.8 is taken from Kac-Peterson [1984 A]. Proposition 5.9 is proved in Kac [1978 A]. Most of the results of §5.11 are due to Vinberg [1972], [1975], who developed 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 (Vinberg [1972], [1975], Vinberg-Kaplinskaya [1978]). Recall that H2@nE8 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 matrix A, the set of positive roots A+(J4) describes the set of dimensions of indecomposable representations of the graph S(A), equipped with some orientation. Moreover, the number of absolutely indecomposable representations of dimension a G A+(J4) over a finite field F^ is given by a polynomial qN + a\qN~l + • • • + as, where at- G Z, N = 1 — {ot\a) (see Kac [1982 A]). In these papers several conjectures are posed; the most intriguing of them suggests that a^ = mult a. Many of these conjectures (but not the latter) have been solved by Schofield [1988A-C]. Ringel [1989]

78

Real and Imaginary Roots

Ch. 5

found a connection of this field to quantized Kac-Moody algebras via Hall algebras. The nature of the root multiplicities in the indefinite case still remains mysterious: there is no single case when the answer is known explicitly. Asymptotic behavior of root multiplicities was studied in Kac-Peterson [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 KacMoody algebras. In Lepowsky-Moody [1979] one can find a detailed study of the root systems in the hyperbolic rank 2 case by making use 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

§6.0. The results of Chapter 4 show that a Kac-Moody algebra 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-Car tan theory of 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 indecomposable). A Kac-Moody algebra associated to a generalized Cartan matrix of affine type is called an affine (Kac-Moody) algebra. We describe in detail the standard bilinear form, the root system, and the Weyl group of an affine algebra 0 in terms of the "underlying" simple finite-dimensional Lie algebra 0. In particular, we show that the Weyl group of 0 is the so-called affine Weyl group of 0; this explains the term "affine" 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. §6.1. Let A be a generalized Cartan matrix of affine type of order £ + 1 (and rank t), and let S(A) be its Dynkin diagram from Table Aff. Let ao, a i , . . . . a* be the numerical labels of S(A) in Table Aff. Then CIQ = 1 unless A is of type A^t , in which case ao = 2. We denote by a( (i = 0 , . . . , £) the labels of the Dynkin diagram 5(M) 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)

a£ = 1. 79

80

Affine Algebras: the Structure Theory

Ch. 6

The numbers

/. = £ > , and /»v=£a,V $=0

t=0

are called, respectively, the Coxeter number and the dual Coxeter number of the matrix A. We list these important numbers below:

4* B'V

C,

hw

h

A

(1)

4 G(2X)

l+l 21

1+ 1 2/-1

A

h

4(2) 2l 4(2)

2/+1

2/+1

2/-1

2/

D

/+ 1

2/

A

2/

/+ 1

2/-2

2/-2

9

12

12

12

4

6

18

18

30

30

12

9

6

4

l+1

Another important number is r, the number of the Table AfF r containing A. Remark 6.1. The dual Coxeter number of the afiine matrix X}$' is independent of r. The matrix A is symmetrizable by Lemma 4.6. Moreover, we have (6.1.2)

A = d i a g ( a o a ^ 1 , a i a ^ - 1 , . . . , a / a ^ - 1 ) S , where B = %B.

Indeed, let « = *(a 0 ,... ,at) and 6V = *(<#,... ,(#); if ,4 = JD£ where £> is diagonal invertible and B = tB1 then B6 = 0 and hence t6B = 0. On the other hand, t6s/A = 0 implies (t6v)DB = 0, and we use the fact that dimkerB = 1. §6.2. Let 0 = #(A) be the affine algebra associated to a matrix A of affine type from Table AfF r, let I) be its Cartan subalgebra, II = { a o , . . . , a / } C fj* the set of simple roots, IIV = { c $ , . . . , a ) f } C b the set of simple coroots, A the root system, Q and Q v the root and coroot lattices, etc. It follows from Proposition 1.6 that the center of &(A) is 1-dimensional and is spanned by

t=0

Ch. 6

Affine Algebras: the Structure Theory

81

The element K is called the canonical central element Recall the definition of the element 6 (cf. Theorem 5.6):

t=0

Fix an element d € t) which satisfies the following conditions: (oti,d) = 0 for i = 1 , . . . , / ;

(ao,d) = 1.

(Such an element is defined up to a summand proportional to K.) The element d is called the scaling element It is clear that the elements c*o , • . . , <*i, d form a basis of f). Note that

We define a nondegenerate symmetric bilinear C-valued form (.|.) on f) as follows (cf. (2.1.2, 3, and 4) and (6.1.1)):

(6.2.1)

r(ay|aV) = a i a y - 1 a l i

(,\j = 0 , . . . ,*);

I (a?W = 0

(t'=l,...,*);

lK

= 0.

By Theorem 2.2 this form can be uniquely extended to a bilinear form (.|.) on the whole Lie algebra 0 such that all the properties described by this theorem hold. From now on we fix this form on 0. This is, clearly, a standard form. We call it the normalized invariant form. To describe the induced bilinear form on I)*, we define an element Ao G (A 0) a?) = fa for i = 0 , . . . , / ;

(Ao,
Then { a o , . . . , a / , Ao} is a basis of I)* and we have

(6.2.2)

] l

1

(A0|A0) = 0.

The map v : f) \-* fj* defined by (.|.) looks as follows: (6.2.3)

82

Affine Algebras: the Structure Theory

Ch. 6

We also record some other simple formulas: (6.2.4)

(6\ai)

= 0(i = 0,...,£);

{6\6) = 0;

(6.2.5) (K\a?) = 0 (i = 0 , . . . , / ) ; 0

(K\K) = 0;

(6\A0) = 1; (K\d) = o 0 .

0

Denote by f) (resp. ()R) the linear span over C (resp. R) of a ^ , . . . , a / . The 0

0

dual notions f)* and f)g are defined similarly. Then we have an orthogonal direct sum of subspaces:

f) = () e (CK + Cd);

f)* = 5* 0 (C6 + CA0).

We set f)R = f)R + R # + Rd, ^ = ^ + RA0 + R6. 0

0

Note that the restriction of the bilinear form (.|.) to f)R* and J)R (resp. 0

0

f)R 4- R£ and f)n 4- R/C) is positive-definite (resp. positive-semidefinite with kernels R6 and RK) by Proposition 4.7 a) and b). o

For a subset 5 of I)* denote by 5 the orthogonal projection of 5 on f)*. (This should not be confused with the sign of closure in metric topology.) Then we have the following useful formula for A Gfy*such that X(K) ^ 0:

A - A = (A, # ) A 0 + (2A> ^)" X (I A I 2 " \M2)6-

(6.2.6)

Indeed, A — A = &1A0 4- &2^. Taking inner product with 6, we obtain, by (6.2.2, 3, and 4) that &i = (A, K). As |A|2 = |A|2 4- 26x62, we are done. We also have another useful formula: A = A 4- (A, tf)A0 + (<MAo)$.

(6.2.7)

Define p 6 ¥ by (p,aV) = 1 (t = 0 , . . . ,t) and (pyd) = 0 (cf. §2.5). Then (6.2.7) gives (6.2.8)

p

§6.3. Denote by g the subalgebra of fl generated by the a and /,- with i = 1 , . . . , £ By the results of Chapter 1 it is clear that g is a Kac-Moody 0

algebra associated to the matrix A obtained from A by deleting the Oth row and column. The elements e,-, /,- (t = 1 , . . . , I) are the Che valley generators of 0, and f) = g PI f) is its Cartan subalgebra; n = {a\,..., at) is the root 0

basis, and IIV = {ot{,...

0

, a / } is the coroot basis for g. Futhermore, by

Ch. 6

Affine Algebras: the Structure Theory

83

Proposition 4.9, 0 = &(A) is a simple finite-dimensional Lie algebra whose 0

Dynkin diagram S(A) is obtained from S(A) by removing the Oth vertex. oo

0

The set A = A n f)* is the root system of 0; it is finite and consists of real 0 0 roots (by Proposition 4.9), the set A + = A D A+ being the set of positive 0

0

roots. Denote by A, and A/ the sets of short and long roots, respectively, in A. Put Q = ZA. Let W be the Weyl group of A. Recall that the sets of imaginary and positive imaginary roots of 0 are as follows (Theorem 5.6 b): Airn = {±6, ± 2 6 , . . . } ,

A f = {6,26,...}.

The following proposition describes the set of real roots A r c and positive 0

real roots A+e in terms of A and 6. PROPOSITION 6.3.

a) A r c = {a + n6 \ a G A,n

e 1} ifr

= 1.

re

b) A = {a + n8 | a G A s , n G Z} U {a + nr6 \ a G A*, n G Z} if r = 2 or 3, but A is not of type A^e . c) A r c = { I ( a + (2n - 1)6) \ a G A*, n e Z} U {a + n6 \ a G A 5 ,n € Z} U{a + 2n6 \ a G Ati n G Z} if A is of type d) A

rc

A$.

rc

+ r6 = A .

e) A; c = {ae

A r e with n > 0} 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 6 denote the square lengths of a short and a long root, respectively, and let A£c , AJC be the sets of short and long real roots. First, suppose that A is not of type A$• Then 6 = ao 4- aic*i + • • • • If 0

t

now a = YLfc»a«€ A«c> then a = |a| 2 = |a —Jbo6|2 and hence a — ko6 G A 5 «=o by Proposition 5.10 a), which gives the inclusion C in the relation (6.3.1)

Ar,' = {a + n6 \ a G A., n G Z} if A /

/$.

The reverse inclusion also follows from Proposition 5.10 a). If r = 1, then ao is a long root and the same argument as above gives, using Proposition 5.10 b): (6.3.2)

A[e = {a + n6 \ a G A*, n G Z} if r = 1.

84

Affine Algebras: the Structure Theory

Ch. 6

If r = 2 or 3, then e*o is a short root, hence, by Proposition 5.10b), a = fco<*o + • • • G AJC only if ko is divisible by r. Therefore we obtain, by Proposition 5.10b), (6.3.3)

AJC = {a + nr6 | a G A*, n G Z} if r = 2,3; ,4 # 4 ? -

Formulas (6.3.1, 2, and 3) prove a} and b). Finally, let A be of type A^t • Then short (resp. long) real roots have square length 1 (resp. 4), and the roots from A™ := A r e \ (A™ U AJ C ) have square length 2 (A™ ^ 0 if £ > 1). We have to show that 0

(6.3.4)

A; c = {!(« + ( 2 n - 1)«) | a G A , , n G Z},

(6.3.5)

A ^ = { a + n « | a 6 A , , n 6 Z},

(6.3.6)

AJC = {a + 2n6 \ a G A/,n € Z}.

By Proposition 5.10 b), a =fco<*o+ • • • G AJC only if to is divisible by 4. Now the same argument as above proves (6.3.6). A similar argument gives (6.3.4 and 5).

• 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: 0

(6.3.7)

Q v = Q v 0 IK (orthogonal direct sum),

and that (due to (6.2.1) and (6.2.2)) we have the isomorphism of lattices equipped with bilinear forms (6.3.8)

Qv(,4) ~ Q(M).

Warning. A = A\{0} in all cases except A%i , in which case A\{0} is 0

a non-reduced root system, and A is the associated reduced root system. §6.4.

Introduce the following important element:

0 = 6- aoao = ^2

Ch. 6

Affine Algebras: the Structure Theory

85

It is easy to deduce from the formulas of §6.2 that \0\2 = 2a 0 .

(6.4.1)

Hence \0\2 is equal to the square length of a long root if A is from Table Aff 1 or is of type A^i. , and it is equal to the square length of a short root otherwise. One deduces now from Proposition 5.10 a), b) that in all cases 0

0 G A+. Again, from the formulas of §6.2 we deduce: v =

OLQV\U

)\

\U

J

=

ZQQ

)

O(Q = 1 /

yO — U)

=

J\

— OLQU

Furthermore, one has PROPOSITION 6.4. a) If A is from Table Aff 1 or is of type A$, 0

^

.

then

0

0 G (A+)/ and 0 is the unique root in A of maximal height (= h — ao). b) If A is from Table Aff 2 or 3 and is not of type AK2l}', then 0 G (A+), 0

and is the unique root in A, of maximal height (= h — 1). Proo/ It is easy to check that all simple roots of the same square length 0

0

0

0

0

in A are W-equivalent, hence both A, and A/ are orbits of W. Also, (0,c*y) = — ao(ao,a^) > 0 for all i = 1 , . . . , £ Now a) and b) follow from Proposition 3.12 b).

• Unless otherwise stated, in the case of a finite type matrix A, we shall normalize the standard invariant form (.|.) on 0(A) by the condition (6.4.2)

(a|a) = 2 if a G A*,

and shall call it the normalized invariant form. We deduce from (6.4.1) and Proposition 6.4 the following: COROLLARY 6.4. Let g be an affine algebra of type X%\ Then the ratio of the normalized invariant form of 0 restricted to 0 to the normalized Q

invariant form ofyis equal to r.

• Note that we have the following description of II and IIV: II = {ao = CLQX{6 - 0 ) , a i , . . . , a * } ,

86

Affine Algebras: the Structure Theory

Ch. 6

§6.5. Now we turn to the description of the Weyl group W of the affine algebra g. Recall that W is generated by fundamental reflections r0, ri, . . . , ry, which act on ft* by ri(A) = A-(A,a, v )" i . >

A € ft*.

As (5,aV) = 0 (i = 0, . . . , * ) , we have w(<5) = j for all u i G K Recall also that the invariant standard form is W-invariant. 0 Denote by W the subgroup of W generated by r i , . . . , r y . As r,-(Ao) 0

= Ao for i = 1 , . . . ,£, we deduce that W operates trivially on CAo + C6; it 0

0

0

is also clear that ft* is W^-invariant. We conclude that W operates faithfully o

o

0

on ft*, and we can identify W with the Weyl group of the Lie algebra 0, 0

0

operating on ft*. Hence (by Proposition 3.12 e)) the group W is finite. Recall that for a real root a we have a reflection rQ G W defined by

LEMMA

6.5. Let a G A+c be such that 0 := 6 - aa G A!Jf for some a.

Then rarfi(X) = A + (A, K)u(^)

- ((A,/? v) + ||^ V | 2 (A, K))S

for A G ft*. Proof. First, we compute modC6: = ra(A + (A|tf)2aa|aa|~2 - (A,a v )a)modC6

To compute the coefficient of 5, we use the equality |ra r^(A)| 2 = |A|2.

a 0

Applying Lemma 6.5 to 0 = 6 — ao<*o G A+ C AJjf, we get (6.5.1)

r ao r,(A) = A + (A,

Ch. 6

Affine Algebras: the Structure Theory

87

Motivated by this formula, we introduce the following endomorphism tQ 0

of the vector space (3* for a € | ) * :
(6.5.2)

In the case when m := (A, K) ^ 0 we can rewrite this as follows using (6.2.6):

ta(\) = mA0 + (A + ma) + J-(\X\2 - |A + ma\2)6.

(6.5.3)

As (Ao,i^) = 1, we obtain, in particular: (6.5.4)

t a (A 0 ) = Ao + a -

\\a\26.

Note also that (6.5.2) implies ta(\) = A - (\\a)6, if (A, K) = 0.

(6.5.5)

Now we can easily deduce the additivity property of ta: (6.5.6)

tatp=tQ+i3.

Indeed, by (6.5.5) it is sufficient to check (6.5.6) for A = A o . But then, by (6.5.4 and 5) we have tatfi(A0) = * tt (A 0 + P- \\P?*) = *«(Ao) +

ta{fi - \\p\26) = Ao + a - I|a|

\

Ao + a + /? - I|or + f3\26 = t We also have (6.5.7)

tw(a) = wtQw~ l

0

for w G W.

= w(w~l(X) + (w~l(X),K)a - ((w^(X)\a) +

Indeed, wia(w' (X)) \\a\2(w~l\ (A), #))£). W- invariant.

Now (6.5.7) follows since w(K) = K and (.|.) is 0

0

Now we introduce the following important lattice M c d Ji. LetZ(W^-^ v ) 0

denote the lattice in J)R spanned over Z by the (finite) set W -^ v , and set 0

W • 0 V )). Here is a description of the lattice M: 0

(6.5.8)

M = Q = Qifylis symmetric or r > ao; 0

M = i/(Q v ) = v(Qv) otherwise.

88

Affine Algebras: the Structure Theory

Ch. 6

Indeed, if r = 1, then 0V is a short root of A v , hence W • 0V = ( A v ) , . It is well known (cf. Exercise 6.9) that in the finite type case the root lattice is 0

spanned over Z by the short roots, hence in this case M = ^((? v ), giving (6.5.8) for r = 1. Equivalently, M = Q (resp. M = lAt) if r = 1 and A is symmetric (resp. nonsymmetric). 0

V

0

v

Similarly, if aor = 2 or 3, then 0 is a long root of A , hence W • 0V = ( A v ) / , and we get M = Q. Finally for A$

one has i/(0 v ) = \0, hence

0

M = | Z A / , which is equivalent to (6.5.8) in this case also (see (6.3.6)). Note also the following useful fact: (6.5.9)

Q D J/(Q V ) if r = 1

and

Q C v(Qw)

if r > 1.

The lattice Af, considered as an abelian group, operates faithfully on I)* by formula (6.5.2). We denote the corresponding subgroup of GL(f)*) by T, and call it the group of translations (formula (6.6.3) below explains why). Now we can prove 0

PROPOSITION 6.5.

W = W K T.

Proof By (6.5.1), ta G W for a = i/(0 v ), hence ^ ( a ) G W for w G W" by (6.5.7). Now it follows from (6.5.6) that ta eW

for all a G M. Since W

0

is finite and M is a free abelian group, W D T = 1. It follows from (6.5.7) that T is a normal subgroup in W. Finally rao = tu^^re, and therefore 0

the subgroup of W generated by T and W contains all rt- (i = 0 , . . . ,£) and hence coincides with W.

n 0

Since ty^v) = raor$ and T is generated by wtv^y)W"1 (w G W), we have (6.5.10) defy, w = 1 for w G T. Remark 6.5, A vertex i of the diagram 5(^4) is called special if 0W := 6 — aja,- is a positive root. For example, t = 0 is a special vertex. Denote by W^ the subgroup in W generated by all r,- (j ^ i), and denote by Af(*) the lattice spanned over Z by the set i/(W
Ch. 6 §6.6.

Affine Algebras: the Structure Theory

89

For s e R set

t

Note that f)$ = J2 " a * an(^ that *he hyperplanes \)* are W^-invariant. Furthermore, the action of W on fjjj k faithful by (3.12.1). Consider now the affine space f)J mod R6. Since the action of W on f)j$ is faithful, its action onfyg/RJ~ 1)5 an<^ ^ n u s o n ^l niodR^ is also faithful. The affine action of W on the affine space fjj modR6 has the following 0

simple geometrical meaning. We identify IjJ modR<5 with f)g by projection, thus obtaining an isomorphism from W onto a group of affine transforma0

tions Waf of 1)R*. We denote this isomorphism by af, so that

0

Now we describe the action of W*s on f)£. It is clear that (6.6.1)

af(u;) = w for w G W.

Furthermore, (6.6.2)

af(rO0)(A) = r,(A) + u(0^)

(A € ! t ) .

= J» h e n c e (A»'ao) = (»,K - a o 0 v ) = Indeed, if n € Ifi, t h e n ^K) l - a o ( / i , r > , and r O0(^) = /i-(l-a o
af(
So, the group W^ is none other than the so-called affine Weyl group of g. Introduce the fundamental alcove: Car = {A G fc | (A|a<) > 0 for 1 < » < £ , and (A|0) < 1}.

90

Affine Algebras: the Structure Theory 0

PROPOSITION

unique point

Ch. 6

0

6.6. a) Every point of f)J is W-equivalent modM to a ofC^. O

0

b) The stabilizer of every point of Caf under the action ofW on f)ji/M is rat}. generated by its intersection with {r$, rQl,..., c) For every A £ fyj one has Bt(WX) = and W\ fl T = e, where W^ denotes the stabilizer ofX. 0

Proof. Consider the projection map 7r: fyj —• l)g. Then it is clear that IT is surjective and that af(iu) o TT = TT O W for tu G W. Furthermore, (cf. §3.12):

But by Proposition 3.12b), C v fl f)J ^s a fundamental domain for the action of W on 1)1 C Xv. This together with (6.6.1 and 3) proves a). We also deduce from the above that &f(W\) = (Waf)x- Now b) follows from Proposition 3.12a). Finally WXC\T = 1, since af(WA) = (H^af)x contains no nontrivial translation.

• §6.7. As we have seen, the root and coroot lattices, the root system and the Weyl group of an afiine algebra 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 algebras. Let Rn be the n-dimensional real Euclidean space with the standard basis v\,..., vn and the bilinear form:

All the lattices below will be sublattices of Rn with the inherited bilinear form (.|.). All indices are assumed to be distinct.

At : Q = Qv = {£>«,- € R<+1 | ki € Z, J > = 0}, A = {vi-Vj}, II = { a i = vi -v2ict2

= v2-V3,...,<*/

=

Vt

~

Ch. 6

Affine Algebras: the Structure Theory

91

0 = vi W = {all permutations of the v,}.

£

G 2Z},

A = {±vi±vj], II = {ax = vi - t ; 2 , . . . , a * _ i = v/_i - v/,a/ = v/_i + vt}, 0 = vi H- v2, i y = {all permutations and even number of sign changes of the i;,}.

Bt : Q = {J2 kM e Rl | k{ € Z}, Qv t

A = {±Vi±Vj,±Vi},

W = {all permutations and sign changes of the v,-} = AutQ. A = {-^(±v» ± vj),

n = {ax = -J=(VI

-

W = W(Bt). Q A = {^=(v{ -

Vj),

II = {aX = - ^ ( - ^ l

0(

+

W = ±{all permutations of the v,-} = AutQ. Q = {^A:^ G R4 | all fc^Zor all kt e ^ A = {±v i ,±t; l -±t; i ,-(±t; 1 ± v2 ± t;3

II = {a! = v2 -v 3,or 2 = t;3-t;4,a3 = t;4,a4= - ( ^ - v2 -

v3-v4)},

92

Affine Algebras: the Structure Theory

Ch. 6

0 = vi + t>2,

W = AutQ. € R8 | all *, € Z or all *,• € \ + Z, £Jfc, G 2Z}, A = {±v,- ± Vj, T(±VI ± t>2 i

#

• • i t>s) (even number of minuses)},

II = {ot\ = v2 - v3, a2 = v 3 - v4i a3 = v4 - v5, a 4 = v5 - v 6 ,a 5 = t;6 - v 7 ,a 6 = t;7 - v8 , <*7 = 5(^1 - t;2 - . . . - v7 + v8),or8 = ^7 + v$}, 0 r= Vi + t;2,

R8 | all ib,- € Z or all *, € 5 + Z , £ > = 0}, A = {v{ — Vj, | ( ± v i db • • • ± vs) (four minuses)}, II = {<*i = v2 - v3i a2 = v3-

v4, a 3 = v4 - t;5,

a 4 = Vs - W6, « 5 = V6 - V7, Qf6 = V7 - V 8 , a? =

0 =

2^~ t ; i "" V2 - ^3 ~ ^4 + v5 + v6 + v7 -h

tf2-«1,

: Q = Q v = {*lWl + . . . + Jk6t;6 + V2k7v7 e R 7| all jfe,- G Z

o r a l l * , e i + Z,Jfc1 + ... + Jfc6 = 0 } , A = {vi -

Vj(i,

j < 6), -((ciwi + • • • + €6v6) ± y/2v7)(ei = ±1,

{ai = vi - v2,a2 = v2- v3,a3 = v3 0:5 = ^5 ~ ^6, &6 = g ( ~ v l -

V

2 - ^3 + ^

0 = \/2t; 7 ,

= AutQ. In order to prove these facts we first check directly that II is a basis of Q over Z, and that the matrix (2(ai\ctj)/(ai\oti)) is the Cartan matrix of the corresponding type. Furthermore, one easily checks that A = {a \ (a|a) = 2} for At, Dt and £"/, and A = (resp. C) {a | (<*|a) = 1 or r} for Bi, G 2 , and F4 (resp. C/), where r = 2 for Bt,CtiFA, and r = 3 for G2.

Affine Algebras: the Structure Theory

Ch. 6

93

Using Proposition 5.10 a), b), we see that A is the set of all roots. The computation of W follows easily from Corollary 5.10 and Remark 5.9 b). Remark 6.7. It follows from the above discussion that if A is of finite type, then AutQ = Aut(A) K W, except for A of type C 4 . In the latter case, W is a group of index 3 in AutQ (cf. Exercise 5.8). We can list now the lattices M for all affine algebras:

M = Q(Di); (2)

u

M = y/2Q(B£y, M = Q(£> 4 );

M = Q{A2)\ A §6.8.

{2)

M = Q(B€).

Exercises.

6.1. Check that the square length (<*|a) of a root a from A/ (resp. A,) is equal to 2r (resp. 2r/s, where s = max aji/dij). 0

6.2. Let r = 1. Show that h is the Coxeter number of the root system A and that /i v = ^ ( M ) " 1 = 1 + (p\6)> w h e r e * t h e Killing form of g and p is the half sum of the roots from A+. Show that ^(x, y) = 2hSf(x\y) for g. o

[Note that 4>(6 + 2 P,9) is the eigenvalue of the Casimir operator associated to the Killing form 0, hence it equals 1, and use (6.2.3).] 6.3. Let A be a Cartan matrix of type X# from Table Aff r, let £ — rank A, and let h be the Coxeter number. Let Ao be the finite root system of type XN. Check that rth = |A 0 |. 6.1 Let A be of type A^\ i.e., A = (_22 " / ) . Then A + = {(Jb - l ) a 0 -I- Jbai, ita0 4- (k - 1 ) ^ , ifea0 + Jfcc*i, where k = 1 , 2 , . . . } . tf. J. Let A be of type

A =

( 2 -1

-1 2

0 -1

-1

0

0

-1

2

94

Affine Algebras: the Structure Theory

Ch. 6

Show that A + = {k(a0 + • • • + a , - i ) + (* ± l)(at- + • • • + otj-i) + *(a; + • • • + at)}, where k = 0 , 1 , 2 , . . . ; 0 < t < j < £ + 1, and only + is allowed if Jfc = 0. 6.6. Let A be of type A(*\ i.e., ^ = (_\ ~ 4 ) . Then A+ = {4na 0 + (2n - l)c*i, 4(n - l ) a 0 + (2n - I)*!, (2n - l ) a 0 (2n - l ) a o + (n — l)ori, 2na 0 + nari; n = 1 , 2 , . . . } . 0

tf.7. Let s0, 8i,...,8i be the reflections in the vector space t) with respect to the hyperplanes 0 = 1, ari = 0 , . . . , a* = 0, respectively, and let Wa be 0

the group generated by *o> • • • > ££• Show that Wtf = W K i/~ 1 (M) and that the map s,- i-+ r a . defines an isomorphism : Wa —•• W. Show that the image under <j> of a translation by a £ i/" 1 (M) is tfa. 0

0

6.8. Let f)a denote the vector space of affine linear functions on f). Consider 0

the isomorphism if; : \JQ —* \ja defined by ^ ( a 0 = 5 f for i = l , . . . , 4 0

The action of Wa on f) defined in Exercise 6.7 induces a linear action of Wa = W on ^ a . Show that the morphism (<^, V 1 ) : (Wa, *)a) -* (W, f)J) is equi variant. 6.9. Show that for a finite or affine type matrix Ay the set of short real roots A£ C (J4) span the lattice Q over Z. [The set Q' = ZA£C is W-invariant. We can assume that A is not symmetric. If A ^ J4^ , then there exists a short simple root a and a long simple root (3 such that (a\/3) ^ 0; then rp(a) = a + /? G Q'. Hence, /? G Q' and AJC C (?'. So II C A r c C Q' and Q = Q'. The same argument works in the case A$ as well.] 6.10. Let r = 1 and y G M. Show that

[Use Exercise 3.6.]

Ch. 6

Affine Algebras: the Structure Theory

95

6.11. Denote by F n the lattice in Rn defined by the same conditions as the root lattice of E% in R8. Show that if n is divisible by 8, then F n is an even unimodular lattice. 6.12. Show that an integral lattice Q (i.e., (x\y) G Z if x, 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 At, Di, Ee, £7, ^86.13. Show that the group of all automorphisms of f) preserving the bilinear form (.|.) and fixing K is the semidirect product of the group of all 0

orthogonal automorphisms of f) (fixing K and d) and the group of all tQ,

a el*. §6.9. Bibliographical notes 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 [1972], Kac [1978 A], Kac-Peterson [1984 A]. The "quadratic" 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

§7.0. In this chapter we describe in detail a "concrete" construction of all "nontwisted" affine algebras. It turns out that such an algebra 0 can be realized entirely in terms of an "underlying" simple finite-dimensional Lie algebra 0. Namely, its derived algebra [0, 0] is the universal central extension (the center being 1-dimensional) of the Lie algebra of polynomial maps from C x into 0. 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 finitedimensional simple Lie algebras. §7.1. Let C = Cp,*" 1 ] be the algebra of Laurent polynomials in t. Recall that the residue of a Laurent polynomial P = JZ cktk (where all *€Z

but a finite number of c* are 0) is defined by ResP = c_i. This is a linear functional on C defined by the properties: Res*"1 = 1;

R e s — = 0. at

Define a bilinear C-valued function

Then it is easy to check the following two properties: (7.1.1)

(7.1.2)


(P,Q,ReC).

§7.2. The affine algebra associated to a generalized Cartan matrix of type X\ ' (from Table Aff 1) is called a nontwisted affine algebra. Here we describe an explicit construction of these Lie algebras. Note that the generalized Cartan matrix A of type X^ (where X = J 4 , J 3 , . . . , G ) is

96

Ch. 7

Affine Algebras as Central Extensions of Loop Algebras

97

nothing else but the so-called extended Cartan matrix of the simple finite0

o

o

dimensional Lie algebra 0 := 0(J4), whose Cartan matrix A is a matrix of finite type Xt (obtained from A by removing the Oth row and column). Consider the loop algebra This is an infinite-dimensional complex Lie algebra with the bracket [ , ]o defined by

0

It may be identified with the Lie algebra of regular rational maps C x —* 0, so that the element ^(f 1 ® X{) corresponds to the mapping z i—• £2 z*x,-. Fix a nondegenerate invariant symmetric bilinear C-valued form (.|.) on 0; such a form exists (e.g., by Theorem 2.2) and is unique up to a constant multiple. We extend this form by linearity to an £-valued bilinear form (.|.)i on £(5) by Also, we extend every derivation D of the algebra £ to a derivation of the Lie algebra £(g) by D(P x) = D(P) x. Q

Now we can define a C-valued 2-cocycle on the Lie algebra £(fl) by

Recall that a C-valued 2-cocycle on a Lie algebra 0 is a bilinear C-valued function ip satisfying two conditions:

(Col) (Co 2)

i>(a,b) =-tP(b,a) 1>([a,b],c) + WV>,c],a) + 1>([c,a],b) = Q

(a,b,ce&).

It is sufficient to check these conditions for a = P ® x , 6 = Q®y, c = where P, Q, ReC and x, y, z £ 0. We have

Hence, (Co 1) follows from (7.1.1) and the symmetry of (.|.). Property (Co 2) follows from (7.1.2) and the symmetry and invariance of (.|.). Indeed, the left-hand side of (Co 2) is

([x,y]\z)
98

Affine Algebras as Central Extensions of Loop Algebras "** 0

Ch. 7

0

Denote by £(0) the extension of the Lie algebra £(0) by a 1-dimen"

0

0

sional center, associated to the cocycle \j>. Explicitly, £(0) = £(0) 0 CK (direct sum of vector spaces) and the bracket is given by (7.2.1)

[a + AX,6-h,i^] = [a,6]o + V(a,6)A:

(a,6e£(0); A,/i€C).

Finally, denote by £(0) the Lie algebra that is obtained by adjoining to £(0) a derivation d which acts on £(0) as t-jjj and which kills K (see §7.3). A

0

In other words, £(0) is a complex vector space

with the bracket defined as follows (x, y G 0; A, /1, Ai, /ii E C): (7.2.2)

[*m ® x 0 \K 0 /id, tn ® y © \XK [x, y] + /intn ® y - yLXmtm ®x)®

m6m-n(x\y)K.

We shall prove that £(0) is an affine algebra associated to the affine matrix A of type Xp\ §7.3. Here we check that d is a derivation of the Lie algebra £(0). More generally, denote by d$ the endomorphism of the space £(0) defined by (7.3.1)

d,|£(.) =

_ r ^ ;

d.(K) = 0,

so that do = —d. PROPOSITION

7.3. d8 is a derivation ofC(^).

Proof. Since D := ds is a derivation of £(0), we deduce that D([a + \K, b + 11K]) = D([a, b]0) = [D(a)f 6]0 + [a, D(b)]0. But (a), 6] = Hence, one has to check that (7.3.2)

Ch. 7

Affine Algebras as Central Extensions of Loop Algebras

99

Set a = P ® x, b = Q y; then the left-hand side of (7.3.2) is:

=

(x\y)(-
D Note that

is a Z-graded subalgebra in der£(g) with the following commutation relations: 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 1-dimensional center, say Cc, called the Virasoro algebra Vir, which is defined by the following commutation relations (see Exercise 7.13): [d^dj] = (i - j)
(7.3.3)

(ij

e Z).

The semidirect product of Lie algebras Vir +£(fl) defined by (7.3.3), (7.2.1), (7.3.1), and the equation [c, £((j)] = 0, plays an important role in the representation theory of affine and Virasoro algebras and in the quantum field theory. 0

0*

0

§7.4. Let A C I) be the root system of the Lie algebra g, let {c*i,..., at] be the root basis, {Hi,..., Hi} the coroot basis, £",-, F{ (i = 1 , . . . ,£) the 0

Chevalley generators. Let 0 be the highest root of the finite root system A (see Remark 4.9). Let 9 = 0 fla be the root space decomposition of g. o

a£AU0

Recall that (a|a) ^ 0 and dimg a = 1 for a G A (there are no imaginary roots). Let u be the Chevalley involution of 0. We choose Fo € %e such that (F0\u(F0)) = -2/(0|0), and set EQ = -u>(F0). Then due to Theorem 2.2 e) we have (7.4.1)

[Eo,Fo) =

-0".

100

Affine Algebras as Central Extensions of Loop Algebras

Ch. 7

The elements E{ (t = 0, . . . , £ ) generate the Lie algebra 0, since in the 0

0

0

0

adjoint representation we have 0 = U($)(Eo) = U(n+)(Eo) (recall that 0 is simple). Now we turn to the Lie algebra £(0). It is clear that CK is the (1dimensional) center of the Lie algebra £(0), and that the centralizer of D in £(0) is a direct sum of Lie algebras: CK 0 C ( f ® ( l ® 0 ) . In particular, l ® 0 ^ 0

0

is a subalgebra of £(0); we identify 0 with this subalgebra by x i—• 1 ® x. Furthermore,

t) := 5 + CK + Cd is an (£ + 2)-dimensional commutative subalgebra in £(0). We extend 0*

0*

A G fy to a linear function on f) by setting (A, K) = (A,
We deduce from (7.4.1) that (7.4.2)

[eo

Now we describe the root system and the root space decomposition of £(0) with respect to f): A = {jS + 7, where j G Z, 7 G A} U {j6, where j G Z\0},

= {,0(0£(0) a ), where

^ ® 5We set II = {a0 := < 5 - ^ , a i , . . . , a / } ,

Note that our 6 is the same as the one introduced in §6.4 (for r = 1). Then Proposition 6.4 a) implies

(7.4.3)

A = «<*,-, a?»fj=o-

Ch. 7

Affine Algebras as Central Extensions of Loop Algebras

101

In other words, (f), II, IIV) is a realization of the affine matrix A we started with. (Indeed, II and IIV are linearly independent, i.e., (1.1.1) holds and 2n - rank A = 2(£ + 1) - £ = £ + 2 = dim &, i.e., (1.1.3) holds.) Now we can prove our first realization theorem. 7.4. Let 0 be a complex finite-dimensional simple Lie algebra, and let A be its extended Cartan matrix. Then C(&) is the affine KacMoody algebra associated to the affine matrix Aff) is its Cartan subalgebray II and IIV the root basis and the coroot basis, and eo, . . . , e*, /o, . . . , ft the Chevalley generators. In other words, (C(&), f), II, II v ) is the quadruple associated to A. THEOREM

Proof. We employ Proposition 1.4 a). Some of the hypotheses of this proposition have already been checked. The relations (1.4.2) are clear. As for relations (1.4.1), they evidently hold when t, j = 1, . . . , £ because E{} F{ (t = 1, . . . , £) are Chevalley generators of 0. The relations [eo,/»] = 0 and [et-,/o] = 0 for i = 1, . . . , £ hold since 0 is the highest root of 0. This together with (7.4.2) proves all the relations (1.4.1). ~ 0

Furthermore, £(g) has no ideals intersecting f) trivially. Indeed, if i is a nonzero ideal of £(g) such that i f! f) = 0, then by Proposition 1.5, i H C(g)Q ^ 0, for some a G A. Hence t7 x E i for some j G Z and o

6

o

x e 9 7 , x ^ 0, 7 G A U 0. Taking y G S- 7 such that (x|y) ^ 0, we obtain o

[tj <S> x,t~j

0 2/]= j ( x b ) ^ + [^y] € f) n i- As [x,2/] € I), we deduce that o

j = 0. But then 7 ^ 0 , and hence [x, 2/] is contained in f) Pi i and is different from zero. This is a contradiction. Finally, it remains to show that e,-, /,- (i = 0, . . . , £), and f) generate the Lie algebra £(0). For that purpose we denote by £1(0) the subalgebra of £(g) generated by them. Since £,-, F,- (1 = 1, . . . , £) generate the Lie 0

0

•*•

0

**

0

algebra 0, we obtain that 1 0 0 C £1(0). Furthermore, t0i?o G £1(0); since [t 0 x, 1 0 y] = t 0 [x, y] for x, y G 0 and since 0 is simple, we deduce that < ® g C £1(0). Since [t 0 x,f* 0 y] = <*+1 0 [x,y], it follows by induction on Jb that t 0 0 C £1(0) for all Jb > 0. A similar argument shows that tk 0 0 C £1(0) for all k < 0, completing the proof.

• The following important corollary of Theorem 7.4 is immediate. 7.4. Let Q(A) be a nontwisted affine Lie algebra of rank £+1. Then the multiplicity of every imaginary root ofg(A) is £.

COROLLARY

102

Affine Algebras as Central Extensions of Loop Algebras

Ch. 7

Remark 7.4- Given a simple finite dimensional Lie algebra 0, the Lie algebra £(0) is usually referred to in the literature as the affine algebra associated to 0, or the affinization of 0. §7.5. One can also describe explicitly the rest of the notions introduced in the previous chapters. The normalized invariant form (.|.) of 0 (introduced in §6.2) can be Q

described as follows. Take the normalized invariant form (.|.) on 0 and extend (.|.) to the whole £(0) by (751)

(CK + Cd\C(i)) = 0; (K\K) = (d\d) = 0; (K\d) = 1. It is clear that this is a nondegenerate symmetric bilinear form. We check the only nontrivial case of the invariance property: ([
side is (d\PQ 0 [*, y] + (Res ^Q)(x\y)K)

= (Res

the right-hand

fQ)(x\y).

Finally, the restriction of (.|.) to f) coincides with the form given by (6.2.1). Indeed, for both forms, (d\d) = 0, and hence it is sufficient to compare them on one element. We have (ao|e*o) = (6 — 0\6 — 0) = (0\0) = 2. Note also that the element K of £(0) is then the canonical central element and that the element d is the scaling element. §7.6. Let 0 = n_ ©fj©n+ be the triangular decomposition (1.3.2) of the Lie algebra 0. Then the triangular decomposition of £(0) can be expressed as follows: £(0) = n . 0 ^ 0 n + , where

n_ = ( r lc[ n + = (tC[t] 0 (n_ + 5)) + C[t] 0 The Chevalley involution u> of £(0) can be expressed in terms of the Chevalley involution a; of 0 as follows:

u(P(t) ®x + \K + fid) = P(rx) 0 u>(x) -\K-

fid,

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103

where P(t) G £ , x G 0; A, /i G C. Indeed, we obviously have u;(e t ) = —/,-, u>(/i) = —et- (i = 1, . . . , ^) and u;|j, = —id. Furthermore, u>(e0) = u{t i£ 0 ) = t~

l

0

1

® u(Eo) = — f"" 0 Fo = —/o and similarly, u;(/o) = —e 0 .

Analogously the compact involution u>0 of C(g) can be expressed in terms 0

0

of the compact involution o;o of 0: + XK + fid) = Tit'1)

2>o(x) - \ K -

pd,

where for P(t) = X^ci^; w e ^ ^ ( 0 = 5Z^i^ a n ^ <* denotes the complex conjugate of a G C. Hence the compact form of £($) = (space of polynomial maps from the unit circle to the compact form of g) + iRK + iRd. Finally, we have (P(t) ® x\P(t) ® x)0 = -(P(t)

x\P(t

One deduces that the Hermitian form (.|.)o is positive-definite on the 0

A

0

0

subspace C(g) of £(fl), using the fact that it is positive-definite on 0 (see Remark 7.9 d) below). A more general approach will be developed in Chapter 11. §7.7. We briefly explain here the physicists' approach to affine algebras. Let g be a simple finite-dimensional Lie algebra with the normalized invariant bilinear form (.|.). Let xt- be a basis of g, and let *j £

C

We denote the element tn ®x of £(g) by x
(7.7.1)

[x] k

Physicists associate to x G 0 the generating series:

n€l

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called the current of x. Here z is a formal parameter. In order to write down the commutation relations between currents, we need to introduce the formal 6-function:

n€l

2

Its basic property is (7.7.2)

Res, l=0 /(*i)«(*i - z2) = f(z2).

Here, for f(z) = J^/;**7, one defines Resj = o/(z) = / - i . It is straightforward to check that [*(*i),y(*2)] = [«,y](*i)«(*i - *a) - # ( z | y ) ^ ( * i - z2).

(7.7.3)

In this form the algebra C(g) is called the current algebra. The following lemma is very useful for calculations: 7.7. One has the following equality of formal power series in zfl and zf whenever both sides make sense: LEMMA 1

(7.7.4)

/ ( * ! , z2)S(z1 - z2) = f(z2i

(7.7.5) f(zuz2)6'Zl(zl

- z2) = f(z2,z2)6'Zl(Zl

Z2)S(z1

- z2),

- * a ) - fMl(zuz2)

| J l M a «(*! - * 2 ).

Prx?a/. Multiply both sides by zj* and check the equality of the ResZl=o using (7.7.2).

• One also easily checks the following relation:

(7.7.6)

dm(x(z)) = zm(z± + (m + l))s(z).

Given two Vir-modules V and Vi, and a sequence of operators Fj : V —> Vi, j G Z, the generating series F(z) = ^ ^ i 7 yz~ J ~ A is called a primary conformal weight A if [*», F(z)] = *m(z-£- + A(m + 1))F(*), az

m € Z.

Equation (7.7.6) shows that currents are primary fields of conformal weight 1.

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105

§7.8. As we have seen, the (nontwisted) affine algebras can be described entirely in terms of the corresponding simple finite-dimensional Lie algebras. In this section we give an explicit construction of the "simply-laced" simple finite-dimensional Lie algebras, i.e., those of type At, Dt> and Ei (£ = 6,7,8). The remaining cases will be treated in §7.9. Let Q be the root lattice of type Ai, Di, or Et and let (.|.) be the bilinear symmetric form on Q such that (see §6.7): A = {a G Q | (a|a) = 2}. Let e : Q x Q —• {±1} be a function satisfying the bimultiplicativity condition

(7.8.1)

e(a,0 + ^') = e(a,0M«.P)

*« «,«'>/*>P € Q,

and the condition e(a, a) = (-1)*M«)

(7.8.2)

for

aeQ.

(Recall that (a\a) G 2Z for o; G Q.) We call such a function an asymmetry function. Since (a\/3) e Z for a,/? G Q, replacing a by a + /? in (7.8.2), we obtain (7.8.3)

e(a,/3)e(/?,a) = ( - l ) W )

fora,/?GQ.

An asymmetry function e can be constructed as follows: choose an orientation of the Dynkin diagram, let (7 8 4^

e a

( i>aj)

= -1

if

« = J or

if

%o

-*3°

£r(a,-,a;) = 1 otherwise, i.e., o o or o«— o, and then extend by bimultiplicativity. It is clear that then (7.8.2) holds for all a G Q. (More generally, choose a Z-basis j3\,...,/?/ of Q, let e(/?,-, /?,-) = ( _ l ) i ( * l * ) f £(/?,-,^) = ( - i ) ^ l f c ) if i < j and = 1 if i > j , and extend by bimultiplicativity. Thus, e "breaks the symmetry" of (.|.)). Now let f) be the complex hull of Q and extend (.|.) from Q to f) by bilinearity. We shall identify I) with fj* using this form. Take the direct sum of f) with 1-dimensional vector spaces CEai a G A:

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Define a bracket on 0 as follows:

if he ^ a G A , (7.8.5) [E«,EP) = ^ [Ea,Ep] = e(aiP)Ea+/3

if <*,/?,<* + P G A.

Define the symetric bilinear form (.|.) on 0 extending it from f) as follows: (7.8.6)

(h\Ea)

= 0iihe\)1aeA,

(Ea\E0)

= -6a,-0

if a, (3 e A.

7.8. IfQis the root lattice of type At, Dt, or Et and € is an asymmetry function on Q xQ, then g is the simple Lie algebra of type At, Di, or Eiy respectively, the form (.|.) being the normalized invariant form. PROPOSITION

Proof. Let x, y, z G 0 be from either f) or one of the Ea. If one of these elements lies in f), the Jacobi identity trivially holds. Let now x = EQ, y=Ep,z = Ey. If all a + /?,<* + 7,/? + 7 g A U {0}, the Jacobi identity trivially holds. Thus, we may assume that a + /?G AU{0}. Note that given a,/? G A we have: (7.8.7)

(7.8.8)

a ± p e A iff (a|/?) = ^1,

c(a,a) = - 1 ,

e ( a f / W , a ) = (-l)< a '«.

If a+P = 0, consider four cases: 1) both a±y £ AU{0}, 2) a+j = 0 or a —7 = 0, 3) a + 7 G A, 4) a —7 G A; the Jacobi identity trivially holds in cases 1) and 2), and reduces to conditions ^(7,0^)^(7 + a,—or) = (of|of) in case 3) and e(—a,i)e{—a + 7, a) = (a|a) in case 4), which holds due to bimultiplicativity of e, (7.8.7) and (7.8.8). Thus, we may assume that all a + p,a + 7, P + 7 € A, i.e., that (a|/?) = (a| T ) = (^7) = - 1 . Then \a + /? + 7|2 = 0, hence a + P + 7 = 0 (this is the place where we use that (.|.) is positive-definite). The Jacobi identity reduces to the condition: e(a, p)(a + p) + e(P, y)(p + 7) + e(7,«)(« + T) = 0, or e(a, /?)(a + /?) - e(/J, - a - /?)a - e(-a - /?, a)/? = 0,

Ch. 7

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107

which holds due to bimultiplicativity of e, (7.8.6) and (7.8.7). Thus, g is a Lie algebra. Let (see §6.7) n = II v = { a i , . . . , a * } , e i = Ea., fc = - £ L a . . It is easy to see that (g, f), II, IIV) is the quadruple associated to the matrix A of type At, Di, or Ei, respectively, hence, by Proposition 1.4 a), Q is of type Ai, Dt, or Et, respectively. It remains to check that (.|.) is invariant; this is straightforward.

• §7.9. In this section we give an explicit construction of the non-simplylaced simple finite-dimensional Lie algebras, i.e., those of type Bt, Ct, F 4 , and G2. Consider the following (simply-laced) root lattices Q{Xs) and their automorphisms Ji of order r = 2 or 3. (We use the construction of Q(XN) given in §6.7.) Case 1. Q(Dt+i) Case 2. Q{A2t-\)

: /I(i>,) = ««• for 1 < i < I, ? ( ^ + i ) = -vt+x\ : ]I(vi) = - v ^ + i - i j

Case 3. Q(E6) : p(t;t-) = -v 7 -,- for 1 < i < 6, ^(VT) = v7; Case 4. Q(I>4) : /i(afi) = «3, ^ 3 ) = »4, ^ 4 ) = « i , ^(^2) = at2.

We denote here by (. |.)' the normalized invariant form on Q(XN), by A' the set of elements of Q(XN) of square length 2, by II' = {a£,..., a^} the set of simple roots, etc. Note that in all cases, /Z(II') = II', hence JL e Aut(Xiv) is the diagram automorphism of Q(XN). In all cases, fix an orientation of the Dynkin diagram S(XN) which is invariant under JJ, and let £(<*, /?) be the corresponding asymmetry function (defined by (7.8.4)). Let 0 CE'Q) be the corresponding simple Lie algebra, as defined in §7.8. It is clear that the automorphism /I of Q(XN) induces an automorphism /1 of defined by (7.9.1)

/,(<*)= ?(*),

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This automorphism has the following characteristic property: (7.9.2)

M««) = «WQ'

Such an automorphism is called a diagram automorphism of Introduce the following notations:

%(XN).

A< = {« = a' € A' | /!(<*') = a'}; A, = {a = r-'Cpia') + ••• + 7(<*') I cl € A'.tfa') # a'}; A = A, UA/;

Q = ZA;

£ # ) = J^^ ( a /) + • • • + VrjE^r-(a,}, where 77 = exp2-rri/r; Ea = E'a if a € A,,

£a

e ar€A,

(&CEa). We can now state the result. PROPOSITION

7.9. Let (XN,r)

= (£>/+i,2), (i4 M -i,2), (£76,2), or (£>4,3).

a) &(XN) = QQ&W (resp. = g 0 g^1) 0 g(2>) if r = 2 (resp. r = 3), where g is the fixed point set of /i and Q^ (resp. gW), j = 1 or 2) is the eigenspa.ce of /i with eigeuvaiue —1 (resp. exp27rf,;/3). b) g is the simple Lie algebra of type Bif Ci, F^, or G2, respectively, and its commutation relations are as follows:

(7.9.3)

[Ea,E-Q] = -a(resp. - ra)

if a G A/ fresp. a G A,), ifa,/?GA, a + ^ A U { 0 } ifa,0,a

+ Pe A,

wAere p is tAe maximal positive integer such that a — p/? G A. c) The normalized bilinear form (.|.) on g is given by the following formulas (h\h') = (h\h')'\ti1

(h\Ea) = o

(Ea\Ep) = ~^a,-/3 (resp. - r6ai_^)

ifh,h'et>,*eA, if a, ft G A^ (resp. G A,).

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109

d) A is the set of roots of $ with respect to the Cartan subalgebra f), A , (resp. At) being the set of short (resp. long) rootsf and Q is its root lattice; A v = A*UrA,. e) The sets of simple roots II {numbered as in Table Fin) are these: Case 1: <*i = <*[,.. .,<**_ i =<x't_vai Case 2: <xx = %{<*[ + a'u^)%...,

= \{ot't + a'l+1);

a / - i = §(<*/_ x + o j + 1 ) , or* = a't;

Case 3: ax = e*'6,a2 = a^,a 3 = \{ot'2 + a i ) » a 4 = |(<*i + <*;5); Case 4: o^ = a ^ , ^ = | ( ^ i + <*3 + ^i). f) Letting n^ = II H A^, II, = II fl A,, we have: IIV = n^ U rll,. g) [fl,0^] = Q^\ and the g-module g^ is irreducible with highest weight #o (= —lowest weight) listed below: Case 1: 0o = c*i + c*2 H

h a/;

Case 2: 0O = <*i + 2a 2 + • • • + 2a/_i + a/; Case 3: 0O = a x + 2a 2 + 3a 3 + 2a 4 ; Case 4: 0O = «i + 2a 2 . Proof. The proof of a), c), d), e), f), and g) is straightforward; the first part of b) follows from d), e), and Proposition 1.4 a). All commutation relations (7.9.3) except for the last one are clear. If one of the a, /? £ A is long and a + j3 £ A, we clearly have: (7.9.4)

[Ea,Ep] =

e(a'J')Ea+p,

and a — (3 $. A, hence the last formula in b) holds in this case. It remains to consider the case when both a and /? are short and a + /? £ A. First, let r = 2; then p = 1, a + /? £ A*, and we have to show that (7.9.5)

[E

This follows from the following fact: (7.9.6) if r = 2 and a', ff £ A ' ^ , a' + /?' £ A', then a' + T^/?') g A ; . Since the Z-span of a', (3',~p(a'), and /i(/3') can be only one of the root lattices Q{An), n = 2,3, or 4, it suffices to check (7.9.6) in these cases only, + which is straightforward. Hence, we have: [Ea,Ep] — [E'a, + EL,,yE'^ Finally, let r = 3. Then either a' + ft' £ A , , p = 1, and exactly one of a1 + /!(/?'), a1 + /J2(/?') lies in A ; , or a1 + /?' £ A/,p = 2, and none of

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Affine Algebras as Central Extensions of Loop Algebras

Ch. 7

a' + /!(/?')>a' + F2(/?') lies in A. Hence, in both cases the last equation of (7.9.3) holds.

• 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 = 0. b) Commutation relations (7.9.3) can be extended to the whole Q(XN) as f o l l o w s . L e t E i 0 ) = E a = E'Q i f a G A t a n d l e t a<-» = rf fi(a') + ••• + t f > n r ( a ' ) if " € A , . T h e n ( 0 ), /»(i) € if /,«) e h«), a G A,

= 0 (ftW|tt)'£4<+i) 4°, ^ - « ] = -a ( > + j ) (resp. - a)

4

if a G A,(resp. a G A/),

= 0

if a , 0 G A, a + /? g A U {0},

= (p + l)£(a', ^ ) ^ >

if a, /?, a + /3 G A,

where p is as in (7.9.3). c) The Chevalley involution u and the compact involution u>o of Q(XN) are given by (a G A): u>(Ea) = E-ay

u(a) = - a ;

u;0(£a) = i^-a,

^ 0 (a) = - a .

d) It follows from c) and Proposition 4.9 that the Hermitian form (.|.) 0 is positive-definite on g(X/), where Xi is of finite type. §7.10. For applications in Chapter 8, we shall now take care of the remaining diagram automorphism: Case 5: Q(A2i) : Jl(vi) =

-v2i+2-i.

In this case there is no //-invariant orientation; we consider the orientation o—o->> • • • —»o instead. Then the diagram automorphism // of Q(A2I) is defined by (7.10.1)

/i(a) = Ji(a), ?(&„) =

(-l)1+htaEL(a).

Let

At (resp. A,) = {|(a' + Ji(a')) | a' # Ji(a') and (a'|/J(a'))' = « (resp. # 0)},

Ch. 7

Affine Algebras as Central Extensions of Loop Algebras

111

Ea = E'a, - (-l) h t o r *£(„,) if a € At, Ea = V2(E'a, - (-l)htaEL{a>))

if a € A,, = {h€ V

a€A

Then we obtain the following extension of Proposition 7.9. PROPOSITION 7.10. a) 0(^2/) = 0 0 0

(1)

.

b) g is the simple Lie algebra of type Bt, and its commutation relations are given by (7.9.3). c) The normalized bilinear form (.|.) on 0 is given by (h\h') = 4(h\h')% {Ea\Ep) = -46ai-p(resp.

(h\Ea) = o - 8«a,_^)

if a € A* (resp. e As).

d) Same as Proposition 7.9 d) (with r = 2). e) The set of simple roots II is this:

f) Letting II* = II D A*, II, = II fl A,, we have IIV = 211/ U 411,. g) [0> 0 ^ ] = 0 ^ and the %-module 0 ^ is irreducible with highest weight (= -lowest weight) 0Q = 2c*i + 2a 2 + • • • + 2a/.

• iiemar* 7J0. Letting £ i 0 ) = £?ai f^0 = £^, + ( - l ) h t a ^ ( a 0 if a € A/, Eix) = V2(E'Q, + ( - l ) h t a £ J ! ( a 0 ) if a G A,, A m = {a G A' I jl(a) = a } , Ea = E'a if a G A m , formulas (7.9.3) extend in the same way as described in Remark 7.9.

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Affine Algebras as Central Extensions of Loop Algebras

Ch. 7

§7.11. A generalized Cartan matrix of infinite order is called an infinite affine matrix if every one 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): AQQ , A+oo, BQO ) Coo

an

d -^oo •

Let A be an infinite affine matrix, and let Qf(A) be the associated KacMoody algebra (defined in Remark 1.5 for an infinite n). These Lie algebras are called infinite rank affine algebras. For these Kac-Moody algebras we have A = Are; given a G A+, we denote by a v the element of f)' such that a v G [0a,0-a] and ( a , a v ) = 2, and we put A V = { a v G V\a G A} to be the set of dual roots. Here we give an explicit construction of these Lie algebras, which generalizes the usual construction of classical finite-dimensional Lie algebras. We also construct certain completions and central extensions of them, which play an important role in the theory of completely integrable systems (see Chapter 14). Denote by gloo the Lie algebra of all complex matrices (aij)ij€i» s u c n that the number of nonzero a,-7- is finite, with the usual bracket. This Lie algebra acts in a usual way on the space C°° of all column vectors (a,) t ^i, such that all but a finite number of the a,- are zero. Let Eij G gloo be the matrix which has a 1 in the t, j-entry and 0 everywhere else, and let V{ G C°° be the column vector which has a 1 in the i-th entry and 0 everywhere else, so that Let A = Aooi then g'(A) = sloo := {a G
fi = 25i+i,i (i G Z),

so that

n v = K = Eiti - £,+ M+1 (• G z)} is the set of simple coroots. Then fj' consists of diagonal matrices, and n+ (resp. n_) of upper- (resp. lower-) triangular matrices. Denote by e,- the linear function on t)' such that ti(Ejj) = 6^ (j G Z). Then the root system and the root spaces of 0'(J4), attached to nonzero roots, are

A = fa - cj (i / i, ij G Z)};

Q€i.€j =

€i — €j being a positive root if and only if i < j .

Ch. 7

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113

The set of simple roots is H = {a,- = €> - e t + i (t'GZ)}. The set of positive dual roots is (7.11.1)

A* = n v U {a? + aV+i + ••• + « / , where t < j ; ij

G Z}.

The description for A = >l+oo is similar, replacing Z by Z+. The remaining infinite rank affine algebras, of type XQO, where X = J3, C o r D, are subalgebras of the Lie algebra
^(Xoo) = {ae Qf(Aoo)\X(a(u),v) + X(u,a(v)) = 0 for all ti, v G C°°}. These bilinear forms are as follows:

We describe below the Chevalley generators e,-, /,- (i G Z + ), the set of simple coroots IIV, the root system A, the set of simple roots II, and the set of positive dual roots A^ of Qf{A)9 where A = JBQO> COO or DOQ. In all cases, ty consists of diagonal matrices and n+ (resp. n_) consists of upper(resp. lower-) triangular matrices.

/o = 2(£*i,0 + £o,-i), fi = ft+i,i + £ L t > t _ i ( i = 1 , 2 , . . . ) ; nv = K

= 2 ( £ - i , - i - ^1,1), aV = Eiti + ^ - . - i , - , - !

A = {±€i ± €t (1 ^ i, i , i € Z + ), ±e< (t G Z+)}; _ej = C(JS?,j - ( - l ) ' + ' £ - i f - , ) , • # i, i , i G Z; 11 = {ofo = - e i , a i = e f - e t > i (t = 1 , 2 , . . . ) } ; J = n v U{a, v +a f y +1 + . . . + a / (t < j , ij

G Z+),

Here the e,- are viewed restricted to f);, so that ct- = —c_t*.

/o = ^i,o> ft = ^t+i,t + ^-t+i.-t (« = 1,2,...); II = {a0 = #00 ~ #11» «,• = Eiti + -E1-,,-! — -^-t-+il_<+1 ( t = 1,2,...)};

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Affine Algebras as Central Extensions of Loop Algebras

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A = {±et ± eit ±2e, (i # j, U = 1,2,...)}; 11 = { a o = - 2 € i ,

a j = € < - € < + i (t = 1 , 2 , . . . ) } ;

v

A ; = n U { a y + a y + 1 + . . . + ay ( i < j , i ! , i € Z + ) , 2
— ^ - i , i i c,- = ^i,<+i — -S—»,—»+i,

/o = J?2>o - £ 1 , - 1 , /« = ^»+i,« ~ £ - « + i , - » (*' = 1 , 2 , . . . ) ; n v = { « Q = ^o,o + ^ - 1 , - 1

a^ = E i j +

E i i

A = {±e,- ± cj (t # j ; i , i = l , 2 , . . . ) } ; -ey = C(^?y -

£_j+i,_,+i);

II = { a 0 = - c 1 - f2, "i = e< - U+i

(» = 1 , 2 , . . . ) } ;

<*o+<*2

Here the e,- are viewed restricted to f);, so that et- = —e_t-|_i. §7.12. Now we turn to the description of a completion and its central extension for an infinite rank affine algebra. More generally, let A = (a>ij)i,jei be an infinite generalized Cartan matrix, such that every row (and hence column) contains only a finite number of nonzero entries. Let #'(A) = 0 g^ Of

be the associated Kac-Moody algebra. We denote by ~§(A) the subspace of Y\ 0^ consisting of the expressions of the form u = ^aaeQ, where aa G C, a

a

e

<* ^ 0a» a n d s u c h t h ^ the set {j\j = hta with aa ^ 0} is finite. It is clear that we can extend the bracket from %'{A) to ^(A) by linearity. The Lie algebra ~&(A) contains I) := ]~[ Ca)', the completed Cartan subalgebra.

Denote by gl^ the Lie algebra of all complex matrices (ciij)ijez, such that aij = 0 for |t — j \ sufficiently large, with the usual bracket. It acts in a usual way on the vector space C°° of columns (a,-),-gi, where all but a finite number of the a,- are zero.

Ch. 7

Affine Algebras as Central Extensions of Loop Algebras

115

If A is an infinite affine matrix of type XQQ = Aoo (resp. Boo, Coo or A » ) , w e denote ~&(A) by Zoo. Then, clearly, Zoo is isomorphic to gl^ (resp. the subalgebra of gl^, which consists of matrices preserving the bilinear form B, C or D). A completed infinite rank affine algebra, denoted by Xoo, is the central extension of Zoo defined as follows. The Lie algebra a©o has a 2-cocycle tfr defined by: xl>{EihEji) = 1 = -iKEjitEij)

if t < 0 and j > 1,

(7.12.1) ip(Eij,Emn)

= 0 otherwise.

One easily checks that this is a cocycle (see Exercise 7.17). Then if A = AQO (resp. Boo, Coo or Doo), we put r = 1 (resp. | , 1 or | ) , and let Zoo = XQO © CK be the Lie algebra with the following bracket: [a@\K,b®iiK] = (ab-ba)®rtl;(a,b)K

(a,ft€Xoo; A,/iGC).

The elements e,-,/t- £ C^XQO) C XQO (defined above) are called Chevalley generators of XQQ, and f) = f) 4- C/^ is called the Cartan subalgebra. The elements of the set fiv = {a% =a%+K, a^ = aV for i ^ 0} are called simple coroots. It is easy to see that we have the usual relations: [5)^,5/] = 0, [eijj]

= Sij&V, [ai,ej] = a^e,-,

[5n/i] = - « 0 7 i , (ade.O^^e,- = 0, ( a d / O ^ ^ / i = 0. Remark 1.12. The Chevalley generators {e,-,/i}, € / generate a subalgebra ( 0 Ca)0 © ( © 0a) of Zoo, which is isomorphic to &'(A). This follows from Corollary 5.11b). Note that the principal gradation of &'(A) extends in a natural way to a gradation of ZQO, called its principal gradation. We also have the following expressions for the canonical central element:

K = • >0

t>2

116 §7.13.

Affine Algebras as Central Extensions of Loop Algebras

Ch. 7

Exercises.

7.1. Let p be a Lie algebra, i) C p a finite-dimensional diagonalizable subalgebra, p = 0 p a the root space decomposition. Show that every a

C-valued 2-cocycle %j> on p is equivalent to a cocycle Vo (i.e., ip(x,y) — ^ 0 (x, y) = /([x, y]) for some / G p* and all a?, y G p) such that ^o(Pa, P/?) = 0 for a + $ / 0. [p operates on the space of all 2-cocycles in a natural way, so that f) is diagonalizable. Show that an eigenvector with a nonzero eigenvalue is equivalent to 0.] 7.2. Let p be a Lie algebra with an invariant symmetric bilinear C-valued form (.|.) and let d be a derivation of p such that (d(x)\y) = —(x|d(y)), x, y G p. Show that ip(x,y) := (d(x)\y) 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 p', so that we obtain the Lie algebra p = p' + Cd. 7.3. Let p be a Lie algebra with a nondegenerate invariant bilinear C-valued form (.|.). Let R be a commutative associative C-algebra with unity, and / a linear functional on R. Extend the form (.|.) to the complex Lie algebra p = R 0 c P by (ri
Ch. 7

Affine Algebras as Central Extensions of Loop Algebras

117

7.6. Let p be a Lie algebra with a nondegenerate invariant bilinear form (.|.) and let tii,t«2,... be its orthonormal basis. Let %l> be a C-valued 2cocycle on p such that for each i only a finite number of ^(m^Uj) are nonzero. Show that %l> is of the form described in Exercise 7.2. 7.7. Prove that every 2-cocycle on the Lie algebra £(0) is equivalent to a cocycle A^, where A € C, and \l> is described in §7.2. [Use the action of do on the space of 2-cocycles and apply Exercises 7.6, 7.4, 7.3.] 7.8. Show that £(0) is the universal central extension of the Lie algebra £(0) (cf. Exercise 3.14). [Use Exercise 7.7.] 7.9. Show that a C-valued 2-cocycle on the complex Lie algebra C p i , ^ 1 , . . . , ^ , ^ 1 ] 0 c 0, where 0 is a finite-dimensional simple Lie algebra, is equivalent to a cocycle c*£>, where D = £^ ^t^"> with J2 Ifr = ^» t

t

defined by a I ? ( P ( g ) x , Q 0 y ) = (coefficient at (fi...*,)"" 1 in £>(P)Q)(x|y). 7.10. Consider the bilinear form (x\y) = tr(xy) if 0 = sln or spn, and

(*|y) = 5 t r ( ^ ) i f * =

SOn

> n > 4-

Check that |0| 2 = 2, so that the "normalized" cocycle %j? on £(0) is given by V>(a,6) = Restrf — b\ if 0 = s^n or spni ^(a,6) = - R e s t r f - ^ 6 j if 0 = so n , n > 4. Show that the normalized standard bilinear form on £(0) is given by (a\b) = constant term of tr(a6) if 0 = sln or sp n , (a|6) = - constant term of tr(a6) if 0 = son, n > 4. o

7.11. Let 9= s£i+i(C). We keep all the notation of Exercise 1.13, and add ° on the top: 0, £, II, and IIV. We have £(0) = 5^+1 (£). Show that

118

Affine Algebras as Central Extensions of Loop Algebras

Ch. 7

© CK © Cd with the bracket

the space

[a(t) + XK + fid, ai(t) + XtK + md] dt

^

dt

dt 0

is the affine algebra of type A\ }. Set f) = f) © Clf 0 Cd, extend et- from f) to f) by e,(X) = €i(d) = 0, and define 6 G f)* as in §7.4. Set

Set II = {a o ,II}, IIV = {a^,II v }. Show that (h,II,II v ) is the realization of the matrix (of type A^ '):

and / 2 -1 0

-1 2 -1

0 -1 2

... ... ...

0 0 0

0 V-l

0 0

0 0

... ...

2 -1

-1 0 0 -1 2 /

The root space decomposition of C(&) is

Show that £(g) = g(^l), with the Chevalley generators e,-, /j (i = 0 , . . . ,€). Show that the set {ct- - e;- + s6 (i £ j , s e Z); s6 (s G Z\0)} is the root system of $(A). Show that its subalgebra n_ (resp. n+) consists of all matrices from «
if a(<) G £(g),

Ch. 7

Affine Algebras as Central Extensions of Loop Algebras

119

7.12. Introduce the following basis of the Lie algebra sli(C):

0)'

Then [U^Lj] = cyL,-+,- (i, j G Z), where C{j = —1, 0, or 1 according as j — i = —1, 0, or 1 mod 3. 7.13. Show that the Lie algebra 0 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 $ such that %l>(d^dj) = 0 whenever i + j ^ 0. Denote by CJ the vector space (over C) of such cocycles. Show that = 2 and that the cocycles ip\ and ^2> defined by

form a basis of C\. Deduce that Vir is the universal central extension of D. 7.14. Let E1 (7 € A), c^ (t = 1, . . . , * ) be a ChevaJley basis of a finitedimensional simple Lie algebra 0, i.e., 7)£_7]

= 7V;

[7v,7'v]=0;

, J57] = n^j7^_(_7,

[
where n^)7 G Z, n/3j7 = —n_7) _^.

(cf. §7.8 and §7.9 for an explicit construction of such a basis.) Then the elements =tk®Ey

(k e Z, 7 G A);

OJY (z = 1 , . . . , t)\

k and d,

form a basis of >C(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 0 be a simple finite-dimensional Lie algebra and V a finitedimensional 0-module. Then we may in a natural way define an £(g)module V = C 0 c V. Fix A G C and define an action w of the affine algebra £(g) on V by: x | £ ( 0 ) unchanged;

* ( # ) = 0;

ir(d) = < ~ ® 1 + A l v .

Show that this is an integrable £(g)-module.

120

Affine Algebras as Central Extensions of Loop Algebras

Ch. 7

o

7.16. Let Q be a simple finite-dimensional Lie algebra, (.|.) the normalized o

o

o

invariant form, J) a Cartan subalgebra, A+ a system of positive roots, P o

their half-sum, 6 the highest root. Choose a basis u\,..., U£ of \) and the dual basis u1,..., ul. For each a £ A+ choose root vectors eQ and e_ a such o o that (e a |e_ a ) = 1. Let fi be the Casimir operator of 0. Finally, let be the nontwisted affine Lie algebra associated to 0, and set p = hv Ao + p (where Av is the dual Coxeter number of £(g)). Show that

and that the Casimir operator Q for the affine algebra C(&) (see Chapter 2) can be written as follows:

t=l

7.i7. Let Ln = J^Zt;,- be the standard lattice in the Eucildean space Rn. »=i

Define on Ln x Ln a bimultiplicative function e by letting e(vj,vk)

= 1 if j < k, and = - 1 if j > k.

Show that by restricting to root lattices Q(Ai) C i / + i and Q(Di) C (described in §6.7), we obtain an asymmetry function. 7.18. Let | L n = y^Zti,-, where it,- = |v,-, and define on (|^n)

x

i\Ln) a

•=i

bimultiplicative function e (extending the one from Exercise 7.17) by: £(tfpti*) = 1 if j < ib, and = e^ if j > k. Show that by restricting to the root lattices Q{E8) and Q(E7) ( c \L%), we obtain an asymmetry function. Similarly, taking the lattice ^L6 + Zu in R 7 , where u = -7^7, and extending e by £(u, Uj) = e(u, u) = e ^ , S(UJ,U7) = 1, we obtain an asymmetry function for Q(E6). 7.19. Let p be a subalgebra of £(fl) of finite codimension, where 0 is a simple finite-dimensional Lie algebra. Then there exists a nonzero Q G £ such that p D Q£ ®c 0. [Show that for a subspace V of finite codimension in £, the space V2 contains a nonzero ideal of C]

Ch. 7

Affine Algebras as Central Extensions of Loop Algebras

121

1.20. Show that the adjoint representation of the affine Lie algebra 0 = sln(C) + CK + Crf induces the adjoint representation of the loop group G = SLn(C) on 0 given by the following formula: (Ada)(x + XK + pd) = (axa~l - / i t a ' a - x ) +(A + Res trCa'xa"1 - \nt(a'a~l)2)K

+ fid),

where a £G, o! denotes the derivative of a by t, x G sln(C), and A,/i G C. [Use the following facts: a) Ada preserves (.|.); c) (Ada)x = axa'1 mod CK] d) (Ada)K = K]

b) [a, £] = —a';

0

7.21. Let G be a connected simply-connected complex algebraic group op0 erating faithfully on a finite-dimensional complex vector space V. This 0

0

action extends to the action of the loop group G := G(C) on V := £<S>c V. oo

0

oo

Let 0 be the Lie algebra G, fy a Cartan subalgebra, Q v C f) the dual root lattice; let V = 0 VA be the weight space decomposition with respect to A

f). Given 7 G Q v , define V G End £ V by:

0

Show that V G G, thus giving an injective homomorphism / : Q v —• G. Show that, via the adjoint action of G on £(0), we have: v

f(f) • (tk ® ea) = ik+^) ® ea (ea € L); (Thus, we obtain a canonical embedding of the group of translations T 0

into G.) Similarly, define an injective homomorphism / : P v —• AdG (where P v = {h G b | (a,/i) G Z for all a G A}). Show that the group ~

0

^

0

~

T := / ( P v ) acts on £(0)J>y the same formulas. Show that the group T normalizes W and that W := W K T preserves Q v and Q. Show that the group W+ = {w € W \ w(A+) C A+} acts simply transitively on 7.22. Let 0 be a simple finite-dimensional Lie algebra and let R be a commutative associative algebra with unity. Let Q^ denote the space of all formal differentials over R, i.e., expressions of the form fdg, where / , g G -R, with relation d(fg) = fdg + yd/. Then the universal central extension of the complex Lie algebra R ®c 0 is 0 — Q}RldR — 0 H := 0 H © (fiJt/di?) -^ 0H — 0,

122

Affine Algebras as Central Extensions

of Loop Algebras

Ch. 7

where the bracket on $R is defined by: [ n <8> g\, r 2 02] = r i r 2 [01,
mod dR

(This is a generalization of Exercise 7.9.) d n z~ n ~ 2 be the generating series for Vir. Show that

7.23. Let T(z) = ^ n€Z

[d m ,T(z)] = zm(z£ + 2(m + l))T(z) + *^z™-2c. (T(z) is called the energy-momentum tensor. It is a primary field if and only if the "conformal anomaly" c is 0.) 7.24- Consider the representation of the Lie algebra D (= representation of Vir with c = 0) on the space Ua^ of "densities" of the form P{t)ta(dtf, where a,/? are some numbers and P(t) G C. Show that in the basis v* = <*+Qf(dt)/? this representation looks as follows: dn(vk) = - ( * + a + P +

pn)vn+k.

Show that F(z) = £ ^ F*z~*~ A , where F* : V - • Vi, is a primary field of *€Z

conformal weight A if and only if the map F i t ^ - i . i - A ® ^ — * V\ defined by F(vk 0 u ) = Fjc(v) is a Vir-module homomorphism. 7.25. Let 0 be a simply-laced simple Lie algebra as constructed in §7.8. Given a G A, let Ra = (exp - ad £_ a )(exp - ad EQ)(exp - ad £ L a ) . Show that Ra\h = r o , i? a (^/j) = -e(aip)EraW = JE1^ otherwise.

if (a|/?) ^ 0 and

7.i?tf. Let 6:(a,y9) be a 2-cocycle on Q with values in {±1}, i.e., e(a, /3)e(a + /?, 7) = e(i8,7)e(«, ^ + 7) for a, /? e Q, ^(0,0) = 1 (and hence e(0,a) = e(a,0) = 1 for a G Q); suppose that in addition e ( a , - a ) = (-l)*(°l a ) and e(a,P)e(P,a) Then Proposition 7.8 still holds.

= ( - l ) ( a | / 3 ) , a,/? G Q.

Ch. 7

Affine Algebras as Central Extensions of Loop Algebras

123

7.27. Let Q be the root lattice of a simply laced simple Lie algebra. Let B : Q x Q —• Z be a bilinear form such that B(a,/?) + £(/?, a) = (c*|/?). Then e(a,/?) = ( - l ) B ( a ^ ) is an asymmetry function. Write B(a,0) = (Ra\(3). Then the conditions on B are equivalent to R(Q) C Q* and R + R* = I. Show that if w G Aut Q is such that Q C (l-w)Q*, then R = 1-w satisfies these conditions. Show that the Coxeter element has the above property. 7.28. Let V = VL 0 V+ be a vector space over C (in general infinite dimensional) represented as a direct sum of two subspaces. Let gl+(V) denote the Lie algebra of endomorphisms of V which have the form a = ( a* a *) with respect to the above decomposition, where a$ : V_ —• V+ has a finite rank. Show that /(a, b) := trv_a2h ~ try_b2a^ is a 2-cocycle on the Lie algebra gl+(V). Show that the restriction of this cocycle to the subalgebra of finite rank endomorphisms is trivial. Show that for V = ® Cv,-, VL = £ Cv<, V+ = X! Cw,-, t h e cocycle / restricted t'€l

»>0

t<0

to gl^ coincides with the cocycle ip defined in §7.12. 7.29. Let V = C[t,t~l]n be the natural s* n (C[M-^-module. This extends to the module over the Lie algebra of differential operators a = ^ n ( C [ M ~ \ ^ ] ) - Set V+ = (tC[t])n,VL = ( q t - 1 ] " C V. Show that the restriction of the cocycle / of Exercise 7.28 to the subalgebra a C gl*(V) is given by the following formula (£ € Z + ; m, m' e Z; a, a! e g£n(C)):

§7.14. Bibliographical notes and comments. Except for the explicit formula for the central extension, the realization of nontwisted affine Lie algebras was given by Kac [1968 B] and Moody [1969]. The formula for the cocycle has been known to physicists for such a long time that it is difficult to trace the original source. Proposition 7.8 (in the form given by Exercise 7.26) is due to Frenkel-Kac [1980]. The Virasoro algebra (first studied by Virasoro [1970]) plays a prominent role in the dual strings theory (see, e.g., Mandelstam [1974], Schwartz [1973], [1982]). Mathematicians started to develop a representation theory of the Virasoro algebra quite recently (Kac [1978 B], [1979], [1982 B],

124

Affine Algebras as Central Extensions of Loop Algebras

Ch. 7

Prenkel-Kac [1980], Segal [1981], Feigin-Fuchs [1982], [1983 A, B], [1984 A, B], 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 connection with statistical mechanics and a revival of the dual strings theory. The conformally invariant field theory, which is the foundation of both of these remarkable developments, originated in papers by Belavin-PolyakovZamolodchikov [1984 A, B]. The notion of a primary field, briefly discussed in §7.7, is a key notion of this theory. A survey of recent developments in string theory may be found in the two volume book by Green-SchwartzWitten [1987]. Exercise 7.2 is due to Kupershmidt [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 Z-form of the universal enveloping algebra of an affine Lie 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 x of the group of polynomial (or analytic, etc.) maps of C x to a complex simple finite-dimensional Lie group. The corresponding "compact form" is a central extension by a circle of the group of polynomial (or analytic, or C°°, etc.) loops on a connected simply-connected compact Lie group. Thus, there is a whole range of groups associated to an affine Lie algebra (or rather a certain completion of it). The group of polynomial maps is, naturally, the minimal associated group; this is a special case of groups discussed in §3.15. Various aspects of the theory of loop groups may be found in Garland [1980], Frenkel [1984], Pressley [1980], Segal [1981], Atiyah-Pressley [1983], Goodman-Wallach [1984 A], Kac-Peterson [1984 B], [1985 B], [1987], Freed [1985], Pressley-Segal [1986], Mitchell [1987], [1988], Kazhdan-Lusztig [1988], Mickelsson [1987], and others.

Chapter 8. Twisted Affine Algebras and Finite Order Automorphisms

§8.0. Here we describe a realization of the remaining, "twisted" affine algebras. This turns out to be closely related to the Lie algebra of equivariant polynomial maps from C x to a simple finite-dimensional Lie algebra with the action of a finite cyclic group. As a side result of this construction we deduce a nice description of the finite order automorphisms of a simple finite-dimensional Lie algebra, and in particular the classification of symmetric spaces. §8.1. Let 0 be a simple finite-dimensional Lie algebra and let a be an automorphism of 0 satisfying crm = 1 for a positive integer m. Set € = exp 2£i. Then each eigenvalue of a has the form eJ, j G Z/mZ, and since a is diagonalizable, we have the decomposition (8.1.1)

0= i€Z/mI

where 0;- is the eigenspace of a for the eigenvalue eJ. Clearly, (8.1.1) is a Z/mZ-gradation of 0. Conversely, if a Z/mZ-gradation (8.1.if is given, the linear transformation of 0 given by multiplying the vectors of 0;- by e7 is an automorphism a of 0 which satisfies
Proof. Given x € 0,-, y € 0j, we have (x\y) = (
126

Twisted Affine Algebras and Finite Order Automorphisms

Ch. 8

where f) is a Cartan subalgebra of 0 containing f)^ 0 a are root spaces with respect to F) and the summation is taken over a G A such that a\^ = 0. It follows that 3 = t) + 6, where * is a 0 and x G «n - Then (ad z)r*t- C *nr+t• Select a positive integer r such that n(r—1) < m —t < nr. Then nr-|-i = m+t with 0 < f < n, so by the inductive assumption, finr+i = *t = 0. Thus adx|, is nilpotent; similarly, ady| f is nilpotent if y G *- n - But [fi n ,*-n] C «o = 0, so ada: and adj/ commute on « and by the nilpotency, tr,(adxady) = 0. Now Lemma 8.1a) (applied to a) implies 0 = 0, proving b).

• It follows from Lemma 8.1b) that 1)Q contains a regular element, say x, of 0. Hence, the centralizer f) of x in 0 is a ^-invariant Cartan subalgebra, and the sum of eigenspaces of adx with positive eigenvalues (we say that a G C is positive if either Re a > 0 or Re a = 0 and Im a > 0) is a maximal nilpotent cr-invariant subalgebra n+. Let A+ be the corresponding set of positive roots. Thus, a induces an automorphism of A+; let /i be the corresponding diagram automorphism of 0. It is clear that crfi"1 = exp(adA), where h G fj(f. Since Cartan subalgebras of 0 are conjugate, we have proved the following 8.1. Let g be a simple finite-dimensional Lie algebra, let t) be its Cartan subalgebra and let U = {<*[,... ,<*'N} be a set of simple roots. Let a G Aut0 be such that orm = 1. Then a is conjugate to an automorphism of 0 of the form PROPOSITION

(8.1.2)

jiexpfad^A), ^

h G %,

where p is a diagram automorphism preserving I) and W, f)^ is the fixed point set offi in t), and (a(«, h) G Z,i = 1 , . . . , N.

D

§8.2. We associate a subalgebra £(0,
Ch. 8

Twisted Affine Algebras and Finite Order Automorphisms

(8.2.1) £(0,
£(0,(7,^1),, where £(0, (7, m), = *' 0 0;

127

m o dm.

J€Z

The decomposition (8.2.1) is clearly a Z-gradation of Note that £(0, <7, m) is the fixed point set of the automorphism
(; 6 Z , x G 0).

Hence, £(0,<7, m) may be identified with the Lie algebra of equivariant maps (with respect to the action of Z/mZ): (C x ;

multiplication by e""1) —> (0; action of a).

Recall the Lie algebra £(0) = £(0) 0 CK' 0 Cd' defined in §7.2 (here we write K1 and d! instead of K and d for reasons which will become clear later on). We set £(0, (7, m) = £(0, (7, m) 0 CK1 0 Cd'. This is a subalgebra of £(0), which is the fixed point set of an automorphism a of £(0) defined by cr\c(9}<7tm) = ^, ^ ( ^ 0 = K1 > &(
§8.3. The structure of the Lie algebra £(0, (7, m) for arbitrary (7 will be studied in the following sections. Here we consider some very special examples, which give us an explicit construction of all twisted affine algebras, i.e., those listed in Tables Aff 2 and Aff 3. Let 0 = f)' 0 ( 0 C ^ ) be a simple finite-dimensional Lie algebra of type XJV, as constructed in §7.8 and 7.9 with the normalized invariant form (. I.). Let /I be an automorphism of the Dynkin diagram of 0 of order r (= 1, 2 or 3), and let /i be the corresponding diagram automorphism of 0. Case 0. For XN = At, B^ . . . , E$ and r = 1 let II = {e*i,..., a^} be the set of simple roots of Q enumerated as in Table Fin. Let Ei — E'a., Fi =

-E'_ai,

Hi = a t (i = l , . . . , e ) , E o = ELe, Fo = - E ' e , H 0 = 0 = e l

128

Twisted Affine Algebras and Finite Order Automorphisms

Ch. 8

Cases 1-5. Let now XN = #*+i, Ait- I, #6, &4 or Au with the ordering of the index set as in Table Fin, and let r = 2, 2, 2, 3 or 2 respectively (these are, clearly, all the cases when /i ^ 1). We have the corresponding Z/rZ-gradations (described by Propositions 7.9a) and 7.10a) in a slightly different notation): if r

(8.3.1)

g = 0o © 0 1 i f r = 2>

and

0 = Go

(8.3.2)

I)' = J)o © &Tif r = 2>

and

*)' = *)o © ^T © ^2 i f

= r

= 3-

Here and further, "s e Z/rZ stands for the residue of s m o d r . Let II' = {a[,..., a'N} e \) be the set of simple roots of g (enumerated as in Table Fin) and let E[ = E'a., F[ = -ELa. (i = 1,.. -, N) be its Chevalley generators and H[ = a™ simple coroots. Introduce the following elements

Case 1: XN = Dt+u


r

= 2:

1),H£ = H't + H'i+1,H0 =

-60-

< i < / - 1), £ / = ^ + JES+1,£?o = # - , o -

Case 2: A^y = A2/-1, r = 2:

^ = i/; + H'v-t (1 < * < / - 1), ^ = # , ^0 = - « ° E{ = £^,' + E'2t_i (1 < ii < I — 1), S/ = £"/, f^o = ^l^o — Fi = F! + FM-i (1 < « < I - 1), Ft = F;,

FO

= -^. +

Case 3: XN = E6, r = 2: 0° = c*i + 2<4 + 2a^ + «4 4- a'5 + a^; Fx = H[ + H's,,H2=H'2

+ H'4,H3 = H'z,H4 = 1%,Ho = -9°

EI = E[ + E'S,E2 = E2 + E'4, E3 = E'3,E4 = E'4, Eo = E'_g0 Ft = F{ + Fi, F2 = F^ + Fi, F3 = F^, F4 - Ft, Fo = -E'g0 + £ Case 4: XN = D4, r = 3, r) = exp2*-i/3:

0° = oi + a'2 + a'3;

Ch. 8

Twisted Affine Algebras and Finite Order Automorphisms H'4t H2 = H'2, Ho = -6° - Ji(60) 4 E4i E2 =

129

Ji2(00);

= E_QO 4 TJ E^—^o^ 4-

F1 = F[ 4 ^ 4 Fl F2 = F^ Fo = -E'eo - r)E'n0O)

-

Case 5: XN = A2i, r = 2: /9° = ai -f • • • 4- a^; JTi = F ; + JT^_ i+1 (l < t < £ - 1), Ho = 2(H'£ + H'i+1 ) , Ht = Ei = E\ 4- ^ _ i + i (1 < * < ^ - l ) , £ o = \ / 2 ( ^ 4- E'i

1)rn,Ei

=

E'_e0.

Fi = F[ + i ^ _ . + 1 (1 < i < I - 1), F o = y/2{F't 4 F; + ), Ft = - ^ 0 . Let «o = "(PC* 0 ) + • • • + ? r ( ^ 0 ) ) i n Cases 1-4, and 0Q = 6° in Case 5. Let e = 0 in cases 0 - 4 , and € = I in case 5, and let 7 = { 0 , 1 , . . . , ^ } \ { ^ } . P R O P O S I T I O N 8 . 3 . a) The elements E{ (i = 0 , . . . , ^ ) generate the Lie algebra 0. b) Elements £",-, Ft- with i £ I are Chevalley generators of the Lie algebras 0Q-; elements a,- = 2/f,/(#,-1 H{)} i G /, being simple roots. c) [Et,Ft] = # £ , (Ee | F£) = r/a 0 , and (0O | *o) = 2a o /r, where a0 = 1 in Cases 1-4 and ao = 2 in Case 5. d) The representation of 0Q- on 0j- is irreducible, and is equivalent to the representation on 0_j. e) Fo (resp. EQ) is the highest (resp. lowest) weight vector of the ^-module 0j- (resp. Q_j) with weight 0Q (resp. —0Q) (i.e. $o 4 a,- is not a weight). The types of&Q and the decompositions $o = ^ ata,- are listed in the following

table:

r

0

2 2 2

Au-u * > 3 £>M-i, * > 2

2 2 3

<*•

Bo Bt

O — O

Ct

1 2 O — O

Bt

o—

2

2

1

1

O

Ft

G2

2

—O

O

2

1

O

O

1

—

1

—O

=>

3

2

2

o

o

A\ #6

2

—

1

2

o --

O

1

o

O — O

2

O

Proof follows immediately from the results of §§7.9 and 7.10.

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Ch. 8

Remark 8.3. Letting ae = —0Q and ae = 1, we can write: z •=O

where the a,- are given by Table Aff. The restriction of (. | . ) to J)Q = f)' n g$ is nondegenerate, and hence defines an isomorphism v : J)Q —> J)jj. Let A j (s = 0, . . . , r — 1) be the set of nonzero weights of (}Q on gj, and let Qj = 0 #s, a be the weight A{0}

space decomposition. Proposition 8.2 implies that (a\a) ^ 0, dim07 (a = 1 and [g 7|a , fl-j.-a] = Ciz-^a) if a 6 A 7 . Now we turn to the Lie algebra £(g,/i,r), which we denote by £(0,/i) for short.. Set f) = ^ + CK' + Cd' and define 6 G *)* by «|ft-+cir' = 0, (5, <*') = 1. Set ee = t 0 £ 6 , / c = r 1 ® F e , c< = 1 ® ^ , /< = 1 ® ^ (t € / ) . Then we have: [tu fi] = I® Hi (i€ I);

[«e> /.] = rao 1 /C' + 1 ® F e .

We describe the root system and the root space decomposition of £(g,/i) with respect to f): (8.3.3) A = {jS + 7, where j G Z, 7 G A j , j = 5 mod r, 5 = 0 , . . . , r — 1} U{j6,

(8.3.4)

where j G Z, j ^ 0};

A«./0 = *©(©£(«,/0«).

where (8.3.5)

£(g,/i),*+ 7 = t9 ® 07,7,

We set

Using Proposition 8.3 we obtain that if g is of type XN and r (= 2 or 3) is the order of /x, then the matrix

is of type X^ and the integers do, . . . , a* are the labels at the diagram of this matrix in Tables Aff2 and Aff3. Now we can state the second realization theorem. Its proof is similar to that of Theorem 7.4.

Ch. 8

Twisted Affine Algebras and Finite Order Automorphisms

131

8.3. Let g be a complex simple finite dimensional Lie algebra of type XN = Di+ij ML-\, EG, D 4 or A2i and let r = 2, 2, 2, 3, or 2, respectively. Let \i be a diagram automorphism of g of order r.1 Then the Lie algebra £(g, /i) is a (twisted) affine Kac-Moody algebra g(A) associated to the affine matrix A of type X$ from Table Affr (r = 2 or 3), f) is its Cartan subalgebra, A the root system, II and IIV the root basis and the coroot basis, and eo, . . . , ei, /o, . . . , ft the Chevalley generators. In other words, (£(g,/x), f), II, IIV) is the quadruple associated to A. D THEOREM

We can summarize the results of Theorems 7.4 and 8.3 as follows. Let A be an affine matrix of type X# , let g be a simple finite-dimensional Lie algebra of type Xjq and let /i be a diagram automorphism of g of order r ( = 1, 2 or 3). Then the Lie algebra £(g,/i) is isomorphic to the affine Lie algebra &(A). Note that £(g,/i) is isomorphic to $'(A) and C(Q,fi) to 8.3. Let g(A) be an affine algebra of rank I + 1 and let A be of type Xff. Then the multiplicity of the root jr6 is equal to £, and the multiplicity of the root sS for s ^ Omodr is equal to (N - £)/(r — 1). COROLLARY

Proof. If r = 1, then multj6 = £ (for j £ 0) by Theorem 7.4. If r = 2 or 3, then by Theorem 8.3, mult s6 (for s ^ 0) is equal to the multiplicity of the eigenvalue exp2nis/r of \i operating on f)', which gives the result.

• The Chevalley involution u/, the compact involution UQ and the triangular decomposition of the Lie algebra £(g,/i) C £(g) are induced by those from £(g). The normalized invariant form on £(g,/i) is given by (P^x\Q^y) = r

**

j

(Ctf' + Cd'|A<M))

Res(rPQ)(x\y)(xiyeQ,

0 {K'\K')

P,Q G £);

(J\f) = 0; (K'\f) = 1,

where ( . | . ) is the normalized invariant form on g. The proof is similar to that of (7.5.1). It is also easy to see that K = rKf is the canonical central element, and that d = aor"1cP is the scaling element. 1

For r = 3 there are two such automorphisms which are equivalent. We choose one of them.

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Twisted Affine Algebras and Finite Order Automorphisms

Ch. 8

Warning. The Lie algebra 9^ is isomorphic to the Lie algebra 0 introduced in §6.3 in all cases except A"2t*\ in the latter case 0 is of type Ct whereas 0o is of type Bt. §8.4.

Here we present another application of realization theorems.

8.4. Let Q(A) be an affine algebra. a) Set t = CK + J^ 0,$. Then t is isomorphic to the infinite-dimensional PROPOSITION

'€1

Heisenberg algebra (= Heisenberg Lie algebra of order 00; see §2.9) with center CK. b) The Hermitian form (x|y)o = —(vo(x)\y) is positive semidefinite on g'(A) with kernel CK. Proof. By the realization theorem, Q'(A)/CK ~ £(0,/i). The gradation of 0 which corresponds to /i induces the gradation of the Cartan subalgebra f)' of 0 (see §8.2): f)' = © f)'.. We obtain the following isomorphism: i€Z/n

It follows that t/CK is a commutative subalgebra. It is easy to see that the restriction of the cocycle ip to this subalgebra is nondegenerate. This proves a); b) follows from the remarks at the end of §7.6.

• The subalgebra t is called the homogeneous Heisenberg subalgebra of the affine algebra 0(^4). It plays an important role in representation theory of affine algebras. §8.5. Let 0 be a simple finite-dimensional Lie algebra, let m be a positive integer and let e = exp - 1 . Let 7^) : C* —• Aut 0 be a regular map (we view Aut0 as an algebraic group over C); we may regard 7(2) as an element of Aut£(0) (by pointwise action of y(t) on a(t) £ £(&))• We shall view
Ch. 8

Twisted Affine Algebras and Finite Order Automorphisms

133

if and only if (8.5.1)

0y(e"1t) = y(t)a

forallteC*. D

If a, /? G Aut g and 7 : C x —• Aut g (a regular map) are such that (8.5.1) holds, we write a —• 0. The following relations are immediate: a 1+0 implies 0 7~^ a;

(8.5.2)

(8.5.3)

a

2+0 2*01 imply a 7-3 ft,

in particular (8.5.4)

a 1+0 implies a C yftjT1 for y G Aut 0.

PROPOSITION 8.5. Let a be an automorphism of a simple finite-dimensional Lie algebra g of the form (8.1.2). Denote byth the regular map C x —* Aut g such that th on 0 a is an operator of multiplication by t^Q*h). Then

(8.5.5) Proof follows immediately from Lemma 8.5, since thji = fith.

Remark 8.5. It follows from Proposition 8.5 that the isomorphism class of £(g,{r, m) depends only on the connected component of Autg containing a. It is not difficult to show that this statement still holds if g 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 said let a be an automorphism of g such that
134

Twisted Affine Algebras and Finite Order Automorphisms

Ch. 8

(i) the bilinear form on £(g, $/), where Sj are nonnegative integers which satisfy the relation (8.5.6)

(iii) $(1 f)o + C ^ ' + C c 0 * t A e Cartan subalgebra (iv) m$(K') is the canonical central element K of g(A); (v) m~1^(d0)

= a,Q1d+u- ^(u\u)K, where d is the scaling element

o

of Q(A) and u e I) is defined

b y ( a * , u ) = r s i / m (i = l , . . . , £ ) .

Proof. Due to Proposition 8.1, Lemma 8.5 and (8.5.4), we may assume that
In other words, according to Theorems 7.4 and 8.3, we have an isomorphism, denoted by

The Z-gradation (8.2.1) of £(0,
The isomorphism y?"1 can be lifted to a (unique) linear isomorphism 9 : C(Q,crf m) —• 0(J4), which satisfies (iv), (v) with s,- replaced by s(, and the following condition (and hence (iii)): 1

(8.5.7)

*'- 1 (aV) = y > K ) + r(a,«;/ay)A: /

(i = 0 , . . . , * ) .

Ch. 8

Twisted Affine Algebras and Finite Order Automorphisms

135

One easily checks that this is a Lie algebra isomorphism. Furthermore, it is clear that the Z-gradation (8.2.2) of £(g, cr, ra) induces via the isomorphism $' the gradation of type sf of g(A). The property (i) of $' is also straightforward. It remains to show that the s^ can be made non-negative. For that pick v € \) (the Cartan subalgebra of Q(A)) such that (oti,v) = 5^; note that 6(v) = m/r > 0. Hence, by Lemma 3.8, Proposition 3.12b) and Proposition 5.8b), there exists w G Wad (see Remark 3.8) such that (ai,w(v)) = Si € Z+. Now $ = WQ1®' is the desired isomorphism.

§8.6. We deduce from Theorem 8.5 a classification of finite order automorphisms of a simple finite-dimensional Lie algebra 0 of type XN • Let /i be a diagram automorphism of 0 of order r. Let £ , , i^-, Hi (i = 0, . . . , £) be the elements of 0 introduced in §8.3 and let c*o, . . . , (*i be the roots attached to the #,-. Recall that the elements E% (i = 0 , . . . ,t) generate 0 (by Proposition 8.3a)) and that there exists a unique linear dependence ]H cuoii = 0 such that the a,- are positive relatively prime integers (see »=o Remark 8.3). Recall also that the vertices of the diagram X^' are in one-to-one correspondence with the E{ and that the a,- are labels at this diagram. In order to derive an application to finite order automorphisms, we need the following fact. LEMMA 8.6. Every ideal of the Lie algebra £(0,/i) is of the form P(f r )£(0,/i), where P(t) 6 C. In particular, a maximal ideal is of the form (1 - (at)r)C(8,l*), where a G C x . Proof. Let i be a nontrivial ideal of £(0, JA) and x = J^ VPjs(t) 0 aj8 G i, _ 7,* where 0 < .; < r is such that j = j mod r, Pj s(t) G £ , Pjs / 0 and djtS G 0; are linearly independent. We show that Q(tr)Pj $(t)C(&,ii) C i for all Q(t) G £ . Let 1)Q be a Cartan subalgebra of 0Q-; we can assume that x is an eigenvector for ad f)Q with weight a G f)£. If a ^ 0, taking [x,t ; ® a_j] with a_j of weight —a, instead of ar, we reduce the problem to the case

136

Twisted Affine Algebras and Finite Order Automorphisms

Ch. 8

a = 0 and J = 0, i.e., ajt E l)o- Let 7 6 J)£ be a root of 0^ such that (7,aj,,) ^ 0. Then the element y = [[x,Q( e 7 ],e_ 7 ] € i, where e± T is a root vector with root ± 7 , has the following form: y =

Q(tr)(P®h+tPT®hT+---

+ tr-lPF-i®hr_l),

where p

= ^ ( 0 . ^ € £,

/i G ^', /i / 0, and the /»? € 0j have zero weight with respect to f). Since [y> e j] € * f° r all r o °^ vectors ej € 0o"> w e conclude that Q(tr)P® I) C i and therefore Q(tr)PC(g,n) C i. For x e 0j let io = {p(<) G £ U J p ( 0 ® x € i}. It follows from above that io is an invariant ideal (with respect to transformation t —• r)~li) of £ , independent of j and x, and that i = io£(0,/i). Since all ideals of £ are principal and io is invariant, we deduce that i = PC(&,fi) for some

• THEOREM

8.6. Let s = ($o>...,«/) be a sequence of nonnegative relatively 1

prime integers; put m = r ^ a,-s,-. Then »=o a) The relations (8.6.1)

< r, ;r (£- > )

= c2'^/m£:>

(i = 0,...,<)

define (uniquely) an m-th order automorphism
To prove a) note that the root space decomposition (8.3.4) of £(0,/i) induces a gradation £(0,/i) = £(0,/i)/CA' / = © £ a . Define the automorat

phism
where a = ^ Ar,at-.

Ch. 8

Twisted Affine Algebras and Finite Order Automorphisms

If £(g,/i) =

0£(#,AOJ Z

is the gradation of type s, then £(g,fi)j

137 and

, /i)j+ m lie i n the eigenspace of
Let now a be an m-th order automorphism of 0. Theorem 8.5 gives us an isomorphism such that the Z-gradation of £(0, of Q and a GC X :

Hence tpa^~x = (7,>r, proving b). The proof of c) also uses the covering map. Suppose that a — aS]r and a' — as'-y are conjugate, i.e., rcrr" 1 = a' for some r G Autg. Note that by Propostion 8.6b) (see below), ^ (the Cart an subalgebra of g^) is the Cartan subalgebra of Qa and Qa'. Since r(Qa) = Qa', T(^Q) is another Cartan subalgebra of g*7 . Let T\ be an inner automorphism of Q° such that TI(T(^Q)) = f)^. Replacing r by ^ r , we may assume that r leaves f)o invariant and that the sets of positive roots of Q° and ga correspond to each other under r (by Proposition 5.9 applied to g*7). The extension f of r given by f(P r{a), is an isomorphism of Z-graded Lie algebras: £(g, a, m) —• £(g, (7;, m), which maps positive roots onto positive roots. Hence the sequences s and s' correspond under an automorphism of the diagram Xj$\

• Given a nonzero sequence s = (so, • • •, st) of nonnegative integers and a number r = 1, 2 or 3, we call the automorphism
138

Twisted Affine Algebras and Finite Order Automorphisms

Ch. 8

the automorphism of type (s; r). Let 0 = 0 0j(s; r) be the Z/mZ-gradation i associated to it. Here are some of its properties. 8.6. a) r is the least positive integer for which
b) Let t'i, . . . , ip be all the indices for which s t l = • • = s fp = 0. Then the Lie algebra 0Q(S; r ) I S isomorphic to a direct sum of the (£ — p)-dimensional center and a semisimple Lie algebra whose Dynkin diagram is the sub diagram of the affine diagram X# consisting of the vertices t'i, . . . , ip. c) Let ji,..., j n be all the indices for which Sjx = • • • = Sjn = 1. Then the &o(s;r)-module Qj(s;r) (resp. Q_j(s;r)) is isomorphic to a direct sum of n irreducible modules with highest-weights — a 7 l , . . . , —OLJTI (resp. otj1,...,

Proof a) follows from the (easy) fact that a finite order automorphism (7 of g is inner if and only if there exists a Cartan subalgebra which is pointwise fixed under 1. Using the Weyl complete reducibility theorem proves c).

•

§8.7. Later we will need the following reformulation of Theorem 8.5 (which is a generalization of Theorems 7.4 and 8.3). THEOREM 8.7. Let A be an affine matrix of type X%\ Let g be a simple finite-dimensional Lie algebra of type XN, and let (. | . ) be its normalized invariant form. Let p be a diagram automorphism of order r of 0 and E{, Fi, Hi (i = 0, . . . , £) the elements of 0 introduced in §8.2. Let a8]r be an automorphism of type (s; r) of 0, and let 0 = 0 0 ; (s; r) be the associated 3

t

Z/mi-gradation,

where m = r^2 at-5t-. Define the Lie algebra structure on •=o

Ch. 8

Twisted Affine Algebras and Finite Order Automorphisms

139

( £ V ® 8; mod m(s; r)) 0 Ctf by

[(/MO ® «/i) e =

P1(t)P2(t)

7/1

Then this is isomorphic to the derived afGne algebra Qf(A), with Chevalley generators e, = tSi 0 E{, /, = t~8i 0 F, (i = 0, . . . , ^), the coroot basis

aV = (10 Hi) 0 (aiSi/aYm)K (i = 0, ..., £) and canonical central element K. Extending this Lie algebra by Cd', where d'(P(t) 0 g) = ^ ^ W ® ^

m at and [^',/C] = 0, we obtain a Lie algebra which is isomorphic to the affine / algebra &(A) with the scaling element d = ao(d — H — ^(//\H)K), where H G 53 C^« is defined by (a,-, 7/) = s,-/ra (i = 1, . . . , £). The normalized invariant form is defined by 92) = r

(C/f + Cd'|P(0 0 g) = 0,

(/if |iif) = 0,

(<*V) = 0 and

Finally, setting degt = 1, deg<7 = 0 for g G 0 and deg K = degc/' = 0 defines the Z-gradation of type s.

The realization of the afBne algebra of type X)$' provided by Theorem 8.7 is called the realization of type s. Using Corollary 6.4 and Exercise 6.2, we obtain COROLLARY 8.7. Let £(g,/x,r) be an affine algebra of type X#\ Then (x,y) = 2 r / i v ( x | y ) for x,y € 0, where <j> is the Killing form on g. In particular:

J2

|a) = 2r/iv(A|/i) for A,/i G ff,

where f) is a Cart an subalgebra of 0 and A9 C I)* the root system of 0.

§8.8.

Exercises.

In Exercises 8.1-8.4 we sketch another proof of Theorem 8.5.

140

Twisted Affine Algebras and Finite Order Automorphisms

Ch. 8

8.1. Choose a Cartan subalgebra ^ of QQ and set J) = 1 (g> J)Q -f CK"' + Cd' C £(g, cr, m). With respect to \) we have the root space decomposition; denote by A the set of roots and by Ca the root space attached to a G A. Denote by A the restriction of A G J)* to 1 ® f)o and set (A|/x) = (A|/Z) for A, /i G J)*. Set A° = {a G A|a = 0}. Show that a d £ a is locally nilpotent provided that a £ A°. 8.2. Let a £ A\A°. Then (i) dim Ca = 1 and (a\a) ^ 0; (ii) for p € A, the set of /? + i a € A U {0} is a string p - pa, . . . , p + qa, where p and q are some nonnegative integers such that p—q = 2(a|/?)/(a|a). (iii) [ £ , , £ , ] # 0 if /?, 7, ^ + 7 € A, p i A 0 . 5.5. Let A o C A be the set of roots of 1 (g) Q-Q C £(g, cr, m) and let A o + be a subset of positive roots in Ao; let A+ = Ao+ U {a G A|(a,d) > 0}. A root a G A4. is called simple if it is not a sum of two members of A+. Let II = {ai, c*2, • • •} be the set of all simple roots. Then (i) each a 6 A+ can be written in the form a = ^2 ifc»art, where ki £ Z+; t

( i i ) n c A\A°; (iii) aT, a j , . . . span f)^ and the bilinear form (. | . ) is positive definite on

(iv) there exists a nontrivial linear dependence of 07 with nonnegative coefficients; (v) for % ^ j we have a ti := 2(a t |a ; )/(a t |a f ) € - Z + ; (vi) if a G A + is not simple, then a — a,- £ A + for some a,- £ II. 5.^. Deduce the existence of the isomorphism $ . 5. J. Let G be a connected semi-simple complex algebraic group and let G be the group of regular maps C x —• G. Let a £ AutG be such that am = 1;

Ch. 8

Twisted Affine Algebras and Finite Order Automorphisms

141

we extend a to an automorphism of G D G by letting a(y(t)) = a Show that, given g G G, there exists 7 G G such that (8.8.1)

9 = 7-^(7)

if and only if (8.8.2)

\

8.6. Taking for granted that any finite order automorphism of G leaves invariant some connected semisimple subgroup G' of maximal rank (see Kac-Peterson [1987]), deduce from Exercise 8.5 that given g G G, there exists 7 G G satisfyng (8.8.1) if and only if g satisfies (8.8.2) 8.7. Show that every ideal of the Lie algebra £ ( 0 , ( T ) is of the form P(* m )£(0,
be its Che valley generators. Let 0 = ^ ajar,- be the decomposition

of the highest root of 0 via simple roots. Define an involution
8.10. Show that an automorphism a 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 a. Give an example of two non-conjugate automorphisms of order 5 of A
142

Twisted Affine Algebras and Finite Order Automorphisms

Ch. 8

8.11. Show that the minimal order of a regular automorphism (i.e., an automorphism with an abelian fixed point set) of a simple finite-dimensional Lie algebra 0 is the Coxeter number h ( = (height of the highest root) +1). Show that such an automorphism is conjugate to the automorphism of type ( 1 , 1 , . . . , 1 ; 1 ) . Show that every regular automorphism of 0 of order h + 1 is conjugate to the authomorphism of type ( 2 , 1 , . . . , 1; 1). 8.12. An automorphism cr of order m of a simple finite-dimensional Lie algebra 0 is called quasirational if for the associated Z/mZ-gradation 0 = ® 0j one has: dim0, = dim0j if (t,m) = (j, m), or, equivalently, the j

characteristic polynomial of a on 0 has rational coefficients. Show that the automorphisms of type ( 1 , 1 , . . . , 1; r) and ( 2 , 1 , . . . , 1; 1) are quasirational. Classify all quasirational automorphisms of A\ and A2 up to conjugation. 8.1S. An automorphism a of order m of 0 is called rational if ak is conjugate to
Show that \i is the diagram automorphism of g. Show that the subalgebra Z(g,/i) C £(g) is an affine Lie algebra of type An , K being the (resp. twice the) canonical central element if n — 2£ (resp. n = 2£ - 1). Describe the root space decomposition. 8.16. Introduce the following basis of the Lie algebra £(s^3(C), /i) (s G Z): 0 f

L8, =

Is.+2

•

•

Ch. 8

Twisted Affine Algebras and Finite Order Automorphisms

LS$+A

•c = j

0 0

0 -2*2'*1 0

/O =10 \0

*2'+1 0 0

0 L 8 ,+ 7 = < 2 ' +2 \ 0

0 0 * 2 '+ 2

0 \ 0 , *2'+1/

'0 0 < 2 ' + 1 \ 0 0 0 , , 0 0 0 /

L8,+5 /

143

0> 0 0,

Show that [LifLj] = dyLi+j (t,j 6 Z), where d,j G Z depend on t, j mod 8; compute them. 8.17. Construct a "Chevalley basis" of £(g,/i) (cf. Exercise 7.14). §8.9. Bibliographical notes and comments. The realization of the "centerless" twisted affine Lie algebras was given in Kac [1968 A, B]; so was the application to the classification of symmetric spaces. The application to the classification of all finite order automorphisms was found in Kac [1969 A]; a detailed exposition of this is given in Helgason [1978]. The present proof of Theorem 8.5 is simpler than the original proof. It resulted from conversations with G. Segal and D. Peterson. Exercises 8.1-8.4 are taken from Kac [1969 A] and Helgason [1978]. = As B. Weisfeiler pointed out, Exercise 8.6 means that H1(Z/mZiG) {1}; note also that #°(Z/mZ,G) = Ga. Steinberg [1965] proved that if 1 (Z/mZ,G(C((t))) = 1. Exercises 8.12-8.14 are taken from Kac [1978 A]. Levstein [1988] has classified the involutions of all affine algebras. For a classification of finite order automorphisms and conjugate linear automorphisms see Bausch-Rousseau [1989], Kac-Peterson [1987], Rousseau [1989]. There is an intriguing connection of the material of Chapter 8 with invariant theory. Namely, given a Z/mZ-gradation of a simple Lie algebra 0 = 0 0 ; corresponding to an m-th order automorphism (T, we get a go3

module fli; the corresponding connected reductive linear group G acting on the space 0i is called a (T-group. These groups have many nice properties (see Kac [1975], Popov [1976], Vinberg [1976], Kac-Popov-Vinberg [1976]), but the most remarkable thing is that
144

Twisted Affine Algebras and Finite Order Automorphisms

Ch. 8

I believe that this is an indication of a deep connection between the theory of infinite-dimensional Lie algebras and groups, and invariant theory. We have already discussed one aspect of such a connection in §5.14.

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 O, which is roughly speaking the category of restricted ()-diagonalizable modules (the precise definition is given below). We study the "elementary" objects of this category, the so-called Verma modules, and their connection with irreducible modules. We discuss the problems of irreducibility and complete reducibility in the category O. We find, as an application of the representation theory, the defining relations of Kac-Moody algebras with a symmetrizable Cartan matrix. At the end of the chapter we discuss the category a for the infinite-dimensional Heisenberg algebra and the Virasoro algebra. §9.1. 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 decomposition: Q(A) = n_ e f j 0 n + . We have the corresponding decomposition of the universal enveloping algebra: (9.1.1)

U(g(A)) = t/(n_) <8> U(t))<S> U(n+).

Recall that a g(i4)-module V is called fy-diagonalizable if it admits a weight space decomposition V = 0 V\ by weight spaces V\ (see §3.6). A nonzero vector from VA is called a weight vector of weight A. Let P(V) = {A G f)*|Vx T£ 0} denote the set of weights of V. Finally, for A G f)* set D(X) = {/i G G'l/i < A}. The category O is defined as follows. Its objects are g(>l)-modules V which are f)-diagonalizable with finite-dimensional weight spaces and such that there exists a finite number of elements A i , . . . , A, G f)* such that

The morphisms in O are homomorphisms of g(i4)-modules. 145

146

Highest-Weight Modules over Kac-Moody Algebras

Ch. 9

Note that (by Proposition 1.5) any submodule or quotient module of a module from the category O is also in O. Also, it is clear that a sum or tensor product of a finite number of modules from O is again in O. Finally, remark that every module from O is restricted (see §2.5). §9.2. Important examples of modules from the category O are highestweight modules. A g(j4)-module V is called a highest-weight module with highest weight A Gfy*if there exists a nonzero vector vA G V such that (9.2.1)

n + (v A ) = 0; h(vA) =

A(A)I;A

for h G f); and

U(a(A))(vA) = V.

(9.2.2)

The vector vA is called a highest-weight vector. Note that by (9.1.1), condition (9.2.2) can be replaced by (9.2.3)

U(n-)(vA) = V.

It follows from (9.2.1 and 3) that (9.2.4)

V=®VX\

VA = CvA]

dimV A
Hence a highest-weight module lies in 0 , and every two highest-weight vectors are proportional. A $(A)-modu\e M(A) with highest weight A is called a Verma module if every g(j4)-module with highest weight A is a quotient of M(A). PROPOSITION 9.2. a) For every A E fj* there exists a unique up to isomorphism Verma module M(A).

b) Viewed as a U(n_)-module, A/(A) is a free module of rank 1 generated by a highest-weight vector. c) M(A) contains a unique proper maximal submodule Af'(A). Proof. If Mi (A) and M2(A) are two Verma modules, then by definition there exists a surjective homomorphism of g(>l)-modules ij> : Mi (A) —• M 2 (A). In particular, ^ ( M I ( A ) A ) = M 2 ( A ) A and hence dimMi(A)x > dimM2(A)A- Exchanging Mi (A) and M 2(A) proves that ^ is an isomorphism. To prove the existence of a Verma module, consider the left ideal J(A) in U(Q(A)) generated by n+ and the elements h — A(A) (h Gfy),and set M(A) = U(n(A))/J(A).

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The left multiplication on U(g(A)) induces a structure of C/(g(>l))-module on M(A). It is clear that Af(A) is a g(j4)-module with highest weight A, the highest-weight vector being the image of 1 G U(g(A)). If now V is a g(i4)-module with highest weight A, then the annihilator of VA C V is a left ideal J\ which contains J(A). Hence, V ~ U{${A))/J\ and we have an epimorphism of 0(A)-modules Af(A) —> V,which proves a). b) follows from the explicit construction of M(A) given above and the Poincare-Birkhoff-Witt theorem, c) follows from the fact that a sum of proper submodules in M(A) is again a proper submodule (since, by Proposition 1.5, every submodule of M(A) is graded with respect to the weight space decomposition and does not contain M ( A ) A ) .

• Remark 9.2. One can also obtain M(X) 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 i/> : a -+ b. Recall that the induced b-module is defined by

U(b) ®u(a) V := (U(b) ® c V)/ £

C(ty(a) t; - b ® a(v)),

a,b,v

where the summation is over 6 G U(b), a 6 fl, v G V, and the action of b is induced by left multiplication in U(b). Define the (n+ + f))-module C^ with underlying space C by n+(l) = 0, h(l) = (A, &>1 for h G f). Then

§9.3. 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 some other purposes, we introduce the following notion. Let V be a g(^4)-module. A vector v G VA is called primitive if there exists a submodule U in V such that v i U;

n + (ti) C U.

Then A is called a primitive weight. Similarly, one defines primitive vectors and weights for a 0'(A)-module. A weight vector v such that n+(t;) = 0 is obviously primitive.

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PROPOSITION 9.3. Let V be a nonzero module from the category O. Then a) V contains a nonzero weight vector v such that n+(v) = 0. b) The following conditions are equivalent: (i) V is irreducible; (ii) V is a highest-weight module and any primitive vector of V is a highest-weight vector; (iii) V ~ L(A) for some A G I)*. c) V is generated by its primitive vectors as a Q(A)-module.

Proof To prove a), take a maximal A G P{V) (with respect to the ordering >). Then one can take v to be a weight vector of weight A. Let V be an irreducible module; then a weight vector v is primitive if and only if n+(v) = 0. Take a primitive vector v of weight A. Then U(&)(v) is a submodule of V, hence V = U(Q)(V) and V is a module with highest weight A. In particular, P(V) < A and dimV^ = 1. Hence every primitive vector is proportional to v, which proves the implication (i) =» (ii) of b). If V is a highest-weight module and U C V is a proper submodule, then U contains a primitive vector by a). This proves the implication (ii) => (iii) and the assertion b). Let V be the submodule in V generated by all primitive vectors. If V ^ V, then the 0(j4)-module V/V contains a primitive vector v by a). But a weight vector in V which is a preimage of v is a primitive vector.

• Thus we have a bijection between I)* and irreducible modules from the category (9, given by A i-+ L(A). Note that L(A) can also be defined as an irreducible g(j4)-module, which admits a nonzero vector v\ such that (9.3.1)

n + K ) = 0 and h(vA) = A(h)vA for fcef).

Remark 9.3. A module V from the category O is generated by its primitive vectors even as a n_-module. Indeed, a weight vector v G V is not primitive if and only if v G £/(n_)[/o( n +) v (= the submodule generated by n+(v)). Here and further i/o(0) denotes the augmentation ideal QU(Q) of (7(0). We have the following "Schur lemma." LEMMA

9.3. Endfl(i4)L(A) = C/ L(A ).

Proof If a is an endomorphism of the module L(A) and v\ is a highestweight vector, then by Proposition 9.3b), we have a(v\) = Xv\ for some A G C. But then a(u(v\)) = Xu(v\) for every u G i/(0), hence a = A/X,(A).

•

Ch. 9 §9.4.

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Let L(A)* be the g(A)-module contragredient to L(A). Then

L(A)* = U(L(A)xy.

The subspace

A

is a submodule of the g(yl)-module L(A)*. L*(A) is irreducible and that for v G (L(A)A)

It is clear that the module one has:

n.(v) = 0; fc(v) = -(A,ft)t> for A G f). Such a module is called an irreducible module with lowest weight —A. As in §9.3 we have a bijection between f)* and irreducible lowest-weight modules: A >-• L*(—A). Denote by TTA the action of &(A) on L(A), and introduce the new action v\ on the space L(A) by (9.4.1)

xX(j)» =

*A(w(g))v,

where LJ is the Chevalley involution of &(A). It is clear that (L(A),7r^) is an irreducible g(A)-module with lowest weight —A. By the uniqueness 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) such that (9.4.2)

B(g(x),y)

= -B(zM9)(v))

f ° ra « 9 € « ( A ) , x , y e L ( A ) .

A bilinear form on L(A) which satisfies (9.4.2) is called a contravariant bilinear form. PROPOSITION 9.4. Every $(A)-module L(A) carries a unique up to constant factor nondegenerate contravariant bilinear form B. This form is symmetric and L(A) decomposes into an orthogonal direct sum of weight spaces with respect to this form.

Proof. The existence of B was proved above, the uniqueness follows from Lemma 9.3. The symmetry follows from the uniqueness. The fact that B(L(A)x, I(A),,) = 0 if A # /i follows from (9.4.2) for g G f), x G L(A)A ,

• A more explicit way to introduce the contravariant bilinear form is the following. Let V be a highest-weight g(j4)-module with a fixed highestweight vector v\. Given v G V, we define its expectation value (v) G C by v = (v)vA + ]jT VA-or, where v A - a G V\- a-

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Extend the negative Chevalley involution —u; to an anti-involution u of U(Q(A)). Due to (9.1.1) we immediately see that (9.4.3)

<"(«K> = (avA),

a G U(g(A)).

It follows that (u(a)a'v\) is symetric in a, a' G £/(0(A)), hence the formula B(avAia'vA)

= (u>(a)a'vA)

gives a well-defined symmetric bilinear form on V, which is contravariant, and normalized by B(vA, vA) = 1. §9.5. The underlying statement of most of the complete reducibility theorems is contained in the following lemma. 9.5. Let V be a &(A)-module from the category O. If for any two primitive weights A and pofV the inequality A > \JL implies A = fif then the module V is completely reducible (i.e., V decomposes into a direct sum of irreducible modules). LEMMA

Proof. Set V° = {v G V|n+(t;) = 0}. This is ^-invariant, hence we have the weight space decomposition V° = ® V\°, where all elements from L are primitive weights. Let A G L and v G V^, v ^ 0. Then the g(A)-module U(&)(v) is irreducible (and hence isomorphic to L(\)). Indeed, if this is not the case, then by Proposition 9.3b), we have £/(n_)(i;) n Vjf / 0 for some fi < A. This contradicts the assumption of the lemma. Therefore, the g(i4)-submodule V' of V generated by V° is completely reducible. It remains to show that V — V. If this is not the case, we consider the 0(i4)-module V/V. Then there exists a weight vector t; G V of weight /z such that v £ V but e,(t;) G V and ^ 0 for some i. But then, since V is from the category O, there exists A G L such that A > /i + a,-, and hence A > /i, which contradicts the assumption of the lemma.

• §9.6. Unfortunately, a module V G O does not always admit a composition series (i.e., a sequence of submodules V D V\ D V2 D ... such that each K'/K'+i is irreducible) (see Exercises 10.3 and 10.4). However, one can manage with the following substitute for it.

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LEMMA 9.6. Let V € O and A 6 d*. Then there exists a filtration by a sequence of submodules V = V% D Vt-i D • • D V\ D Vo = 0 and a subset J C { l , . . . , 0 swch that; (i) ifj e J, then Vj/Vj-i ~ L(AJ) for some A, > A; (ii) ifj $ J, then (Vj/Vj-i)^ = 0 for every /x > A.

Proof. Let a(V, A) = J^ dimVJ,. We prove the lemma by induction on a(V,A). If a(V,A) = 0, then 0 = Vo C Vx = V is the required filtration with J = 0. Let a(K, A) > 0. Choose a maximal element /i € P(V) such that ^i > A, choose a weight vector v G V^, and let U = C/(0)(v). Then, clearly, U is a highest-weight module. Proposition 9.2c) implies that U contains a maximal proper submodule (7. We have

OCUCUC

V;

U/U~L(fi),

fi>\.

Since a(£/,A) < a(V, A) and a(V/U,X) < a(V, A), we use induction to get a suitable filtration for U and K/C/. Combining them we get the required filtration of V.

• Let V € O and /i € G*. Fix A E f)* such that p > A and construct a filtration given by Lemma 9.6. Denote by [V : L(fi)] the number of times /i appears among {Xj\j £ «/}. It is clear that [V : £(/*)] is independent of the filtration furnished by Lemma 9.6 and of the choice of A; this number is called the multiplicity of L([A) in V. Note that L{JJL) has a nonzero multiplicity in V if and only if p is a primitive weight of V. §9.7. Now we will introduce and study the formal characters of modules from 0 . For that purpose, we define a certain algebra S over C. The elements of £ are series of the form

where c\ e C and c^ = 0 for A outside the union of a finite number of sets of the form D(fi). The sum of two such series and the multiplication by a number are defined in the usual way. E becomes a commutative associative algebra if we decree that e(A)e(/x) = e(A 4- fj) and extend by linearity; here the identity element is e(0). The elements e(A) are called formal exponentials. They are linearly independent and are in one-to-one correspondence with the elements A of fy*.

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Ch. 9

Let now V be a module from the category O and let V = © VA be its weight space decomposition. We define the formal character of V by

Clearly, ch V e€. First, we prove the following PROPOSITION

9.7. Let V be a g(A)-module from the category O. Then

(9.7.1)

ch V =

Proof. Denote by the map which associates to each V G O the difference <£(V0 £ £ between the left- and right-hand sides of (9.7.1). Then (L(\)) = 0, and given an exact sequence of modules 0 —• V\ —• V2 —* V3 —• 0 we have ^(Va) = *(Vi) + ^(V8). Using Lemma 9.6, we deduce that given A G f)* there exist modules M i , . . . , Afr G 0 such that (M,)^ = 0 when /1 > A, and ^(7) = £ ^(Af,-). In particular, for every A G f)* the coefficient at e(A) in (V) is zero.

D Let us compute the formal character of a Verma module M(A). Using Proposition 9.2b) and the Poincare-Birkhoff-Witt theorem, one can construct a basis of the space M(k)\ as follows. Let /?i,/?2>-- be all the positive roots of the Lie algebra Q(A), and let e-/?,,tt be a basis of 0-/3.(1 < l* ^ mult/?, = m,). Let VA be a highest-weight vector of Af(A). Then the vectors

such that (ni,i 4h nitmi)Pi -h (n 2| i + h n2|m3)/?2 + • • • = A - A and n,j G Z + , form a basis of M(A)\. Therefore

chM(A) = e(A) JJ ( H - c ( ~ a ) - h e ( - 2 a ) - h . . . )multa Hence, we have

(9.7.2)

chM(A) = c(A) JJ (1 - e(~a))" m u l t a .

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§9.8. Assume now that A is a symmetrizable matrix and let (.|.) be a bilinear form on $(A) provided by Theorem 2.2. Then the generalized Casimir operator ft acts on each module from the category O (see §2.6). LEMMA

9.8. a) IfV is a #(A)-module with highest weight A, then

b) If V is a module from the category O and v is a primitive vector with weight A, then there exists a submodule U C V such that v £ U and ft(t,) = (|A + p| 2 -|/>| 2 )t;

modU.

Proof, follows immediately from Gorollary 2.6.

• PROPOSITION

(9.8.1)

9.8. Let V be a &(A)-module with highest weight A. Then

ch V =

Y,

c

* c h M ( A )>

where

CA € Z, cA = 1.

A
Proof. Using (9.7.1), it suffices to prove (9.8.1) for V = L(A). Set J3(A) = {A < A||A + p\2 = |A + p\2}, and order the elements of this set, Ai, A2,... so that the inequality A^ > Xj implies i < j . Then the proof of Proposition 9.7 and Lemma 9.8 imply the following system of equations: chM(A,)= A,€B(A)

The matrix (c tJ ) of this system is triangular with ones on the diagonal. Solving this system proves (9.8.1).

• §9.9. Now, using the Casimir operator, we can investigate irreducibility and complete reducibility in O.

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PROPOSITION

9.9. Let A be a. symmetrizable

Ch. 9

matrix.

a) If 2(A 4- p\0) # (P\p) for every /? G Q+,£ # 0, Men tie 0(A)-modu]e M(A) is irreducible. b) If V is a 0(j4)-module from the category O such that for any two primitive weights A and /i of Vf such that A — \x = /? > 0, one has 2(A + p|/?) ^ (/?|/?), then 7 is completely reducible. Proof. Proposition 9.3b) implies that if Af(A) is not irreducible, then there exists a primitive weight A = A — /?, where /? > 0. But then Lemma 9.8a) gives: 2(A + p\fi) = (/?|/?), proving a). To prove b) we may assume that the 0(>l)-module V is indecomposable. Since, clearly, ft is locally finite on V, i.e., every v G V lies in a finitedimensional Q-invariant subspace, we obtain that there exists a G C such that Q — al is locally nilpotent on V. Hence Lemma 9.8b) implies |A+p| 2 = |/i-f p\2 for any two primitive weights A and /i. Now b) follows from Lemma 9.5.

• §9.10. Here we consider the subalgebra &'(A) = [g(j4),0(A)] of instead of Q(A). Recall that &(A) = 0'(^) + ^ ^ t h a t ^ = E C a « Y = ^'(yljn ^. Recall the free abelian group Q and the decomposition U(tf(A)) = © U'a (see §2.6 for details). ct€Q

A 0'(yl)-module K is called a highest-weight module with highest weight A G (VY if V admits a Q+-gradation V = 0 VX-o such that C/"i(VX-a) Q

C VA-Q+/3y dim7 A = 1, fc(v) = A(h)v for A G (^)*i « € VA, and V = [/(0 ; (J4))(VA). In other words, this is a restriction of a highest-weight module over 0(J4) to $'(A). In the same way as in §9.2, we define the Verma module M(A) over $!{A) and show that it contains a unique proper maximal graded submodule M'(A). We put L(A) = M(A)/M ; (A). This is, of course, a restriction of an irreducible 0(yl)-module to LEMMA

9.10. The %'{A)-module L(A) is irreducible.

Proof. Let V C £(A) be a nonzero 0/(j4)-submodule.

We choose v =

TO

V{ G V such that i;,- G L(A)\iy

V{ / 0 and ^2ht(A — A,) is minimal.

If A,- ^ A for some i, then ^-(VAJ / 0 for some j (by Proposition 9.3b). Hence v G L(A) A and V = L(A). D

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155

We shall sometimes describe A Gfy*by its labels (A, a*) (i = 1 , . . . , n). If A, M G f)* have the same labels, they may differ only off I)'. Then it is clear from Lemma 9.10 that the modules L(A) and L(M) when restricted to &'(A) are isomorphic as (irreducible) g'(yl)-modules, and the actions of elements of g(A) on them differ only by scalar operators. Note that (9.10.1)

dimL(A) = 1 if and only if A|y = 0.

Indeed, if A|^/ = 0, we can consider the 1-dimensional g(A)-module C which is trivial on &'(A) and is defined by h —• (A,/i) on f); by the uniqueness of L(A) (see §9.3), 0(A)-modules L(A) and C are isomorphic. The "only i f part follows from the computation: 0 = e,/,(t;) = (A,a^)t; if v is a highest-weight vector. If A is a symmetrizable matrix, then &'(A) carries a symmetric invariant bilinear form (.|.) which is defined on f)' by (2.1.2), and whose kernel is c (by Theorem 2.2). We put (/%) = ( ^ ( / ^ K H T ) )

f

°r for /?, 7 G Q (see

§2.6). We can also define p G (ty)* by (/>, a?) = \ai{ (i = 1 , . . . , n). In the sequel we will use the following version of Proposition 9.9 for the Lie algebra 0 ; (J4), where A may be infinite. PROPOSITION

9.10. Let A be a symmetrizable matrix, possibly infinite. l

a) If 2(A+p, v- (P)) # {(5\P) for every 0 G <3+\{0}, then the tf{A)'module M(A) is irreducible. b) Let V be a gf(A)-module such that the following three conditions are satisfied: (i) for every v G V, e,(t;) = 0 for all but a finite number of the e,-; (ii) for every v G V there exists k > 0 such that e tl • • • e,-f (v) = 0 whenever s> k; (iii) V = 0 Vx, where Vx = {v G V\h(v) = (A, % for all h G f)'}; (iv) if A and \i G (f)')* are primitive weights such that A — /i = /?|^ for some /? G Q+\{0}, then 2(A H - p , ^ 1 ^ ) ) # (/'I/')Then V is completely reduciblef i.e., is isomorphic to a direct sum of$'(A)modules of the form L(A), A G (f)')*. Proof. To prove the proposition, we employ the operator Qo instead of Q (see §2.5). It is clear that Qo 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 G V\ be such that (Qo — alv)k(v) = 0 for some Jb G Z+ and a G C, and let i/ G t^(t>) (/? G Q); then (see (3.4.1))

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Ch. 9

(Uo - (a + 2(A + p, i/" 1 ^)) - (/?|/?))/ v )V = 0.

(9.10.2)

To prove a), suppose that J is a proper nonzero submodule of M(A). Take a nonzero element t/ = ]P v,- G «/, where v,- 6 ULpfa), v is a highest-weight vector of M(A) and /?,- G Q+\{0}, such that £ h t / ? f is minimal. Then n+(v,-) = 0 and therefore Qo(v{) = 0. Note also that Jio(v) = 0. Applying (9.10.2) to V = M(A) and t/ = v,-, we arrive to a contradiction with the hypothesis of a). To prove b), put V° = {v G V|n + (w) = 0} and V" = tf(0'O4))V°. Note that V° is graded with respect to the decomposition (iii). Using (9.10.2) and (iv) we show that every v G V° n V\ generates a Qf(A)module L(X). Indeed, first we show that the g'(j4)-module U(&'(A))(v) is Q+-graded. In the contrary case, there exists a linear combination of nonzero vectors Y^vi — 0, where Vi G U^p^v),^ e Q + and Pi ^ fy for i

i ^ j , with ]jTht/?t- minimal. By the same argument as in proof of a) t

we arrive at a contradiction with (iv). Second, we show that this module is irreducible. By Lemma 9.10 it suffices to show that it has no Q+graded ideals. In the contrary case, there exists a nonzero 1/ G ULp(v), where f} G Q+\{0}, such that n+(t/) = 0. Again, (9.10.2) gives us a contradiction with (iv). Hence the g'(>l)-module V1 is completely reducible. Now we prove that V1 = V. Suppose the contrary; then there exists a vector v G V\\V such that n + (v) C V and (fi 0 - alv)k(v) = 0, for some k G Z+ and a G C. Since, clearly, £2o(v) G V7, we have a = 0 and hence fioC^) = 0. But then there exists /? G Q+\{0} and u EUp such that n+(ti(t;)) = 0 (by (ii)). Since Q0(u(v)) = 0, using again (9.10.2), we arrive at a contradiction with (iv).

• COROLLARY 9.10. Let A be a symmetrizable matrix with nonpositive real entries and let V be a &'(A)-module satisfying conditions (i), (ii) and (iii) of Proposition 9.10b). Suppose that for every weight A ofV one has (A, a / ) > 0 for all i. Then V is a direct sum of irreducible Q'(A)-modules, which are free of rank 1 when viewed as U(n~)-modules.

Proof. We may assume that A = (a tJ ) is a symmetric matrix (replacing

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157

the elements a* by proportional ones). For /?,= ][2 *«**«' € Q + \ { 0 } and a weight A of V we have

Now we apply b) and then a) of Proposition 9.10.

§9.11. Here we give an unexpected application of the results of the representation theory developed in this chapter, to the defining relations of the Lie algebras &(A) with a symmetrizable Car tan matrix A. As before, (.|.) denotes a bilinear form on $(A) provided by Theorem 2.2. Let &(A) be the Lie algebra introduced in §1.2, so that Q(A) = fl(A)/r and r = t_ ®t+. Set t a = g a n t. 9.11. The ideal t+ (resp. t_) is generated as an ideal in n+ (resp. fL) by those t a (resp. r_a>) for which a G Q + \ n and 2(p|a) = (a|a).

PROPOSITION

Proof. We define a Verma module Af (A) over &(A) by J7(g(>l))/J, where J j s the left ideal generated by fi+ and h - X(h) (h£ f>) (cf. §9.2). Let M'(A) be the unique (proper) maximal submodule of M(A). Then we have an isomorphism of 0(A)-modules (9.11.1)

This is due to the fact that fL is a free Lie algebra (see Theorem 1.2b)) and hence U(n~) is freely generated by / i , . . . , fn. The isomorphism (9.11.1) gives us the following isomorphism of 0(j4)-modules:

(9 U 2)

' '

~ 1/(0) «!,(» 1=1

«=i

Here and further we write 17(0),... in place of U($(A)),... for short. Let 7T : 0(J4) —• 0(^4) be the canonical homomorphism. We define a map

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Ch. 9

Ai : t_ —* (7(g) ®u(g) Af'(O) by X\(a) = 1 0 a(t>), where v is a highestweight vector of M(0). This is a g-module homomorphism; indeed, for x G 0, a G t_ we have

= TT(X) ® a(v) = ^(A^a)) since ir(a) = 0. Similarly, we get Ai([t_,t_]) = 0, so that we have a g(A)-module homomorphism: A:

t./[t_,t_]-®M(-a,) 1= 1

by (9.11.2). More explicitly, A is described as follows: write a G t_ in the form a = £ t i , 7 i , where m G tfo(fi-); t h e n ^( a + [ r -» t -]) = ZMW»)V«> where t;,- is a highest-weight vector of M(—a,). We deduce that A is injective. Indeed, X(a + [t_,t_]) = 0 implies w(ui) = 0 for all i, hence Ui G t_[/ 0 (n_), hence a G r_t/ 0 (fi-) H t_. Therefore, a G [t_, r_] by the following general fact: given a Lie algebra n and a subalgebra t e n , one has (9.11.3)

tnttf o (n) = [t,t].

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 [ / 0 ( t ) 2 = [t,t].

This follows by passing to (7(t/[c, t]), which is a polynomial ring. As a result we have an imbedding A : t _ / [ t _ , t _ ] —• ©M(—a,) in the category O of fl(i4)-modules. Now let —a (a G Q+) be a primitive weight of the g(j4)-module t _ / [ t _ , t _ ] . Note that a £ U since no /,- lies in c. Using the embedding A we conclude by Lemma 9.8 that | - a,- + p\2 = | - a 4- p\2 for some i and therefore 2(p\a) = (e*|a) (since 2(p|a f ) = (a,-|at-)). Applying Remark 9.3 we deduce that r . is generated as an ideal in n_ by those r_ a for which a G Q + \ n and 2(/?|a) = (a|a). This completes the proof of the proposition for t_. The result for t+ follows by applying the involution Q of &(A). D In the case of a Kac-Moody algebra we deduce the following theorem.

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THEOREM 9.11. Let g(A) = a(A)/x be a Kac-Moody algebra with a symmetrizable Cartan matrix A. Then the elements

(ad*)l-"ej9

(9.11.5)

i # j (ij

= 1 , . . . ,n),

(9.11.6) generate the ideals t+ and t_, respectively. Proof. Denote by Qi(A) the quotient of &(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 Q\(A) = © 0«. Let t± denote the image of t± in cr€Q

01 (A). Suppose the contrary to the statement of the theorem: t+ ^ 0 (the case of ti. is obtained by applying u>). Choose a root a of minimal height among the roots a G Q+\{0} such that t+ p|0^ ^ 0, and let a = ]T fcta,-. It t

is clear that t^ fl gfa must occur in any system of homogeneous generators oft^. It follows from the proofs of §3.8 (which used only the relations (3.3.1)) that there is f* G Aut0i(j4) such that r,(0o) = ^rda) anc^ ^•( t +) = t + for all i. Hence htr t (a) > hta for all i = l , . . . , n , and hence (a|a,-) < 0 for all i, where (. | . ) is a standard form. Therefore (a\a) < 0. But 2(p\a) = 2J2ki(p\&i) — J2 ki(&i\oti) > 0 and we arrive at a contradiction i

i

with Proposition 9.11.

• Theorem 9.11 implies the following definition of a Kac-Moody algebra in terms of generators and relations. Let A = (flij)" J=1 be a symmetrizable generalized Cartan matrix and let (f),II,II v ) be a realization of A (see §1.1). Then g(A) is a Lie algebra on generators e,-,/,- (i = 1 , . . . ,n), f) and the following defining relations: ,

[h,ei] = (at-,/i)et-,

[A,/,] = -(ar,-,A)/.-,

M'] = 0 ( M ' € f)), (ade.O^^e,- = 0,

Furthermore, fl'(j4) has the presentation described in the Introduction (see §0.3).

160

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Ch. 9

§9.12. Given below are some other important applications of Proposition 9.11. PROPOSITION

9.12. Let A = (a#) be a symmetrizable n x n-matrix,

the associated Lie algebraf A+ the set of positive roots, $(A) = n_ © F)©n+ tie triangular decomposition, etc. a) If (ty are nonzero real numbers of the same sign for all i,j = 1 , . . . , n, then n + (resp. n_) is a free Lie algebra on generators e x , . . . , c n (resp. fi,...,fn). b) Let L C A+ be such that (i) (a|/?) are nonzero reai numbers of the same sign for all a, /? G £; /3eL. (ii) a,/3 e L and a - j 8 € A + impiy that aLet n£ be the subalgebra of n± generated by 0

0± a ; set J)L =

£ Oz-H*), and aeL (9.12.1)

0 L = n£ © f)L © n£.

For a e L set g>°±Q = {x € 8±a|(*|y) = 0 for aii y G [n£,n£]}. Let I be a (possibly infinite) set containing a G L with multiplicity dim 0°, and consider the matrix B = ((a|/?)) a ^ 6 / and the associated Lie algebra &'(B). Then 0 L is a Lie algebra isomorphic to a quotient of $!{B) by a central ideal (which lies in the Cartan subalgebra of$'(B)). Furthermore, (9.12.1) is induced by the triangular decomposition of $'(B), and n£ (resp. n f j is a free Lie algebra on a basis of the space 0 0° (resp. 0 0?.a>).

Proof. First we prove a). Changing the elements of the dual root basis by proportional ones, we may assume that the matrix A is symmetric and has positive entries. But then for a = £^ J w € Q+ we have

2(p\a) - (a|a) = £>,(*, - *?) - £>;*,*,- < 0 if a / 0 and a £ II. Now we can apply Proposition 9.11. To prove b) we may assume that L is finite. We use induction on |L| and the statement a) to prove b) and also to prove that (.|.) gives a nondegenerate pairing of n£ and n£.

•

Ch. 9

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161

9.12. a) Let Abe a symmetrizable matrix and let a € A + ( J 4 ) be such that (ot\ot) ^ 0. Then 0 $*« is a free Lie algebra on a basis of

COROLLARY

k>i

the space © gjja , where gla = {x € 0*o|(^|y) = 0 for all y from the *>i

subalgebra generated by g _ a , . . . , 0_(*_i) a }b) Let A be a generalized Car tan matrix and let a £ A + ( J 4 ) be an isotropic root. Then Ci/~ 1 (a) © ( © Qja) is an infinite Heisenberg Lie algebra. Proof, a) follows from Proposition 9.12 b) by setting L = {ka\k 6 Z + } . If a is an imaginary root of an affine Lie algebra, then b) amounts to Proposition 8.4. Applying Proposition 5.7 and Lemma 3.8 proves b) in the general case.

• §9.13. Following is an application of the results of §9.10 to representation theory of the infinite-dimensional Heisenberg algebra B (= Heisenberg Lie algebra of order oc; see §2.9). Recall that this is a Lie algebra with a basis Pi,qi(i = 1,2,...) and c, with the following commutation relations: (9.13.1)

[pi, qi) = c(i = 1,2,...), all the other brackets are zero.

This is a nilpotent Lie algebra with center Cc. It is well known that for every a G C x , the Lie algebra 8 has an irreducible representation
a

a{qi) = Xi,

aa(c) = aIR.

Here and further the operator of multiplication by a polynomial P is denoted by the same symbol. We denote this s-module by Ra. Introduce the following commutative subalgebras of *:

A vector v of an a-module is called a vacuum vector with eigenvalue A £ C if *+(i>) = 0 and c(v) = \v.

162

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Ch. 9

Note (cf. §2.9) that 8 can be viewed as the Lie algebra g'(0)/ti, where ti C t is a central ideal, so that n+ (resp. n_) is identified with s+ (resp. * _ ) . Since (R% N. Then V is isomorphic to a direct sum ofs-modules of the form Ra,o> ^ 0. LEMMA

Proof. We can assume in b) that c = aly with a ^ 0. V may be viewed as a (^(Oj-module for which ctjf = aly for all i. But then for every weight = aht/?. Since (/3\f3) = 0 and p = 0, A and for /? G Q we have (\ti/-l(0)) we have 2(A + />, u'\P)) = 2aht/3 ^ (0\P) for /? G Q+\{0}. Now a) and b) follow from Proposition 9.10 a) and b).

• 9.13. Let V be an irreducible 8-module which has a nonzero vacuum vector with a nonzero eigenvalue A. Then the 8-module V is isomorphic to R\. COROLLARY

The Lie algebra 8 is often extended by a derivation do defined by [
where where Given ^a,6,A

[dOipj] =

-

nij are some positive integers. The Lie algebra a = (« + Cdo) 0 a0, o 0 is a finite-dimensional central ideal, is called an oscillator algebra. b G C and A G OQ, we can extend the s-module Ra to the a-module as follows:

do ^b + y^rrijXj-?—, ax

;-

Letting 8Q = Cc 4- Cdo + &o, w e

a n (A,a)/for ae a0.

j

nave

^ ne trianglur decomposition

t = 8_ ©S o ©*+. The following proposition is immediate by Lemma 9.13.

Ch. 9

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163

9.13. Let V be an a-module such that «o is diagonalizable and c has only nonzero eigenvalues. a) If there exists VQ G V, i>o / 0, such that PROPOSITION

than V is isomorphic to an a-module RQtb,\ b) If for every v e V there exists N such that pix ... ptri (v) = 0 whenever n> N, then V is isomorphic to a direct sum of a-modules Ra,b,\,a ^ 0. Note that the monomial x[l ... x£n G Ra,b,x is an eigenvector of do with eigenvalue ^mkjk- Hence, for the a-module R = i2a,6,A with a / 0, we k

have: (9.13.2) Here, as usual, for a diagonalizable operator A on a vector space V with eigenvalues Ai,A2,... counting the multiplicities, one defines try QA = As in §9.4, it is easy to see that the a-module Ra carries a unique ("contravariant") bilinear form B such that S ( l , 1) = 1 and pn is an operator adjoint to qn, provided that a G R. It is also easy to see by induction that distinct monomials are orthogonal with respect to B and that

(9.13.3)

B(x\> ... «*-, **» ... 4 - ) = as*' TJ *; '• i

As in §9.4, B can be written also in the following form: (9.13.4) Note finally that for the a-module Ra,b,x the operators do and s G *o are selfadjoint with respect to B. §9.14. The following basic notions of the representaion theory of the Virasoro algebra Vir (see §7.3), very similar to that of Kac-Moody algebras, will be used later. Define the triangular decomposition of Vir as follows: Vir = Vir> 0 V i r o e Vir + , where Vir± = @ C d ± j , Vir0 = Cc0C
164

Highest-Weight Modules over Kac-Moody Algebras

Ch. 9

Given c, h E C, define a Vir-module V with highest weight (c, h) by the requirement that there exists a nonzero vector v = vCth such that (cf. §9.2): Vir+(?;) = 0, U(Vii-)v

= V, do{v) = hv, c(v) = cv.

(Here and further we follow the usual physicists practice to denote the eigenvalue of a central element by the same letter, hoping that this will not cause confusion.) The definition of the Verma module M(c, h) over Vir, as well as the proof of its existence and uniqueness are given in the same way as in §9.2. It is clear that c acts on M(c>h) as el. The number c is called the conformal central charge. As in §9.7, we easily show that the elements (9.14.1)

d_jn .. . d - f c d - ^ K * ) .

where

0 < ji < j 2 < • • •,

form a basis of M(c,h). Since [cfoj^-n] = nd_ w , we see that do is diagonalizable on M(c, h) with spectrum H Z + and with the eigenspace decomposition (9.14.2)

M{c,h)=($M(c,h)h+J,

where M(c,h)h+j ji+-+jn=jIt follows that

is spanned by elements of the form (9.14.1) with

dimM(c,h)h+j=p(j),

(9.14.3)

where p(j) is the classical partition function. (Thus, the Kostant partition function for the Virasoro algebra is just the classical partition function.) Equation (9.14.3) can be rewritten as follows: oo

t*M(cth)q

do

:=^2dimM(c,h)xq X

x

h

= q JJ(1 -

qt)'1.

; =1

d

As in §9.7, the series try q ° is called the formal character of the Vir-module V. As in §9.2, one shows that there exists a unique irreducible Vir-module L(c, h) with highest weight (c, h). The Chevally involution u of Vir is defined by (9.14.4)

u(dn) = -d- n > u/(c) = - c .

The contravariant bilinear form B on a Vir-module is defined as in §9.4. In other words this is a symmetric bilinear form with respect to which dn and
Ch. 9 §9.15.

Highest-Weight Modules over Kac-Moody Algebras

165

Exercises.

9.1. Show that every nonzero homomorphism M(Ai) —* M^X?) is an imbedding. 9.2. Prove that for V G O there exists an increasing filtration (in general infinite) by submodules 0 = Vb C V\ C . . . such that K+i/K' is a highestweight module. 9.3. Let V be a module with highest weight A. Then chV = Y^, CAchM(A), where c\ = 1,CA G Z, and c\ = 0 unless A is a primiA
tive weight of a Verma module M(/i), where ft is a primitive weight of V. Show also that if cx / 0, then [Af (A) : I(A)] ^ 0. tf.^. Let A be a generalized Cartan matrix of finite type, and V G O a finitely generated module over &(A). Show that V admits a Jordan-Holder series, i.e., a filtration by submodules V = Vo D V\ D - — D Vt = 0 such that all the modules K'/K+i are irreducible. Describe the Jordan-Holder series for Verma modules over 5 9.5. Let g'(0) be the derived algebra of the Lie algebra g(0) associated to the n x n zero matrix. Given c i , . . . ,c n G C, the following formulas define the structure of a 0;(O)-module on the space V = C[a?i,..., x n ]: e

* "*• c»"s—) /» "~* multiplication by x,-, a* —* c,/v. ax,-

Prove that 7 ~ M(A), where ( A , ^ ) = ct- (i = 1 , . . . ,n). Show that V is irreducible if and only if all c» are nonzero. 9.6. Let M(0) be the Verma module with highest weight 0 over &(A). Then if v is a highest-weight vector, /i(v) is killed by n+, and one has an imbedding M ( - a t ) C M(0). 9.1. Assume that A is symmetrizable. Show that M(—<*{) is an irreducible module if 2(/>|7) ^ (7I7) for all nonzero 7 G Q + \ n . 9.8. Using Exercise 9.7, show that if 2(p|7) ^ (7I7) for all nonzero 7 G Q + \ n , then the submodule 5^M(—a t ) C M(0) is, in fact, a direct sum © M ( - a f ) . Deduce that in this case \[

(l-e(-a))multOf = 1 -

P.P. Use Exercise 9.8 to show that if A is symmetrizable and 2(/?|7) ^ (7I7) for all nonzero 7 G Q + \ n , then the subalgebra n + (resp. n_) of #(A) is a free Lie algebra on generators e% (resp. /,) i = 1 , . . . ,n. (This is a special case of Proposition 9.11, but this alternative proof is simpler.)

166

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Ch. 9

9.10. Prove that the Lie algebra on generators e,-,/f- (t = l , . . . , n ) and h and defining relations [e,-,/,] = 6ijh, [fc,c,-] = e f , [/*,/•] = —/i («, j = 1 , . . . , n) is simple, and therefore ci, . . . , e n generate a free Lie algebra. 9.11. Prove that if A is an indecomposable symmetrizable generalized Cartan matrix of indefinite type, then &(A) contains a free Lie algebra of rank 2, and hence has exponential growth, i.e., lim log dim 0,(1)/ | j | > 0. j—oo

9.IS. Let p be a Lie algebra, o an ideal in p,J> = p/a. following sequence of p-modules is exact:

Show that the

0 - o/[a, a] - Uo(p)/aUo(p) - U0(p) - 0. 9.13. Using Exercise 9.12 construct the following exact sequence of &(A)modules: a.) - M(0) - C - 0, 1=1

where C is viewed as a trivial module. 9.14- Let T be the space of all functions on f)* which vanish outside a finite union of sets of the form D(X). Then one can define product (convolution) of two functions / , g 6 T by: ( / * g)(X) = £ — /i). Define the delta function #A(AO = e(A) >-• 6A gives an algebra isomorphism E-^T.

^A,/I-

Show that the map

9.15. Let A be an affine matrix of type A[X\ Show that the g'^-module M{—p) has an irreducible quotient which is not isomorphic to L(—p). [Let v b e a highest-weight vector of M(—p). Show that t; — (/1/2 + /2/Ov generates a proper submodule of M(—/>).] P.itf. Suppose that we have two filtrations ; and " of V G O satisfying conditions of Lemma 9.6. Then there exists a bijection a : Jf —• J" such that \j = A^/x for all j £ Jf. Furthermore, if v^ is a primitive vector of V, then // = Xj where j is the least integer such that v^ E Vj. 9.17. Consider the associative algebra a on generators an(n G Z), and commutation relations

Ch. 9

Highest-Weight Modules over Kac-Moody Algebras

167

Let V be an a-module such that c*o ^ fi/ and for every v G V,ocn(v) = 0 for all but a finite number of n > 0. Let

Ln =

o 5 3 a - i a i + n + «Anan for n ^ 0.

Show that [L m ,L n ] = (m - n ) L m + n + 6 m _ n ^ 1 - ^ ( 1 + 12A 2 )/, producing thereby a representation of Vir with conformal central charge c = 1 + 12A2. 9.18. Consider an associative algebra b on generators a m ,aj^ and bm(m £ Z) and the following defining relations: [am>an] = f>mt-n,[bm,bn] = m6 m > - n , all other brackets are 0. Let V be a b-module such that &o »-+ 0 and for every v G V, an(v) = 0, €Ln(v) = 0 a n d 'n(v) = 0 for all but finitely many n > 0. The sign :: used below of normal product of generators, as usual, means that the generators with non-negative indices should be moved to the right of the rest of generators. Let

Hn = £ : an_ia; :, N = ;€l

Given numbers X,fi € C, let = an, in)

) = (»(^ 2 - 2) -

AK

-

Show that 7TA,p is a restricted representation of the affine algebra onF.

168

Highest-Weight Modules over Kac-Moody Algebras

Ch. 9

9.19. If in Exercise 9.17 (resp. 9.18) there exists a a- (resp. b-) cyclic vector |0) e V such that an |0) = 0 for n > 0 (resp. an |0) = 0 = bn |0) for n > 0 and a* |0) = 0 for n > 0) then the character of V is equal to the character of the Verma module Af(l + 12A2, §(A2 + /i 2 ))(resp. M ( ( ^ 2 - A - 2)A0 + AAi)), but V is not necessarily isomorphic to a Verma module. 9.20. In Exercise 9.18 let fen = 0 for all n and let /x = 0. Show that if there exists a 6-cyclic vector |0)€ V such that a n | 0 ) = 0 for n > 0 and < | 0 ) = 0for n > 0 , then

9.21. (Open problem) Let V be an irreducible module over the Virasoro algebra such that do is a diagonalizable operator whose eigenvalues have finite multiplicities. Then either these multiplicities are < 1 (then V is an irreducible subquotient of a module UQtp discussed in Exercise 7.24), or else V is a highest- or a lowest-weight module, i.e., there exists t; ^ 0 such that di(v) = 0 for all t > 0 or for all i < 0. §9.16.

Bibliographical notes and comments.

The category O of modules over a finite-dimensional semisimple Lie algebra was introduced and studied in Bernstein-Gelfand-Gelfand [1971], [1975], [1976]. There is a vast literature on the subject, which is summarized 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 this filtration is based on a formula due to Shapovalov [1972] for the determinant of the contravariant form on M(A), lifted from L(X). The study of the category O and the highest-weight modules over KacMoody algebras was started in Kac [1974]. There have been several developments of this in the papers Garland-Lepowsky [1976], 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 [1979] (generalizing Shapovalov's formula) and Jantzen's filtration. This technique was applied by Kac [1978 B], [1979] to initiate a study of highest-weight modules over the Virasoro algebra by computing the determinant of the contravariant form on Verma modules M(c, h). There have been several published proofs of the determinant formula: see FeiginFuchs [1982], Thorn [1984], Kac-Wakimoto [1986], and others. The computation of chL(c, h) consists of two steps, based on this formula. First,

Ch. 9

Highest-Weight Modules over Kac-Moody Algebras

169

one finds all possible inclusions of Verma modules, and all cases when [M{c, h) : L(c, h')] ^ 0, using the Jantzen filtration (see Kac [1978 B]). Second, one shows that [M(c, h) : L(c, h')] < 1. This more difficult fact, special for the Virasoro algebra, was conjectured by Kac [1982 B] and proved by Feigin-Fuchs [1983 A, B], [1984 A, B]. The explicit character formulas for all L(c, h) are easily derived from these facts: see Feigin-Fuchs [1983 B], [1984]. (The cases c > 1 and c = 0 were examined previously by Kac [1979] and Rocha-Wallach [1983 A].) Using the determinant formula, Kac [1982 B] proved that L(c, h) is unitarizable for c > 1, h > 0. By a detailed analysis 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-KentOlive [1985], [1986], Kac-Wakimoto [1986] and Tsuchia-Kanie [1986 B]. The solution to the problem of integrability of unitary L(c, h) was given by Segal [1981], Neretin [1983] and Goodman-Wallach [1985] (see PressleySegal [1986] for an account of this). As a result, the only problem from Kac [1982 B] that remains unsolved is Exercise 9.21. Kaplansky-Santaroubane [1985] 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.* The remarkable link between the theory of highest-weight modules over the Virasoro algebra and conformal field theory and statistical mechanics was discovered by Belavin-Polyakov-Zamolodchikov [1984 A], [1984 B]. Conformalfieldtheory has become by now a hugefieldwith many remarkable ramifications to other fields of mathematics and mathematical physics: see the reviews Goddard-Olive [1986], Ginsparg [1988], Fur Ian-Sot kov-To dor ov [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 [M(X) : L(\')]. In the finite-dimensional case, KazhdanLusztig [1979] came up with a remarkable conjecture, which was soon after proved by Beilinson-Bernstein [1981] and Brylinski-Kashiwara [1981]. Kazhdan-Lusztig conjectures were generalized by Deodhar-Gabber-Kac [1982] to arbitrary Kac-Moody algebras. Quite recently these conjectures were proved by Casian [1989] and by Kashiwara [1989]. (Kazhdan-Lusztig's [1980] cohomological interpretation of the multiplicities was extended to arbitrary Kac-Moody algebras by Haddad [1984].) The exposition of §§9.1-9.8 is a simplification of that in Deodhar-Gabber-Kac [1982]. A more complete statement than Proposition 9.9 a) can This problem has been solved recently by O. Mathieu "Classification of HarishChandra modules over the Virasoro algebra" and by C. Martin and A. Piard "Indecomposable modules over the Virasoro algebra and a conjecture of V. Kac."

170

Highest-Weight Modules over Kac-Moody Algebras

Ch. 9

be found in Kac-Kazhdan [1979]. 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 [1981]. 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 [1971]. Exercises 9.6-9.10 follow Kac [1980 C]. 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 [1970]. Exercises 9.18 and 9.20 are due to Wakimoto [1986]. These "free field" constructions of representations of the Virasoro algebra and the affine algebra £(s^2(C)) (and, more generally, C(s£n(C))) play an important role both in the study of representations (Feigin-Prenkel [1988], [1989 A-C]) and in the conformal field theory (Feigin-Fuchs [1984 A], Dotsenko-Fateev [1984], Felder [1989], Bernard-Felder [1989], BershadskyOoguri [1989], Distler-Qiu [1989]), and others. One last comment concerns Exercises 9.10 and 9.11. The main result of the paper Kac [1968 B] is the following. Let g = ® $j be an infinitedimensional Z-graded Lie algebra which satisfies the following conditions: (i) linij-ioo ofOgpiflj < oo, i.e., dim Qj grows polynomially as \j\ —> oo; (ii) there are no nontrivial graded ideals; (iii) 0 - i + g o + 0i generate 0 and the 0O-module 0_i is irreducible. Then 0 is isomorphic either to gf(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 C n : W n , 5 n , 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 (iii), the only algebra which should be added to the list is the "centerless" Virasoro algebra d (see §7.3) and its subalgebra D+ — Yl Cdj. A special case of this problem when > Qj < 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 dim0j < const.** He has shown that any Z-graded Lie algebra g = 0^- with only trivial graded ideals, is either gf(A)/CK with gradation of type 5, or d or D+. Recently Mathieu completely solved the problem in the paper "Classification of simple graded Lie algebras of finite growth."

Chapter 10. Integrable Highest-Weight Modules: the Character Formula

§10.0. 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 "^-dimension" of L(A). §10.1. Let &(A) be a Kac-Moody algebra of rank n and let f) be its Cartan subalgebra. Set P = {A e l ) ' | ( A , « V ) € Z ( i = l , . . . . n ) } , P + + = {A 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 lattice Q. Let V be a highest-weight module over &(A)> and v a highest-weight vector. It follows from Lemmas 3.4 b) and 3.5 that the module V is integrable if and only if /^'(v) = 0 for some N{ > 0 (i = 1 , . . . , n) (see §3.6 for the definition of an integrable module). LEMMA

10.1. The %(A)-module L(A) is integrable if and only if A G P+.

Proof. Formula (3.2.4) implies the "only if" part and the following formula:

e,/i A ' ar)+1 W = Oif(A,a, v )€Z +) where v is a highest-weight vector. It follows (since [e ; ,/,] = 0 for ./' ^ i) that if the vector f\ > a *' + (v) is nonzero, it is a primitive vector of L(A), which is impossible (by Proposition 9.3b). Hence, for A 6 P+, we have (10.1.1)

//A'^>+1(t;) = 0

( t = l

n),

which proves the "if" part (by the remark preceding this lemma).

• Denote by P(A) the set of weights of L(A). It is clear that P(A) C P if A G P . The following proposition follows from Lemma 10.1 and Proposition 3.7a). 171

172

Integrate Highest-Weight Modules: the Character Formula

PROPOSITION

Ch. 10

10.1. I / A G P + , then mult£,(A) A = multx,(A) w(A) for w G W.

In particular, P(A) is W-invariant.

• 10.1. If A G P+ then any A G P(A) is W-equivalent to a unique p e P+C\ P(A).

COROLLARY

Proof. Take /i G W • A such that ht(A — /i) is minimal; Lemma 3.12 b) implies the uniqueness.

• §10.2. We let the Weyl group W act on the complex vector space £ of all (possibly infinite) linear combinations of formal exponentials by cxe(X)) = 2LfCXe(w(X)) (w G W). X

A

The space £ contains £ as a subspace. However, product of two elements P\>P2 € £ doesn't always make sense, but if it does, then w(P\P2) = w(Pi)w(P2). Proposition 10.1 says that (10.2.1)

w(ch L(A)) = ch L(A) for w G W and A G P+

Consider now the element (cf. §9.7) R = Fix an element p G ()* such that (cf. §2.5) (p,a,y) = l FoiweW

(i=l,...,n).

set e(u;) = ( - 1 ) ^ ) . By (3.11.1) we have c(w) = detj,* w.

Furthermore, one h a s (10.2.2)

w(e(p)R)

= e(u;)c(p)iJ for

weW

Ch. 10 Integrable Highest-Weight Modules: the Character Formula 173 Indeed, it is sufficient to check (10.2.2) for each fundamental reflection r,-. Recall that by Lemma 3.7, the set A+\{a,} is r,-invariant and, by Proposition 3.7, we have mult r,(a) = mult a for a G A+. Hence,

= e(p - a,)r,(l - e(-at))rt

(1 - e(-a)) mult °

JJ o€A+\{ori}

= e(/>)c(-aO(l - e(aO) =

(1 - e(-a)) mult «

U

e(ri)e(p)R.

§10.3. From now on we assume that the generalized Cartan matrix A is symmetrizable. Let (.|.) be a standard invariant bilinear form on $(A)\ recall that (a,-|a,-) > 0 (i = 1 , . . . , n). The following is the key fact in the proof of the character formula, complete reducibility theorem and other results. LEMMA 10.3. Let A, A € P be such that A < A and A + A e P+. Then either (A + A,aV) = 0 for i £ supp(A - A) or (A|A) - (A|A) > 0. In particular, if A G P++, A G P+ and A < A, then (A|A) - (A|A) > 0.

Proof We have A = A - /?, where /? = ]£*,•<*,•,ifc,- > 0. Hence (A|A) (A|A) = (A + \\p) = E ±Jbf(af|at)(A + A,aV). Since (a,-|a.-) > 0 for all i, s

the lemma follows. D

§10.4. Now we can prove the following fundamental result of our representation theory. THEOREM 10.4. Let g(A) be a symmetrizable Kac-Moody algebraf and let L(A) be an irreducible &(A)-module with highest weight A G P+. Then

£ e(w)e(w(A + p) - p) / 1 f\ ji 1 \

(10.4.1) v

'

i_r/A\

ChL(A)=

W € "^

==—;

;

rr

j-

I! (l-e(-a))multQf

.

Proof Multiplying both sides of (9.8.1) by e(p)R and using (9.7.2) we obtain (10.4.2)

e(p)RchL(A) = A
174

Integrable Highest-Weight Modules: the Character Formula

Ch. 10

where cA = 1, cA G Z. By (10.2.1 and 2) the left-hand side of (10.4.2) is jy-skew-invariant (an element L G S is called W-skew-invariant if w(L) = e(w)L). Hence the coefficients in the right-hand side of (10.4.2) have the following property: (10.4.3)

c\ = e(ti;)c/i if w(X + p) = \L + p for some w G W.

Let A be such that c\ ^ 0; then by (10.4.3) we have cw(\+pyp ^ 0 for every w G W\ hence, it follows from (10.4.2) that w(X + p) < A + p. Let /i G {w(X + p) — p(w € W)} be such that ht(A — fi) is minimal. Then, clearly, fi + p £ P+ and |/i + p| 2 = |A -f />|2. Applying Lemma 10.3 to the elements A + p G -P++ and fi + />, we deduce that /i = A. Thus, cA ^ 0 implies that A + /? = it;(A + p) for some w G W and in this case, CA = e(iy) (see (10.4.3)). But A + p G ^++, hence, by Proposition 3.12b), the equality tu(A + p) = A + p implies that w = 1. Hence, finally, we obtain e(p)/ichL(A)= ^

c(«i)e(ui(A + p)),

which is (10.4.1). D Now set A = 0 in (10.4.1). Since L(0) is the trivial 1-dimensional module over &(A) we deduce the following "denominator identity":

(10.4.4)

I ] (1 - e(-a)) mult « = £

e(u;)e(ti;(p) - p).

Substituting (10.4.4) in (10.4.1) we obtain another form of the character formula:

(10.4.5)

ch L(A) = £

e(ti;)e(u;(A + />))/

Of course, in the case when &(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(i4)-module with highest weight A E 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(\) = M(A)/£(t/(n_)// A '^> + 1 t, A ) if A € P+

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Integrable Highest-Weight Modules: the Character Formula

175

This can be stated also as follows: COROLLARY 10.4. Let A be a symmetrizable generalized Cartan matrix and let A £ P+. Then the annihilator in U(&(A)) of a highest-weight vector of the &(A)-module L(A) is a left ideal generated by the elements e*, fjA>a^+1 and h - (A, h), where i = 1 , . . . , n and h £ J). In particular, an integrable highest-weight module over &(A) is automatically irreducible.

• In the finite-dimensional case, (10.4.6) is a result of Harish-Chandra. Another corollary is the following PROPOSITION 10.4. Let A be a symmetrizable generalized Cartan matrix and let A £ f)'* b e such that ( A , a V ) £ Z + for i = l , . . . , n . T h e n t h e $!(A)-module L(A) is characterized by the properties that it is irreducible and that there exists a nonzero vector v £ L(A) such that

c*y(v) = (A, <*y)v and e,-(t/) = 0,

(10.4.7)

i = 1 , . . . , n.

In particular, the definitions of a g'(A)-module L(A),A £ P+, given in the Introduction and in Chapter 9, are equivalent. Proof. Let V be an irreducible g'^J-module which has a nonzero element v satisfying (10.4.7). Considering the gradation of V by eigenspaces of c*y, we see that the element /J >Qf*' (v) generates a proper submodule of V and hence is zero. Due to (10.4.6), we have a surjective 0;(^4)-module homomorphism L(A) —• V. Lemma 9.10 completes the proof.

• §10.5.

Consider the expansion

(10.5.1) defining a function /C on t)* called the (generalized) partition function (K in honor of Kostant). Note that K((3) = 0 unless /? E Q+; furthermore, /f (0) = 1, and K(f3) for /? £ Q+ is the number of partitions of ft into a sum of positive roots, where each root is counted with its multiplicity. The last remark follows from another form of formula (10.5.1):

£

tf(/?)e(/?)=

J J (l + e(a) + e(2a) + . . . ) m u l t o .

176

Iniegrable Highest-Weight Modules: the Character Formula Ch. 10

Note that (9.7.2) can be rewritten as follows: (10.5.2)

mult M(A) A = K(A - 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

£(multL(A)A)e(A)= ]T e(w)e(w(A + p) - p) £

K{p)t{-p)

A
=E Comparing the coefficients at e(A), we obtain the multiplicity formula: (10.5.3)

mult L(A) A = ] T €(w)K(w(A + p) - (A 4- p)).

In the finite-dimensional case, this is Kostant's formula. Warning. The series e(—p) J^ K((3)e(/3) is not W-skew-invariant, though this is the inverse of the W-skew-invariant element e(p)R. This is possible since the inverse in £ is not unique. §10.6. Here we adopt a less formal point of view toward the characters, replacing the formal exponential e(A) by the function ex on f) defined by ex(h) = e for h e &. We define the character chv of a g(,4)-module V from the category O to be the function

defined on the set Y (V) of the elements h 6 f) such that the series converges absolutely. Note that chv(h) = t r r exp/* for h 6 y(V). Let us introduce some additional notation. Define the complexified Tits cone Xc by Xc = {x 4- ty|* 6 X , y € ()•} .

Ch. 10

Integrable Highest-Weight Modules: the Character Formula

111

Set

Y = {ft G f)| ] £ (multa)|c-| < oo}, YN = {he f)| Re(at-, ft) > JV (i = 1 , . . . , n)} for TV G R+. Note that Y lies in XQ (by Proposition 3.12 c)). We also have (10.6.1)

Xc = | J ti;(Y0).

10.6. Let V be a highest-weight module over &(A). Then a) Y(V) is a convex set. b)Y(V)DYnY0. LEMMA

c)Y(K)DY,ogn. Proo/. a) is clear from the convexity of the function |e A |. Furthermore, since V is a quotient of a module M(A), we deduce the following estimate from (10.5.2): multy A < K(A - A), which gives

But (10.5.1) implies that for ft G YQ we have

|

|

(l-|c" (flfi * ) ir multor .

The product part of this formula converges if ft G Y, proving b). Part c) follows from b) by the estimate (1.3.2).

• For the proof of the proposition below we need the following Remark 10.6. Let T C XQ be an open convex W-invariant set. Then TC

convex hull ( ( J

w(Tf)Y0)).

Indeed, To := (J w(Yo\Yo) is nowhere dense in Xc> Hence every ft £T lies in the convex hull of T\T0 =

\J w(T D Yo). D

For a convex set it! in a (real) vector space denote the interior of R in metric topology by Int R.

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Integrable Highest-Weight Modules: the Character Formula

Ch. 10

PROPOSITION 10.6. Let Q(A) be a Kac-Moody algebra and let L(A) be an irreducible $(A)-module with highest weight A G P+. Then a) Y(L(A)) is a solid convex W-invariant set, which for every x G IntXc contains tx for all sufficiently large t G R+. b) cli£,(A) is a holomorphic function on Int Y(L(A)). c)y(L(A))DlntY. d) Tie series J2 €(w)ew^A^ converges absolutely on Int Xc to a holowew morphic function, and diverges absolutely on f)\ IntXc. e) Provided that the Cartan matrix A is symmetrizable, cli£(A) can ^ e extended from Y(L(A)) H Xc to a meromorphic function on IntXcProof. Set T = Int Y; it is clear that T is open, convex (see the proof of Lemma 10.6a)), and W-invariant; by Lemma 10.6b), we have Y(L(A)) D Y fl YQ. 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' := {x G IntXc|*£ G Y(L(A)) for sufficiently large t G R+} coincides with IntXc. But again, X' is W-invariant and convex and contains Yo by Lemma 10.6c). Again, we can apply Remark 10.6. The convexity of |e A | implies that the absolute convergence is uniform on compact sets, which proves b). In order to prove d) remark that w(A+p)—(A+/>) are distinct for distinct w G W by Proposition 3.12b). It is clear from the proof of Proposition 3.12d) that (A + p) - w(A + p) G Q+. Hence we have for h G Yo:

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 G f)\Int X c . Then the set A o := {a G A^c| Re (a, h) < 0} is infinite by Proposition 3.12c) and f), and for every a G Ao, we have |c{ro,(A+p),fc)| > |e(A+p^)|} p r o v i n g the divergence at h. Finally, e) follows from d) and (10.4.5) (or rather (10.6.1))

• We shall give a more precise description of Y(L(A)) in Chapter 11. Denote by Oc the full subcategory of the category O of g(yl)-modules V such that chv converges absolutely on YN for some N > 0. Also denote

Ch. 10

Integrable Highest-Weight Modules: the Character Formula

179

by £c the subalgebra of the series from £ which converge absolutely on YN for some N. By Lemma 10.6, every highest-weight module lies in Oc. We have a homomorphism ip of £c into the algebra of functions, which are holomorphic on YN for some TV, defined by V> • e(A) i—• e \ Applying ip to both sides of formulas (10.4.5) and (10.4.4), we obtain on Y: (10.6.2)

ch L(A) = wew

wew

a multa

(10.6.3)

Yl (l-e- )

§10.7.

=

J2 e(w)ew{p)~p-

Here we prove the complete reducibility theorem.

THEOREM

10.7. Let A be a symmetrizable generalized Car tan matrix.

a) Suppose that a %'(A)-module V satisfies the following two conditions: (10.7.1)

for every v 6 V there exists N such that e

«'i ' *' e»fc (v)

(10.7.2)

=

0 whenever k > N.

for every v G V and every i there exists N such that f?(v) = 0.

Then V is isomorphic to a direct sum of Qf(A)-modules L(A) such that (A,aV)€Z+, for alii. b) Every integrable $(A)-module V from the category O is isomorphic to a direct sum of modules L(A), A £ P+. Proof. Thanks to Proposition 9.10 b) and Remark 3.6 in the case of statement a) (resp. Proposition 9.9b) in the case of statement b)), it suffices to check that if A and /i are primitive weights and /? E Q+\ {0} is such that A — fi = /?, then Since the module V is integrable, we have by (3.2.4): (10.7.3)

(A,a?) > 0 (i = 1 , . . . , n) for every primitive weight A.

But then we can write 2(A + p, v-\(J))

- (/?|/?) = (A + (A - 0) + 2p, v

using (10.7.3) and the fact that (^i/" 1 ^)) > 0 for all /? € <9+\{0}. D

180

Integrable Highest-Weight Modules: the Character Formula

Ch. 10

10.7. a) A &(A)-module V from the category O is integrable if and only ifV is a direct sum of modules L(A) with A E .P+. COROLLARY

b) Tensor product of a finite number of integrable highest-weight modules is a direct sum of modules L(A) with A G -P+. Theorem 10.7 contains as a special case the classical Weyl complete reducibility theorem for finite-dimensional representations of semi-simple Lie algebras. §10.8. Let s = ( « i , . . . , sn) be a sequence of integers. In §1.5 we introduced the Z-gradation of type s:

A particular case of this is the gradation of type 1 = ( 1 , . . . , 1) called the principal gradation. Note that if Si > 0 (i = 1 , . . . , n), we have

dim $j(s) < oo. Similarly we have the Z-gradation of type s of the dual Kac-Moody algebra

Fix elements A, £ I)* and h* £ f) which satisfy (A,,<*,y) = «.-, (A',ar<) = «,- (* = l , . . . , n ) . Note that A ! = p and h1 = pv are the elements p for the Kac-Moody algebras Q(A) and 0(*-A), respectively. Warning: j/(p v ) / 2p/(p|/>). Provided that all st- > 0, the sequence s defines a homomorphism Fs :

(10.8.1)

F.(e{-<*i)) = «'•

(* = 1, ••••»)•

This is called the specialization of type s. Note that (10.8.2) PROPOSITION

F . ( e ( - a ) ) = «<*'•">. 10.8. Let

Q(A)

be a symmetrizabie Kac-Moody algebra.

Then dim f l i ( I ) = d i r n d l ) .

Ch. 10

Iniegrable Highest-Weight Modules: the Character Formula

181

Proof. Note that both sides of identity (10.4.4) are elements from the algebra C[[e(—c*i),... ,e(—<*n)]]. Applying the homomorphism F% to both sides of this identity, we deduce (10.8.3)

^

^

Similarly for 0(M) we have (10.8.4) Since W^(J4) and W^A) are contragredient linear groups, the right-hand sides of (10.8.3 and 4) are equal.

• Comparing (10.8.3 and 4), we also deduce

J J (1 - g(Pv>"

(10.8.5)

Remark 10.8. In the sequel we use the specialization of type s when some of the Si are 0. Then F, is not defined everywhere and we have to check that Fs is defined on a given power series. §10.9. The specialization of type 11 is called the principal specialization. The following proposition gives a product decomposition of the principally specialized character. PROPOSITION 10.9. Let g(A) be a symmetrizable Kac-Moody Let A G P+ and set s = ((A, a^>,..., (A, c^)). Then

(10.9.1)

F!(e(-A)chL(A)) =

Proof. By (10.4.5) we have

(10.9.2)

e(-A)ch

For A € P4--1- set

J2 c(w)e(w(p) - p)

algebra.

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Integrable Highest-Weight Modules: the Character Formula

Ch. 10

Note that Nx G C[[e(-c*i),..., e(-a n )]] (by Proposition 3.12 b), d)). We have

where r = ( ( A , ^ ) , . . . , (A,a*)). Applying identity (10.4.4) for g(*A) we obtain Fi(Nx)

= Fr

Hence (10.9.3)

n

Fx (Nx) = TT

- -<*•«> v» uIt «

So, by (10.9.2 and 3) we have

(10.9.4)

F. (e(-A)ch I(A)) = J ] ( l

It is clear that (10.9.4) is an equivalent form of (10.9.1).

• §10.10.

Let V =

© VA be a g(A)-module with highest weight A. A
Again, fix a sequence of nonnegative integers s = ( « i , . . . , s n ). Let deg(AThen setting A:degA=j

defines the gradation of V of type s:

Note that dimV}(s) < oo if all «,• > 0. The gradation of type I of V is called the principal gradation. Provided that dimV}(s) < oo, we have

We call the formal power series ^2 dim V}( I )qj the q-dimension of V and denote it by dim^ V'. Then Proposition 10.9 and formula (10.9.4) can be restated as follows (here we use also Proposition 10.8):

Ch. 10

Integrable Highest-Weight Modules: the Character Formula

183

10.10. Consider the principal gra.da.tion L(\) = ) . Then under the hypotheses and notation of Proposition 10.9,

PROPOSITION

0

Lj{\

one has: dinifl L(A) = (10.10.1)

• The following corollary is a classical result of Weyl. COROLLARY 10.10. Let A be a finite type matrix, so that &(A) is a simple finite-dimensional Lie algebra. Let A £ P+. Then L(A) is finitedimensional and

dimL(A)= JJ (A + p,a)/(p,a).

Proof. Let q tend to 1 in (10.10.1) and apply l'Hdpital's rule.

• §10.11.

Exercises.

10.1. Show that

ch L(mp) = e(mp) J J (1 + e(-a) + • •. + e(-ma))m u l t a [Replacing e(—at) by e(—(m + l)at-) we deduce from (10.4.4):

l)w(p)) = e((m + l)/») ft (I - e(-(m + l) i^.5. Show that p - w(p) (it; G W) is equal to the sum of all positive real roots a such that w"1(a) < 0; the number of such roots equals £(w). [See Exercises 3.6 and 3.12.]

184

Integrable Highest-Weight Modules: the Character Formula

Ch. 10

10.3. Show that w(p) — p (w G W) are primitive weights of the module M(0) over a Kac-Moody algebra $(v4); deduce that this module has no irreducible submodules if dim Q(A) = oo. 10.4- Let UJ be the Chevalley involution of Q(A) and let V be a g(A)-module from the category O. Set V" = { / G F*|/(F A ) = 0 for all but finitely many A}. The algebra Q(A) acts on V" by (x-f)(v) = -f(u(x)v)(x G 0(A), v G V, / G V™). Show that V1" G 0 , ch V = ch V", L(A) - ^(A)". Show that M(0)w has no proper maximal submodules if &(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 A G *)* such that (A,<*V) = s, let Fx stand for F9. Show that Flt(Nx) = Fx(Nli)9\,tL€ll\ 10.7. Let A = (flij) be a symmetrizable generalized Cart an matrix and let n be a Lie algebra on generators e,- (i = 1 , . . . , n) and defining relations (ade,-)1"'a^e by rt(p) = ^> — a,- and put s(w) = p — w(p) for w G W. Show that s(t/;) is the sum of all a G Q+ such that n a / 0 and -w-x(a) G Q+. Show that €(w)e(s(w)). ifl.^. Let g(A) be a Kac-Moody algebra with the Cartan matrix A = [ j " 2 j , where a6 > 4. For a given pair of nonnegative integers m and n define the sequences aj(m}n) and 6^(171,11) (j G Z) by the following recurrent formulas (j G Z): ,n) = ai ; (m,n)-f m

and ao(m, n) = ai(rn, n) = 6o(m, *0 = ^i(^,n) = 0. Show that for A G such that (A,a)f) = m, (A,a^) = n, one has

- A) = J^-iycf-a^m, n)©! - 6;+i(m, n)a2).

Ch. 10

Integrable Highest- Weight Modules: the Character Formula

185

10.9. Show that if we set u = e ( - a i ) , v = e(-c*2), the identity (10.4.4) for A = ( J 2 ~ 2 J turns into the famous Jacobi triple product identity: - u n t; n )(l - ti n - 1 v n )(l -

unvn-x)

n=l

m€Z

and if ^4 = ( _^ ~* 1, the identity (10.4.4) becomes the important quintuple product identity: oo

JJ(l-ti 2n t;n )(l - ti* 1 -V- 1 ) - u4n~

i0.i0. Let L(A) be an integrable A^^module, so that Jbt- := (A,a/) € Z + (i = 1,2); put s = Jfci + Jb2 + 2. Check that if ifci ^ ib2, then

n^0,±(iki-|-l) mod*

(For (ki,k2) = (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 0 its highest root and let h = ht 0 + 1 be the Coxeter number. Let G be the associated complex Lie group. 10.11. Show that { L ( A ) } A C P + exhaust, up to isomorphism, all irreducible finite-dimensional modules over 10.12. Set E = £ e,-. Define constants cu . . . , cn by p v = £ c.aV, and put t

t

# = 2/?v and F = 2 ]£c,/ t -. Show that { £ , H, F } form a standard basis of t

«^(C) (this is the so-called principal 3-dimensional subalgebra). Using the representation theory of s/2(C) show that for A G P+ the expression 1 „ q(A+P,«)

is a polynominal in q with positive coefficients do = 1> ^i>..., d m , and that this sequence is unimodal, i.e., it increases up to d[m/2], and
186

Integrable Highest-Weight Modules: the Character Formula

Ch. 10

10. IS. Show that the sequences of the coefficients of the following polynomials are unimodal:

[Apply Exercise 10.12 to the ib-th symmetric power of the natural representation of An and the spin representation of Bn]. 10.14. Let r be a real number such that \r\ ^ 0 , 1 , . . . , h - 1, and let L(A) be an irreducible g(yl)-module with highest weight A G P+. Show that sin

Prove a similar formula for tr£,(A)exp(2?riV~1(/?)/r). 10.15. Let m = A (resp. m = A + 1) and let L = Q (resp. L = P). For it; G W K mL put c(iy) = e(iy/), where w' is the canonical image of it; in W. Show that for A G P, either (A, a) = 0 mod m for some a G A v or else there exists a unique element w\ G W K mL such that I/;A(A) = p. Let A G P+, and let s G Z be relatively prime torn. In the case m = h + 1, assume that s is divisible by det A (which divides h). Show that

{

0 if (A + p, a) = 0 mod m for some a G A v , e(ti;A+/>)(= ±1) otherwise.

[We have

hence, this is unchanged (resp. changes only the sign) when A+p is replaced by A -I- p + ma (a G L) (resp. by r ; (A + />), i = 1,.. .,£)] 10.16. Let (7^+1 = exp 7 -p4 G G and let C be the cyclic group gener/i+ 1 ated by cr^+i. Show that the order of C is h + 1. Let A + 1 be a prime integer. Deduce from Exercise 10.15 that the G-module L(A) (A G P+), restricted to C, is a direct sum of several copies of the regular representation of C plus at most one 1-dimensional or ft-dimensional representation ofC. [Show that C is defined over Q.]

Ch. 10

Integrable Highest-Weight Modules: the Character Formula

187

10.11. Let ft + 1 be a prime number and let A G P + . Show that dim L(A) = 0 or ± 1

mod (ft + 1),

(in particular, dimL(A) = 0 or ±1 mod 7, 13, 13, 19, 31 if A is of type G2, F4, #6, £7, #8, respectively), where 0, 1 or — 1 appear according to Exercise 10.15 (in the case m = A - f l ) . 10.18. For A of type J4 P _I, p a prime number, and the element exp(2?ripv/p) G G one has statements similar to those of Exercises 10.16 and 10.17. Furthermore, show that: -i)p)exp(27rip v/p) = ( - 1 (the Legendre symbol). Deduce the quadratic reciprocity law. Compute tiCh^x in the spin representation of Bi. Deduce that 2 " ^ = (—l)"""^1 mod I for an odd £. 10.19. Let M C f)* be the lattice spanned by long roots, $(•!•) the Killing form on 0(^4), v : f) —• f)* the corresponding isomorphism, g = ^(fllfl)"1, and set k = 2 for type Bt,Ct>F\, Jb = 3 for type G2 and Jfc = 1 for the rest of the types. Show that exp4irii/""1(p) is an element of order kg, which coincides with exp(27ripv//i) if Jb = 1. Show that for A G P, either 2(A|a) G Z for some a G A or else there exists a unique element w'x G W KtfAfsuch that u^(A) = i^" 1 ^)- Deduce that for A G P+ one has

(

0 if $(A + p|a) = 0 mod kg for some a G A

^ A + p ) otherwise. 10.20. Show that the conjugacy class of the element ah = exp 2nips/ /h in G consists of all elements 0

enveloping algebra of a Kac-Moody algebra Q(A), and let GvU(g(A)) be the associated graded algebra. Let AnriA be the annihilator of a highestweight vector of L(A) in U(g(A)) and let RA = Gr U(Q(A))/GT Ann A. Using Exercise 10.21, show that # A has a unique maximal ideal if and only if

188

Integrablt Highest-Weight Modules: the Character Formula Ch. 10

10.23. (Open problem). Classify irreducible integrable modules over a Kac-Moody algebra. §10.12.

Bibliographical notes and comments.

Theorem 10.4 as well as other results of §§10.1-10.5 are due to Kac [1974]; the exposition closely follows this paper. Formula (10.4.1) is usually referred to as the Weyl-Kac character formula. The results of §10.6 are due to Kac-Peterson [1984 A] (see also Looijenga [1980]). The first version of the complete reducibility theorem, which is Theorem 10.7 b), was obtained in Kac [1978 A]. In its present form, Theorem 10.7 is taken from Kac-Peterson [1984 A]. Note that this refinement is important for the proof of a Peter-Weyl type theorem in Kac-Peterson [1983]. The exposition of §§10.8-10.10 follows Kac [1978 A]. The trick employed in the proof of Proposition 10.9 goes back to Weyl and may be found in many textbooks (e.g. Bourbaki [1975], Jacobson [1962]). The only new thing is the use of the dual root system. The results of §§10.8 and 10.9 were generalized by Wakimoto [1983]. Exercises 10.3 and 10.4 are taken from Deodhar-Gabber-Kac [1982]. 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 Rogers-Ramanujan identities, as announced at the 83rd AMS summer meeting in 1979, abstract 768-17-1, and at the 775th AMS meeting in 1980, abstract 775-A14 (see Lepowsky-Wilson [1982] for an exposition of these results). For further development see Lepowsky-Wilson [1984], Misra [1984 A,B], [1987], [1989 A,B], Meurman-Primc [1987], Lepowsky-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 trJL(A)(exp 2iripw/h) and tr£(A)(exp 4TTIV~1(P)) is 0,1 or —1 is due to Kostant [1976] (his proof is more complicated). The rest of the material of Exercises 10.14-10.19 is taken from Kac [1981]. Exercise 10.22 is essentially due to Feigin-Fuchs [1989]. Theorem 10.7 implies that H1 (gf(A), L(A)) = 0 for A <E P+. Duflo (unpublished) and Kumar [1986 A] showed that Hk($'(A), L(A)) = 0, k > 1, for every A e P + , A / 0. Kumar [1984] showed that H*(Q'(A),C) is isomorphic to the cohomology of the associated topological group G{A)\ KacPeterson [1984 B] computed previously the Poincare series of G(A). The complexified Tits 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 K 27rtQv acting on the interior of Xc, which

Ch. 10

Integrable Highest-Weight Modules: the Character Formula

189

plays an important role in the deformation theory of singularities. Further connections of the theory of Kac-Moody algebras and groups to the theory of singularities may be found in Slodowy [1981], [1985 A, B]. Chari [1986] and Chari-Pressley [1987], [1988 A], [1989] have made a significant progress in classification of irreducible integrable modules over affine algebras. In particular, they showed that such a module with finitedimensional weight spaces is either L(A) or L*(A) (A G P+) or is of the type considered in Exercise 7.15. In a remakable recent development, Kumar [1987 A] and Mathien [1987] proved Demazure's character formula for an arbitrary Kac-Moody algebra, using the geometry of Schubert varieties. (In the finite-dimensional case this is due to Demazure [1974] and Andersen [1985].) A corollary of this is the Weyl-Kac character formula 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 Uq(Q(A)), and it remains irreducible for generic values of q (Lusztig [1988]). A remark on terminology. The integrable highest-weight modules are sometimes 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 B], [1989 A] shows that not only integrable L(A) are of interest.

Chapter 11. Integrable Highest-Weight Modules: the Weight System and the Unitarizability

§11.0. 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 &(A). We establish the existence of a positive-definite Hermitian form on L(A), invariant with respect to the compact form l(A) of 0(^4). Finally, 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 cli£,(A). §11.1. Fix A G -P+. From the results of Chapter 3 one easily deduces the following statement. PROPOSITION

11.1. Let A G P(A), a 6 A r e and mt = mult L( A)(A -Ma).

Then a) The set of t G Z such that A + ta G P(A) is the interval {t G Z| — p < t < q}, where p and q are nonnegative integers and

b) For ea G 0 a \ {0} the map eQ : L(A)\+ta —> L(A) A+(t+i)a is an injection if —p
^ 0.

Proo/. By Lemma 10.1 and the results of §3.6, the proposition holds for a simple root a. Applying Proposition 3.7a), we deduce that it holds for an arbitrary real root a.

• §11.2. Fix A G P+. Recall that P(A) is W-invariant (by Proposition 10.1). An element A G P is called nondegenerate with respect to A if either A = A or else A < A and for every connected component 5 of supp(A — A) one has (11.2.1) 190

Ch. 11

The Structure of Integrate Highest-Weight Modules

191

11.2. Every weight A of the g(A)-module L(A) is nondegenerate with respect to A.

LEMMA

Proof. Suppose that A G P(A)\{A}. Let 5 be a connected component of supp(A - A). Denote by n_(5) the subalgebra of n_ generated by the /,such that i G 5. Then L(A)X C

(11.2.2)

tf(n_)n_(S)L(A)A.

If (11.2.1) were false, then n_(S)L(A) A = 0 for some 5 and hence, by (11.2.2), L(A)A = 0, a contradiction.

• Now we can describe explicitly the set of weights of the g(A)-module L(A). PROPOSITION

11.2. L e t A G P + . Then

a) P(A) = W • {A G P+1A is nondegenerate with respect to A}. b) If {i|(A,c*V) = 0} C S(A) is a disjoint union of diagrams of finite type, then

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 A G P+ and A = A - /?, where /? G Q+, then (/?,a,y) < 0 for i such that (A, a)') = 0, which implies that A is nondegenerate with respect to A under the hypothesis of b). It remains to show that if /x = A — a G P+, where a = ]T} Jfc,a,-, t

^t > 0, Yl *i > 0, and 5 fl {i|(A, a*) ^ 0} ^ 0 for every connected compot

nent 5 of suppa, then fi G P(A). Let Qa = { 7

G Q+|T

< a and A -

T

G P(A)} .

The set Q a is finite and the union of supports of its elements has a nonempty intersection with each connected component of supp a. Let /? = £^ mta,be an element of maximal height in Q a . It follows from Proposition 11.1a) that (11.2.3)

supp /? = supp a.

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The Structure of Integrable Highest-Weight Modules

Ch. 11

Suppose that /? / a. Then (11.2.4)

A - p - a{ £ P(A) if ki > mi.

Set 5 = {j G S(A)\kj = m ; } . Let A be a connected component of (suppa)\S. Since /i G P+, we deduce from (11.2.4) and Proposition 11.1a) (11.2.5)

1fi,a() > (A,*)') and (a,c%) < (A,c%) if i € R.

Set

/?' = J^ mfat,

a' =

Then (11.2.3 and 5) imply (11.2.6)

{/^ f ay>>0if i € i i f

(11.2.7)

{a',aV)
It follows from (11.2.7) that R and hence S(A) are not of finite type. In particular, for every A G P(A) there exists a,- such that A — at- € ^*(A). (Otherwise, dimL(A) < oo and dimg(>l) < oo). Hence 5 ^ 0 and, by the properties of /i, we can choose R so that it is not a connected component of suppa. But then, in addition to (11.2.6), we have: (/?',aj) > 0 for some j G R. Hence R is a diagram of finite type. This is a contradiction.

• §11.3. We proceed to study the geometric properties of the set of weights P(A)forAGP+. 11.3. a) P(A) coincides with the intersection ofA + Q with the convex hull of the orbit W • A.

PROPOSITION

b) If \,fi Gfy*are such that A — \i G Q and fi lies in the convex hull of W • A, then mult L(A) (/i) > mult L(A)(A). Proof. First we prove by induction on ht(A — A) that a weight A of L(A) lies in the convex hull of W • A. If A = A, there is nothing to prove. If A < A, there exists t such that A + a,- G P(A). Take the maximal s such that /i := A + sat- G P(A). Then fi lies in the convex hull of W • A by the inductive assumption; since A lies in the interval [^,r,(/i)] (by Proposition 11.1), it also lies in the convex hull of W • A.

Ch. 11

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193

Let now A = ]Hcww(A) G A + Q, where cw > 0 and $ 3 ^ = 1. Then w

(11.3.1)

w

A-A

Replacing A by w(X) with minimal ht(A — A), we may assume that A G .P+. Finally it is clear from (11.3.1) that A is nondegenerate with respect to A. Hence, by Proposition 11.2a), A G ^(A), proving a). To prove b) we may assume that A G P+. Then we apply a) to L(A) to obtain /i G ^(A). We prove b) by induction on ht(A — [A). If A = /i, there is nothing to prove. Otherwise, \i + a,- G P(X) for some i. Let s > 0 be such that /* + sat- G P(A) but fi + (s + l)at- £ P(A). By a), /i + satlies in the convex hull of W • A and in -P(A). Hence, by the inductive assumption, multj,(A)(^) < mult^(A)(/i + sa t ). On the other hand, /i lies in the interval [/i+m,-, r,(/i+«a t )] and hence mult^( A )(/i+«a,) < mult/;(A)(/i) by Proposition 11.1b), c). Combining these inequalities proves b).

• §11.4. In the rest of the chapter we will assume that A is a symmetrizable generalized Cartan matrix; we fix a standard invariant bilinear form PROPOSITION

11.4. Let A G P+ and A,ji G P(A). Then

a) (A|A) - (A|/i) > 0 and equality holds if and only if\ = ^eW2

A.

2

b) |A + p| - |A + p| > 0 and equality holds if and only ifX = A. Proof. Since both (. | . ) and P(A) are W-invariant, we can assume in the proof of a) that A G P+. Since /? := A — A and /?i := A — /i lie in Q+, we have (A|A) - (A|/i) = (A|>9) + (A|/?i) > 0 (cf. the proof of Lemma 10.3). In the case of equality we have (A|/?) = (A|/?i) = 0. Since A is nondegenerate with respect to A (by Lemma 11.2), we deduce that /? = 0, i.e., A = A. But then (A|/?i) = 0 and, by the same argument, /i = A, proving a). To prove b) we write: (A + p|A + p) - (A + p\\ + p)= ((A|A) - (A|A)) + 2(A - \\p) > 0, since (A|A) — (A|A) > 0 by a) and A — A G Q+. Clearly, equality occurs if and only if A = A.

•

194

The Structure of Integrable Highest-Weight Modules

Ch. 11

§11.5. Let V be a g(j4)-module. A Hermitian form H on V is called contravariant if H(g(x),y)

= -H(x,vo(g)(v))

for all g € B(A), *,y € K

For example the Hermitian form (. |. )o on &(A) is contravariant (see §2.7). 11.5. Let A G f)J. Tien the &(A)-module L(A) carries a unique, up to a constant factor, nondegenerate contravariant Hermitian form. With respect to this form L(A) decomposes into an orthogonal direct sum of weight spaces. LEMMA

Proof. Denote by &(A)u the real subalgebra of &(A) generated by e,-, /,and f)R, and let £ ( A ) R = U($(A)i)v\. Then H is the Hermitian extension of B (see Proposition 9.4) from L(A)R to I(A).

• As in §9.4, it is easy to see that the fomula H(gvAi9'vA)

(11.5.1)

=

(&O(9)9'VA)

where g,g' G U(&(A)) and u>0 is the extension of — u>Q to an antilinear antiinvolution of U($(A))y defines a contravariant Hermitian form on L(A) (provided that A G fy£), normalized by H(v\yv\) = 1. Note that, by definition, the operators g and — u*o(g) are adjoint operators on L(A) with respect to H. Thus, the compact form t(A) is represented on L(A) by skew-adjoint operators. The Q(A)-module L(A) is called unitarizable if the Hermitian form iJ defined by (11.5.1) is positive definite. §11.6. We proceed to prove the positivity of (. |. )o on n_ + n+ and of H on L(A). First we prove the inequality (11.6.1)

2{p\a) > (a\a) if a € A + \n. 1

If a € Ay , this is clear, since then (a|a) < 0, but (p\a) > 0 for all a > 0. If a e A?\n, then a v G ( A X r \ n v , and hence 2(p\a)/(a\a) = (/>,av) > 1. This and (c*|a) > 0 imply (11.6.1). By analogy with the "partial" Casimir operator QOi we define an operator Qi on n . as follows:

Here, as before, {c« } and {e_^} are dual bases of 0 a and 0 - a with respect to the bilinear form (. | . ) , 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.

Ch. 11 LEMMA

The Structure of Integrable Highest-Weight Modules

195

11.6. If a E A+ and x £ g_ a , then = (2{p\a)-(a\a))x.

Proof We calculate in M(0) the expression Qox(v)i where v is a highestweight vector, in two different ways. By (2.6.1), we have (11.6.2)

Oox(y) = (2(p\a) - (a\a))x(v).

On the other hand, by the definition of QQ> w e have

Putting 5 = {/? G A + |/? < a } , we may write

Using (2.4.4), we obtain

Comparing this with (11.6.2) gives

«, As M(0) is a free C/(n_)-module (see Proposition 9.2 b)), the lemma follows.

•

196

The Structure of Integrable Highest-Weight Modules

§11.7. result.

Ch. 11

Now we are in a position to prove the following fundamental

THEOREM 11.7. Let g(A) be a symmetrizable Kac-Moody algebra. Then a) The restriction of the Hermitian form (. |. )o to every root space Qa (a G A) is positive-definite, i.e., ( . |. )o is positive-definite on n _ 0 n + .

b)Every integrable highest-weight module L(A) over g(A) is unitarizable. Conversely, if L(A) is unitarizable, then A G P+. Proof. We first prove a). Using u;o, it suffices to show that ( . | . ) o is positive-definite on 0_ a with a G A+. We do it by induction on hta. The case hta = 1 is clear by (2.2.1). Otherwise, put 5 = {/? G A+|/? < a} and use the inductive assumption to choose, for every /? G 5, an orthonormal -uo(e^)y basis | e ^ | of g_^ with respect to (. |. ) 0 . Then, setting e ^ = we have (ep \elp) = 6^. Now we apply Lemma 11.6 with this choice of ey and e^/p (the choice for the /? G A + \ S is arbitrary). For x G 0 - a we have (2(p|a) - (o|a))(«|*) 0 = (Oi(«)|«)o

By the inductive assumption, the last sum is nonnegative; using (11.6.1), we get (x|x)o > 0. Since ( . | . ) o is nondegenerate on g^ a , 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(A)A is positive-definite. We prove this by induction on ht(A — A). Let A G P(A)\ {A} and v G £(A) A . Thanks to a), we can choose a basis {e&\ of g a such that I — u>o(ea ) f is the dual (with respect to (. | . ) ) basis of 0_ a . Then we have 1

£ • - 2

and hence: (11.7.1)

Q(«) = (A + 2/»|A)e - 2

Ch. 11

The Structure of Iniegrable Highest-Weight Modules

197

Computing /y(Q(v),t;) in two different ways by making use of Corollary 2.6 and (11.7.1), and equating the results we obtain

By the inductive assumption, the right-hand side is nonnegative. Using Proposition 11.4 b) we deduce that H(v,v) > 0. Since H is nondegenerate on L(A)\i we conclude that it is positive-definite. To prove the converse, note that by (3.2.4), we have:

= Jfc((A,

Hence(A,a t y)€Z+.

Warning. The restriction of (. | . )o to t) and even J)' is in general an indefinite Hermitian form: the matrix ((Q^| a /)o) is a symmetrization of the matrix A. In fact it is positive-definite (resp. positive-semidefinite) on fy if and only if A is of finite (resp. affine) type. As was just mentioned, if A is a matrix of finite type, then the restriction of (.|.)o to t) is positive-definite and hence, using Theorem 11.7a), the Hermitian form (. |. )o is positive-definite on g(A), so that &(A) carries a positive-definite l(;4)-invariant Hermitian form. Thus, Theorem 11.7 is a generalization of a classical result of the finite-dimensional theory. §11.8. We deduce from Theorem 11.7 b) another complete reducibility result. For that we first prove LEMMA 11.8. Let h e I n t X c . Then for every r e R the number of eigenvalues (counting multiplicities) X of h in L(A) (A G P+), such that Re A > r, is finite.

Proof. This follows from Proposition 10.6d).

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The Structure of Integrable Highest-Weight Modules

Ch. 11

PROPOSITION 11.8. Let a c 0(A) be an u>0-invariant subalgebra which is normalized by an element h G Int XQ (i.e., [A, a] C a). Then with respect to a, the module L(A) (A G P+) decomposes into an orthogonal (with respect to H) direct sum of irreducible h-invariant submodules.

Proof. Put ai = 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 u>o-invariance of cti, that for every cti-submodule V C L(A), the subspace V1 is also an ai-submodule and L(A) = V 0 V*~. Hence L(A) decomposes into an orthogonal direct sum of irreducible Oi-modules. Let U C L(A) be an irreducible ai-submodule. It remains to show that U remains irreducible when restricted to o. Let U\ denote the Aeigenspace of h in U and let Ao be the eigenvalue of h with maximal real part. Let aA denote the A-eigenspace of ad A in a; we denote by a0 (resp. a + or a_) the sum of all aA with Re A = 0 (resp. > 0 or < 0). Then a = a_ 0 a0 © a+, and it is clear that U\Q is an irreducible a0-module and that {x G U\\a+(x) = 0} = 0 if Re A < Re Ao. Hence U is an irreducible a-module.

a §11.9.

Fix a G A+, and set

Then ^ is a subalgebra of &(A) (by Theorem 2.2e). It follows from Proposition 5.1 and Chapter 3 that if a is a real root, then g( a ) ^ sl2(C) and module L(A) restricted to g(a) decomposes into a direct sum of irreducible finite-dimensional modules. If a is an imaginary root, then $(Q) is an infinite-dimensional Lie algebra as described in §9.12. Now we can describe the restriction of L(A) to g(a) for a G A ^ . PROPOSITION 11.9. Let a e A1^71 and A G P+. Introduce the following two subspaces ofL(A):

L(A)^=

0 A:(A|a)=0

L(A)X; I(A)f> =

0

L(A)X.

A:(A|a)>0

a) Considered as a $(a)-module, L(A) decomposes into a direct sum of ( submodules L(A) = L(A)(Qa) 0 \

Ch. 11

c) L(A)+

The Structure of Integrate Highest-Weight Modules

199

is a free (7(ni ')-module on a basis of the subspace

d) The QW-module L(A) is completely reducible. Proof. Recall that, by Proposition 9.12 b), we can view g(a) as a quotient of a Lie algebra &'(B). Using Proposition 11.8, L(A) decomposes into a direct sum of f)-invariant irreducible g^-submodules. Each of these submodules is clearly generated by a nonzero vector v\ G L(A)\ such that tv+(v\) = 0. If (A|a) = 0, then Cvx is g^-stable by (9.10.1). If (A|a) > 0, then Cvx generates a Verma module over g( a) by Proposition 9.10 a). To complete the proof note that (11.9.1)

(A|a) > 0 if A 6 P(A) and a G A j * .

Indeed, by Corollary 10.1 and because A+" is W- in variant, we may assume that A G P+. But then (11.9.1) is obvious.

• Statements a), b) and c) of Proposition 11.9 imply COROLLARY 11.9. Let a G A1^1 and A 6 P + . Let A be a weight of the Q(A)-module L(A). Then either a) (A|a) = 0; then A — ka is not a weight of L(A) for k ^ 0; or else b) (A|a) ^ 0; then (A|a) > 0 and one has the following properties: (i) the set of t G Z such that A — ta G P(A) is an interval [—p, +oo), where p > 0, and t i-> mult£(A)(A — to) is a nondecreasing function on this interval; (ii) if x G 0-a, £ ^ 0, then x : L(A)A-*a - • ^(A)A-(t+i) a JS an injection. c)If A G P++ and v is a weight vector of weight A, then the map n_ —• L(A) defined by n n n(t;) is injective.

§11.10. Here we use the results of the preceding section to describe explicitly the region of convergence of chL ( A ). 11.10. Let A be an indecomposable symmetrizable generalized Cartan matrix, and let L(A) be an irreducible g(A)-module with highest weight A G P+, such that (A,a)') / 0 for some i. Then the

PROPOSITION

200

The Structure of Integrable Highest-Weight Modules

Ch. 11

region Y(L(A)) C I) of absolute convergence fl/ch^^) is open and coincides with the set

Y = {h G I)| ^2 (™»lta)|c-| < oo}. 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£i)\Re(6)h)>0}.

This follows from the description of the root system A(^4) given by Proposition 6.3 and the fact that the multiplicities of roots are bounded (by £) by Corollaries 7.4 and 8.3. Now the inclusion in question follows since mult L(A )(A - sS) ^ 0 for all s G Z+ by Corollary 11.9b(i). Finally, if A is of indefinite type, then, by Theorem 5.6 c), there exists a G A1™ such that suppa = S(A) and (a, aV) < 0 for all i. But then A - a G P(A) by Proposition 11.2a). Moreover, by Proposition 11.1b) and Corollary 11.9c), for every nonzero v G L(A)A-C* the map tp : n_ —• L(A) defined by V>(j/) = J/(v) is injective. This completes the proof of the inclusion Y(L(A))CY. To see that Y is open, note that (11.10.2)

a G A t m implies {h G Int X\a(h) = 0} = 0.

This is clear since we can assume that h G 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 := ]la€A im (^ ~ e ~ ° ) m u l t a . For h G Int X consider the meromorphic function f(z) = B(zh)~l. By a standard property of Dirichlet series with positive coefficients (see e.g. Serre [1970]), the set {t G R\th G Y} of convergence of the series obtained by multiplying out the product B(zh)~l, which represents f(z), is an open segment (c, +oo). It follows that if th G Y, then (t — e)h G Y for some e > 0. But the argument in the proof of Proposition 10.6a) shows that Y is convex and that with every h! it contains t'h! for sufficiently large tf > 0. Hence Y fl fa is open in fa, so that Y = (V fl fa) + *'fa is open in f).

•

Ch. 11

The Structure of Integrate Highest-Weight Modules

201

§11.11. In this section we deduce a specialization formula of the denominator identity (10.4.4). Let f)o be a subspace of f) such that (11.11.1)

f)onlntXc#0.

Let A »-+ A denote the restriction map I)* —• f)5; denote by p the homomorphism of S to the completed group algebra of fjjj defined by p(e(A)) = e(A). Put A 0 = {e*G A|a = 0 } ;

Ao+ = A o n A + ;

Ro

By Proposition 3.12, the set Ao is finite; hence Ao C A"5. It is clear that A o satisfies the usual axioms for a finite root system (see Bourbaki [1968]). Denote by Wo the (finite) subgroup of W generated by reflections ra (a G Ao), and let p0 (resp. p%) be the half-sum of roots from A o + (resp. A Q + ) . Define a polynomial D(X) on f)* by

LEMMA

11.11. ForXefy* we have ) =

D(\)e(\).

Proof. Let IIo C Ao+ be the set of simple roots of the root system Ao. Define the homomorphism F : C[e(—a); a G IIo] —• C[tf] by F(e(—a)) = q for all a G n 0 . Then (11.11.2)

p(f) = lim F(/);

F(e(-a)) = q^o >°).

But by (10.9.3) we have

(11.11.3)

F( £

e(w)e(w(X) - A)) = JJ (l '

^^])'

Formulas (11.11.2 and 3) together with (10.8.5) prove the lemma.

• Now we can deduce the specialization formula:

(11.11.4)

JJ (1 - e(-a)) multOf e{w)D{w{p))e{Mfi)---p).

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The Structure of Integrable Highest-Weight Modules

Ch. 11

Here and further, WbXW denotes a set of representatives of left cosets of Wo in W. Indeed, dividing both sides of (10.4.4) by i?o, we deduce that the left-hand side of (11.11.4), with a replaced by a, is equal to c(uOe(u
Applying p and using Lemma 11.11, we get (11.11.4). Now we consider a very special case of identity (11.11.4). Fix a sequence of nonnegative integers s = ( $ i , . . . , sn) such that the subdiagram {i G S(A)\si = 0} C S(A) is a union of diagrams of finite type. Fix an element h* € t) such that: (<*.•,*')=«.-

(i=l,...,n).

The subspace Ch' of f) satisfies condition (11.11.1) by Proposition 3.12. Define A e (C/is)* by (A, hs) = 1, and set q = e(-A). Then p(e(-ai)) = Si q (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)

Here D.(\)

=

fl

(A,av)/(p,,av),

where A f +

=

{a

G

A+

|(a, ft*) = 0} and p8 is the half-sum of roots from A,+ ; W* is a system of representatives of left cosets of the subgroup Wt generated by r a , a G A,+, in W, so that W = WSW8; Q(A) = © 0 J ( « ) is the Z-gradation of Q(A) of i type s. §11.12. In this section we shall discuss the unitarizability problem for the oscillator and the Virasoro algebras (cf. §§9.13 and 9.14). A contravariant Hermitian form H on a a (resp. Kir)-module V is a Hermitian form with respect to which the operators pn and qn are adjoint and the operators c and a G a0R are self adjoint (resp. dn and
(resp. # ( 1 ^ , ^ , 0 = 1).

Ch. 11

The Structure of Integrable Highest-Weight Modules

203

Recall that a module is called unitarizable if H is positive-definite. It is immediate by (9.13.3) that the o-module i2a,6,A is unitarizable if and only if a > 0. The unitarizability problem of the Vir-modules L(c, h) is quite nontrivial. We shall make here only a few simple remarks (which will be used in Chapter 12). 11.12. a) If the Vir-module L(c,h) is unitarizable, then h > 0 andc> 0. PROPOSITION

b) If V = L(0, h) is unitarizable, then h = 0, and hence V is the trivial 1-dimensional Vir-module. c) Let V be a unitarizable Vir-module such that do is diagonalizable with finite-dimensional eigenspaces 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 v be the highest-weight vector of a unitarizable Fir-module L(c,h). Then H(d^(v)%d^(v)) = ff(Mi<*-i(tO) = H(vy2dQ{v)) + H(v,d-idi(v)) = 2h. Hence unitarizability implies h > 0. Looking at H(d-j(v)yd-j(v)) as j —+ oo similarly implies c > 0, proving a). To prove b), consider the subspace Un of L(0, h) spanned by d_2n(u) and dl n (v). Since H is positive-definite on Un we have: detun H > 0, which, after a simple calculation gives 4n3ft2(8A — 5n) > 0 for all integral n > 0. This implies h = 0. To prove c) note that if U is a Vir-submodule of V, it is graded with respect to the eigenspace decomposition of d0 (by Proposition 1.5), hence V = U 0 UL. Since (7 X , the orthocomplement to [/, is clearly also a Virsubmodule, we deduce that V is an orthogonal direct sum of unitarizable irreducible Kir-modules (7, such that do is diagonalizable on {/, with specter bounded below. Thus the U% are highest-weight modules L(c, h) and the positivity of the spectrum of do on V follows from a).

• §11.13. In conclusion of this chapter, we shall sketch a theory of generalized Kac-Moody algebras. This is a Lie algebra g(A) associated to a real n x n matrix A = {a%j) satisfying the following properties (cf. §1.1): (Cl ; )

either ai{ = 2 or an < 0;

(C2 7 )

dij <0iii^j,

;

(C3 )

and a{j e Z if an = 2;

aij = 0 implies a,,- = 0.

204

The Structure of Integrable Highest-Weight Modules

Ch. 11

Let 11™ (resp. IF m ) = {at- 6 II|a t t = 2 (resp. < 0)}, let W be the subgroup of GL(t)*) generated by the fundamental reflections rf- such that a,- e 11™ (cf §3.7), and let C = {h € fa | (a,-, A) > 0 if a,-t- = 2}, C v = As in §3.3 we have: (11.13.1)

(11.13.2)

(ade,-)1""""^ = 0; (ad/,) 1 """/* = 0 if an = 2 and i / j ;

[e,-,^] = 0, [/,-,/>] = 0 if at;- = 0.

A f)-diagonalizable g(j4)-module V is called integrable if e,- and / , are locally nilpotent when an = 2. All the results of Chapter 3 hold with obvious modifications for generalized Kac-Moody algebras—one just restricts attention to the i with an = 2. The case of i with a,-,- < 0 is treated using the following LEMMA 11.13.1. Let V be an integrable 0(^4)-moduie, let an < 0, let X £ f)* be such that (A,a^) < 0 and let v G V\ be a nonzero vector such that fi(v) = 0. Then e{{v) ^ 0 for all j = 1 , 2 , . . . .

Proof This follows from the formula (cf. (3.6.1)):

• COROLLARY 11.13.1. If an < 0, a e A + \ { a J and supp(a + a j is connected, then a + jcti G A+ for ali j eZ+.

Proof. We may assume that a — ai $ A+. Note that ( a ^ a j ) < 0 for all j . Since supp(a -f ai) is connected we conclude that (a, a*) < 0 and apply Lemma 11.13.1.

• All the results of Chapter 4 hold for generalized Kac-Moody algebras, except that there is one additional affine matrix—the zero l x l matrix (cf. §2.9). In order to describe the set of roots A, let A re = W ^ i r ) , A' m = A\A r e , K = {a € Q+ | a € - C v and suppa is connected}\ (J jUim.

Ch. 11

The Structure of Integrate Highest-Weight Modules

205

Then we have As+m = ( J

(11.13.3)

w(K).

The proof is the same as that of Theorem 5.4 using Corollary 11.13.1. All other results of Chapter 5 hold for generalized Kac-Moody algebras with obvious modifications. From now on in this section we will assume that g(A) is a generalized Kac-Moody algebra with a symmetrizable Cartan matrix. T H E O R E M 1 1 . 1 3 . 1 . 0(J4) is a Lie algebra on generators et-, /,- (i = with defining relations (1.2.1), (11.13.1) and (11.13.2).

l,...,n)

Proof. Let 0I(J4) be the quotient of &(A) by the ideal generated by all elements (11.13.1 and 2), and let Ai be the set of roots of Qi(A). By the argument proving Lemma 1.6, the support of any a G Ai is connected, and since Ai is W-invariant, we obtain, using (11.13.3), that Ai = A. Now the theorem follows from Proposition 9.11 and Lemma 11.13.2b) below.

• LEMMA

11.13.2. a) Let a = J2^iai

^e

sucn

that the k are positive

iei

integers and a e - C v . Then 2(p | a) > (a | a) with equality if and only if {oLi \oti) < 0 and (a^ | oij) = 0 when i ^ j and i,j e I, and (a* | a*) = 0 when hi > 1. b) Inequality

(11.6.1)

holds.

Proof. 2(p\a) - (a|a) = £ *,(at|at- - a). If at- € 11^, then (a t |a t ) > 0 and t

(a,-| - a) > 0, so ki(ai\ai - a) > 0. If a,- G n t m , then t,-(a,-|a,- - a) > 0 since a — a,- G Q+. Hence 2(p|a) — (<*|a) > 0, and if the equality holds then all the a,- with i € I are in IF m and satisfy (a,|at- — a) = 0. This completes the proof of a). To prove b) note that we can keep on strictly reducing (p\a) by fundamental reflections while keeping a positive until either a G II or a G —C v . In the second case a G II im by a) since suppa is connected.

• It is clear by Lemma 11.13.2 b) that Theorem 11.7 a) holds for generalized symmetrizable Kac-Moody algebras. It turns out that this property characterizes them. More precisely, let A be a symmetrizable matrix, possibly infinite but countable, satisfying (Cl'HCS'), and let Q'(A) be the associated algebra, i.e. the quotient of the Lie algebra defined in Remark 1.5

206

The Structure of Integrable Highest-Weight Modules

Ch. 11

by relations (11.13.1 and 2). Let c (C V) be the center of g'(A) and let g be a central extension of tf(A)/c (described explicitly by Borcherds [1989 A]). Define a Z-gradation of 0 by deg(center) = 0, dege,- = — deg/t- = s,-, where Si is a collection of positive integers with finite repetitions. Then g has the following properties: (gl) g = ®g;- is a Z-graded Lie algebra, [go, go] = 0 and dimgj < oo; 3

(g2) g has an antilinear involution u> which is —1 on gon and o;(gm) = g_ m ; (g3) g carries an invariant bilinear form (. | . ) such that (gt-1 gj) = 0 unless * + j = 0; (g4) the Hermitian form (x\y)0

= — (u>(x) \y) is positive definite on g;- if

i#o. 11.13.2. Any Lie algebra g satisfying (&1-4) can be obtained from a generalized Kac-Moody algebra Qf(A) as described above. THEOREM

Proof. The argument proving Proposition 9.12 b) shows that g can be obtained from some g'(B) as 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 (C1')-(C3') for an invertible diagonal matrix D (see Borcherds [1988] for details).

• We turn now to the study of g(A)-modules L(A) with A satisfying (11.13.4)

( A , ^ ) € Z+ if a« = 2; (A,ctf) > 0 for all i.

We denote by P+ the set of such A's. LEMMA

11.13.3. Proposition 11.4 holds for a generalized Kac-Moody al-

gebra. Proof. The proof of part a) of Proposition 11.4 for a generalized KacMoody algebra needs no modifications. The proof of its b) part is different. Let A G P(A). It is clear that (A|ai) > 0 for all a» G I P m . We can act on A by fundamental reflections each time strictly increasing |A -f p\2 until A e C v , so we may assume that (\\cti) > 0 for all c^ G II r e . Thus: (A|a f) > 0 for all a,- G II. If A ^ A, let i 6 supp(A-A). If a,- G IT6, then, of course (a t |A+A+2/>) > 0. If a,- G IF m we have: (a,|A -f A + 2/>) = (a t |A) + (at|A -f a,-) > 0 since (A|a t ) > 0 and A+at- = A-)9, where /? G Q+. Hence (A-A | A+A-l-2/?) > 0,

Ch. 11

The Structure of Integrable Highest-Weight Modules

207

which is equivalent to |A+p|2—|A+/>|2 > 0. Equality implies that (A|a t ) = 0 for any t G supp(A — A), i.e. that A = A.

• It is clear by Lemma 11.13.3 that Theorem 11.7b) holds for generalized symmetrizable Kac-Moody algebras. Finally, we find a character formula. THEOREM

11.13.3. Let A e P+, and let SA = e(A-\-p)J2e(s)e(-s), where s

s runs over all sums of elements from II tm and e(s) = (—l) m if s is the sum ofm distinct pairwise perpendicular elements perpendicular to A, and

e(s) = 0 otherwise. Tien /

chL(A)= £ e(w)w(Sx)/e(p) JJ (1 - e(-a))mult a Proof Following the argument of §10.4 we find that (11.13.5)

c(/>)

where both sides are antisymmetric under W, the c\ are integers and A G A - Q+ satisfy |A + p\2 = |A + p| 2 . Let 5 A be the sum of the terms on the right of (11.13.5) for which A 4- p e C v . If A + p £ IntC v , then £ e(w)e(w(X + p)) = 0, hence the right-hand side of (11.13.5) is equal to ^

e(w)w(S\),

and it remains to

evaluate 5A. If e(A + p) occurs in 5A, we write A = A — £ ib,a,- where Jfc,- are positive integers. Then |A + p\2 = |A + p\2 implies:

<€/

But (A|a t ) > 0 and (A + 2/>|a,) > 0 (the second inequality is proved in the same way as in Lemma 11.13.3). It follows that (11.13.6)

(A|a,) = 0 and

(A + 2p|a,-) = 0.

Since (A + p\oti) > 0 and {p\a^ > 0 for a. G n r e , it follows that a» G Uim for i £ I. The second equality of (11.13.6) can be written as follows:

•€/

208

The Structure of Iniegrable HighesU Weight Modules

Ch. 11

Hence (oti\otj) = 0 for i, j G I unless i = j and Jb-t = 1. If e(A — J2 kiOLi) occurs in ch L(A), then (A|c*i) ^ 0 for some i, hence the only terms on the right-hand side of (11.13.5) which give a contribution to 5 A are those coming from e(A + p) f ] C1 - e(-a)) m u l t O f . If (at-|a,-) = 0 A

for i / j , then the coefficient of e(A + p — £ i,**,) in this expression is 0 if some of the &*, are greater than 1 and is (—I)22** otherwise.

• Since chL(O) = 1, we deduce the following COROLLARY

11.13.2. Let S = e(p) ]£e(s)e(«), where s runs over all sums

of elements from Uim and e(s) = (—l) m if s is the sum of m distinct pairwise perpendicular elements, and e(s) = 0 otherwise. Then e(w)w(S). D §11.14.

Exercises.

i i . i . Show that if A 6 P+ and A € P ( A ) \ { A } , then ^

c(ti;) mult L(A) (A -f p - u;(p)) = 0.

(This formula allows one to compute mult/,(A) by induction on ht(A - A)). [Rewrite (10.4.5), multiplying by the denominator, as:

and note that, by Proposition 11.4, the equality A + p = w(A + p) implies A = A.] 11.2. If A is of finite, affine or strictly hyperbolic type, and A 6 P+, then every dominant A < A is nondegenerate with respect to A. U.S. Show that if A € P+, then P(A) D P+ = {A G P|A > w{\) for every u; G W and A is nondegenerate with respect to A}. 11.4- Show that if A G P+, then the set of asymptotic rays for the set of rays through {A - /i|/i G P(A)} lies in Z. [Use Exercise 3.12 and Lemma 5.8.]

Ch. 11

The Structure of Integrable Highest-Weight Modules

209

11.5. Let A, A', ••• G P+. Show that the Cf(g(j4))-submodule generated by v\ v\t . . . of the g(i4)-module V = L(A) L(A') ® . . . coincides with the eigenspace of il corresponding to the eigenvalue CA+A'+...> where 2 2 CM := \M + p\ — \p\ . Moreover, CA+A'+... > CM for every M such that L(M) C V. 11.6. Let fl(i4) be an infinite-dimensional Kac-Moody algebra and let A, M G P+, AW ± 0- Show that L(A) ® L(M) ^ L(0). 11.7. Let 0(^4) = © 0j be a Z-gradation of type (^i,t,... >$n,i) of a symi€i

metrizable Kac-Moody algebra &(A). Prove that the 0o-module 0_i is isomorphic to the 0o-module L(—a$). [See the proof of Proposition 8.6c).] 11.8. Let A be a symmetrizable generalized Cartan matrix and a G A t m . Show that Ma) := lim l o * m u l t n a exists. Show that V»(a) = 0 if and only if (a|a) = 0 and that V(<*) > 0.48 if (a\a) ^ 0. Show that Y C Int XQ coincides with Int XQ if and only if A is a direct sum of matrices of finite and affine type. 11.9. Let 0 and 0° be Kac-Moody algebras with symmetrizable Cartan matrices. Let f) and f)° be their Cartan subalgebras, W and W° their Weyl groups, etc. Let ir : 0° —• 0 be a homomorphism such that

*(&°) C &; *r(IntX%) nIntX c # 0;

(±n) n **{Z) = 0.

Let A G P+. Show that the 0-module L(A) is isomorphic as a 0°-module to a direct sum of integrable highest-weight modules L°(fi) with finite multiplicities, which we denote by (A : /i). For fi = w(n' + p°) — p 0 , where t i ; 6 ^ 0 and / e P l , set (A : fi) = e(u;)(A : fi'); for all other ^ G *)°* set (A : fi) = 0. We define A o , VTo, />o, £>(A), etc., for f)° as in §11.11. Show that

11.10. Let m be asubalgebraof 0 such that m+n_ (resp. m-|-n+ ) has finite codimension in 0(J4) and let A G P+. Show that dim {v G L(A)\m(v) = 0} < oo (resp. dimL(A)/mi(A) < oo). [Let C/+ = (exptade,| t G C, t = l , . . . , n ) . span n+.]

Show that UC/+(ef)

210

The Structure of Integrable Highest-Weight Modules

Ch. 11

11.11. Let g(>!) be a symmetrizable Kac-Moody algebra. Set (l-e(-a))multa;

F =

Fix an orthonormal basis {u,*} of I), and define derivations d,* of £ by di(e(\)) = (A,ut)e(A). Show that

Deduce that =(p\p). 11.12. For 0 e Q+ set C/? = £ n"1 mult(/?/n). Using Exercise 11.11 show n>l

that under the hypotheses of this exercise one has

(This formula allows one to compute the multiplicities of roots by induction on the height, thanks to (11.6.1). Note that if/? € Ky then, due to Exercise 5.20, all summands on the right-hand side are non-positive.) 11.13. Consider the following generalized m > 3: / 2 - 1 - 1 - 1 2 0 A= -1 0 2 \-l Set fi = 2c*x + a 2 + a 3 +

0

0

Cartan matrix of order m where ... ... ...

-1\ 0 0

...

2 /

h a m G A(>1). Show that

mult/? = 2 m - 2 - ( m - l ) . H.14. Let A : V\ -+ V2 and J3 : V2 —> Vi be linear maps of finitedimensional spaces. Show that txAB — trJ3A Let V be a module from the category O over a Kac-Moody algebra with a symmetrizable Cartan matrix. Deduce that for e a G g a and e_ a € Q-a such that (e Q |e_ a ) = 1, a G A+j, one has J ^ + .;a|a) dim KA+;a.

Ch. 11

The Structure of Integrable Highest-Weight Modules

211

Let V be a module with highest weight A. Deduce the following generalization of Freudenthal's formula:

J2 £

a\a) dimVx+ja.

11.15. Let V = © L ( A t ) , where A, G P+ satisfy (At-,aJ) = ^

(j =

1 , . . . , n). Show that the group associated to the 0(j4)-module V is a central extension of every group G* constructed in Remark 3.8. 11.16. Show that if a module L(\) over a generalized Kac-Moody algebra is unitarizable then A G P+. 11.17. Let A, A' G P+ and ) mult L(A) (M + p - ii;(A; + />)).] €W

11.18. Using Exercise 9.17, show that the V't'r-module L(c, h) with c > 1 and h > (c — l)/24 is unitarizable. 11.19. Show that the fixed point set of a diagram automorphism of a generalized Kac-Moody algebra is a central extension of a generalized KacMoody algebra. 11.20. Prove an analogue of Corollary 10.4 for generalized Kac-Moody algebras. 11.21. Let g(A) be a generalized Kac-Moody algebra. Show that A+ = is a subset of Q+\{0} defined by the following properties:

A+(J4)

(i) a{ € A+,2orf- g A + , (ii) if a £ A+, a,- G II

i = l,...,n;

lm 1

and supp(<* + a,) is connected, then a - f a , G A + ;

(iii) if a G A + , a,- G II * and a^ai}

then [a,r,(a)] f l Q + C A + ;

(iv) if a G A + then suppa is connected. 11.22. Let Qf(A) be a generalized Kac-Moody algebra. Show that the C/(0'(yl))-module 0 L(A) is faithful. [Use Exercise 11.20 to show that C/(n_) acts faithfully. Then 0 < iJ(ti_VA> U_VA) = (^o(w-)ti_t;A) for some A G P+, depending on u_ G C/(n_). Hence U(n+) also acts faithfully.]

212

The Structure of Integrable Highest-Weight Modules

Ch. 11

§11.15. Bibliographical notes and comments. 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]. Of course, Proposition 11.4 is a standard fact in the finite-dimensional case. Proposition 11.3 a) in thefinite-dimensionalcase, is due to Steinberg (see Bourbaki [1975]). It seems that Proposition 11.3 b) was not previously known even in thefinite-dimensionalcase. Lemma 11.6 and the positivity Theorem 11.7 are due to Kac-Peterson [1984 B]. The proof of Theorem 11.7 b) is a direct generalization of that of Garland [1978] in the affine case (as soon as Theorem 11.7 a) is established). Theorem 11.7 is important since it allows one to apply powerful Hilbert space methods, see e.g. Goodman-Wallach [1984 A], Kac-Peterson [1984 B]. (Note that the Hilbert completion of L(A) is the space of a unitary representation of the "compact form" of the group associated to a KacMoody algebra.) Jakobsen-Kac [1985], [1989] classified the unitarizable highest-weight modules over affine algebras for all natural choices of antilinear involutions and "Borel subalgebras"; it turned out that there exists only one essentially new family of unitarizable highest-weight representations, that of the Lie algebra of maps of the circle with a positive-finite measure into su(n, 1). There is an interesting connection of these representations when n = 1 to the representations constructed in Exercise 9.20 (cf. Bemard-Felder [1989], Feigin-Frenkel [1989 B]). The material of §§11.9-11.11 is taken from Kac-Peterson [1984 A]. The proof of Proposition 11.12 b) is taken from Goddard-Olive [1986]. Most of the material of §11.13 (on generalized Kac-Moody algebras) is taken from Borcherds [1988 A]. The new material is the description of the root system and Theorem 11.13.1 (that fills a slight gap in his paper). Exercise 11.1 in the finite-dimensional case is attributed to Racah. Exercises 11.4, 11.8 and 11.9 are taken from Kac-Peterson [1984 A], and Exercises 11.5 and 11.6 are from Kac-Peterson [1983]. Exercise 11.10 is due to Peterson-Kac [1983]. This property has important applications to the conformal field theory. A similar "admissibility" property plays a prominent role in representation theory of p-adic groups. Exercises 11.11, 11.12 and 11.14 are due to Peterson [1982], [1983]. The proofs indicated here were communicated to me by Peterson. These recurrent multiplicity formulas are very convenient for computations of root and weight multiplicities. Another formula for root multiplicities, which is also a formal consequence of identity (10.4.4), was found earlier by BermanMoody [1979]. Exercise 11.13 is taken from Kac [1983 A]. It gives some nontrivial

Ch. 11

The Structure of Integrate Highest-Weight Modules

213

evidence in support of the conjecture mentioned in §5.13. Exercise 11.15 is taken from Peterson-Kac [1983]. Exercise 11.17 is taken from KacWakimoto [1986]. This is a special case of the Parthasaraty-RaoVaradarajan 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 all roots of an indefinite-type Kac-Moody algebra are not known explicitly in any single case (in some cases explicit formulas are known for low "level" roots, see e.g. Feingold-Frenkel [1983], Kac-Moody-Wakimoto [1988]). Nevertheless Borcherds [1989 B], [1989 C] managed to find the root multiplicities of some generalized Kac-Moody algebras intimately related to the Monster simple group, and used this to establish the modularity properties of the Monster. Given below are a few computations of the root multiplicities done on a computer by R. Gross, using Peterson's recurrent formula. Here (Jki, Jb2,...) denotes the root a = Jbiai + kioti + We may assume that a e —Cv and that (a | a) := \ Y^^ijh^j ^ 0. If A is symmetric of rank 2, we can assume that k\ > &2, since then mult(A:i, A^) = mult(/c2, fci). In Table H2 (resp. H3) all such roots are listed with fci < 21 (resp k\ < 14). A part of Table H% was computed by Feingold-Frenkel [1983].

214

The Structure of Integrable Highest-Weight Modules

A =

-3

TABLE

-H«)

a ( 1, 1) ( 2, 2)

(3, 2) (3, 3) ( 4, 3) ( 4, 4) ( 5, 4)

(6, 4) (5, 5) (6, 5) ( ( ( ( ( ( (

7, 6, 7, 8, 9, 7, 8,

5) 6) 6) 6) 6) 7) 7)

(9, 7) (10, 7) ( 8, 8)

(9, 8) (10, 8) (11, 8) (12, 8)

(9, 9) (10, 9) (11, 9) (12, 9) (10, 10) (13, 9) (11, 10) (12, 10) (11, 11) (13, 10) (14, 10) (15, 10) (12, 11) (13, 11) (12, 12) (14, 11)

1 4 5 9 11 16 19 20 25 29 31 36

41 44 45 49 55 59 61 64 71 76 79 80 81 89 95 99 100 101 109 116 121 121 124 125 131 139

144 145

mult or

a

1 1 2 3 4 6 9 9 16 23 27 39 60 73 80 107 162 211 240 288 449 600 720 758 808 1267 1754 2167 2278 2407 3630 5130 6559 6555 7554 7936 10531 15204 19022 19902

(15, (16, (13, (14, (13, (15, (16, (17, (18, (14, (15, (14, (16, (17, (15, (18, (19, (16, (15, (17, (18, (16, (19, (20, (21, (17, (16, (18, (19, (17, (20, (21, (22, (18, (17, (19, (20, (18, (21,

11) 11) 12) 12) 13) 12) 12) 12) 12) 13) 13) 14) 13) 13) 14) 13) 13) 14) 15) 14) 14) 15) 14) 14) 14) 15) 16) 15) 15) 16) 15) 15) 15) 16) 17) 16) 16) 17) 16)

-(o|a) 149 151 155 164 169 171 176 179 180 181 191 196 199 205 209 209 211 220 225 229 236 239

241 244 245 251 256 261 269 271 275 279 281 284 289 295 304 305 311

mult a 23750 25923 30865 45271 55853 60654 74434 84121 87547 91257 135861 165173 185526 233487 271860 271702 292947 409725 492420 569358 732180 815214 874650 972117 1006994 1242438 1476973 1752719 2298090 2458684 2808958 3207547 3426450 3783712 4456255 5411212 7217527 7453376 9005900

Ch. 11

Ch. 11

The Structure of Integrable Highest-Weight Modules

215

2 - 2 0 -2 2 -1 0 - 1 2

A =

TABLE c

(2, 2,1) 3,1) 4,2) 4,1) 4,2) 5,1) (4, 5,2) (6, 6,1) (5, 5,2) (7, 7,1) (5, 6,2) (8, 8,1) (5, 6,3) (6, 6,2) (9, 9,1) (6. 6,3) (6, 7,2) (10, 10,1) ( 7, 7,2) (11. U,l) (6. 7,3) (7, 8,2) (12, 12,1) (6. 8,4) ( 7, 7,3) (8, 8,2) (13, 13,1) (8, 9,2) (14, 14,1) (7, 8,3) (9, 9,2) (7, 8,4) ( 8, 8,3) (9, 10,2) ( 7, 9,4) (3, (3, (4, ( 4, ( 5,

-H a) mult a 1 2 3 3 4 4 5 5 6 6

7 7 8 8 8 9 9 9 10 10 11 11 11 12 12

12 12 13 13

14 14 15 15 15 16

2 3 5 5 7 7 11

11 15 15 22 22 30 30 30 42 42 42 56 56 77 77

77 101

101 101 101 135 135 176 176 231 231 231 297

a

— (a|a) mult a

( 8, 8, 4) (10, 10, 2) ( 8, 9, 3) (10, 11, 2) ( 9, 9, 3) (11, 11, 2) ( 8, 9, 4) (11, 12, 2) ( 8, 10, 4) ( 9, 9, 4) ( 9, 10, 3) (12, 12, 2) ( 8, 10, 5) (10, 10, 3) (12, 13, 2) (13, 13, 2) ( 9, 10, 4) (10, 11, 3) (13, 14, 2) ( 9, 10, 5) ( 9, 11, 4) (10, 10, 4) (U, 11, 3) (14, 14, 2) (10, 10, 5) (14, 15, 2) ( 9, 11, 5) (11, 12, 3) ( 9, 12, 6) (10, 11, 4) (12, 12, 3) (10, 12, 4) (11, 11, 4) (10, 11, 5) (12, 13, 3) (11, 11, 5)

16 16 17 17 18 18 19 19

297 297 385 385 490 490 627 626

20

792 792 792

20 20

20 21 21 21

22 23 23 23 24 24 24

24 24 25 25 26 26 27 27 27 28

28 29 29 30

791 1002 1002 1001 1253 1574 1574 1571 1957 1957 1957 1956 1953 2434 2429 3007 3005 3712 3713 3710 4557 4557 5593 5587 6826

a

-Ha) mult or

(13, 13, 3) (10, 12, 5) (U, 12, 4) (10, 12, 6) (11, 13, 4) (12, 12, 4) (13, 14, 3) (10, 13, 6) (14, 14, 3) (11, 12, 5) (11, 12, 6) (12, 15, 5) (12, 13, 4) (14, 15, 3) (11, 13, 5) (12, 12, 6) (12, 14, 4) (13, 13, 4) (11, 13, 6) (11, 14, 6) (12, 13, 5) (13, 14, 4) (11, 14, 7) (13, 13, 5) (14, 14, 4) (13, 15, 4) (12, 13, 6) (12, 14, 5) (13, 13, 6) (12, 14, 6) (13, 14, 5) (12, 14, 7) (14, 14, 5) (12, 15, 6) (13, 14, 6) (13, 14, 7) (14, 14, 6) (14, 14, 7)

30 31 31 32 32 32 32 33 33 34 35 35 35 35 36 36 36 36 38 39 39 39 40 40 40

40 41 41 42

44 44 45 45 45 47 48 48 49

6818 8326 8322 10111 10108 10107 10096 12266 12246 14821 17892 17893 17886 17861 21525 21526 21514 21515 30993 37083 37080 37053 44258 44247 44217 44219 52753 52741 62719 88255 88230 104456 104415 104450 145513 171355 171337 201527

Chapter 12. Integrable Highest-Weight Modules over Affine Algebras. Application to ^-Function Identities Sugawara Operators and Branching Functions

§12.0. 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 the denominator identity (10.4.4) for affine algebras is nothing else but the celebrated Mac don aid identities. Historically this was the first application of the representation theory of Kac-Moody algebras. The basic idea is very simple: one gets an interesting identity by computing 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 g-series of modular forms, the simplest ones being Macdonald identities for the powers of the Dedekind ij-function. We study the structure of the weight system of an integrable highestweight 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 identity. This identity involves classical theta functions and certain modular forms called string functions, which are, essentially, generating functions for multiplicities of weights in "strings." Furthermore, we consider branching functions, which are a generalization of string functions when instead of the Car tan subalgebra a general reductive subalgebra 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 forms and theta functions. §12.1. Let $(A) be an affine algebra of type xff (here X = A, B, C, £>, E,F or G; r = 1,2 or 3; N as in Table Aff r). We keep the notation of Chapter 6. Since the multiplicities of the roots of g(A) are known (by 216

Ch. 12

Integrable Highest-Weight Modules over Affine Algebras

217

Corollary 8.3), we can write down the denominator identity (10.4.4) explicitly. Recall that the left-hand side of (10.4.4) is

Introduce the following polynomials in x (cf. Proposition 6.3):

L(x) = (1 - x)1 Yl (1 - xe(a)) for A of type X$l); 1—

/ \\ TT /i __ r ( \

for A of type Xfr' with aor = 2 or 3, where s =

—;

L{x) = (1 - «)' I ] (1 - xe(a)) ft (I - xe{\{6 - a)))(l - .:2e(a))

for 4 of type Ft C1 - e ( ~ a ) ) -

Set R=

Then

» b y Proposition 6.3 and Corollary 8.3

0

a€A +

we have (12.1.1)

Furthermore, the right-hand side of (10.4.4) is

where p is as defined in §6.2. By Proposition 6.5, we can write w = utQ, o

where u 6 W, a G M. Using the equalities (p,K) = Av and \p\2 = |p| 2 , we deduce from (6.5.3):

(12.1.2)

uta(p) -p = u(p + /«va) - p+ ±(\p\2

- \p + h"a\2)6.

218

Integrable Highest-Weight Modules over Affine Algebras

Ch. 12

Hence we obtain (12.1.3)

Comparing (12.1.1 and 3) we deduce

(12.1.4) e(-Jg£<

These are the Macdonald identities. Example 12.1. Let A be of type A^\ Then II = {e* 0 ,ai}; A + = { ( n - l ) a 0 + nai; na 0 + (n - l)ai; na 0 + nc*i(n = 1 , 2 , . . . ) } ; mult a = 1 for a e A + ; M = Zau p = \a^ (ai|ai) = 2; /i v = 2; W = { ± 1 } . Put u = e(—ao), t; = e(—ai). Then using the expression for i?ieft in (10.4.4) and that given by (12.1.3) for flrighti we obtain:

(12.1.5)

JJ(1 - unvn)(l - t i n - V ) ( l - unvn~l)

This is one of the forms of the Jacobi triple product identity. For A € f)* set

Note that if (A,aV> G Z + for i = 1 , . . . ,t, then x(A) is nothing other than the formal character of the 0-module L(X) (where | is a simple finiteo

dimensional Lie algebra with the root system A). Dividing both sides of (12.1.4) by R we obtain another form of Macdonald identities:

(12.1.7) e ( - | D ) I ] L(e(-n6)) = £ $(Ava)e(--^\p +

Ch. 12

Integrable HighesU Weight Modules over Affine Algebras

219

Consider now the particular case of a nontwisted affine algebra g of type X} '. In this case g is of type Xi. Recall that one has the "strange" formula of Freudenthal-de Vries: 1

Later (see formula (12.3.7)) we will prove a much more general "very strange" formula. Recall also that M is the lattice spanned over Z by long roots of g. Setting q = e(—6), we can rewrite (12.1.7) as follows:

(12.1.9)

qdim9'24l[((l-qn)1 n>l

§12.2. The specialization of type ( 1 , 0 , . . . ,0) is called the basic specialization] we denote it by F. Note that (12.2.1)

F(e(-a)) = ql*'*. o

In particular, F(e(a)) = 1 if a 6 A. Hence, by Lemma 11.11 we obtain

(12.2.2)

Fft(\)) = d(A), where S(A) = f [ (\ + p,a)/Q>,a).

Introduce the Euler product (12.2.3) n=l

and the Dedekind ij-function: (12.2.4)

i?

Applying F to both sides of (12.1.9) and using the equality dim g = t+ |A|, we obtain Macdonaid's 77-function identities

(12.2.5)

i/ dim8 =

Y, a€M

220

Integrable Highest-Weight Modules over Affine Algebras

Ch. 12

§12.3. Here we derive a generalization of identity (12.2.5) using an arbitrary specialization of type s. We keep the notation of §8.2. Let 0 be a simple finite-dimensional Lie algebra of type XN , let /i be a diagram automorphism of 0 of order r and 0 = 0 Qj the corresponding Z/rZgradation. Introduce elements £,-, /*}, i/ t (t = 0 , . . . , £) of 0 as in §8.2. Given a nonzero sequence of nonnegative integers s = (so, . . . , £ / ) , we set m = r ^ a«5« (where at- are given by Table Aff r). Let e = exp 2ii. •=o Recall the definition of the automorphism crs>r of type (s;r) of 0:

Let 0 = © 0y(s; r) be the associated Z/mZ-gradation. Put 3

dj(s; r) = dim 0j(s; r)

(j G Z),

where j G Z/rnZ denote jmod m. By the last assertion of Theorem 8.7, we have 12.3. Let Q(A) be the affine algebra of type X^ 0;(s) be its Z-gradation of type s. Then

LEMMA

0

and let &(A) =

D Hence, the left-hand side of (11.11.5) is (12.3.1)

i?left =

Recall that the right-hand side of (11.11.5) is bright =

We rewrite i?right in terms of the Lie algebra 0.

Let A, be the sub-

0

set of A which consists of linear combinations of the roots from the set {a<\\si = 0 (i = 0, . . . , < ) } . Let Wt be the subgroup of the Weyl group W o

generated by reflections in the roots from A,. Let W* be the set of representatives of left cosets of WST in W, so that W = W8TW*. We may

Ch. 12

Iniegrable Highest-Weight Modules over Affine Algebras

221

choose W = TW*. Then the set {tQw}, where a € M,w € W9, is a set of representatives of left cosets of W, in W. Using (6.2.8) and (6.5.3) we obtain (12.3.2)

taw(p) = /»VAO + w(p) + A v a - ( y (a|a) + (w(p)\a))6.

Hence we have (p-taw(p),h')

V) + y( y We define 7, € f>S *>y (12.3.3)

Mat)

= rsi/m

(t=l,...,/).

Then the last formula can be rewritten as follows: (12.3.4)

(p-ta

2^

Using (12.3.1 and 4) we deduce from (11.11.5) the following identity: (12.3.5)

Note that the 77-function identity (12.2.5) is a special case of this identity for s = (1,0,...,0), r = 1. As in that special case, one can express the first factor in the left-hand side of (12.3.5) entirely in terms of the dj(s,r). Namely, one has the following "very strange formula," which is a generalization of the "strange formula" (12.1.8):

(12.3.6)

^-yp-h^A^-^

The proof of this formula uses identity (12.3.5) and some elements of the theory of modular forms. It will be given in Chapter 13.

222

Integrable Highest-Weight Modules over Affine Algebras

Ch. 12

§12.4. Recall that the center of an afBne algebra g(A) is 1-dimensional and is spanned by the canonical central element (see §6.2):

t=0

It is clear that K operates on a g(j4)-module L(A) by the scalar operator (A,AT)/L(A)- In particular, (A, AT) = (\,K) for every A G P(A). The number (12.4.1)

*:= ff) = s=0

is called the level of A G fj*, or of the module L(A). If A G -P+, then the level of A is a nonnegative integer; it is zero if and only if all the labels (A,a^) of A are zero. Hence, by (9.10.1), an integrable g(j4)-module L(A) has level 0 if and only if dimL(A) = 1; if L(A) is integrable and dimL(A) ^ 1, then the level of L(A) is a positive integer. Note that the level of p is equal to (12.4.2)

(p,K) = s=0

the dual Coxeter number. Let A,- (t = 0, . . . , € ) be the fundamental weights: (A,-,a/) = 6ih j = 0 , . . . , / , and (A,-,
Af = A t-+ <*)%,

where Ao = 0 and A i , . . . , A/ are the fundamental weights of 0. Note also that (12.4.4)

P+rr t=0

It is clear that the level of A from P+ is 1 if and only if A = A, mod C6 and i is such that a? = 1; in particular, the level of Ao is always 1. A glance at Table Aff gives that if A is symmetric or r > 1, then (12.4.5)

level(Af) = 1 if and only if i G (Aut 5(^1)) • 0.

Finally, the following observation, which follows from Propositions 3.12 b) and 5.8 b), is useful:

Ch. 12

Integrable Highest-Weight Modules over Affine Algebras

223

LEMMA 12.4. If Re(A,AT) > 0, then (W - A) n {/* G *)* | Re(/x, < ) > 0 for ail i} consists of a single element. In particular, if A G P has a positive level, then the set P+ D W • A consists of a single element.

n We let P*(resp. P*) = {A G P(resp. P + ) | (A, tf) = Jfc}. §12.5. We collect here some facts about the weight system P(A) of an integrable module L(A) over an affine algebra 0(^4), proved earlier in the general context of Kac-Moody algebras. PROPOSITION 12.5. Let L(A) be an integrable module of positive level k over an affine algebra. Then

a)P(A) = W.{AGP+|A mult L ( A ) A. d) P(A) lies in the paraboloid {A G f£||A| 2 + 2Jb(A|A0) < |A| 2 ; (A, K) = Jb}; the intersection of P(A) with the boundary of this paraboloid isWA. e) For A G P(A) the set oft e l such that A - tS G P(A), is an interval [—p, -foo) with p > 0, and f >-• mult^(A)(A — tS) is a nondecreasing function on this interval. Moreover, ifx G 0-6, # / 0, then the map x : L(A)\-t6 —• L(A) A . ( t + i)6 is injective. f) Set n(_^} = 0 g_ n 6 ; then L(A) is a free C/(n(_6))-module. n>0

Proo/. a) follows from Proposition 11.2 b), while b) and c) are special cases of Proposition 11.3 a) and b). d) follows from Proposition 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).

• §12.6. We continue the study of the weight system P(A) of an integrable module L(A) of positive level Jb over an affine algebra. A weight A G P(A) is called maximal if A + £ £ P(A). Denote by max(A) the set of all maximal weights of £(A). It is clear that max(A) is a Winvariant set (since P(A) is W-invariant and 6 is M^-fixed) and hence, by Corollary 10.1, a maximal weight is VF-equivalent to a unique dominant

224

Integrable Highest-Weight Modules over Affine Algebras

Ch. 12

maximal weight. On the other hand, it follows from Proposition 12.5 e) that for every /i G P(A) there exists a unique A G max(A) and a unique nonnegative integer n such that /i = A - n6, i.e., we have (12.6.1)

P(A)=

(J

{A-n%€Z+}

(disjoint union).

A€max(A)

Here is a description of dominant maximal weights. PROPOSITION 12.6. Tie map A »-+ A defines a b\jection from max(A) fl P+ onto JbCa/ fl (A + Q). in particular, the set of dominant maximal weights of L(A) is finite.

Proof. Straightforward using Proposition 12.5.

• The following lemma describes explicitly the weight system of certain particularly important highest-weight modules. 12.6. Let A be an affijie matrix of type X%\ where X = A, D or E. Let A G P+ be of level 1. Then LEMMA

(12.6.2)

(12.6.3)

max(A) = W - A = T • A,

P(A) = {Ao + \ |A|2 « + a - ( £ |a| 2 + j)«, where a E A + Q, s G Z + } .

Prt>o/. Since level(A) = 1, A is a fundamental weight A,- mod C6. Using (12.4.5), one easily sees that t is a special vertex, and hence W • A^; = T • A,(see Remark 6.5). Hence (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 = { < o | a € Q } (see §6.5). Let A G max(A); we have A = A - /?, where /? G Q. Since /? = ]5 mod C£, we have: t^(A) = A mod C£ by (6.5.2). This proves (12.6.2).

D

Ch. 12 §12.7.

Integrable Highest-Weight Modules over Affine Algebras

225

Let A 6 P+. If follows from Proposition 11.10 and (11.10.1) that converges absolutely to a holomorphic function in the region

In fact, Y is the region of convergence of di£(A) if level(A) > 0. Note also that for any highest-weight module V over an affine algebra, chy converges absolutely in the domain YQ (see Lemma 10.6 b). For A G max(A) introduce the generating series oo

n=0

This series converges absolutely in the region Y since it is majorized by e" A ch L(A) . Since W\ DT = 1 for A G P(A) (see Proposition 6.6 c)) and W(6) = 6, we deduce using (12.6.1):

(12.7.1)

chL(A)=

£ A€max(A)

eM=

£ A€max(A) A mod T

This simple formula reduces the computation of the character of L(A) to the computation of the functions a*. The following case is especially simple. LEMMA

12.7. Let A and A be as in Lemma 12.6. Then

7€Q+A

Proof. Use (12.6.3) and the fact that every a£ is W-conjugate to a£ = a*°o due to (12.4.5) and Lemma 12.6.

• The function aA° will 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 functions. For A G I)* such that level(A) = Jfe > 0 set (12.7.2)

0A = c - ^

226

Integrable Highest-Weight Modules over Affine Algebras

Ch. 12

Using (6.5.3) we obtain

e A = e*A°

(12.7.3)

W

which is a classical theta function (see Chapter 13 for details). It is clear that this series converges absolutely on Y to a holomorphic function. Note also that Q\ depends only on A mod kM + C6. Using theta functions, we can rewrite the denominator identity (10.4.4) in yet another form. Recall that (p, K) = hv (the dual Coxeter number) and p = p + AVAO (formula (6.2.8)). Using the decomposition W = WK T, we get e(u>)

Hence, (10.4.4) can be rewritten as follows

(12.7.4)

Ap=

Introduce the following number for

P*:

For reasons which will become clear later we call this number the modular anomaly o/A, and we introduce the normalized character

x A = e "- A V cn L ( A ) For a weight A € ^(A) introduce the number |\|2

(12.7.6) It is clear that

m AiA = mA _ L L . TTIA,A

is a rational number.

Ch. 12

Integrable Highest-Weight Modules over Affine Algebras

227

For A G f)* set 4 = e-m*.*« £

(12.7.7)

multL(A)(A -

nS)e'nS.

n€l

Just as the series a£, this series converges absolutely to a holomorphic function on Y. Note that c£ = e~mA'A'a£ if A G max(A). The function c^ is called the string function of A G I)*. Note that c£ (A) = c^ for to € W.

(12.7.8)

Since W = W K T, we use (6.5.2) to obtain (12.7.9)

ctw+ky+ai = C${OT \eV,

wew,

yeM,

aec.

Note also that 4 = c$+aS for a € C.

(12.7.10) 0

Using W = W K T and (12.7.2) we can rewrite the character formula (10.4.5) as follows: (12.7.11)

xA =

On the other hand, we have by (12.7.1) and the definitions of Q\ and

A modT

Using (12.7.9) we can rewrite this as follows: (12.7.12)

xA= A€P*mod (kM+C6)

Comparing (12.7.11 and 12) gives the theta function identity (12.7.13)

AA+P = AP A€P*mod

We shall use this important identity in the next chapter to study and compute the string functions.

228

Integrable Highest-Weight Modules over Affine Algebras

Ch. 12

§12.8. We turn now to the Sugawara construction. For the sake of simplicity, we consider only the case of a non-twisted affine algebra tf = g'(X)1^), leaving the general case as an exercise (see Exercise 12.20). Recall that 0' = £(g) = Cft,*"1] ®c 0 + CK with commutation relations [«<»>, y(">] = [x, y]C"+") +

(12.8.1)

mSm,.n(*\y)K.

Here § is a simple finite-dimensional Lie algebra of type X*, (x\y) is the normalized (by (6.4.2)) invariant bilinear form on 0, and x^n^ stands for $ Let {ti,} and {ti*} be dual bases of 0, i.e. (ti,- \u*) = 6{j. Recall that

t

the Casimir operator of 0, is independent of the choice of these dual bases. In particular, ^Ttijti* = J^ti'ti,-. This proves the following useful formula: » (12.8.2) Furthermore, note that

(12.8.3)

fi

= 2(tf + /»v)d+ft + 2'

oo

n=l

i

is the Casimir operator of the affine algebra 0 = g(JT£(1)) = *' + Cdy where [d, x^] = nx^n>. Since Q is independent of the choice of the dual bases {ti,} and {ti1}, we can check this for a special choice of these bases. Consider the root space decomposition 0 = f) © ( © Ce a ), such that (eole-a) = 1, and 0

choose dual bases hi,..., hi and ft1,..., h* of f). Then we have: fi = Y^hih1 + X) e_ a e Q . Using that [e a ,e_ a ] = i / ' ^ a ) (see Theorem 2.2e)), aGA

and that 2 p= 2J a we obtain the form of Q given in §2.5: a€A+

Ch. 12

Integrable Highest-Weight Modules over Affine Algebras

229

Hence (12.8.3) can be rewritten in the same form as Q, defined in §2.5, 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 the restricted completion Ue(tf) of the universal enveloping algebra U(tf). Consider the category of all restricted g'-modules, i.e. modules V such that for any v eV, x^\v) = 0 for all x G 0 and all j > 0 (cf. §2.5). Consider all oo

series ^2 Uj with Uj G U(tf) such that for any restricted g'-module V and any v G V, tij(v) = 0 for all but finitely many tij. 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(gf) (since, by Exercise 11.22, 0 L(A) is a faithful [/(g')-inodule) and acts on every restricted A(EP+

g'-module. Note that the derivations dn (see §7.3) extend from U(Q') to derivations of Uc(gf). Now we introduce the Sugawara operators Tn (n G Z):

To = 1 > V + 2 (12.8.4)

Tn

n=l

<

*

= H X i «4~ m)« <(m+n) if n ^ 0. m€Z

t

Due to (12.8.2), these operators are contained in Uc{tf). Note also that these operators are independent of the choice of dual bases of g. We have the following basic lemma. LEMMA 12.8.

a) For x € 0 and n, m G Z one

has:

[*( m \T n ] = 2(K + A v )m*( m + n ). b) Let V be a restricted ^-module said let v £V and h(v) = (A, h)v for some A G G*. Then

be such that n+(v) = 0

Proof, a) We perform the calculations in the semidirect product 0 x Uc(&'). First, note (12.8.5)

To = -2(K

+

230

Integrable Highest-Weight Modules over Affine Algebras

Ch. 12

Also, it is straightforward to check using (12.8.2) that 7) = ^ , To]

ifj#O.

Now we have, using Theorem 2.6 and (12.8.5):

Finally, for ,; ^ 0, we have: [ ,

i

]

[ , [

2(K + A v )(i[mx( m + »,T 0 ] + *[<*;, m* (m) ]) = 2{K b) follows immediately from (12.8.5) and (2.6.3).

• Now we can calculate [T m ,T n ]. One should be careful about staying within the algebra Uc(tf). We may assume that m > n. Let first ra+n / 0, m / 0, n ^ 0. Then we have:

= 2(K + A by Lemma 12.8. Replacing j by j + n in the second summand, we obtain: (12.8.6)

[TmiTn]

=

Thus, we have, provided that m + n ^ 0, m ^ 0, n ^ 0: (12.8.7)

Cr m ,r n ] = 2(AT + A v )(m -

n)Tm+n.

A similar calculation shows that (12.8.7) holds when m + n ^ 0 but m or n = 0. Let now m+n = 0, m > 0. Then the right-hand side of (12.8.6) does not lie in Ue(Qf). We proceed as follows. Since Q is independent of the choice

Ch. 12

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231

of dual bases we have E , u ^ V W+m) = E , liCj+~)lim(-i) w h e n m ^ 0 . Hence we can write (here and further we drop the sign of summation over i, but assume that it is present):

We have:

4- (m - ,><-'V«>) v

A )(mT 0 + E i t i $ - i + m ) u ' « - m ) - E iti^- m ) ti««+m)). i>o i>o Replacing .; by ,; -f m in the first summation and .; by j — m in the second summation, we obtain:

+ ftv)(T + £ (j + ) j - ; ) ' O ) + ^ (m Av)(2mT0 +

E 171 — 1

E i(m i=o m-l

(

)(

E i(m m-l

by (12.8.2). Since E Km - i ) = ( m 3 -

m

)/ 6 > combining with (12.8.7),

we obtain the final formula: (12.8.8) [Tm,Tn] = 2(/C + / » v ) ( ( m - n ) T m + n We immediately obtain from (12.8.8) and Lemma 12.8 the following COROLLARY 12.8. Let V be a restricted tf-module such that K is a scalar operator kl, k / —fcv. Let

Tn

h ' (12.8.10)

c(Jfe) =

neZ)

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Ch. 12

a) Letting extends V to a module over 0 ; +Vir (the semidirect sum defined in § 7.3). In particular, V extends to a module over 0 (= 0 ; + Cd) by letting d »-• — Lob) If K is tie 0-moduie L(A), tAeji L o = hAI - d. D The number c(k) is called the conformal anomaly of the g'-module V, and the number hA is called the vacuum anomaly 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 £ f)* of level k: (12.8.12)

m A = hA -

Another useful property of Sugawara operators is given by the following PROPOSITION 12.8. Let V be a restricted $'-module with a non-degenerate contravariant Hermitian form (see §11.5). Then the operators Tn and T l n are adjoint with respect to this form. o

o

Proof. Let I be the compact form of 9, i.e., the fixed point set of u0. Then o

the bilinear form (.|.) restricted to I is negative definite (see Theorem o

11.7a)). Hence we may choose a basis {VJ} of t such that (v, \VJ) = —6t;-. Let Uj = y/—lvj\ this is an orthonormal basis of § such that WO(UJ) = —Uj, hence w o(tij(n)) = —ti;(—n). The proof of the proposition is now immediate.

• Remark 12.8. Let 0R be a finite-dimensional vector space over R with a positive definite bilinear form ( . | . ) - Let 0 be its complexification and extend (. | . ) to 0 by bilinearity. Viewing 0 as an abelian Lie algebra, we have: § = 5, A = 0, Q = 0, Q v = 0, P = 5*, W = {1}, p = 0. We let also /i v = 0. Then all results of this section (and their proofs) hold in this case as well. In particular, the conformal anomaly c(k) = dim§ is independent of ifc, the vacuum anomaly h\ = ^ J p , where Jfc = level(A) ^ 0, and we let mA = hA -

^c(k).

Ch. 12

Iniegrable Highest-Weight Modules over Affine Algebras

Finally, for the 0-module L(A), A 6 A

= e /v?(e~"*)

dim

*)*, of level k ^

233

0, ch L ( A)

8, where

x A := e" m A ' ch L(A) = e-

(12.8.13)

where t] = e~6/24
(12.9.1)

where 0(O) is the center of 0 and 0(,) with i > 1 are simple. We fix on 0 a non-degenerate invariant bilinear symmetric form (. |.) so that (12.9.1) is an orthogonal decomposition. We shall assume that the restriction of (. |.) to each 0(t) with t > 1 is the normalized invariant form, and that 0(Q) and the form (. |.) restricted to it are as described in Remark 12.8. We shall call such form a normalized invariant form on 0. We let

= 48(0) t>0

We also let (cf. §7.3): A8) = £(8) + Cd9 where d\l(Q(i)) = - * ~ , d{K{) = 0. The Lie algebras C(Q) and C(g) are called affine algebras associated to the reductive Lie algebra g. The subalgebras C(Q^) (resp. C(Q^) 4- Cd) are called components of C(g) (resp. C(g)). Note that c = 0(O) + £ CKi is the center of C(Q) and C(g). As before, i>i

_

we identify 0 with the subalgebra 1 ® 0. Let I) be a Cartan subalgebra of 0 and let 0 = n"_ 0 t) © TT+ be a triangular decomposition of 0. The subalgebra f) = J) + c + Cd is called the Cartan subalgebra of £(0). The triangular decomposition

£(0) = n.eden +

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Integrable Highest-Weight Modules over Affine Algebras

Ch. 12

is defined in the same way as in §7.6. For A € f)*, we denote (as before) its restriction to f) by A. As before, define 6 £ f)* by

Given A G ^*, we denote (as before) by L(A) the irreducible £(&)module which admits a non-zero vector v\ such that n + (v A ) = 0 and h(v\) = (A, h)v\ for h € f). Using uniqueness of L(A), we clearly have: (12.9.2) • >0

where A(t) denote the restriction of A to f)(t) := f) fl £(0(i)) and L(A(t)) is the £(0(,))-module with highest weight A(,-). We let ki, the eigenvalue of K{ on £(A), be the i-
A

A =

EAA(O»

m

*

=

EmA(i)-

and

(12.9.3)

Let V be a restricted £(g)-module such that jfe, acts as Jb,-/ and Jfe, ^ — h* (where hY is the dual Coxeter number of £(0(t-)). Let T£ be the Sugawara operators for £(0(i)), and let (n G Z): (») —

The operators L^ are called the Virasoro operators for the 0-module V. Then letting dn »-* L^, c i-* c(ib) extends V to a module over £(g) + Vir. Note also the following useful formula (cf. (12.8.5)):

where fl,- is the Casimir operator for

Ch. 12

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235

§12.10. In the remainder of this chapter, we let 0 be a reductive finitedimensional Lie algebra with a normalized invariant form ( . | . ) and let 0 be a reductive subalgebra of 0 such that ( - l O l g ls non-degenerate. Let 0 = 0 0/$x and 0 = 0 0( t ) be the decompositions (12.9.1) of 0 and 0. t>0

t>0

Let (. | . ) be a normalized invariant form on 0, which coincides with (. | . ) on 0(o). Due to uniqueness of the invariant bilinear form on a simple Lie algebra we have for x, y 6 0(,), 8 > 1:

where X(r) denotes the projection of x on 0( r ) and j s r is a (positive) number independent of x and y\ we let jor = 1. The numbers j s r ( s , r > 0) are

called Dynkin indices. The inclusion homomorphism i\> : 0 —• 0 induces in an obvious way the inclusion homomorphism £(0) —• £(0). This lifts uniquely to a homomorphism V> : £(0) —* £(0) by letting rj>(Ks) = "52ri$rKr, which extends to a homomorphism rp : £(0) —• £(0) by letting tj>(d) = d. Here and further the overdot refers to an object associated to 0. Let V be a restricted £(0)-module such that K{ acts asfct-J,fct-^ —h?. Via V>, this is a £(0)-module with K{ acting as Jt,-7, where (12.10.1)

it, :

We shall assume that it,- ^ -h?. Let (see (12.9.4)):

PROPOSITION

12.10. a) The operators L*>* commute with C(g).

b) The map dn h-> L%9, c —• c(it) — c(it) defines a representation of Vir on V. Proof, a) is immediate by Lemma 12.8. Furthermore, we have:

[££', L™] = [L™,L*)

(since L* € Ue(C(o)))

- [Li,L»]

(since L-, € Uc(C(g)))D

236

Integrable Highest-Weight Modules over Affine Algebras

Ch. 12

The Vir-module defined by Proposition 12.10 is called the coset Vir~ module. Choose Car tan subalgebras f) and f) of 0 and g such that f) C f). Choose a triangular decomposition 0 = n_ 0 f) © n+; then we have the induced triangular decomposition 0 = n_ © f) © TT+1 where TT-t = IT± fl 0. We have the associated triangular decompositions: C(g) = n_ 4- 1} 4- n+, £(0) — n_ 4-fy4- A+, etc., and we have: 1, and A | j,nfl(0)R i s r e a l and (\,Ko) > 0} be the set of dominant integral weights for C(Q). Let P = {AeP Fix A e P£. Using (12.9.2), Theorem 11.7 and §11.12, we see that the £(0)-module L(A) is unitarizable. It follows, by Proposition 11.8, that viewed as a £(0)-module, the module L(A) decomposes into a direct sum of £(0)-modules L(X) with A G P+. Since the eigenspaces of d on L(A) are finite-dimensional, it follows that the multiplicity of occurrence of L(A) in this decomposition is finite. We denote this multiplicity by multA(A;0). §12.11. The following notion will play an important role in the sequel. The pair M € P$ and /i E P(M)|$ n P$ such that hM = /i^ is called a vacuum pair of level k (cf. Proposition 12.12b), below). Denote by ifo the set of all vacuum pairs of level k. PROPOSITION

(12.11.1)

12.11. Let (M;fi) e Rk- Then mult M (/i;0)=

£

mult L(M )(/i) > 0.

In particular, (M;/i) € #* if and oniy if /ijv/ = h^ and multM(/i; 0) / 0. Pnoo/. Let ft G P(Af) be such that fi\^ = /i. Let v be a non-zero vector from L(M)fi. We have to show that n + (v) = 0. In the contrary case, there exists P e Q+\{0} such that mult M (/i + f3\ 0) > 0. But then

») + /?(<) + P(»)l2 ~ 1^(0 + />(012 + ^(0) + ^(0 + 2ft I Ao)

0

Hence /IM
Ch. 12

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237

Remark 12.11. Let M e P+. Since the module L(M) is completely reducible with respect to £(g), we see from (12.9.5) that LQ' 0 is diagonalizable on L(M). It is clear from the proof of Proposition 12.12b) that its spectre is non-negative. Each of its eigenspaces is £(g)-invariant (by Proposition 12.10a)). Finally, its zero eigenspace (the vacuum space) decomposes into a direct sum of £(g)-modules L(/x) such that (M; /i) is a vacuum pair, with multiplicities given by (12.11.1). §12.12.

For A € 1% and A G 6*, let

multA(A - n«; fie"1'. This series converges absolutely to a holomorphic function on Y, called a branching function. Note that string functions are essentially special cases of branching functions:

<£ = tfti)iT£.

(12.12.1)

This follows from (12.8.13). The branching functions have a simple representation theoretical meaning. To explain this, let t/(A, A) = {v e L(A)\n+(v) = 0 and h(v) = ( A , % for h G t } . Note that, due to Proposition 12.10, the subspace f/(A,A) is a coset Virsubmodule (with c = c(k) - c(k)). Comparing Corollary 12.8b) with (12.8.12), we obtain the following interpretation of branching functions:

(12.12.2)

tf(j)

= t r y ( A A)

^••'-£( e (*)-c(i)) j

where q = e~6. Thus, we have obtained the following decomposition of L(A) with respect to £(g) 0 Vir (direct sum of Lie algebras): (12.12.3)

L(A)= A6PJ mod C6

which immediately implies an equation for normalized characters:

(12.12.4)

XA

=

£

*X(*).

A€P{ mod C*

Now we can prove the following important proposition.

238

Integrable Highest-Weight Modules over Affine Algebras

Ch. 12

12.12. a) The module L(A), A G P+, viewed as a coset Vir-module decomposes into an orthogonal direct sum of unitarizable irreducible highest-weight modules. PROPOSITION

b) If A G P+ and multA(A; g) ^ 0, then

hA>hx.

c) If Jfco > 0 and Jbt- G Z+ for i > 0, then c(k) > c(k). Proof. By Proposition 12.8, the coset Vir-module L(A), A G .P+ is unitary, hence all i/(A, A) are unitary. Also, all eigenspaces of L*'* on £/(A, A) are finite-dimensional and its spectrum is bounded below. Now a) follows from (12.2.3) and Proposition 11.12c). Let v G L(A) be a highest-weight vector of a £(g)-submodule L(X) of L(A). Using Corollary 12.8b) we obtain: £••*(„) = (hA -

hx)v.

b) and c) follow now from Proposition 11.12.

• Now, if 0 is semisimple, then the sum in (12.12.3) is clearly finite. This is, of course, not the case in general, as we have seen on the example 0 = fj. We shall transform this sum to a finite one using the same trick as in §12.7. For this we shall assume that 0(o) C\Q spans 0(O) over C,

(12.12.5)

—v — • where Q C ^ is the coroot lattice of 0. Introduce the lattice Mo = ^~1(0(o) H Q v ), which is a sublattice of the lattice M. We let i)' = 0 ()(,•). Then we have: f) = f)(0) + f)', d(o) H f); = Cd, hence * = i)(0) + d'* and 6?o)n6'* = CS. Given A G f)*, we have a decomposition (12.12.6)

A = A(o) + A^1), where A(o) G ^ 0 ) ,

which is unique up to adding multiples of 6. Due to (12.9.3) and (12.8.3) we have for A G / $ : (12.12.7) Here \

{l)

xA = = flXx

XAO)^^"

1

^'/^

0

)!

w

^ r e / 0 = dimfl (0 ).

is * n e normalized character of the £(0 ; )-module

), where 0 ; is the derived algebra of 0.

Ch. 12

Integrable Highest-Weight Modules over Affine Algebras

239

It is clear that for a G Mo we have: ta(xxm) = XA
(12.12.8)

xA=

J2

*\(i)xxm(9im/iiio),

A6PJ mod (C6+* o Mo)

where @\

=e

2^

o

Y] e*«(A) ^s ^ie theta function associated to the

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. §12.13. We apply here the developed machinery to calculate the functions a^ of level 1, 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)

dim Q(XN) = W(ftv + 1) if X = A, D or E.

Recall also that (12.13.2)

a£ = 1 + axq + ..., where q = e~6.

Consider the coset Vir-submodule U(A, A) of L(A) (for the twisted case use Exercise 12.20). We have the following formula for its conformal central charge: (12.13.3) Hence, by Proposition 11.12b) and c), the coset Vir-module f/(A, A) is trivial. Hence, comparing (12.13.2) with (12.12.2), we obtain

Comparing this with (12.13.2) and (12.12.1) in the non-twisted case (resp.

240

Integrable Highest-Weight Modules over Affine Algebras

Ch. 12

Exercises 12.20 and 12.21 in the twisted case), and using Lemma 12.7, we deduce 12.13. Let A 6 P+. Then for affine algebras of type where X = A,D or E, we have:

PROPOSITION

oo

a£ = J ] ( l - e-nS)~

(12.13.4)

multn

X%\

«,

n=l

mult^A^p^aiAp-IAI 2 )/^,

(12.13.5)

(12.13.6)

e

(

)

£

where

J J

7€Q+A

»>1

• Remark 12.13. If A is of type A^\ D[X\ E[X) or A$>, then ] T p A 0 V = i y>(^)~/, so that pA(j) is the number of partitions of j into positive integral parts of £ different colours; in particular, pAS (j) = p(j) (r = 1 or 2) is the classical partition function. Incidentally, using (12.13.3) and Proposition 11.12b), we obtain h\ = AA, hence the following COROLLARY 12.13. Under the hypothesis of Proposition 12.13 we have:

§12.14.

Exercises.

12.1. Show that setting q = e(-<5), z = e(-a0) in (12.1.4) for Q(A) of type ^ ^ and A2 ,one gets the following classical triple and quintuple product identities (which are alternative forms of identities from Exercise 10.9):

n=l

n=l

m€l

Ch. 12

Integrable Highest-Weight Modules over Affine Algebras

241

12.2, Let 0 be a simple finite-dimensional Lie algebra of rank I, let $(x, y) o ° ° be the Killing form of 0, let f) be a Car tan subalgebra, A the root system, Mp the lattice spanned by {<*/$(<*, a), a G A } , A+ a set of positive roots, "p their half-sum. Deduce from (12.2.3) the following identity:

i#.3. In the notation of Exercise 12.2, let c*i,... ,ar/ be the set of simple roots, h the Coxeter number of 0, and let Mq denote the lattice spanned o

by {ha, a G A } . Deduce the following identity from formula (12.1.7):

I

[Use the formula £ $(«,-,<*,-) = 1=1

i5.^. Show that for automorphisms of sli of type ( s , l ; l ) , where s = 0,1,2,3, the identity (12.3.5) turns, respectively, into the following classical identities + l)?2n3+n 1 n

(Jacobi)

) «n3

(Gauss)

2(-l) n « ( 3 i | a + f l ) / 2

(Euler)

n€l n€l

n€l

i^.5. Let W be the Weyl group of an affine algebra. Show that if W = 0

WW\, where H^i is the set of representatives of minimal length of left cosets of W in Wy then w(p)-pe

P+ for w G W\. Show that W^i = {w G

[Use Lemma 3.11 a).] 12.6. We keep the notation of Exercise 12.2 and identify f) with f)* via the o

Killing form. Let P+ be the set of dominant integral weights and let L(X)

242

Integrable Highest-Weight Modules over Affine Algebras

Ch. 12

denote an irreducible g-module with highest weight A G P+. Prove the following identity:

n>l

= J2 tr° ( \eP+

0

[Consider the decomposition W = WWi from Exercise 12.5. We can write (cf. §12.1)

where
0

of square lengths of a long and a short roots, and let A,, A/ be the sets of all short and long roots. Let p v be the half-sum of positive dual roots and h the Coxeter number of §. Set a = h(h + 1) ]£ $(<*,-,**,•). Prove the 1=1

following identity:

?a/24 nft1- «n)'(i - «*n)'~' n c1 - «c(«)) n (! - «*e(a))) n>l

P

[The proof is similar to that of Exercise 12.6. Use Exercise 10.15 for m = A, and the hint from Exercise 12.3]. 12.8. Show that the identities of Exercises 12.2 and 12.3 can be written as follows: diS

] T tro

Ch. 12

Integrable Highest-Weight Modules over Affine Algebras

243

2 tr° In particular, the case g = sli gives another form of Jacobi's identity:

n=0

12.9. We keep the notation of Exercises 12.2 and 3. Let s = (sOy... ,si) be a set of nonnegative integers, not all of them zero. Set m = SQ + ]T} a t s t , »=i

z where the at- are defined by: £^ a,at- is the highest root. Define hs £ f) t=i

by: (a s -,/i 5 ) = s , / m for i = 1, . . . , £ , and set as = exp27ri/i 5 . Prove the following identity: g d im $/24

J J d e t § (1 _ qn
=

n>l

where X°A(^) = ( £

cfu;)^^^/ X) t(w)ew™)(-2irih.).

[Apply the following specialization to formula (12.1.7): * . ( e ( - a 0 ) ) = q,

«.(e(-a«)) = e 2 "'" /m (* = 1, • • . , / ) ] •

i^.iO. Deduce from Exercise 12.6 another form of the identity of Exercise 12.9:

= Vs

tr 0

(exp47rip)tr0

(

12.11. Let 0 be a simple finite-dimensional Lie algebra, and let &' = (see §7.2). Given an irreducible finite-dimensional 0-module L(X) and a o

non-zero complex number 6, we can give a structure of a ^'-module L(\; b) by letting if »-*- 0, tJ ® x »-• &>£ (x £ g). Show that all irreducible finite0

0

dimensional g'-modules are of the form L(X\;b\)
244

Integrable Highest-Weight Modules over Affine Algebras

Ch. 12

12.12. We keep the notation of Exercise 12.11. Every integrable highestweight g'-module of level it can be constructed as follows. Take a g-module L(X) with (A,0V) < Jb, and consider it as a g'+ := CK + (C[t] ® §)module by letting K *-+ kl, tj § h-» 0 if j > 0. Then the g'-module o

U{ti) ®U(Q' ) L(\) has a unique maximal submodule, this submodule is generated by the element (FQ ))*-(*•* )+* ® vA, and the quotient by this submodule is an integrable g'-module with highest weight ibAo + A. 12.13. Let L(A) be an integrable A^-module of positive level. Show that mu ^L(A)(^ — n ^) > P(n)» where p(n) is the classical partition function. 12.14. Let L(A) be an integrable A^-module; set s = (A, a%)y r = (A, a^). Show that all dominant maximal weights of L(A) are either A — ja0, where 0 < j < [«/2]i o r A - i a i» w h e r e 0 < 5 < t r / 2 ]12.15. Prove that all the weights of the Ap-module L(Ao) are of the form Ao — it2ao ~ (^2 —fc)<*i— s6, where k £ Z,s G Z + . i#.i£. Obtain the following decomposition of the tensor square of the ^ module L(Ao):

L(A0) 0 L(A0) = ^2 a"L(2Ao -

-

nS

)

n>0

n>0

where a n and bn can be determined from the equation

n>0

n>0

n>l

[Use the principal specialization to determine an and 6 n ]. 12.17. Let fl(A) be an affine algebra, let A 6 P* be such that (A,d) = 0 and let L(A) = 0 L(A)j be the gradation of type ( 1 , 0 , . . . ,0) (the basic gradation). Then L(A)j is the eigenspace of d attached to the eigenvalue —j, and hence it is g-invariant. For A G P+ put

Pty

of

U*)

in

n>0

where L(A) denotes the integrable g-module with highest weight A. Show that

Ch. 12

Integrable Highest-Weight Modules over Affine Algebras

Deduce that for A of type X#\

245

where X = A, D or E> one has - ^)mUHi*

if X G M

>

and *A O ,A(?) = 0 if A £ Af. More generally, if k = 1 and A is the only maximal weight in P(A) fl P+, then ) if A G A + M,

and $A,A(g) = 0 otherwise. ii?.i#. Show that the partition function K(a) for A\' is given by the following formula: + l)*o - i*i - \i(J + 1)), o where p^3^(i) is defined by:

[Show that for a = Jfeoao+Jbiai one has: K(a)+K(r1(a+p)-p)

= p (3) (*o)]-

IS, 19. We keep the notation of Exercise 7.21 and Remark 3.8. Let (V, TT) be an integrable highest-weight module of positive level k over an affine algebra &(A), where A is the extended Cartan matrix of 0. Show that the group G* constructed in Remark 3.8 is a central extension
Show that if 5, /? € T* are such that
i£.£0. Let 0 be a simple finite-dimensional Lie algebra of type X^- and let 0 = ©0^(5; r) be its Z/mZ-gradation of type (s; r). Consider the realization of type s of the affine algebra 0 / (A'£ ) ) given by Theorem 8.7:

246

Integrable Highest-Weight Modules over Affine Algebras

Ch. 12

and use notation of this theorem. Choose a basis ti»;_j of 0-j(s; r) and the dual basis tiiy of Qj(s]r), and define the following operator: LS>' =

(ff|F) 2r(Jfc + / , v ) Z ^ p ; o -

' " Z - r "<;-*"

/dimg V 24

/

" ' "

2

v 2h"r) 2A r/ k +

For j G Z, j ? 0, let

Show that [*<">, I f ] = ni( n + m '), and that

12.21. Under the hypothesis of Exercise 12.20, let g be a simple or abelian subalgebra of 9, invariant with respect to the automorphism crs;r, and let a = (jg.y. be the induced automorphism of g, so that r J^ d^i = m (si are not necessarily relatively prime). Let Lj = LQ^S — LQ^S'. Show that the Li commute with Z(g, d, m) and that they satisfy Virasoro commutation relations with central charge c(k) — c(k). 12.22. Applying Exercise 12.21 to g = f), & = 1, derive the following formula:

dim 0(XN) 24

\p\2 = (r

2/»vr

24r

?. Show that _n\mult n<

where ii = I (= |?|2/2Av(fc + 1)) if r = 1 and R = |?|2/2/»v(ftv + 1) if r > 1. /^.^. Show that for the £(j|)-module L(A), A e P J , one has (Jb + dim 0

Ch. 12

Integrable Highest-Weight Modules over Affine Algebras

247

§12.15. Bibliographical notes and comments. 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 (= A r e ), is quite lengthy and does not explain the "mysterious" factors corresponding to imaginary roots. (These factors were explained in Kac [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 A] and Lepowsky [1979]. The study of the series a£ has been started by Feingold-Lepowsky [1978] and Kac [1978 A]. Lemma 12.6 for r = 1 is proved in Kac [1978 A]; its present proof is taken from Frenkel-Kac [1980]. The fact that the string functions c% are modular forms is pointed out in Kac [1980 B]. This observation was inspired by the "Monstrous moonshine" of Conway-Norton [1979]. The exposition of §12.4-12.7 closely follows Kac-Peterson [1984 A]. The construction of operators Tn goes back to Sugawara [1968]. The exposition of §12.8 uses Wakimoto [1986], Kac-Raina [1987] and KacWakimoto [1988A]

Proposition 12.10 is due to Goddard-Kent-Olive [1985]. This coset construction was used in Goddard-Kent-Olive [1986] (and independently by Kac-Wakimoto [1986] and Tsuchiya-Kanie [1986 B]) to prove unitarizability of discrete series representations of the Virasoro algebra. This construction has been playing an important role in recent development of conformal field theory. The exposition of §§12.9-12.12 follows Kac-Wakimoto [1988A], [1989A]. Results of §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 Macdonald identities presented in Exercise 12.8 is due to Kostant [1976] (his proof is more complicated). Exercise 12.9 is due to Macdonald (unpublished). Exercises 12.13 and 12.16 are taken from Kac [1978 A]. Exercise 12.17 is taken from Kac [1980 B] and Kac-Peterson [1984 A]. Exercise 12.18 is taken from Kac-Peterson [1980], [1984 A]. A special case of Exercise 12.19 is treated in Frenkel-Kac [1980]; the general case may be deduced using the formula for the central extension via the tame symbol (described in Garland [1980]). Exercises 12.20-12.24 are taken from Kac-Wakimoto [1988A].

Chapter 13. Affine Algebras, Theta Functions, and Modular Forms

§13.0. We begin this chapter with an exposition of a theory of theta functions. Using the classical transformation properties of theta functions, we show that the linear span of normalized characters of given level is invariant under the action of SL,2(2). 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 for these forms. Furthermore, using the theory of modular forms, we prove the "very strange" formula (12.3.6), which in turn, is used, along with the Sugawara construction, to find an upper bound of orders of poles at all cusps for the string and branching functions. Furthermore, we study the "high-temperature limit" of characters and of string and branching functions. All this is applied to find explicit formulas for the weight multiplicities and branching rules for integrable highest-weight modules. §13.1. We develop a theory of theta functions in the following general framework. (Keeping in mind applications to affine algebras, we use notation which is identical to that used in previous chapters.) Let £ be a positive integer and let f)g be an (£ + 2)-dimensional vector space over R with a nondegenerate symmetric bilinear form (.|.) of index {£ + 1,1). We will identify f)« with f)£ via this form. Fix a Z-lattice M in [)R of rank £1 positive-definite and integral (i.e. (a\b) G Z for all a, 6 G M). Fix a vector 6 G *)R such that (6\6) = 0 and (6\M) = 0. 0

0

0

Put f)R = R ®z M C ta, f) = C ®B *)*, f) = C ®R f)R C f) and extend (.|.) to f) by linearity. For A G f) we denote by A the orthogonal projection 0

of A on f). Let

Y = {ve*)\

Re(6\v) > 0} .

0

For a £ f)g let tQ denote the automorphism of f) defined by (cf. (6.5.2)):

ta(v) = v + (v\S)a - ((t;|a) + $(a\a)(v\6)) 6. Note that the automorphism tQ is characterized by the properties: a) ta(6) = 6] b) tQ(v) = v + (v\6)a mod C6} and c) (.|.) is t^-invariant. It follows that tatp = * o +0. 248

Ch.13

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249

For a G()R let pa(v) = v + 27ria, v G fy. All the transformations p a and 0

tp (<*, /? G fj«) of F) generate a group iV, called the Heisenberg group. More 0

0

explicitly, i V : = f ) R x f ) R X i R with multiplication:

(a, /?,«)(«', ft, «') = (« + a', /? + /?',« + «' + ™(( The action of JV on F) is given by (13.1.1)

(a, 0, ti)(v) = *,(«) + 2wia + (u - 7ri(<*

so that (a,0,0)(t;) = Pa(v), (0,/?,0)(t;) = ^ ( v ) , (O,O,ti)(v) = !; + ««, and Y is iV-invariant. Denote by Nj the subgroup of N generated by all (a, 0,0), (0, /?, 0) with a,/3 eM and (0,0, n) with u G 2WZ. Then Ni = {(a,^,ii) G JV | a,/? G M,ti + *i(a\0) G 27riZ}.

§13.2. Fix a nonnegative integer k. A theta function of degree k is a holomorphic function F on the domain Y such that the following two conditions hold for all t; G V: (Tl)

F(n • v) = F(v) for all n G iVz,

(T2)

F(v + aS) = e* a F(v) for all a G C.

Let TAJ. denote the space (over C) of all theta functions of degree k. Then

*>o is a graded associative C-algebra, called the algebra of theta functions. 0

In order to produce examples of theta functions let M* = {A G f)n | (A|a) G Z for all a G M} be the lattice dual to M\ for a positive integer k put Ft = {A G f) | (A|«) = Jfe and A G M*}. Given A G Pk, we define the classical theta function of degree jfc with characteristic A by the series (13.2.1)

6A = e

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Ch. 13

Equivalently, we take A' = A mod C6 such that (A'|A') = 0, and let 0A =

Note that this function is exactly the one defined by (12.7.2) (which naturally arises in the theory of affine algebras). As in §12.7, we can rewrite ©A in another form: (13.2.2)

0 A = e*Ao

As before, q stands for e~*, and Ao G f)R is the unique isotropic vector such that (Ao|«) = 1 and (A 0 |M) = 0. It is clear that the series (13.2.2) converges absolutely on Y to a holomorphic function; one easily sees from (13.2.1) (or (13.2.2)) that ©A satisfies (Tl) and (T2), and hence is a theta function of degree Jb. Note also that (13.2.3)

ex+ka+a6 = 6 A

for a G M, a G C. 0

Choose an orthonormal basis v\,..., (13.2.4)

v = 27rif ^zsvs

vi of f)g and coordinatize f) by -

/

0

so that we shall often write (r, z,ti), where z = ^ z*v* € fy, r, it G C, in 0

place of v G f). Then Y = {(r,z,u) | 2r G ^; r,ti G C,Imr > 0}, and we can rewrite 0 A in its classical form: (13.2.5)

0 A (r, Z, ti) = e2*iku

(where (T|Z) = $^ T«2«» 7

=

£

C«*r(7|7)+2""*<7W

5Z T»vt)- Note also that in these coordinates

we have: q = e~6 = e 2Ttr . Let W = {r G C | Imr > 0} be the Poincare upper half-plane. Note that a holomorphic function in r G W lies in Tho. Conversely, if F G / then F is independent of u and for each fixed T € 7i, F is periodic in

Ch.13

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251

o

with periods in M + TM. Since f)/(M + TM) is compact, we deduce that F is a function in r. So, we have proved LEMMA 13.2. Tho is the algebra of holomorphic functions in r €W.

• The following multiplication formula is very useful for applications. PROPOSITION 13.2. Let 0 A and 6^ be classical theta functions of degree m and n, respectively. Then

a€Afmod (m+n)M

where

_ -A

V^

=

|nA-m7I-mn(a+7)|a/2mn(m+n)

76(m+n)M

Proof. We may assume that |A|2 = |/i| 2 = 0. Then

e A = j2 e*o(A)> % = E et'(/i)^

and

E a€Afmod (m+n)Af a€Mmod (m+n)M

7€(m+n)M

§13.3. Let D be the Laplace operator associated with the form (.|.). We have in coordinates (13.2.4):

Since D(ex) = (A|A)eA we deduce from (13.2.1) that (13.3.2)

D(QX) = 0.

We put Th0 = C,Thk = {F G fhk \ D(F) = 0} for k > 0, Th = ©

Thk.

Note that the subspace (over C) Th of 77i is not a subring (see Proposition 13.2).

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Affine Algebras, Theta Functions, and Modular Forms

Ch. 13

PROPOSITION^13.3. The s e £ { 0 A I A G P* mod (kM + CS)} is a C-basis

of Thk (resp. Tho-basis ofThk)

ifk>0.

Proof. Let F G TA*. Using F(pQ(v)) = F(v) for a G M and (T2), we can, for a fixed r, decompose F into a Fourier series:

By using (Tl), we obtain that a r (7)e~ Ttfc 7 mod kM. It follows that (13.3.3)

T

(i\i)

depends only on

F= kM+C6

Furthermore, fix a positive real number a; then for a G k~lM* we have aA 0 ) = Since the characters of the group (/c~ 1 M*)/M are linearly independent, we deduce { 0 A ( O , Z , 0 ) I A G Pjb mod kM + C6} is a linearly independent (10.0.4)

set over C, where the Q\ are viewed as functions in z.

This completes the proof of the linear independence of the 0\ over C and over Th0. Finally, if D(F) = 0, then, applying (13.3.2), we deduce from (13.3.3) that *

A

Using (13.3.4), we get dc\/dr = 0, hence the 9 A span Thk over C.

• Example IS.3. Let M = Za be a 1-dimensional lattice with the bilinear form normalized by (<x\a) = 2. Then M* = \M and the following classical theta functions form a basis of Thm: 0 n , m (r,z,ti) := e2wimu

]T

a c*r*m(* ''+**)|

n €

Zmod 2mZ.

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253

§13.4. We recall some elementary facts about the group 5L 2 (R) and its discrete subgroups. The proofs may be found in the book Knopp [1970]. The group SL2(R) operates on Ji by

(

a c

b\ )

ar + b T

For every positive integer n define the principal congruence subgroup *£\ G 5 L 2 ( Z ) | a = d = l m o d n ,

6 = c = 0modn},

and the subgroup

Another important subgroup is r , = {l a c

h

^\ e SL2(l)

| ac and bd are even}.

All these subgroups have finite index in F(l) = SL 2 (Z).

5 and T 2 generate IV,

(13.4.1)

S and T generate T(l);

(13.4.2)

T, CTY and - / generate T0(r) for r = 2 and 3.

Recall that the metaplectic group Mp2(R) is a double cover of () defined as follows: Mp2(R) = {(AJ) \ A = ( ° *) € 5L 2 (R) and j is a holomorphic function in r € W such that j 2 = cr + c/}, with multiplication (A,i)(-Ai, ji) = (A-Ai, j(Ai • r)J!(r)). Mp2(R) acts on W via the natural homomorphism Mp2(R) —• 5L 2 (R). We put: M P2 (Z) (resp. Afrf(Z)) = {(A,j) € Mp2(R) | A € 5I 2 (Z) (resp. € r , ) } . Furthermore, we introduce the following action of Mp2(R) on Y (actually, the 5L2(R)-action): ,

far + b

z

c(z\z)

0

where z 6 ^ , r 6 W , « G C . It is clear that the groups N and 5L 2(R) act faithfully by holomorphic automorphisms of Y. One checks that Mp2(R) normalizes N; namely: (13.4.3) Hence, we have an action of the group G := Mp2(R) K iV on Y.

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Affine Algebras, Theta Functions, and Modular Forms

Ch. 13

One checks directly the following LEMMA

13.4. The normalizer of Nz in the subgroup Mp2(R) of G is if the lattice M is even (i.e., all (y\y) are even for y G M), and is if M is odd (i.e., not even).

•

Finally, we define a (right) action of G on holomorphic functions on Y as follows ((AJ) G Mp2(R),n G N): F | ( A J ) (r, z, u) = j((r)rlF(A

• (r, z% II));

F \n (v) = F(n(v)).

(At this point the use of Mp2(R) instead of SL2(R) is essential to have an action.) Obviously: (13.4.4)

D(F) | B = D(F | n ) for n e AT.

We prove now the following important result: PROPOSITION 13.4. a) Th \^Aj^ = Th if the lattice M is even and A G SL2(2) or if the lattice M is odd and AeT$. b) Th \(Aj) = Th if the lattice M is even and A G SL2(1) or if the lattice M is odd and A G IV

Proof, a) follows immediately from Lemma 13.4 and (13.4.4). Due to Proposition 13.3, in order to deduce b) from a), it suffices to check that = 0. The first equation is clear. The second is immediate from the following simple formula (recall that j 2 = r):

We record also the following two simple transformation properties of classical theta functions of degree Jfe, which follow directly from the definitions: (13.4.5)

©A|(aio,o) = ^ H t ; ) © A

for a Gifc^M

(13.4.6)

©A|(0,a,0) = ©A-*a

foi <* £ k'1 M*.

Ch.13

Affine Algebras, Theta Functions, and Modular Forms

255

We have the following corollary of Proposition 13.3 and formula (13.4.5): COROLLARY 13.4. The function Q\ (defined by (13.2.1)) is characterized among the holomorphic functions on Y by the properties (Tl), (T2), (13.3.2), and (13.4.5).

• §13.5. Denote by n = n(M) the least positive integer such that nM* C M and n(7|7) G 2Z for all y G M*. Now we are in a position to prove the following transformation law which goes back to Jacobi. THEOREM

13.5. Let A G P*. Then

(13.5.1)

©(

f ^ ) ( (JbAf+CJ)

(13.5.2)

Furthermore, if A G F(Jkn) (resp. T(kn) fl F^) when M is even (resp. odd), then (13.5.3) where v(AJ)k)

e eC and \v(AJ;k)\

= 1.

Proof. Using that

for g e C, by (13.4.6) it suffices to prove the theorem for A = 0. Note also that when replacing (.|.) by fc(.|.), the theta function Q\(T,Z,U) of degree k transforms to the theta function G fc -i A (r, z, ku) of degree 1. Hence we may (and shall) assume that A = AoBy Proposition 13.4b), we may write for A = (abd) G F$) if M is even (resp. odd): (13.5.4)

eA.|(AJ)=

X)

/(/*)©A.+*..

G 5/>2(Z) (resp.

where/(//) € C.

256

Affine Algebras, Theta Functions, and Modular Forms

Fix aeM*.

Ch. 13

Since, by (13.4.5), 0Aol(a,o,o) = ©AO> we get by (13.4.3): ,j) \(Aj)-l(at0,0)(Aj)

Hence, applying (A,j)~l(a,O,O)(A,j) thanks to (13.4.5 and 6):

=

®Ao\(Aj) \(dat-ca,O) •

to both sides of (13.5.4), we get,

mod M

If A = 5, comparing this with (13.5.4), we get that /(/i + a) = f(fi) for all a,/i 6 M* and hence:

(13.5.5)

e A o | ( sj) = f(S, j)

£

0

Ao+#.,

where

v(5,i) € C.

/i€Af* mod Af

If A e T(r), we get: /(/i) = /(ji + ca) = f(fi)e2iri<(j4,j;m)| = 1 follows from Corollary 13.5 below. (13.5.2) is obvious. Furthermore, 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) G C. In order to compute this constant, notice that (S,j)s = / and that the rows of the matrix of the transformation (5,.;) in the basis {©A} are pairwise orthogonal. It follows that this matrix is unitary. We deduce immediately that \v(S,j)\ = \M*/kM\'i. Finally, using that 0A(O,I',O) > 0, we deduce from (13.5.5)

thatt;(5,i) = H)^K5,i)|. D Since the matrix of the transformation (S,j) in the basis { 0 A } is unitary, and the matrix of 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 of a transformation from Mp2(Z) (resp. MpKZ)) ifM is even (resp. odd) in the basis {0A}A€PfcmodifcAf+c* is unitary.

Example 13.5. Theorem 13.5 gives the following transformation law for theta functions 0 n > m from Example 13.3:

(13.5.6) = (-ir)*(2m)-* n'glmod 2ml

Ch.13

Affine Algebras, Theta Functions, and Modular Forms

257

§13.6. Here we give a brief account of some facts about modular forms which will be used in the sequel. Fix a subgroup F of finite index in F(l), a function x : F —• C x with |X(J4)| = 1 for A € F, and a real number Jb. Then a function / : 7i —* C is called a modular form of weight k and multiplier system x for F if / is holomorphic on H and / ( f ^ ) = x(A){cr+d)k/(r) for all A = ( ° *) G F and r G W. Let / be such a modular form. Then, since F has a finite index in F(1),T* G F for some positive integer s and hence / ( r + s) = e2xiCf(r) for some C G R. Set F(e 2Tt> /') =

e-2HCr/9f(r).

Then F is a well-defined holomorphic function in z = e2*%T/8 on the punctured disc 0 < |z| < 1. Hence F has a Laurent expansion F(z) = ]T a n z n converging absolutely for 0 < |z| < 1. Therefore, we have the Fourier expansion /(r) = ^ a ^ ' C ^ W ' for r G « . We call / meromorphic at too if an = 0 for n < 0 , holomorphic at too if an 5^ 0 implies n-f C > 0, vanishing at too if a n ^ 0 implies n + C > 0. If/ is holomorphic at ioo, we say that the value of/ at too is a_c (interpreted as 0 if C £ Z). If no = min{n | an ^ 0}, we let r = |(no -f C^/sl and say that it has zero (resp. pole) of order r at too if no + C > 0 (resp. < 0). A cusp of a subgroup F of finite index in SL2 (Z) is an orbit of F in Q U {too}, where a/0 is interpreted as too for a G Q,a / 0. Since F(l) acts transitively on Q U {too}, the set of cusps of F is finite. Sometimes we speak of the cusp a G Q U {too} of F; this means the orbit of a under F. For example, F(l) has one cusp too, F$ has two cusps: too and — 1, and Fo(ife) for prime Jfc has two cusps: too and 0. Let / be as above and consider a cusp a of F. Let B = la k

j GF(1)

be such that B(ioo) = a. Then /o(r) := (CT + d)~ f(Br) is a modular form of weight k and some multiplier system xo for B~XTB. We say / is meromorphic, holomorphic, or has zero or pole of order r at a if /o is meromorphic, holomorphic, or has zero or pole of order r at too. We say that / is R-singular if orders of poles of / at all cusps are < R. A modular form of weight k and multiplier system x for F is called a meromorphic modular form, a holomorphic modular form, or a cusp form if it is meromorphic, holomorphic, or vanishes at all cusps of F, respectively.

258

Affine Algebras, Theta Functions, and Modular Forms

Ch. 13

A holomorphic modular form of weight 0 is a constant. This allows one to identify modular forms. In what follows, q will stand, for e 2 x t r , as usual. Using various specializations of classical theta functions, we can construct modular forms. In fact, given a holomorphic function F on Y and 0

<*,/? £ ta> we define a holomorphic function Fa^(r) (13.6.1)

F°*(T)

on 7i by

:= ( F l ^ . o X r , 0,0) = F(r, - a + rp, - | ( / ? | - a + r/?)).

For example, 0 j ' ' ( r ) = «"* (tt|/J)

(13.6.2)

Furthermore, it is clear by (13.4.3) that

A special case of this is (13.6.4) Now it is easy to prove the following PROPOSITION

13.6. Given positive integers s and m, put

jFm>, = {ea>P(r) | 0 € Thmy sa G M, s/3 e M}. Then every function from Tm^ is a holomorphic modular form of weight if the lattice M is even (resp. \l for r(mn)fir(s) (resp. T(mn)nT$nT(s)) odd). Proof. Let A = f**^ G T(mn) (resp. T(mn) D T$). Then by Theorem 13.5 and (13.6.3) we have

where u(i4) € C, \v(A)\ = 1. On the other hand, by (13.6.2) we have

Ch.13

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259

Thus, every function from Tm%s satisfies the required transformation properties. Furthermore, it is clear from (13.6.2), that TmXT€ x = Tm^ for e = 1 or 2, according to whether M is even or odd. Using this, as well as (13.6.4) and (13.4.1), we conclude that the linear span of Tm,s is invariant under Mp2(Z) if M is even. Since all functions from Tm^ are holomorphic at the cusp foo, we deduce, by Lemma 13.12 in §13.12, that all of 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. D 13.6. Let M be a Z-lattice of rank £ and let (,|.) be a positivedefinite Q-valued bilinear form on M. Let € : M —* C be constant on cosets of some sublattice of finite index and let a G Q <S>i M. Then

COROLLARY

2

f(r) =

is a holomorphic modular form of weight ^£ for some T(N) and some multiplier system. Proof. Replacing M by a sublattice of finite index, we can assume that e(y) = 1. Moreover, since replacing (.|.) by £ ( | ) corresponds to replacing T(N) by T(kN), we can assume that M is an integral lattice. But then a

/W = e°;- .

D The most popular examples of holomorphic modular forms are these:

where ra,n £ ^Z, m > 0. Since fn,m(r) (13.5.6), provided that n e Z:

(13.6.5)

1

= 6 n ? m (r,0,0), we obtain from

(^Y

(In fact (13.5.6) implies (13.6.5) only for integral m\ to get half-integral ra, one needs to take (a\a) = 1 in Example 13.3.) Since <7n?m = /n,4m /(n+2m),4m) w e derive from (13.6.5), provided that ra,n e \+1L\

(13.6.6)

fa,m(-r-W^Y *€I

260

Affine Algebras, Theta Functions, and Modular Forms

Ch. 13

A special case of this is

(13.6.7)

Jyf-r-^H^nfr).

Now we consider some examples of modular forms defined by an infinite product, and among them the Dedekind ^function

itfr) = e* ir/12 J J (1 " r G H. Our starting point is the identity (12.7.4) for the affine algebra A\ \ Using (12.1.5) and the information from Examples 12.1 and 13.3, it can be written as follows (we put a = c*i): (13.6.8)

Note that we have for m, r G Z, rn > 0: 9m/2-rtm/*(T) = (6 1>2 - 0 - 1 , 2 ) ° ' ^ Hence, we obtain from (13.6.8) (recall that (a|a) = 2): (13.6.9) 3

A special case of this for m = 3, r = 1 is Euler's identity: (13.6.10)

^r)

Comparing this with (13.6.7), we obtain the classical functional equation for the 77-function: (13.6.11)

i K - i ) = (-»>)^(r).

Using (13.4.1) we deduce that ?7(r) is a holomorphic cusp-form of weight I for F(l) and a multiplier system Xrj such that x?4 = !• Note that: X,(5) = e - " / 4 , X ,(T) = e"/ 1 2 .

Ch.13

Affine Algebras, Theta Functions, and Modular Forms

261

§13.7. We shall apply the theory of theta functions to affine algebras. Let g(A) be an affine algebra of type Xfr* and let M be the lattice introduced in §6.5. Since, by definition, M = 1(W • 0 V ), and (0 V |0 V ) = 2a^1 (see §6.4), we see that the lattice M is always integral; moreover, it is even if and only if the affine algebra is not of type ASH . Using the information about finite root systems given in §6.7, one easily computes for this lattice the constant n = n(M) introduced in §13.5; it is found in the following table.

e

B ? \ D
n = n{M)

odd

2(^+1)

even

e+i

odd

8

2\\e

4

4\t

2

Cf \ E$\ £>&\

4

E<£\ <$\

3

1

F[ \ A%>,

D&

4

2)

2 1

Fix a positive integer Jfe. Using (6.1.1), we can write (cf. §10.1 and §12.4): P* = {A € ft* | level(A) = Jb, (A,a) € Z for a € Q v } . Recall also the definition from §13.2: Pk = {A G V I level(A) = ib, (A|a) G Z for a G M } . Using (6.5.8) and (6.5.9) we obtain: (13.7.1)

PkDP\

(13.7.2)

P* = P*

if r = 1 or a 0 = 2.

13.7. Let ®\, A G P*, be a classical theta function associated with the lattice M. If A G T(Jbn), then

LEMMA

where v(AJ\k) € C, \v(A,j;k)\ = 1.

262

Affine Algebras, Theta Functions, and Modular Forms

Ch. 13

Proof. Since n = 2 is the only case when M is odd, and since Te C the lemma follows from Theorem 13.5.

• Define the space of anti-invariant classical theta functions of degree k by:

(13.7.3)

Tftj ={Fe

Thk | F(w(v)) = e(w)F(v) for w € W, ve h}.

Using Proposition 6.5 and (6.5.10), we see that the space Th^ consists of holomorphic functions F onY, satisfying the following four properties: ^

w e W;

(T2-)

F(X + 2jria) = F(X) for a e M;

(T3~)

F(X + a6) = ekaF(X) for a 6 C;

(T4~) Given A 6 P*, we introduce the anti-invariant classical theta function of degree k and characteristic A: (13.7.4)

Ax =

By (13.7.3), Proposition 6.5 and (6.5.10), we have: (13.7.5)

Ax = e

Let Pib+ = P+ flP k , Pk++ = P + + fl Pk. PROPOSITION

13.7. a) The set {A\ \ X e -Pt++ mod CS} is a C-basis of

Th~h. b) If X e P++,

then:

AX(T + 1, z, u) = e"ffla'*iMr, *,«)

Ch.13

Affine Algebras, Theta Functions, and Modular Forms

263

Proof. By Proposition 13.3, the set {A\ | A G Pk} spans Th^. Thanks to Lemma 12.4 and Proposition 6.5 we have: (13.7.6) Pm

mod(mM + C 6 ) = | J u^Pt+mod C6)

(disjoint union).

But we clearly have: (13.7.7)

Ap = 0 if ri(ii) = /i for some i.

Hence, by (13.7.6), the set {A\ | A G P*++ mod C6} spans T/ij^". These J4A are linearly independent by Proposition 13.3 and (13.7.6), proving a). By Theorem 13.5,

This together with (13.7.6 and 7) proves b).

§13.8.

We turn now to the proof of the following crucial result:

13.8. Let g(A) be an affine algebra of type X^l) or A$. a) Tft; = 0 if k < hv and Th^ = CAP.

LEMMA

b

Then

) M * J > = (-o* l+|A+l ^.

Proof a) follows from (13.7.2) and Proposition 13.7a). By a) and Proposition 13.7b), we have: Ap\(Sj)

= ( - i ) * ' | M 7 f c v * f | - * C J 4 , where c = ^

e(w)e

By Corollary 10.5, \c\ = |M*/h v Af |i. On the other hand, identity (10.4.4), Q

applied to 0, gives

o

It follows that i'A+'c is a positive real number, completing the proof.

•

264

Affine Algebras, Theia Functions, and Modular Forms

Ch. 13

The proof of Lemma 13.8 gives us the following useful identity

(13.8.1)

n 2 s i n ^ J ^ = IMV

since, (13.8.2)

(p\a) < (p\6) <

Recall the formula for the normalized character (see (12.7.11)) for A £

P*: XA(T, Z, U) := qmA ch L ( A ) =

AA+P/AP.

Note that a special case of (12.2.2) for 0 = 0 is (13.8.3)

XA(T,ziu) = e2*ikutTL(A)e2*izqLUo-&(k)

if g(A) =

Proposition 13.7b) together with (13.7.2) and Lemma 13.8.b) imply immediately the following important transformation law for normalized characters: 13.8. Let &(A) be an affine algebra of type X^ let A G P + . Then THEOREM

a) X A ( - 7 , f , u - ^ )

=

XA(T

X)

SAIA>XA'(T,Z,U),

or A$', and

where

+ 1, z, ti) = e2TimAXA(r, z, ti).

b) The linear span of the {XA}ACP* mod ca JS invariant under the following action of SL2(Z): (13.8.4)

The matrix S

:

= ( ^ )

mod

Ch.13

Affine Algebras, Theta Functions, and Modular Forms

265

has a number of remarkable properties. First, we obviously have «S(fc) = S ( t ) .

(13.8.5)

Furthermore, Corollary 13.5 gives us (13.8.6)

S(jb) is a unitary matrix,

and we have from (13.8.5) and (13.8.6): (13.8.7)

S ^ = 5(t) (complex conjugate).

Given A e P+, we let

where *A is the highest weight of the g-module contragredient to L(A). Then

This is immediate from S 2 • /(r,z,ti) = /(r, - z , t i ) . Finally, letting for

and using the Weyl character formula, we can rewrite the formula for as follows: (13.8.9)

5 A ,A' =

where (13.8.10)

Remark 13.8. Due to (13.8.2), a(A) is a positive number since (A|a) < k if ae

A+.

Incidentally, (13.8.6) gives the following generalization of (13.8.1): (13.8.11)

a(A) 2 = l.

£ mod C6

Example 13.8. If ${A) is of type A{1\ then 5( j k _ i ) A o + i A l ) ( i k _ i /) A o + i / A l = ( — — )

sin

266

Affine Algebras, Theta Functions, and Modular Forms

Ch. 13

§13.9. Unfortunately Pk / Ph in the rest of the cases, i.e., when ao = 1 and r = 2 or 3. As a result, Lemma 13.8 and Theorem 13.8 fail in these cases. We shall indicate now how to handle them. For A of type x £ > = A$_v D$x, E^\ and D^ we let A' be of type X%? = D[% A%lv E$\ and D{?\ respectively. Let Aj. = (resp. = ( a ^ x ^ / a ^ ^ ^ A z + i - ; )

if

(aj/afjKj

A = A^_x or L>g\ (resp. = E^ or

L>i3)). Let A(. = % + a^Ao (• = 1 , . . . , / ) , Ai = A o, />' = E AJ, V = *, »=o /

P; = E

ZA

o

i + C *) Af' = Q v , W7 = W. Then P' is the set of integral

t=0

weights for 0(A ; ), etc., and the analogue of (13.7.2) is (13.9.1)

Pk = P'k

We denote by 0^, A'x, \'\ t n e corresponding functions for &(A'). Then the formula (13.5.1) and Proposition 13.7b) can be rewritten as follows: (13.9.2)

M€P'*mod t M '

(13.9.3)

In the same way as Lemma 13.8b), we deduce

(13.9.4) A , ( - i , i > « - ^ ) = (and the following analogue of (13.8.1):

(13.9.5)

J J 2 s i n ^ | ^ = |M7/i v M / | 1 / 2

if aor = 2 or 3.

From (13.9.3) and (13.9.4) we deduce the following analogue of Theorem 13.8.

Ch.13

Affine Algebras, Theta Functions, and Modular Forms

THEOREM 13.9. Let &(A) be an affine algebra of type X^ a0 = 1, and let A G P+. Then:

a)

267

with r > 1,

£

where

XA(T

+ 1, *, u) = e 2 W m x X A (r, z, ii).

b) The linear span of the {XA}A€P*mod cs & invariant with respect to To(r) under the action (13.8.4). Proof. Since *T = ST^S'1,

using (13.4.2) we see that a) implies b).

• In what follows, in order to state all cases in a uniform fashion, in the cases r = 1 or a0 = 2 we let A' = A, A$- = A{, P = P ; , M = M', p = /?', etc., and let r' = 1. §13.10. Now we can prove a transformation law for the string functions for general affine algebras, and branching functions in the nontwisted case. If 0 is semisimple, we let for

If 9 is a reductive subalgebra of 0 satisfying conditions of Theorem 13.10b) below, we let 5 A ( O ) ,A; O )

= iMoVJfeoMol-'e- 2 " 0)

J>o THEOREM

13.10. a) Let g(A) be an affine algebra of rank £+ 1 and let

AePJ. Then *()

lM

268

Affine Algebras, Theta Functions, and Modular Forms

Ch. 13

b) Let 0 be a semisimple finite-dimensional Lie algebra with a Cartan subalgebra f) and the coroot lattice Q C J), and let $ be a reductive subalgebra of 0 such that its center 0(O) is spanned over C by the lattice Mo := Q V H 0(o). Let A G P+, where k = (Jfei, Jb 2 ,...), h > 0. Then: (13.10.1)

b$(-p

0) =

X)

SA,A<SA,A<«#(r; fl).

€P+ mod mod

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 * modC*>

itfa

where we let XA = XA(O(©A O /tirf0) matrix of branching functions:

z u

> ) = (X\)Xep*

mod

( see (12.12.8)), and consider the

|

mod

Then equation (12.12.8) can be written in a matrix form: (13.10.2)

T(r ) *,ii) = B(r)T(r,z > ti),

z e t

Apply the transformation 5 to both sides of (13.10.2), using Theorem 13.8a) and (13.5.1): S(k)lt(T,z,u)

= B ( - - ) 5 ( i } x*(r,*,ti).

Plugging the expression for "x* given by (13.10.2) in this equation, we get: SWB(T)l?(r,

z, ti) = B(~)S{i)

x*(r, z, u).

Since the function {XAIA^P* mod (C6+koMo) a r e ^ n e a f ly independent (see Propositions 13.7a) and 13.3), we deduce: 5(fc)B(r) = B(-^)S^y which, due to (13.8.5) is equivalent to (13.10.3)

* ( - ! ) = 5 ( f c ) £(r)5 ( i ) ,

proving b). D

Ch.13

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269

Remark IS. 10. Conditions on g D g in Theorem 13.10b), simply mean that g is a Lie algebra of a closed reductive subgroup of the semisimple Lie group corresponding to fl. Since obviously US 10 4) Theorem 13.10 implies 13.10. a) If &(A) is an affine algebra and k G Z+, then the linear span of all the string functions c% with A G P+ is invariant under the action of Fo(r') under the (projective) action COROLLARY

C3.10.5, b) Under the assumption of Theorem 13.10b), given k G Z+, the linear span of all the branching functions 6^(g) with A G P+ mod CS is invariant under the action ,13,0.6) Proof, a) for r' = 1 and b) follow from Theorem 13.10, (13.10.4), and (13.4.1). a) for r' > 1 is proved by the same argument as that used in the proof of Theorem 13.9.

• §13.11. In this section Q(A) is an affine algebra of type X# and rank £+1. We give here the proof of the "very strange" formula. Given a,/? G 0

f)n, we define the associated specialization fa*P of Ap as follows. Put: = {7 € A | ( T | A 0 +/?) = 0 and ( T |a) € Z}. It is easy to see that Aa>^ is a finite reduced root system and that

is a set of positive roots. Recalling the definition (13.6.1), we put

- II (l-e-7 )"1) 0' (r). J

270

Affine Algebras, Theta Functions, and Modular Forms

LEMMA

13.11.1. Let a,j9e ^ Qa,. Then fa^(r)

Ch. 13

is a modular form of

weight \{i + |A+>/?|) for T(n), some n. Proof. This is similar to that of Proposition 13.6, using the formula (13 11 1) (13.11.1) where a', ft' are such that Aa'^' = 0.

• We need one more fact. LEMMA 1 3 . 1 1 . 2 . Let 6 1 , 6 2 , . . . be a periodic

sequence

of integers

with

m

period

m , such

that

bj = 6 m _ j for j = l , . . . , m — 1. S e t 6 = 5Z 6 j . For

c G C put

Then / C ( T ) is a modular form (for F(n), some n) if and only if: (13.11.2)

c =

^ _ _

Proof. Recall that the functions gn,r{r) given by (13.6.9) are modular forms. It is easy to see that if c is given by formula (13.11.2), then / C ( T ) can be represented as a finite product of functions of the form <7m,r(r) (1 < r < m — 1) and a power of r)(mr)] hence, /C(T) is a modular form. Conversely, if /C'(T") is a modular form, then qe~c is also a modular form; it follows that c — c'. U Now we are in a position to prove the "very strange" formula (12.3.7). Proof of (12.3.7). Note that /0>7 * is nothing other than the specialization \[ (1 - e" 7 )" 1 . Hence setting e 2 * ir = ^ / r , we of type s of Ap A

+

(7|A0 +7.)=0

obtain (using (12.3.1)): (13.11.3)

Ch.13

Affine Algebras, Theta Functions, and Modular Forms

271

But /°> 7 ' is a modular form by Lemma 13.11.1. Using (13.11.3), we apply Lemma 13.11.2 to complete the proof.

• We have the following important corollary of the "very strange" formula: COROLLARY

13.11. a) For an arbitrary affine algebra of type Xfr* one

has: /in

11 A\

\P\

(13.H.4)

_

b) Ifr > 1 and _o = 1, then (13.11.5)

_

.

_

Proof. The classical "strange" formula (13.11.4) for r = 1 is a special case of the "very strange" formula for s = 0, since |p| 2 = \p\2. The formula (13.11.5) for |p v | 2 /2A v follows from this since p for A is the same as /?v for % A and /i v for XJy is independent of r. The remaining formulae are checked, case after case, by making use of the "very strange" formula for 5 = 0.

• 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 {0,1,...,/}: J = {j | dj (resp. a j ) = 1 if r = 1 (resp. r > 1)} ( = (AutS(_l)).O). THEOREM

(13.11.6)

13.11. LetAeP$.

Then

2*(A|/>)>/i v (A|A),

and the equality holds if and only if A = kAj mod C6 with j £ J. Proof. Inequality (13.11.6) is equivalent to |___.___|2<|___«2

272

Affine Algebras, Theta Functions, and Modular Forms

Ch. 13

which follows from

(13.11.7)

|A_2p<|Z.|2

xz€Cat.

To prove this inequality note that 0 < (<*\z) < 1

if z G Caf for all a G A + .

Hence, in the case r = l we have, using Corollary 8.7:

0 > £ (H*) 2 - H*)) = h*(z\z) - 2(p\z), which is an equivalent form of (13.11.7). We have an equality if and only o

if (ct\z) = 0 or 1 for all a G A+, which means z = A, with as = 1, proving the theorem in the case r = 1. In the case QQ = 1 and r > 1 we have, using Corollary 8.7 (note that

*(*,*) = 2r £ (a|z)2 + 2 £ (a|*)2): 0>r

^

((a|z)2-(a|

which is (13.11.7). We have an equality if and only if (z,<*v) = 0 or 1 0

for all a G A+, which means z = A, with a)f = 1, proving the theorem in this case as well. In the remaining case, A%t\ one checks directly that (13.11.7) holds with equality only for z = 0 (or uses a similar argument 0

with II replaced by II\{a*}).

• PROPOSITION

(13.11.8)

13.11. Let A G f + , Jb > 0, and let A G P(A). Then

mA,A>-Jj£_i_ with equality if and only if A = JbAj mod C6 and a,- = 1 (resp. a^ = 1) in the case r = 1 (resp. r > 1), and X = u;(A), u; G W. Proo/. By definition (12.7.6) of mA,A, (13.11.8) means

Ch.13

Affine Algebras, Theta Functions, and Modular Forms

273

By Proposition 11.4, it suffices to prove this for A = A, in which case this inequality is equivalent to (13.11.6). The cases of equality follow from Proposition 11.4 and Theorem 13.11. D Note that the inequality (13.11.8) in the nontwisted case is a special case of Proposition 12.12b). §13.12. We shall now find levels and upper bounds of orders of poles at all cusps for the string and branching functions. For this we need the following LEMMA 13.12. a) Let V be a space of modular forms of weight k which is invariant with respect to the (projective) action (13.10.5) with £ = — 2k. Suppose that all functions from V have poles of order < R at too. Then orders of poles of these functions at all cusps are < R (i.e., all functions from V are R-singular). b) Let f(r) be a modular form of integral (resp. half-integral) weight for some principal congruence subgroup and some multiplier system. Suppose that f(r + n) = f(r) for some n € Z, n > 0. Then f(r) is a modular form for T(n) with a trivial multiplier system (resp. with a multiplier system with values in ±1).

Proof, a) is clear since 5L2(Z) acts transitively on the set of cusps. For b) see, e.g., Schoenberg [1973]. THEOREM 13.12. a) Let Q(A) be an affine algebra of type xty £ + 1 and let A G P+, k > 0. Let

and rank

Then all string functions C$(T) are modular forms of weight -\l and a multiplier system with values in (±1)* for the group T(s). Furthermore, all of them are R-singular, where

24a0

* + /»v '

b) Under the assumptions and notation of Theorem 13.10, let s = lcm{*,- + AV,n,-,fc,- + h? (t > 1), ko,ho := n(M0),hi

(» > 1),24}.

274

Affine Algebras, Thtta Functions, and Modular Forms

Ch. 13

Then all branching functions t£(r; g) are modular forms of weight 0 with a trivial multiplier system for the group T(s). Furthermore, all of them are (c(Jfe) - c{k))/24-singular. Proof. To prove a), we use identity (12.7.13). Due to Lemma 13.7, this J^ C*(T)Q\ satisfies identity implies that the function F := \£P*

mod (*Af+CJ)

the transformation property F(A(T, Z, U)) = C(A)F(T, Z, it), where C(A) G C and |C(,4)| = 1, for A £ T(nhw) fl T(n(k + A v )). Since the summation in F is taken over a subset of P* mod (JfcM + C£) (see 13.7.1), applying Proposition 13.3 and again Lemma 13.7. we see that c% are modular forms of weight -\l for the group T(nfev) n T(n(k + hw)) n r(nt), and apply Lemma 13.12a). To complete the proof of a), note that ITIA,A > —-ft, 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 QQ = 2, we derive that all c^ are iJ-singular, due to Corollary 13.10a) and Lemma 13.12b). In the rest of the cases the proof is similar using Theorem 13.10a), Corollary 13.11 b), and the fact that To(2) and Fo(3) have only two cusps, ioo and 0. The same argument as above, using (12.12.8) in place of (12.7.13), proves that &£(r;fl)are modular forms of weight 0 for T(s) with a trivial multiplier system. All of them are (c(k) — c(Jb))/24-singular, since due to (12.8.12), Proposition 12.12b) implies that

rnA-mx>~(c(k)-c(k)).

• Note that Theorem 13.12 tells us that there exists an effective integer 6, depending only on 0, 0, and Jfc, such that the first 6 coefficients of a branching function determine the whole function. §13.13. In this section we study the asymptotic behavior of characters as ft := —IT [ 0, of an affine algebra of type X^ and rank £ + 1. We shall write /(/?) ~ <,(/?) if Hm/(/?)M/?) = 1. First of all, we have from the definitions: (13.13.1)

ex(-(i0)-\z,

-i/3(z\z)/2) ~ 1 or 0

depending on whether A 6 Ao + kM + C6 or not, where k — level(A); (13.13.2)

i ? (-l/i/?)- 1 ~e*' i a '.

Ch.13

Affine Algebras, Theta Functions, and Modular Forms

275

Now, S-l{ifi, -iflz,0) = (-(i/3)-\ z, -i/?(z|z)/2), hence, applying (5, r 1 / 2 ) to both sides of (13.5.1), we obtain: (13.13.3) = \M Qx(i0,-i/3z,O) x

*i€Pfcmod (kM+CS)

Hence, by (13.13.1) we have: e A (i/?, -ifiz,0)

(13.13.4)

\M*/kM\-^2l3-t/2-

-

Similarly, from (13.6.11) and (13.13.2) we deduce:

n(ip)-1 ~

(13.13.5)

f'W.

0

0

Let R+ = A + if r = 1 or a 0 = 2, and iZ+ = A^ otherwise. The asymptotics of the ^ is given by the following PROPOSITION 13.13. Let A G Pjb++. Then AxW,-i0z,O) ~ b(\,z

(13.13.6) where (13.13.7)

b{\,z)=\M'lkM\-x'2b{p,z) (13.13.8) b(p,z)=\M'/M\-1/2

Y[ 2sin7r(z|a).

Proo/. We have from (13.13.3):

/i€Pfcmod ( /i regular o

We may take ^x to be regular (i.e., such that W^ = 1), since nonregular \i give a zero contribution (cf. (13.7.7)). Plugging in the explicit expression (13.2.5) for 0 A , we get: (13.13.9) e w

( ) Ai€Pkm /i regular

276

Affine Algebras, Theta Functions, and Modular Forms

Ch. 13

In order to complete the proof we need the following: 1 3 . 1 3 . Let /i G Mk be regular, and £ = p v otherwise. Then we have: LEMMA

and let£

= pifr=l

or a 0 = 2,

l/*|2 > l£|2 w ^ h equality if and only if /* =
• End of the proof of Proposition 13.13. By Lemma 13.13, it is clear from (13.13.9) that the asymptotic behavior of A\(i/3, -i/3z,0) as /? | 0 is deter_

__

o

mined by the terms with y = cr(^)/Jb, ju = °" ^ W* Thus, we have as

/HO:

Hence, from the Weyl denominator identity and the "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 a 0 = 2. In the remaining cases it is equivalent to the identity (13.9.5).

•

Ch.13

Affine Algebras, Theta Functions, and Modular Forms

277

An immediate corollary of Proposition 13.13 is the asymptotics of characters. For A G P+ one has: (13.13.10)

x

where (cf. (13.8.10)) (13.13.11) a(A) = |Ad (13.13.12)

c(k)0

Remark 13.13. Let z G t)m and consider the "Hamiltonian" H = —d — z. Then (13.13.13)

chL(A)(i/?,-i^,0) = t r L ( A ) e - 2 " H ,

hence (13.13.10) has the following interpretation: (13.13.14)

tr L ( A ) e- 2 ^ H -a(A)e i r c W/ 1 2 r / ?

as /? I 0.

In analogy with statistical mechanics, /? = ^r, where T is the temperature. Thus (13.13.13) is the "partition function" and (13.13.14) is its "high temperature limit." It is remarkable that this limit is independent of z. §13.14. We are now in a position to study the asymptotic behavior of string and branching functions. Consider the set Rk of vacuum pairs of level k (see §12.11). If (A; A) G Rk, then clearly (A + a6; A + a6 4- koj) G Rk for any a G C and j G MoSuch pairs are called equivalent. Denote the set of all equivalence classes by Rk. 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 functions we have: Jbo = Jb and Mo = M. In general, the set Rk is quite complicated, but in the case of string functions the answer is very simple. Namely, recalling the definition of J in §13.11, due to Proposition 13.11 and (12.4.5) we have: (13.14.1) THEOREM

Rk = {(kAj\ kAj) | j G J} in the case of string functions. 13.14. a) Under the assumptions of Theorem 13.10a) one has:

278

Affine Algebras, Thtta Functions, and Modular Forms

Ch. 13

b) Under the assumptions of Theorem 13.10b), let
]£

5

A,A'5 A ) A' multA<(A'; g).

Then one of the two possibilities (i) or (ii) holds: (i) d(A, A) = 0 and lim&£(i/?; gJ/e'M*)-**))/ 1 2 ' = 0, (ii) d(A, A) is a positive real number and

Proo/. We first prove b). By the definition of the branching function in §12.12 and since it is a convergent series for Imr > 0, we have: (13.14.2)

6

A'("^;

8) ~ e-2*<mA'-*A'>//>

mu lt A ,(A';

g)

if multA'(A';0) ^ 0, but multA'(A' -f n6;&) = 0 for n > 0. Replacing r by — £ in (13.10.1), and using Proposition 12.12b), we see that the leading term of &£(r; g) as r --» 0, is given by the summands corresponding to the pairs (A'; A') from Rk. Due to (12.8.12) we thus obtain:

£(ftB)/

=
Since the left-hand side is a positive real number as soon as /? is a positive real number, this completes the proof of b). We could prove a) in a similar way by explicitly evaluating d(A, A). We shall give, however, another, more illuminating proof. Note that if A G -P(A), then, by Proposition 12.5, there exists pGZ+ such that A—pS G [W(\)], where [ ] stands for convex hull. Then A - (n +p)6 G [W(X - nS)] for all n G Z+, and since also A — n6 G [W(A — nS)] for all n G Z+, we obtain by Proposition 12.5: — nS) < multA(A — nS) < multA(A — (n + p)6) for all n G Z + . This implies for /? > 0: (13.14.3)

c£(*7?) - c*(i(3)

if A G P(A).

Considering the theta function identity (12.7.13) asymptotically as /? | 0 we obtain, using (13.14.3), (13.13.4), and (13.13.10): (13.14.4)

a(A)e* c W/ 1 2 ^ - 6A|Af

Ch.13

Affine Algebras, Theta Functions, and Modular Forms

279

where 6A = #{A 6 Pf mod (kM + CS) | A = A mod Q}. The proof of a) now follows from (13.14.4) and the following simple formula:

(13.14.5)

ao\Qv'/kM\/\fr*/Q\.

6A =

It is clear that

d(JfcA0, JfeAo) =

£

a(A ; )a(A ; ) mult A /(A/ ; g) > 0

(A',A')€«*

since (fcAo,fcAo) G Rk- On the other hand no example when L(A) C but d(A, A) = 0, is known. It is natural to conjecture that no such example exists. In fact, I would like to propose an even stronger conjecture: CONJECTURE

13.14. If (A'; A')

e

Rk and mult A(A;g)

/

0, then

S\tA'S\t\' > 0. Let (A; A) be a vacuum pair; then we have by (13.14.2):

We deduce from this and Theorem 13.14 as ft [ 0, the following result. PROPOSITION

13.14. The vector (multA(A; g)) ( A A

~ is an eigenvector

with eigenvalue 1 of the matrix (SA,A'SA,A')(A;A)>(A/;A/)6£fc'

• Note that if Conjecture 13.14 is true, then, by the Frobenius-Perron theory Proposition 13.14 determines uniquely the vector, since We conclude this chapter with a very special case of the developed theory, which is important to the string theory. Let g D Q be as in Theorem 13.10b). One checks directly that c(k) - c(k) is a strictly increasing function of k when k G N. Hence, due to Proposition 12.12c), the equality c(k) — c(k) is possible only if k — 1. In this case $ is called a conformal subalgebra of g. It follows from (13.13.10) that a £(g)-module L(A), A G P£, k > 0, viewed as a £(g)-module, decomposes into a finite sum:

(13.14.6)

L(A)= ff)multA(A;JOi(A)

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Affine Algebras, Theia Functions, and Modular Forms

Ch. 13

if and only if c(k) = c(k) and 0 is semisimple (then necessarily k = 1). Looking asymptotically at characters as /? [ 0 in (13.14.6), we deduce from (13.13.10): (13.14.7)

a(A) = £

d(A)multA (A;g).

Thus, the (generally irrational) number a(A) plays a role of a dimension. It is called the asymptotic dimension of L(A). Finally, notice that, due to Proposition 11.12b), the coset Vir-module is trivial if c(Jfe) = c(k), hence (13.14.8)

multA(A;g) # 0 => (A;A) 6

ft,

if c(Jfc) = c(Jfe),

and we may use Proposition 13.14 to compute these multiplicities. §13.15.

Exercises.

18.1. Show that 0(/)(r) = £

e ™ 3 r (where n 2 = £ > ? ) is a holomorphic

modular form of weight \l for T$ and a multiplier system vi such that Using Gauss identity (Exercise 12.4) show v/(T 2 ) = 1, vt(S) = e-*u/4.

that »(l)(r) = ^ ^ p . 5.^. Let / l ( r )

_ ,-1/48 TT

(1

.

Show that these are modular forms of weight 0 for F(48) with a trivial multiplier system. Check that / i ( - l / r ) = / i ( r ) , / 2 ( - l / r ) = \/2/ 3 (r), and deduce that the linear span o f / i , / 2 , and / 3 is £L2(Z)-invariant under the action (13.10.6). 13.3. Let d i , d 2 , . . . be an m-periodic sequence of integers. Put (13.15.1)

nk :

Ch.13

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281

where fi is the classical Mobius function (/i(l) = 1, /*(n) = (—1)* if n is a product of s distinct primes, fi(n) = 0 otherwise). Suppose that (t,m) = (s,m) implies dt = ds.

(13.15.2)

Show that then n* = 0 unless fc|m. Prove a converse statement. [If there exists a prime p such that Jb = p*ti, p does not divide Jbi, p* does not divide m, then

a|*i

i3.^. Let g be a simple finite-dimensional Lie algebra of rank £y $ the Killing form on 0, f) a Cartan subalgebra, A the root system, A+ a subset of positive roots, and p their half-sum, II = {<*i,... at) the set of simple roots, / ~<*o = ^2 OfQff the highest root, W the Weyl group. Let M be the lattice t=i

spanned by long roots. Let s = («0) « i > . . . , « / ) be a sequence of nonnegative

integers; put m = «o + ^2 aisi-

Define A, E b* by ^(A,,a,) = s t /2m

(i = 1 , . . . ,*). Let A , + = A + n Z{af- | 5t- = 0 (i = 0 , . . . ,£)}, let Ws be the subgroup of W generated by reflections in a G A,+ , and let p8 be the half-sum of the elements from A,+ . Put DS(X) = f ] Wa)/(P*\a) f° r AG f)*. Let ejyfj (j = 1, . . . , £ ) be the Chevalley generators of 0. Define the automorphism >\ (r,(/ ; ) = e-™>ilmfj

(j = 1 , . . . , 0 ,

let 0 = 0 0j be the corresponding Z/mZ-gradation and put dj (s) = dim 0j. i Show that (cf. (12.3.6)): (13.15.3)

gH'-A'l i

and prove the "very strange" formula:

(13.15.4)

|

p

^

Here and further ||A||2 stands for

^

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Affine Algebras, Theta Functions, and Modular Forms

Ch. 13

13.5. In the notation of Exercises 13.3 and 13.4, show that

where d is the determinant of the Cartan matrix of g and /i v = ||ao||~ 2 . 13.6. In the notations of Exercise 13.4, assume that the sequence dj = dj mod m(s) satisfies condition (13.15.2), i.e., that the automorphism
Show that k\m

13.7. Let as be a quasirational automorphism of g and let n* be defined by (13.15.1). Show that

j\m

Deduce from that another type of rj-function identities, using Exercise 12.10 and the "strange" formula. In Exercises 13.8-13.12, g is an affine algebra with a symmetric Cartan matrix, so that g is of type Ai> Di, or Et. 13.8. Let g =

® g ; ( 1 ; 1) be the gradation corresponding to the auto/ morphism of type ( I ; 1), let dj = dim fy( I ; 1) and let n* be defined by (13.15.1). Show that nx = I + 1; nh = - 1 ; nf- = 0 if i does not divide h\ nf- = — rij = ±1 or 0 if ij = h and t / 1. [Use Lemma 14.2d).] 13.9. Show that the application of the specialization $ i in Exercise 12.9 gives the following identity:

769

''"

13.10. Using (14.2.8), deduce from (0.10.1) that ) = dim

Ch.13

Affine Algebras, Theta Functions, and Modular Forms

283

13.11. Apply the principal specialization to the formula given by Lemma 12.7 to deduce that

13.12. Deduce from Exercises 13.8-13.11 the formula:

Using the same method, give an alternative proof of (12.13.4). 13.13. Consider an affine algebra of type X)$\ Let

where a = h if r = 1 and a = /i v if r > 1. Check the following table:

G(r) Xt

or A2l

13.14- Let A G -P+ be such that (A, d) = 0. Then the eigenspace decomposition of L(A) with respect to a^xd is of the form

L(A)= 0L(A)- i f

(13.15.5)

where L(A)_j is the eigenspace with the eigenvalue —j. Show that dimL(A)_;- < oo and that (13.15.5) is the basic gradation of £(A), i.e., the gradation of type ( 1 , 0 , . . . , 0). Provided that A is a symmetric affine matrix, show that

0

1

In particular, in the case 0 = Eg, the right-hand side is (qj(r)) 3, where j(r) is the celebrated modular invariant, the generator of the field of modular forms of weight 0 for

284

Affine Algebras, Theta Functions, and Modular Forms

Ch. 13

13.15. Generalize Exercise 13.14 to the case of twisted affine algebras. 13.16. Show that for B^l\ all A G P+ mod C6 of level 1 are A o , Ai, and A/. Show that, up to M^-equivalence, all the string functions of level 1 are:

Show that (cf. Exercise 13.2): (13.15.6)

c £ = f|(r)-<- l i|(2r) = i|(r

(13.15.7) eft - eft = iKr)-'- 1 ^ Jr) = (13.15.8) eft + eft = [A(T) := ^(T) / + 1 i;(2r)~ 1 c^ is holomorphic at cusps ioo and 0 of Fo(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 — ~ and using Theorem 13.10a). Formula (13.15.8) follows from (13.15.7) by replacing r by r + 1.] 13.17. Show that for the hyperbolic Kac-Moody algebra 0(A), where A is from Exercise 3.8, one has: mult(Jfc(*i + k<*2 + 03) = p(k). [Use Exercise 11.7 and Exercise 13.12 for g of type A[1\] 13.18. Let A G / + + . Using Proposition 13.2, show that

where

i5.iP. Deduce from Exercise 13.18 that for the affine algebra of type X = A, D, or E, one has:

Ch.13

Affine Algebras, Theta Functions, and Modular Forms

285

where G(r) is defined in Exercise 13.13, so that we have the following formula for the branching functions for the tensor product decomposition of the 0(*&))-module L(A) L(A0):

13.20. Let 0 be a simplefinite-dimensionalLie algebra and let 0 C 0 '•— 0 0 0 be the diagonal imbedding. Consider the £(0)-module L(A')
where ftv is the dual Coxeter number of £ ( 0 ) , and that 1 AA'|A' + 2p) (A w |A w + 2p) 2 \ t' + ftv jk" + ftv

r,,j 0

0 \ )b/+jfe"+ftvy1

where Q is the Casimir operator for C(Q). 13.21. Deduce the following decomposition as fl' (^ ( i X) ) 8 Vir-modules (j, kel+,

j


0<s
where ,(*) _ 1 _ 6 " (* + 2)(t + 3)'

_ ((^ + 3)r-(fc + 2 ) S ) 2 - l ''~ 4(Jk + 2)(ib + 3)

h(k) r

and L(c(k\hrJ) is a positive energy Vir-module such that c = c^k\ the lowest eigenvalue of Lo is equal to h)-J and has multiplicity 1, and t*HeMtk<8)*do~*k}/2* = ^ ' '

where

^rf* = - ( 13.22. Deduce that the Vir-module L(c^,h^)) r < k + 1 are unitarizable, and that, letting

with ib G Z+ and 1 < s <

286

Affine Algebras, Thtta Functions, and Modular Forms

Ch. 13

one has for these modules the following coefficient-wise inequality of ^-expansions: (13.15.9)

x(*>

< „<*>.

13.23. Show that (see Exercise 13.2):

13.24- Let 6 = 0 or 1/2 and let Cl$ denote the associative algebra on generators V>j, j £ Z + 5, and the following defining relations:

Let Vs denote the irreducible Cl^-module which admits a nonzero vector |0) such that ^•|0) = 0 f o r j > 0 . Let 1 n~~2

^ Z-r WjVri-J je6+z

0I U z

" r

,

Show that [ipm,Ln] = (ra + n/2)tpm+n and deduce that the map dn »—• Ln defines a representation of Vir in Vs with c = \. 13.25. Show that the elements ipjs - - - i>j2i>ji |0)

with 0 > j \ > J2 > - - - > j ,

form a basis of Vs. Define an Hermitian form on Vs by declaring these elements to be orthogonal and to have square length 1 (resp. 2), if ji < 0 (resp. j \ = 0). Show that V>j and V*-; are adjoint operators and deduce that Vs is a unitarizable Vir-module. Show that

13.26. Let V* (resp. Vf) denote the linear span of elements of even (resp. odd) degree. Taking for granted that for h ^ 0, | , or ^r, the Vir-module L(~9h) is not unitarizable (see Kac-Raina [1987] for a proof), show that V+ ~ L($,0), V£ - L ( i , I ) , V+ - Ko- - L ( J . i ) (cf. Exercise 13.22). Deduce that Xr]l =
Ch.13

Affine Algebras, Thtta Functions, and Modular Forms

287

13.27. Deduce from Theorem 13.10b) the following transformation formula (1 < 6 < a < k + 1):

x<

/

13.##. Deduce the following decomposition of the 0(2£g ^-module (viewed as a o(E^l))® Vir-module): L(A0) 9 L(Ao) = L(2A0) 9 L(\, 0) + L(A7) ® L ( | , 5) + L(AX i5.^P. Show that a((7 • A) = a(A) for any (r G Aut5(i4). Deduce that for A of type XJ^ with X = A,D,OT E one has (see §13.14): a(A)=|J|-*

if A GPJ.

13.30. Let 0 be a simple simply-laced finite-dimensional Lie algebra of rank £ and let 0 be its subalgebra generated by elements £",-, F,, i G { 0 , 1 , . . . ,£}\{j} (see §8.2). Show that a level 1 £(g)-module L(A) viewed as a £(g)-module decomposes into a direct sum of a ; irreducible summands as follows:

L(A) =

0

L(X).

AeP|nP(A) 13.31. Recall that we have a surjective map i/> : M* - • Aut 5 ( ^ X ) ) . Show that

Deduce that the matrix 5(i) in the simply-laced case is the character table of the group Af/Af, divided by \J\l>2. 13.32. Show that for the diagonal embedding g C 0 := 0 0 0 one has &,,»„ = {i'A i t t"A i ; (*' + t w )A i I j G J } , and therefore Conjecture 13.14 holds in this case. Deduce, using Exercise 13.31, that

288

Affine Algebras, Theta Functions, and Modular Forms

18.88. Show that for 0 < z < 1 one has: TT (1 _

Ch. 13

3

Deduce that for z £ Int CW and r = 1 one has:

^(i/?,-* l o)~r' /2 Deduce that for A,/i € P+ one has:

13.3^. Let / be a finite set with a distinguished element 0 and let V be a finite-dimensional vector space with a basis {XA}> indexed by A £ /. Let 5 be an invertible operator on V such that all the entries of the 0-th row of its matrix (S\tfi) are nonzero. Show that there exists unique operators 'x and $(, indexed by A £ / , on V satisfying the following three properties: (i) <j/x are diagonal in the basis {XA}>

Show that (both for ' and "): \? = £ „ N^u (the numbers N^ are called fusion coefficients), where

Equivalently: i/e

i3.35. Let 0 be a simple finite-dimensional Lie algebra, let f) be its Cartan subalgebra, let A C ^ be the root system, A+ a subset of positive roots, p their half-sum, 9 the highest root, Av the dual Coxeter number, let Q be the coroot lattice, M = i/(Q ) for the normalized invariant form (.|.), let P + c F be_the set of dominant weights, let P^. = {A £ P + | (A|^v) < Jkj. Given A £ P+, either ra(X + p) = A + ~p mod (Jb + AV)M for some a £ A and we let e(A) = 0, or else there exists a unique w £ W and unique A £ P + such that tu(A+p) = A+p mod (Jfc+Av)Af and we let e(A) = e(ti;). Now let in Exercise 13.34, / := P + and S\n := S\+kAo,n+kAo- Show that the fusion coefficients, as defined by (13.15.10), are given by the following formula:

where multA®^ v stands for the multiplicity of L(y) in L(A) ® L(/i). Show that Nx^y := 7VA^ is symmetric in all three indices.

Ch.13

Affine Algebras, Theta Functions, and Modular Forms

289

13.36. Show that for 0 = s£2 one has (t,j,s G Z+ = P+): ? fJ

_ f 1 if |t - i | < « < i + i, • + i + « € 2Z and < 2ib, I 0 otherwise.

13.37. (Open problem). Show that for the Kac-Moody algebra from Exercise 3.8 one has multa
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)). §13.16. Bibliographical notes and comments. The theory of theta functions is an extensive subject which has its origin in the works of Jacobi and Riemann. The treatment of the part of the theory presented in §§13.1-13.5 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 §§13.7-13.14 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 thetarfunction proof of the Macdonald identities. An independent theta-function proof of the Macdonald 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(k) (computed by Kac-Peterson [1984 A]), have been playing a fundamental role in recent developments of conformal field theory, see Cardy [1986], Gepner-Witten [1986], Capelli-Itzykson-Zuber [1987], Verlinde [1988], and many others. A special case of the "very strange" formula, reproduced in Exercise 13.6, is proved in Kac [1978 A] by the same

290

Affinc Algebras, Theta Functions, and Modular Forms

Ch. 13

method as (12.3.6). Important cases of the "very strange" formula were found earlier by Macdonald [1972]. Exercises 13.8-13.12 are due to Kac [1978 A]. For A^ Exercise 13.12 was previously obtained by FeingoldLepowsky [1978]. Exercise 13.16 is taken from Kac-Peterson [1984 A]. 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 corresponding generating function is qj(r). In Kac [1980 E] such a module was constructed for the centralizer of an involution of the Monster, by analogy with the Frenkel-Kac [1980] construction. Frenkel-Lepowsky-Meurman [1984] and [1989] (see also Borcherds [1986]) managed to extend a modification of this construction to the whole Monster, recovering the construction of Griess [1982] of the action of the Monster on a 196883-dimensional commutative nonassociative algebra. It would be fair 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 simplest affine Lie algebra, of type A[l\ due to Kac-Peterson [1980], [1984 A]. After a lengthy calculation, they turn out to be certain peculiar "indefinite" modular forms discovered by Hecke around 1925. Another expression for these string functions was found by Distler-Qiu [1989] and Nemeschansky [1989 A] by using the free field construction described by Exercise 9.18. The open problem posed in Exercise 13.19 of previous editions has been solved 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 GoddardKent-Olive [1986] and Kac-Wakimoto [1986]. The inequality (13.15.9) is actually an equality, which follows from Feigin-Fuchs [1984 A, B]. 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 Virmodules L(c,h) with c > 1 are those with h > 0 (Kac [1982 B]). Exercises 13.24-13.26 are well-known. Exercise 13.27 is taken from CappelliItzykson-Zuber [1987 A], and Exercise 13.28 from Kac-Wakimoto [1986]. Exercises 13.29-13.32 are taken from Kac-Wakimoto [1988 A] and Exercise 13.33 from Kac-Wakimoto [1989 A]. Exercise 13.34 is a formalization of the work of Verlinde [1988]. As shown by Kac-Wakimoto [1988 B], [1989] the class of modules L(\) with modular invariance properties is much larger than the class of

Ch.13

Affine Algebras, Theta Functions, and Modular Forms

291

integrable modules. They probably correspond to some non-unitary CFT, and are related to the quantum gravity theory of Knizhnik-PolyakovZamolodchikov [1988]. Highest weights A of all these modules lie in the "positive" half-space {A € J)* | Re(A, K) > -hv} (see the list in Kac-Wakimoto [1989 A]), and on the critical hyperplane {A e J)* | (A, if) = - / i v } . Several authors studied the L(X) corresponding to A on the critical hyperplane: Wakimoto [1986], Hayashi [1988], Ku [1988 B], Malikov [1989], Feigin-Prenkel [1989 A] (these modules are modular invariant if and only if (A 4- p, a v ) ^ N for all a E Aljf. Finally, at least a conjectural connection between the modules from the "negative" half-space of an affine Kac-Moody algebra and the modules over the corresponding quantized finite 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(A<j) over an affine algebra &(A) is called the basic representation of #(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 algebras of infinite rank. In the case of A = A^ this construction is essentially the bosonfermion 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. §14.1. Let g(A) be an affine algebra of type X# and rank £+ 1 (from Table Aff r). Recall that g(A) = &'(A) + Cd, where Q'(A), the derived subalgebra, is generated by the Chevalley generators e,-, /,- (i = 0 , . . . ,^), and d is the scaling element. Recall that the center of 0'(A) coincides with that of &(A) and is spanned by the canonical central element K (see Chapter 6). Let 5(^4) = $'(A)/CKi so that we have the exact sequence (14.1.1)

0 -> CK — g!{A) A %{A) — 0.

Recall that relations deg t{ = — deg /,- = 1 (i = 0 , . . . , £) define the principal gradation $!{A) = © 0 j ( l ) ; it induces the principal gradation Ji(A) = 0

0 ^ I ) so that dim0^ 1 ) = dim0 ; ( I ) for ; / 0.

;€i

The element t=0

is called the cyclic element of ~$(A). Let I = {x G 0 | [ar,e] = 0} be the centralizer of e in ^(A). It is clear that 6 is graded with respect to the 292

Ch. 14

Vertex Operator Constructions and Soliton Equations

293

principal gradation oC#(A):

The subalgebra a = TT" 1 ^) is called the principal subalgebra of #'(A) (or 0(A)). It is graded with respect to the principal gradation of

Note that (14.1.2)

dims,- = dims,- for j ^ 0, «0 =

(14.1.3)

The last relation is clear by Proposition 1.6. The nonzero integers of the set which contains j with multiplicity dim Sj are called exponents of the affine algebra &(A). We will compute the exponents below. §14.2. We study the principal subalgebra a by making use of an explicit construction of 5(yl), discussed in Chapter 8. Let 0 be a simple finite-dimensional Lie algebra of type Xjq and let /z be a diagram automorphism of g of order r (= 1, 2, or 3). Let Eiy Fiy Hi (i = 0,...,^) be the elements of 0 introduced in §8.2. Using the results of §6.2, we see that

(14.2.1) i=0

Here af are the labels of the diagram of the transpose of the affine matrix of type X$ in Tables Aff. Recall that the elements JE7t (i = 0 , . . . ,£) generate the Lie algebra g. Let c*j be the labels of the diagram of the affine matrix X#\ The integer ft(r) = rJ2di

is called the r-th Coxeter number of g. By Theorem 8.6 a),

t=0

the relations i = l,

degi/f=0

(1 = 0 , . . . , / ) ,

294

Vertex Operator Constructions and Soliton Equations

Ch. 14

define a Z/A^Z-gradation 8

(14-2.2)

called the r-principal gradation of 0. The element •=o

is called the r-cyclic element of 0. Denote by S( r ) the centralizer of E in 0. It is graded with respect to the r-principal gradation:

PROPOSITION

14.2. a) dim0;(l;r) = ^ + dimSJr) (j G

b) 5^r^ is a Cartan subalgebra of0. c) The subspaces Sf- and Sj are orthogonal (resp. nondegenerately paired) with respect to a nondegenerate invariant bilinear form on 0 if i + j £ 0 mod h^ (resp. i + j = 0 mod A ^ ) . Proo/. Using automorphisms of g of the form Ei H-> A^^, F^ I-» A^"1^, the r-cyclic element E is conjugate to a multiple of an arbitrary element i

of the form Yl ct^i> where all cf- ^ 0. Therefore, it is sufficient to prove t=0

the lemma for one of the elements of this form, say E'. We take E1 = Yl y/^i^i' »=o

Put F' = — u>o(E')> where uo is the compact involution of 0;

then F' = £ \/°?Fii •=o

and we have

«=o

by (14.2.1). Hence ad£" commutes with its adjoint operator adF', with respect to the (positive-definite) Hermitian form (.|.)o (defined in §2.7). Therefore, E is a semisimple element, the orthogonal complement B = Bj to S^ in 0 is ad /^-invariant, and the restriction of ad E to B is invertible. Since [E,Bj] C # ; + i , we conclude that d i m £ ; = £ (= dim5 0 ) for all j e Z/Mr)Z, proving a).

Ch. 14

Vertex Operator Constructions and Soliton Equations

295

From a) we deduce dim g = dim 5 ( r ) + h^L

(14.2.3)

But one easily verifies by inspection that d i m g - / i < r ^ = JV.

(14.2.4)

Comparing (14.2.3 and 4) gives dimS ( r ) = N = rank 0.

(14.2.5)

Since S^ is the centralizer of the semisimple element £", (14.2.5) proves b). The first part of c) follows from Lemma 8.1; the second part now follows from the fact that the restriction of a nondegenerate invariant bilinear form to any Cartan subalgebra is nondegenerate.

• The nondecreasing sequence of integers m[ r) < m^ < • • • from the interval [1, h^ — 1], in which j appears with multiplicity dimSJ , is called the set of r-exponents of g. They have the following properties. LEMMA

14.2. a) The number of r-exponents is equal to N = rankg, and

we have: 1 = m < r ) < m ( 2 r ) < . . . < m £ > = ft - 1. i

c) Ifi and j have the same greatest common divisor with h^r\ then i and j have the same multiplicity among the exponents. d) dim Qj( I ; r) = 1+ (multiplicity of j among r-exponents). e) Let r = 1, so that N = £. Denote by Cj the number of roots of 0 of height j . Then (14.2.6) (14.2.7) (14.2.8)

d i m g ^ 1 ; 1) = es + c M l ) _ ; . for l < j < ej + c M i , + w =eforl<j<

h™ - 1; h^;

dim 0 , ( 2 , 1 , . . . , 1; 1) = t for all j .

f) Let 0 be of type A2i. Then (14.2.9)

dim 0 , ( 2 , 1 , . . . , 1;2) = I for 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

Vertex Operator Constructions and Soliton Equations

Ch. 14

SUBLEMMA 14.2. Automorphisms cr oftype ( 1 , 1 , . . . , l;r) or ( 2 , 1 , . . . , 1; 1) are rational, i.e., a8 is conjugate to a if the order of a and s are relatively prime. In particular, they are quasirational (see Exercise 8.12 for the definition). Proof. Let a be of type ( 1 , . . . , 1; r). Then the order of cr is h^ and the dimension of its fixed point set is I. The element a8, where s and ft(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 cr of type ( 2 , 1 , . . . , 1; 1) is similar, using (12.4.3). D The statement c) of Lemma 14.2 follows from d) and the sublemma. In order to prove e) consider the automorphism cr^i) = a^i (resp. (JhW+i — <J (2 l i i ) ) of 9 of order h^ (resp. = ft^ + l). We have for m = h^ (resp. &" 1): (14.2.10) This implies (14.2.6) and (14.2.11)

dim 0 , ( 2 , 1 , . . . , 1; 1) = cj + ch(i)+1_j

for 1 < j <

If h,(l) + 1 is a prime number, then, by the sublemma, all the eigenspaces of 0fc(i).j_i have the same dimension t\ this together with (14.2.11) proves (14.2.7 and 8) for an exceptional Lie algebra 0, since for 0 of type G2, F4, E6, E7i E8i respectively, ft*1) + 1 = 7,13,13,19,31 is a prime number. If Q is of classical type At, Bi, Ct} or D/, one checks (14.2.7) directly; using (14.2.11) this gives (14.2.8) as well, proving e). Finally, f) is checked directly.

• Using Lemma 14.2 one easily computes the r-exponents. For instance, if j and Mr) are relatively prime, then .;' is a r-exponent of multiplicity 1 by Lemma 14.2 a) and c) (this takes care of all exceptional Lie algebras except £*6, r = 1, in which case one should check that 2 is not an exponent). A complete list of r-exponents is given in Table E$. Remark H.2. The numbers mj. (j = 1,.. .£) are the ordinary exponents, and ft*1) is the ordinary Coxeter number of a simple Lie algebra 0 of type Xi. Indeed, comparing Lemma 14.2 d) with (14.2.6) gives Cj + C/KI)_J- = 1+ (multiplicity of j among exponents),

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297

which is one of the definitions of the exponents and the Coxeter number of g. Note also that all r-exponents have simple multiplicity with one exception: g = £>/, where I is even, and r = 1; in such case £ — I has multiplicity 2. TABLE Eo

• At

Bt Ci

Dt E6 E7 E8 F4

G2

r

1 1 1 1 1 1 1 1 1 2

A*2,t l 2 Dt+i 2

E6

3 2

4r)....,m«

n

€+1 21

21 21-2 12 18 30 12 6 4^ + 2 At-2 21 + 2 12 18

1,2,3, 1,3,5, ...,2^-1 1,3,5, ...,2t-l 1,3,5, ...,2£-3,e-l 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, ...,2t-l,2t + Z,...,4t+l 1,3,5, ..., 4£ - 3 1,3,5, ..., 21 + 1 1,5,7, 11 1,5,7, 11,13,17

§14.3. Now we return to the affine algebra Qf(A) of type X^ . By Theorem 8.7, we have (14.3.1)

S(A) ~ 0 ( * i 0

9j

m o d fc(r,(

1; r)),

where the isomorphism is defined by (14.3.2)

Tr(fi) ^ t~l ®Fi

x(c,-) ^ t ® Ei,

(i = 0 , . . . ,

Moreover, (14.3.1) is the principal gradation of 5(A). Hence (14.3.3)

e = t ® E is the cyclic element of

(14.3.4)

*i=«'®S

Finally, notice that h^ = rA, where ft is the Coxeter number of tf{A). The following proposition now follows from Proposition 14.2.

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PROPOSITION 14.3. a) dimfy(I) = / + dimflJ- forj e Z\{0}. b) s is a commutative subalgebra ofg(A). c) The set of exponents oftf(A) is {m< r >+jA< r >, where 5 = 1 , . . . , AT; j G Z}.

From Proposition 14.3 a) and Proposition 10.8 we deduce: COROLLARY 14.3. Dual affine algebras have the same exponents.

Using Table £"0 one immediately gets the list of exponents of all afiine algebras as the set of all integers satisfying the conditions given by the following table: TABLE E

i=l i=l S=

i= t= *= i= i= i= t= i=

mod 2 mod 2 1 mod 2, t = / - l mod ( 2 / - 2) ± l , ± 4 , ± 5 mod 12 ± l , ± 5 , ± 7 , 9 mod 18 ± 1 , ± 7 , ± 1 1 , ± 1 3 mod 30 ±l mod 6 ±1 mod 6 1 mod 2 such that i £ 0 mod l mod 2 1 mod 2

i = ±1 mod 6 * = ±1 mod 6 Note that the multiplicities of exponents are 1, except for the case A = D[ \ I even, when the multiplicity of (£ - l)(2s + 1)(« € Z) is 2. l

Finally, recall that by Theorem 8.7, &'(A) can be constructed as the central extension of"§(A) by CK: (14.3.5)

&'(A)

= 0 ( « * ® 0>

m o d M ,,(

1 ; r)) © CA",

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299

with the bracket (14.3.6)

where (.|.) is the normalized invariant bilinear form on 0 and h is the Coxeter number of #(A). §14.4. Recalling the definition of the infinite-dimensional Heisenberg algebra (§9.13), note the following simple lemma: 14.4. The principal subalgebra infinite Heisenberg algebra. LEMMA

B

of &'(A) is isomorphic to the

Proof Proposition 14.3 b) implies [*s,*j] C CK for all i, j E Z. By Proposition 14.2 c) and formula (14.3.6), we have [*s,£j] = 0 for i ^ — j and [a, B-i] = CK for every a € « t , <* / 0, proving the proposition.

• Now we are in a position to make the first step in the "principal construction" of the basic representation. 14.4. Let &'(A) be an affine algebraf where A is of type Xff, with X = A, D, or Ef and let 8 be the principal subalgebra of%'(A). Then the basic #'(A)-module L(Ao), considered as an 8-module, remains irreducible. PROPOSITION

Proof Let L(Ao) = ® L(Ao)j be the principal gradation of L(Ao) and let dirrig L(A 0) = ]C dimL(Ao)jtf; be the ^-dimension of L(Ao) (see §10.10). Then by Proposition 10.10 we have (14.4.1)

dimg L(A 0) =

But the algebras 0(*^4) 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* ty(2,1,..., 1) = / for all j £ 0.

Furthermore, by Proposition 14.2 and formula (14.3.4), we have (14.4.3)

dim ty( 11) = I + dim BJ .

300

Vertex Operator Constructions and Soliton Equations

Ch. 14

Substituting (14.4.2 and 3) into (14.4.1), we obtain dim,L(A 0 ) =

(14.4.4)

On the other hand, the Z-gradation a = ® $; induces a Z-gradation U(*) = 0 U(s)j\ set Uj = £/(*)_, (vo), where i>o G L(AO)A 0 is a nonzero i€i

vector. Then (14.4.5)

lOCL(A0)i.

But by Lemma 9.13 a), the s-module (7 = ® (7j is isomorphic to the canonical commutation relations representation; hence we have (14.4.6) Comparing (14.4.4) with (14.4.6) gives equality in (14.4.5) for all j . Hence the s-module L(Ao) = U is irreducible. D

§14.5. Proposition 14.4 allows us to identify the restriction of the basic representation to the principal subalgebra with the canonical commutation relations representation. In order to extend this to the whole Lie algebra &(A) we need a lemma about differential operators. Let R = C[«i,X2,...] be the algebra of polynomials in infinitely many indeterminates X{ and let R = C[[xi, x2i... ]] be the algebra of formal power series (i.e., algebra of all linear combinations of (finite) monomials in the Xi or, in other words, the formal completion of R (see §1.5)). A differential operator on R is a (generally infinite) sum of the form

A differential 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 P G R. Another example is the operator T\ defined by (Txf)(xux7,...) = / ( * ! + Ai,* 2 + A 2 , . . . ) . Note that (14.5.1) by Taylor's formula.

TA=exp2^A<^-

Ch. 14

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301

LEMMA 14.5. Let D : R -* R be a linear map.

*)If[xiiD]

= \iD, i = l , 2 , . . . , then D = D ( l ) e x p ( - £ A , ^ ) . t

b)Jf [ 5 |- ) £>] = /i,D, » = 1 , 2 , . . . , then D ( l ) = cexp £ > * . - , c € C. t

c) If [xiyD] = A,D and cexpf ]T)/i,z,J

ex

[^-,D]

=

ttD,

• = 1,2,..., tie* £> =

p ( - Z ) * t d | - J for some ceC.

Proof, a) We replace D by DTA; then our statement is equivalent to the following: if [*,-, D] = 0, i = 1,2,..., then £> = £>(1). But D(f) = for all / G iZ by an easy induction on the total degree of / . b) We replace D by exp(— ]C/ii£,)D; then our statement is equivalent to the following: if [ a f r ^ ] = 0, t = 1 , 2 , . . . , then D(l) = const. This is obvious: [af-,^] = 0 implies £-(D(l)) D(l) = const. Part c) follows from a) and b).

= 0, i = 1 , 2 , . . . , and hence

• The operators of the form I e x p ^ A j S , 1 I exp — X^Waf" )a r e vertex operators. These operators were discovered by physicists in the string theory. §14.6. In this section we shall construct, in terms of differential operators in infinitely many indeterminates, the basic representation L(Ao) of an affine Lie algebra &'(A) with the Cartan matrix A of type Ajy , where either r = 1 and A is symmetric or r > 1. This construction is called the principal realization of the basic representation. Let g be a simple finite-dimensional Lie algebra of type XN, let 0 = 0 j ( l ; r ) be its r-principal gradation, E G 0j-(I,r) the r-cyclic element, 5 ( r ) = © S ^ r ) the centralizer of E in 0 (see §14.2). According 3

to Proposition 14.2 b), S^ is a Cartan subalgebra of g (graded by the r-principal gradation). Let A C 5^r^* be the set of roots of Q with respect to S^. For (3 G A pick a nonzero root vector Ap G Q and decompose it with respect to the principal gradation: j

(summation over j €

302

Vertex Operator Constructions and Soliton Equations /

Since S^ D gn(l;r) = 0, the subspace g^r)

t

= £ V

j=i

CH

Ch. 14 \

j

is t h e

J

linear span of the Apto, (3 G A. Therefore, one can choose £ root vectors Apx,..., Ape with respect to S^r\ corresponding to some roots /3i,..., (3e e ( 5 ^ ) * , such that their projections on 0o(I;r) form a basis of this space. Since ad E is a nondegenerate operator on g/S^r\ which shifts the induced gradation by 1, we obtain that the images of the elements Aptj in Q/S^ are linearly independent. For all /? and j we have [E,Apj] = such that Tj € 5 ^ r ) and (Ti\TN+i-j)

= Wy for all i, j = 1 , . . . , N (see Proposition 14.2 c)). We set

(14.6.1)

AW = .

Then, clearly, we have [l

(14.6.2)

(here j + my is viewed as an element of Z/h^Z). Note that by (14.2.4), all the elements Apitj and Tk form a basis of g, since they are linearly independent. Note also that (Apj\Tk) = 0. We use the "principal" realization of the Lie algebra 0; (A) defined by (14.3.5 and 6). The map (14.6.3)

a;* t ® Ei,

fi i - t~x ® fi,

a,y i - i/,- + {aila(h)K

provides an isomorphism with this realization. Let E+ = {61,62,...} be the sequence of positive exponents of g(A) arranged in nondecreasing order. We introduce the following basis of the principal subalgebra s: K, pi = tbi ® Tv> where i = 1,2,

qt =

brlt-hi

Here and further on, i' is defined to be the element of

{ 1 , . . . , N} congruent to t mod N. By (14.3.6) we have (14.6.4)

\pi,qj)=6ijK

for all i , j = l , 2 , . . . .

The degrees of these elements in the principal gradation are (14.6.5)

degp, = 6, = -

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303

Assume now that &(A) is an affine algebra of one of the types listed at the beginning of the section and let L(AQ) be its basic module. Since L(AQ) 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(AQ) with the space R = C[xi,X2,...] so that K operates as an identity, pi as gjfr, and qi as multiplication by x* (i = 1,2,...). Prom (14.6.5) we see that the relations (14.6.6)

degxi=bi

(1 = 1,2,...),

(together with deg PQ = deg P + deg Q) define the principal gradation of £(A 0 ). In order to extend the realization of the basic representation from s to the whole Lie algebra &'(A), we extend, in an obvious way, the identification of L(A0) with R to that of L(Ao) with ii, where L(A0) denotes the formal completion of L(Ao) by its principal gradation (i.e., L(Ao) = Y[L(A0)j). j

Let 1^ Qa be the formal completion of &(A). This space is not a Lie algebra; a

however, the adjoint action of &(A) can be extended to it in an obvious way. We introduce the following elements of this completion, depending on a parameter z G C x :

Then Xp{z) maps L(A0) into L(A0). LEMMA 14.6. The operator X?{z) : R -> R acts as the following vertex operator:

Tp(z)=: (14.6.7)

bj (Ao,A0o)[expy\3j>z xj 1

V Jfri ~

)

Proo/. Using (14.6.2) and (14.3.6), we write: -> (^ ® AfiJ)] jez

304

Vertex Operator Constructions

and Soliton Equations

Ch. 14

Similarly,

Now the lemma follows from Lemma 14.5.

• We expand the vertex operator T^(z) defined by (14.6.7) in powers of z:

Then Tpj are infinite-order differential operators, which map R = L(A0) into itself. Since all the elements Ap.j and Ts form a basis of g, we can reformulate Lemma 14.6 as follows. 14.6. Let %'{A) be an affine algebra such that either A is symmetric (from Table Aff 1) or A is from Tables Aff 2 or 3. Set R = C[xi,X2,...]. Then the identity operator, the operators s^^fr (j = 1,2,...), and Tp.j (i = l , . . . , £ ; j G Z) form a basis of a Lie subalgebra of the algebra of differential operators preserving R. This subalgebra is isomorphic to &'(A), and the representation of it on R is equivalent to the basic representation of &'( THEOREM

The realization of the basic representation given by Theorem 11.6 is called the principal vertex operator construction. §14.7. We write down the explicit formulas for the principal vertex operator construction of the affine algebra &'(A) of type A^lx (n > 2). In this case, g = s^ n (C), and (x\y) = ttxy. Let E^ (i,j = l , . . . , n ) denote the n x n matrix which is 1 in the i,j-entry and 0 everywhere else. We take: EQ = £"n i,

Ei = Eiti+x

(i = 1 , . . . , n — 1),

FQ = E\n,

Fi = 2?i+i,i

(i = 1 , . . . , n — 1),

Z/^o = Enn ~ •E'li>

Hi = En — -Bt+i.t+i

(t = 1 , . . . , n — 1).

The 1-principal Z/nZ-gradation of g is given by setting deg Eij = j — i for i ^ j and deg D = 0 for a traceless diagonal matrix D. We set

n-l

s=0

/0 0

1 0

0 1

... ...

I 0

0 VI

0 0

0 0

... ...

1 0/

Ch. 14

Vertex Operator Constructions and Soliton Equations

305

and let 5 be the centralizer of E in g. Then 5 is a Cartan subalgebra of 0, with basis

We have degT) = ,; mod n, and (TilTn-j) = nSij for ij = 1,... ,n - 1. For two distinct n-th roots of unity, e and 17, define n x n matrices

Then A^€1)) G 0 are root vectors with respect to 5: PS. A'*)] = (*' - 9*) A«.if) for i = 1 , . . . , n - 1. Let ^(c,*;),; be the homogeneous components of the A^ir)y The T* together with the homogeneous components of the -A(e,i) form a basis of 0. In the notation of (14.6.1), we have A

(*,»j),j = ^ - ^ for i = 1 , . . . , n - 1.

We use the principal realization of &'(A):

''s; »odn(i;i))ecir, with the bracket given by

[a(t) 0 \K, 6(<) 0 liK] = a(06(0 - *(')«(<) © £ ( R e s

tr

^^6(o)

K

-

We also have /i = n and E+ = { j € Z + | j ^ 0 mod n}. For m = 1,2,...; m ^ 0 mod n, we set pm=r

r m /,

^fm = m

t

ln-m'-

We can use this simpler indexing than in the general case above, because no exponent has multiplicity greater than 1. We have [ft 14m] = famK for all t, m > 0 not divisible by n.

306

Vertex Operator Constructions and Soliton Equations

Ch. 14

We identify the space of the basic representation with the space of polynomials

R = C[xm\m 6 Z+,m ^ Omod n]. Then 1 is a highest-weight vector, and putting degx m = m defines the principal gradation of R. For i = 1 , . . . , n — 1, we set

Then K, pm, qm (m G Z + , . . . , m ^ 0 mod n) and the homogeneous components PA(Eii)j of X^£'l\z) form a basis of Q'(A). By Lemma 14.6 and Theorem 14.6, the basic representation a of Qf(A) on # is given by:

a(K) = 1, ()

( G Z

= the coefficient of z~J in

Indeed, to compute (Ao,A(Cff?)o) in (14.6.7), we observe that - V

/ J7»

J7I

\

I 1 I/*"

QfQ — yUjfifi — i - ' H ) n

-A.,

^v,V

/ jp

OC^ — \^ii

jp

\_i_ll^

— * - ' » + l l » + l / T —-*v

(Z

1

n

^

yl — 1, . . . , 71 — 1 j .

Hence A(e,,))0 = E^((c/i|) + (e/r/)2 + • • • + (eftf )o/ - ^ / T . Since (A 0 ,aV) = «0,t and (A 0 ,i^) = 1, we obtain (A 0,i4( c ^ )>0 ) = jf^. §14.8. We turn now to the homogeneous vertex operator construction. This construction is based on considering the homogeneous Heisenberg subalgebra t, see §8.4 (in place of the homogeneous Heisenberg subalgebra 6 in the principal vertex operator construction). It is not true that L(Ao) is irreducible as a t-module. One can still prove an analogue of Proposition 14.4 (see Exercise 14.6) and proceed along the lines of the proof in the principal case. This way allows one to "discover" the construction. A shorter way to prove that the construction "works" is to apply some vertex 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 case.

Ch. 14

Vertex Operator Constructions and Soliton Equations

307

Let 0 = fy® ( ® CEQ) be a simple finite-dimensional Lie algebra of type -A/ , Z)*, or £7, with commutation relations defined by (7.8.5). Let (.|.) be the normalized invariant form on 0 (defined by (7.8.6)). Let £(0) = Cp, t" 1] ®c 0 + CX + Cd be the associated affine algebra of type A\ , D\ , and J^ , respectively. Consider the complex commutative associative algebra (14.8.1)

V

where S stands for the symmetric algebra and C[Q] stands for the group algebra of the root lattice Q C f) of #(Xi). Let a »-* ea denote the inclusion Q C C[Q]. As before, ti(n) will stand for tn ® u (n G Z, ti G 0). For n > 0, ti G i), denote by ti(—n) the operator on V of multiplication by i / ~ n ) . For n > 0, u G fy, denote by ti(n) the derivation of the algebra V defined by the formula (14.8.2)

ti(n)(i;(- m ) ® e a ) = n6 n ,_ m (u|i;) 0 e a + 6n ,o(a|ti)t;^ m ) 0 e a .

Choosing dual bases «,- and ti* of f), define the operator Do on V by the formula

(14.8.3)

Do = j ; (iti,(O)ti'(O) + £ m(-n)u<(n)) • t=l^

n>l

^

Furthermore, for a G Q, define the sign operator ca: (14.8.4)

ca(f ® c") = c(a, p)f ® eP.

Finally, for a G A C Q introduce the vertex operator (14.8.5)

TQ(z) =

Here 2 is viewed as an indeterminate. Expanding in powers of z:

we obtain a sequence of operators ! « on V. Now we can state the result.

308

Vertex Operator Constructions and Soliton Equations

Ch. 14

THEOREM 14.8. The map a : C(g) -> End V given by

ti ( n ) !-• u(n), n)|

for u G f), n € Z,

n)

4 ^r( a , for T G A, n€Z, defines the basic representation of the affine algebra C(#) on V. Proof. First, we check that this map is a representation. Since Ea(z) i—• F a (z), according to (7.7.3) and (7.7.4) we need to check the following relations (all other relations trivially hold): [Do,Ta(z)]= (z£ + l)ra(z),

(14.8.6) (14.8.7) (14.8.8)

[r a (*i),r_ a (z a )]= -a(z1)S(z1-z2)

+

6'Zl(zl-z2),

[ T ^ z O . i y z a ) ] = c(a,/?)r cr+/) (z 1 )«(z 1 - z 2 ) if a +/3 e A.

Relation (14.8.6) follows from relations [£>„,«(")] = -««(")>

(14.8.9)

[A),^] = -i(7|7)e7,

which are obvious. In order to check (14.8.7 and 8), we let for 7 € A:

If (z) = exp£ 2M,*i,

I-(,) = e ' ^ ^ .

Then we have:

(14.8.10)

r^(zx)r+(za) = r+(z a)r;(z0 (1 - ^ j

,

(14.8.11) Here and further on, by (1 — Z2/z\)m, m G Z, we mean its power series expansion in Z2A1. The second of these formulas is trivial and the first one, viewed as an identity of formal power series in zfl and zf1, follows from the following two facts: eAeB = eBeAe[A>B] for two operators A and B such that [A, B] commutes with A and B\ (14.8.13)

Ch. 14

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309

The following formula is immediate by (14.8.10 and 11):

(14.8.14)

Using (14.8.14) we obtain for a £ A: (14.8.15)

[ r a ( z ! ) , r - a ( z 2 ) ] = 6'Zi(Zl - z2)

Here we have used that (14.8.16)

S(zi-z2)

= Z2

and therefore (14.8.17)

6'(zi

- z2) = z^2 (1 - —

Applying (7.7.5) to (14.8.15) proves (14.8.7). The proof of (14.8.8) is similar. Thus, a is a representation of £(fl) on V. Furthermore, it follows from (14.8.9) that

Comparing this with (12.13.6), we see that
•

310

Vertex Operator Constructions and Soliton Equations

Ch. 14

§14.9. The basic representation of the infinite rank affine algebra of type Aoo (and Boo) can be constructed using one of the two approaches discussed in §§14.6 and 14.8. In this section and the next, we shall discuss yet another approach, which is of fundamental importance for mathematical physics. This is usually referred to as the boson-fermion correspondence. An infinite expression of the form ii A12 AI3 A • • • ,

where i\, i 2 , . . . are integers such that i\ > *2 > **3 > • • • > ^ d in = i n _i - 1 for n > 0, is called a semi-infinite monomial. Let F be the complex vector space with a basis consisting of all semi-infinite monomials, and let H(.,.) denote the Hermitian form on F for which this basis is orthonormal. Define the charge decomposition F

=

m€Z

by letting I*™) = WL A ro — 1 Aro— 2 A • • • 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 \m) only at a finite number of places. Given a semi-infinite monomial