Proceedings of the International Congress of Mathematicians
Helsinki, 1978
Highest Weight Representations of
Infinite-Dimensional Lie Algebras
V. G. Kac
1. Introduction. In [1], in connection with the Cartan classification of infinitedimensional primitive pseudogroups the following result has been obtained.
THEOREM 1. Let g = 0 / € Z 9 / be a complex infinite-dimensional simple Z-graded
Lie algebra of finite Gelfand-Kirillov dimension (i.e. lim^^lndim g,/ln |/|<oo).
If in addition
(*) Ô-i©9o©9i generates g and the ^-module Q_X is irreducible,
then g is isomorphic (without taking into account the gradation) to one of the following Lie algebras: (a) Cartan type algebras W„, Sn9H„9Kn9 (b) algebras C(p,v) =
©iez''Pi mod*» where p is a simple finite-dimensional Lie algebra, v is an automorphism of order k, induced by an isometry of the Dynkin diagram (k = 1,2 or 3),
defining the Zk-gradation p = ©p / 9 and t is an indeterminate.
My conjecture is that if one throws away hypothesis (#), then only one extra
example occurs: the Witt algebra with commutation relations
K ß j ] = (ì-j)et+j>
UJ€Z.
Recent achievements in the computation of the cohomology of Cartan type Lie
algebras and their applications are well known (see survey [22]). Interest in the
second type of infinite-dimensional Lie algebras rose lately when Macdonald's
identities [2] were interpreted as Weyl denominator formulas for the universal
central extension <?(p, v) of the Lie algebra C(p, v) [3]. More generally, in [3]
a character formula was obtained for any irreducible representation L(A) with
300
V. G. Kac
dominant highest weight X of the so-called Kac-Moody Lie algebras g (A) (their
study was started independently in [1] and [4]). Macdonald's identities correspond
to the case g(/4)^<?(p, v) and >l=0. Finally the representations of the universal
central extension of the Witt algebra (the so-called string algebra) have become
recently a topic of interest in physics in the context of dual models [5]. In this article
I want to discuss some results on the structure of highest weight representations
of the Lie algebras Q(A) (§ 2) and of the string algebra (§ 4). The case of Cartan
type Lie algebras has been studied in [6]. In §3 I discuss some applications of
algebras g 04).
2. The Lie algebras Q(A). Let A=(atj) be a complex («X«)-matrix, r be the
free abelian group with free generators ai9 i£l= {1, ..., «}, and T + be the subsemigroup of r generated by ai9 /Ç/. We define a complex T-graded Lie algebra
9(A) = ffi«er9« by ^ e properties: (a) every graded ideal which intersects g 0 =ï)
trivially is zero, (b) Q(A) is generated by elements ei9 fi9 hi9 *£/, such that
Sa=Cel9 Q-a=Cfi9 hË9s form a basis of I), and: [hi9 hj] = 09 K,/}]=5 y A i ,[A l ,« y ] =
aijej9[hi9fj]=—aijfj9
i9j£L We denote by A+ the set of a£r+\0
such that
EXAMPLES. Let p b e a simple finite-dimensional Lie algebra with Chevalley
generators Ei9 Fi9 Hi9 i= 1,..., n. (a) If A is the Cartan matrix of p, then Q(A)^P
and the F-gradation is the root decomposition V^^Bv^- (b) Let 9 be the highest
root of p and A be the extended Cartan matrix of p. Then the homomorphism
$(A)-+C[t9t-1](g)cy = C(Tp9 id), defined by et^Ei9 ff+Fi9 i=\9...9n9
eQ^tE_B9
/ o ^ f _ 1 l s 0 , is the 1-dimensional universal central extension, rank r=n-\-\9 and
T-gradation is the pull-back of the decomposition C{p9 id) = 0 a k tkpa.
We c a l l a g e - m o d u l e V a module with highest weight X £1)* if: (a) ^ = © l / € r V_n
and 9 a (F_ l / )cF a _ / / , and (b) VQ=Cv09 vQ is a cyclic vector of V and h(u0)=X(h)v0
for Agì). We set ch V=®n (dim F_f/) en (e11 is a "formal" exponential). For each
X there is a Verma module M(X) such that the modules with highest weight X are
the quotients of M(X). M(X) has a unique irreducible quotient L(X) = M(X)/I(X).
The function P(fj) = dimM_tJ is called Kostant partition function. One has
chM(X) = Q-\ where Q=H^A+ (\-e«fim*«.
Let co be the involutive antiautomorphism of g (/I) defined by co(e^=fi9 co(f^)=ei9 co(h^=hl9iÇ.I. M (A)
carries a bilinear form F which is uniquely defined by the properties: F(u09 ^o) — 1
and F(g(x)9y) = F(x9co(g)(y)) for x9y^M(X)9g^(A).
In particular Ker F=I(X)
and F(M_ti9M_tf)=0
if n^n'. We set
Ft=F\M_ir
From now on we assume that A is a symmetrisable matrix i.e. A=D>B9 where
D = diag(rfl9 ...,4i)s detZMO, and B={bij) is a symmetric matrix. To ri=kiaiÇ.r
w e set
we assign a linear function on Ì) by nifi^^i^i61^
K^^O^T^K^ We
introduce a bilinear form ( , ) on T by (a/5 0L^=btj and define ££E)*by ö ( ^ ) = T ß » The following theorem generalizes the well-known results of Sapovalov and Bernstein-Gelfand-Gelfand in the ease of finite-dimensional semisimple Lie algebras.
Highest Weight Representations of Infinite-Dimensional Lie Algebras
THEOREM
301
2 [7]. (a) For the g (A)-module M(X) one has:
vP(»7-Ha)dimfla
a£A+
u€7VV
^
'
In particular M(X) = L(X) iff 2(X + Q)(hu)^n(a9oi)9 for any <x£A+9 n£N = {l9 2, ...}.
(b) Any simple subquotient of M(X) is of the form L(X—tf)9 where n£r+ is such
Jhat there exist ßl9 ...9ßkdA + and nl9 ...9nk£N such that
2{X+Q-n1ß1-...-nl-1ßi_J{hß)
for
= w,(ft, ft)
z = l, ...9k9 and ^ = 2 ? = i " i ß -
Theorem 2 and its analogue for Lie superalgebras provides a new proof of character
formulas from [3], [8], [9] and their generalizations.
THEOREM 3 [3]. Suppose that an=2 and —a}j€Z+9 i9j£.I9 and /LCI)* is such
that X(hi)£Z+9 i£l. Let W be the subgroup of GL(t)*) generated by reflections
ri9i£l9 defined by ri(X)=X—X(hl)ai. Then for the §(A)-module L(X) one has:
chL(X) = Qrx 2 (deU0e*+e-™(A+e>.
In particular, for X=0 one has the "denominator" identity:
Q= 2 (detw)e e - M,(e) .
we IF
It is easy to see that e~*~e'Q'cl\L(X)=2icne~tl>
where c A+e = l and
cfl^0 only for the fi's such that L(JJL — Q) is a subquotient of M (A). Since for
a£A+9 a $ ( J f ^ ( a i ) iff ( a , a ) ^ 0 we obtain from Theorem 2 that these \i have
form W(X+Q)9W€W.
Now the theorem follows from W^-skew-invariance of e"Q>Q
and rF-invariance of e~ A chL(A), which are provided by the hypothesis of the
theorem.
PROOF.
3. Applications. One of the first applications of the Lie algebras g(^4) was the
proof of Theorem 1 (§ 1), where they play the role of "test" algebras. In particular
this gives a simple proof of Cartan classification of primitive filtered Lie algebras
[1], [23]. Another application is a simple method of classification of symmetric
spaces [1]. More generally let o be an automorphism of finite order m of a simple
Lie algebra p. We consider the corresponding Z„,-gradation p = ©p f and construct
the "covering" Z-graded Lie algebra C(p, o-) = 0 , / ' p / m o d j l I . The algebra C(p, cr)
is isomorphic (without taking into account the gradation) to a certain C(p, v)
from Theorem 1. This gives the classification of finite order automorphisms of
p [12] and in particular the description of the p0-modules p x ; the corresponding
to these modules connected algebraic linear groups are called c-groups. These
linear groups have many nice properties : (a) the algebra of invariant polynomials
is free, (b) any level variety of invariant polynomials consists of a finite number
of orbits, etc. [14], [13], Moreover it turns out that almost all the connected algebraic
302
V. G. Kac
linear groups acting irreducibly which satisfy (b) or those which are simple and
satisfy (a) are cr-groups [13], [24]. These results and the dimensions of the correspondence between the root system A+ of §(A) and indecomposable representations
of the corresponding graph [17] (see survey [18] for the background) indicate a deep
connection between the Lie algebras g(^4) and invariant theory. In [25] the Lie algebras g(A) provide infinite families of examples of simplefinite-dimensionalLie
algebras in characteristics 2 and 3. Recently the cohomology of the Lie algebras
C(p9 a) was applied to the study of the topology of various loop spaces [15], [26],
[16]. The "infinite-dimensional" groups corresponding to the Lie algebras g (.4) are
discussed in [21], [19], [20], [8]. Finally, various specializations of the formulas of
Theorem 3 (§ 2) for g(^4)=^^(p, v) produce a number of //-function identities,
Rogers-Ramanujan type identities, etc. These and other applications to combinatorics are considered in detail in s urvey [10]. Here I discuss briefly a few examples
taken from [8].
Setting deg e~ —degft=si9 deg ht=09 i£l9 s^Z+, define a Z-gradation ô04) =
— ©it gfc(^5 *); w e consider the corresponding specialization <ps(eai)=X\ i£L For
the Z-graded Lie algebra q(A) = C(p9 o) the corresponding specialization of the
denominator formula produces a theta-series type expansion of the product
J]i(p(Xi)ni. Here (p(X) = JJk^1(l—Xk) and the sequence nt is the Möbius transform of the sequence dimp /modwi , i£N. This product is finite iff the automorphism
G has a rational characteristic polynomial. In this case we obtain as a consequence
the "very strange" formula [8]:
"* k\m
where Q is the half-sum of the positive roots of p, p. is an element of the dual
to the Cartan subalgebra defined by ([i9ai)=si/2m9 i=\9...,«, and ( , ) is the
Killing form. The simplest case tr=id gives Macdonald's decomposition for
rçdimp p] a n d the Freudental- de Vries "strange" formula: ||ö|]2=(l/24) dim p.
The specialization (pj9 where I = ( l , ..., 1), factorises the character formula:
k
(chL(X))=n
where rk(X)= dim $k(AT9 s+T)-dim Qk(AT91)
9ï
k ^(1 ~X y^\
and Si=X(hi). This allows us to obtain new multivariable identities [8]. Let p be
a simple Lie algebra of type An9 Dn9 2s6, E7 or E89 R be the lattice generated by
the roots, h be the Coxeter number, Ä be the extended Cartan matrix. Let
X0^i)* be defined by X0(h0) = l9 ^0(/zl)=0, i = l, ...,« = rank p and let <5=ao+0.
Then for the g(JT)-module L(XQ) one has:
chL(x0) = (2™vW\y\\2s+y))/<p{esy.
A construction of g (^-modules L{X0) in terms of differential operators is given
in [11] (in certain sense L(XQ) contains almost all simple g(^î)-modules with dominant highest weight).
Highest Weight Representations of Infinite-Dimensional Lie Algebras
303
4. The string algebra S is by definition a complex Lie algebra with basis e'Q9 ei9
i£Z9 with the following commutation relations:
[ei9 ej] = (/—J)CI+J + -J2" 'C2— !)*#. -j«fi. fo> «fl = 0.
We set l)^=Ce0CBCe'09r=Z. In the same way as in §2 we define ^-modules
M(X) and L(X)9 X€l)*9 and bilinear forms Fn9n£Z+;
we set X(e0)—h9 X{e^ = c.
For k9s£N9k?±s9
let (pkt&(h9c) denote the quadratic polynomial in h with the
roots :
-^-((13-c)(/c 2 + s 2 )±l/c 2 -26c + 25(/c 2 -s 2 )-24/c5-2 + 2c)
and let <pKk(h9 c)=h + (l/24)(k2-l)(c-l).
Weset^ H ( Ä » c ) = /7s|n <Ps>n/*- In particular,
* . ( * . 0) = tf (* - ^ ( ( 3 s - 2 W / s ) 2 - 1 ) ) .
4. (a) del Fa(h9 c)=77"E=1 ^f ( "~°, w/te/'e /?(,y) w //?<? classical partition
function. In particular the S-module M(X) is irreducible iff cpks(h9c)^0
for any
k9s£N.(b)L(h+n9c)9n£N9
is a subquotient of M (h9c) iff there exists nl9 ..., nk£N
such that \l/rJ(h+n1+...+ni^l9c)
= 0 for / = l , . . . , / c , and n=2!?1iTHEOREM
COROLLARY, (a) The module M(h9 0) over the Witt algebra is irreducible iff
A^(l/24)(w? 2 -l), m £ Z + . (b) (Goldstone conjecture). The S-module M(h,l) is
irreducible iff h^\m*9
m£Z+. One has chL(~m\ 1) = ^ " 1 ( 1 — e ,,I+1 ), where
e1
v
<P=JJi^i(\~~ )' F° lhc^R9 A>0 5 o l , the S-module M(h9c) is irreducible.
Added in proof. For p = 2i8 the specialization e3 = X9 ey= 1, y£R, of chL[^ 0 ] give
[Xj(X)]lj39 where j(X) is the modular invariant. This is related to recent discoveries
about the Monster simple group.
References
1. V. G. Kac, Simple irreducible graded Lie algebras of finite growth, Math. USSR-Izv. 2 (1968),
1271—1311.
2. I. G. Macdonald, Affine root systems and Dedekind's j]-function, Invent. Math. 15 (1972),
91—143.
3. V. G. Kac, Infinite-dimensional Lie algebras and Dedekind's ij-function, Functional Anal.
Appi. 8 (1974), 68—70.
4. R. V. Moody, A new class of Lie algebras, J. Algebra 10 (1968), 211—230.
5. J. H. Schwartz, Dual resonance theory, Physics Reports 8c (1973), 269—335.
6. A. N. Rudakov, Irreducible representations of infinite-dimensional Lie algebras of Cartan type,
Math. USSR-Izv. 8 (1974), 836—866; 9 (1975), 465—480.
7. V. G. Kac and D. A. Kazhdan, Structure of representations with highest weight of infinitedimensional Lie algebras, Advances in Math, (to appear).
8. V. G. Kac Infinite-dimensional Lie algebras, Dedekind's ?j-function, classical Mobius function
and the very strange formula, Advances in Math. 30 (1978), 85—136.
304
V. G. Kac: Highest Weight Representations of Infinite-Dimensional Lie Algebras
9.
Representations of classical Lie superalgebras, Lecture Notes in Math. 676 (1978),
597—626.
10. J. Lepowsky, Lie algebras and combinatorics, these PROCEEDINGS.
11. V. G. Kac, D. A. Kazhdan, J. Lepowsky and R. Wilson, Construction of basic representation of Euclidean Lie algebras (to appear).
12. V. G. Kac, Automorphisms offiniteorder of semisimple Lie algebras, Functional Anal. Appi.
3 (1969), 252—254.
13.
On the orbit classification of algebraic linear groups, Uspehi Mat. Nauk (6) 30
(1975), 173—174.
14. E. B. Vinberg, Weyl group of a graded Lie algebra. Math. USSR-Izv. 40 (1976), 488—526.
15. H. Garland, Dedekind's n-function and the cohomology of infinite-dimensional Lie algebras,
Proc. Nat. Acad. Sci. U.S.A. 72 (1975), 2493—2495.
16. J. Lepowsky, Generalized Verma modules, cohomology of loop spaces and Macdonald-type
identities (to appear).
17. V. G. Kac, Infinite root systems, representations of graphs and invariant theory (to appear).
18. A. Roiter, Matrix problems, these PROCEEDINGS.
19. V. G. Kac, An algebraic definition of compact Lie groups, Trudy MIEM 5 (1969), 36—47.
20. R. V. Moody and K. L. Teo, Tits' systems with crystallographic Weyl groups, J. Algebra 21
(1972), 178—190.
21. H. Garland, Arithmetic properties of loop groups. I, II (to appear).
22. D. B. Fuks, New results on the characteristic classes of foliations, these PROCEEDINGS.
23. B. Weisfeiler, Infinite-dimensionalfilteredLie algebras and their connection with graded Lie
algebras, Functional Anal. Appi. 2 (1968), 88—89.
24. V. G. Kac, V. L. Popov and E. B. Vinberg, Sur les groupes lineares algébriques dont l'algèbre
des invariants est libre, C. R. Acad. Sci. Paris Ser. A 283 (1976), 875—878.
25. B. Weisfeilerand V. G. Kac, Exponentials in Lie algebras of characteristic p, Math. USSR-Izv.
5 (1971), 777—803.
26. H. Garland and M. S. Raghunatan A Bruhat decomposition for the loop space of a compact
group, Proc. Nat. Acad Sci. USA 72 (1975), 4716—4717.
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
CAMBRIDGE, MASSACHUSETTS 02139, U.S.A.