Graded Lie Superalgebras and the Superdimension Formula

204, 597]655 Ž1998. JA977352

JOURNAL OF ALGEBRA ARTICLE NO.

Graded Lie Superalgebras and the Superdimension Formula* Seok-Jin Kang Department of Mathematics, Seoul National Uni¨ ersity, Seoul, 151-742, Korea Communicated by Georgia Benkart Received August 1, 1997

In this paper, we investigate the structure of graded Lie superalgebras L s LŽ a , a. , where G is a countable abelian semigroup and A is a countable abelian group with a coloring map satisfying a certain finiteness condition. Given a denominator identity for the graded Lie superalgebra L , we derive a superdimension formula for the homogeneous subspaces LŽ a , a. Ž a g G, a g A ., which enables us to study the structure of graded Lie superalgebras in a unified way. We discuss the applications of our superdimension formula to free Lie superalgebras, generalized Kac]Moody superalgebras, and Monstrous Lie superalgebras. In particular, the product identities for normalized formal power series are interpreted as the denominator identities for free Lie superalgebras. We also give a characterization of replicable functions in terms of product identities and determine the root multiplicities of Monstrous Lie superalgebras. Q 1998 Academic Press

[Ž a , a.g G=AA

INTRODUCTION The Kac]Moody algebras were introduced independently by Kac wK1x and Moody wMox as generalizations of complex finite dimensional simple Lie algebras. In wK2x, Kac discovered a character formula, called the Weyl]Kac formula for integrable highest weight modules over symmetrizable Kac]Moody algebras, and showed that the Macdonald identities wMx are equivalent to the denominator identities for affine Kac]Moody algebras. Since then, the theory of Kac]Moody algebras Žand more generally infinite dimensional Lie algebras. has attracted extensive research activities due to its rich and significant applications to many areas of mathematics and mathematical physics. * This research was supported by the Non-directed Research Fund, Korea Research Foundation, 1996. 597 0021-8693r98 $25.00 Copyright Q 1998 by Academic Press All rights of reproduction in any form reserved.

598

SEOK-JIN KANG

The Kac]Moody theory has been extended to the theory of generalized Kac]Moody algebras by Borcherds in his study of the vertex algebras and Monstrous Moonshine wB1]B3, CNx. The structure and the representation theories of generalized Kac]Moody algebras are similar to those of Kac]Moody algebras, and a lot of facts about Kac]Moody algebras can be generalized to generalized Kac]Moody algebras wB2, K5x. For example, in wB2x, Borcherds proved a character formula, called the Weyl]Kac]Borcherds formula, for the unitarizable highest weight modules over generalized Kac]Moody algebras. The main difference is that the generalized Kac]Moody algebras may have imaginary simple roots with norms F 0 whose multiplicity can be ) 1. The most interesting example of generalized Kac]Moody algebras may be the Monster Lie algebra, which played a crucial role in Borcherds’ proof of the Moonshine conjecture wB3x. In wKaK3x, we considered general graded Lie algebras L s [a g G La graded by a countable abelian semigroup G such that every element in G can be expressed as a sum of elements in G in only finitely many ways. The Euler]Poincare ´ principle for the graded Lie algebra L s [a g G La yields the denominator identity

Ł Ž 1 y e a . dim L

agG

a

s1y

Ý dŽ a . e a

with d Ž a . g Z Ž a g G . ,

a gG

from which we derived a dimension formula for the homogeneous subspaces La Ž a g G .. Our dimension formula is a generalization of the root multiplicity formulas for Kac]Moody algebras and generalized Kac]Moody algebras given in wBM, Ka2, Ka3x. The applications of our dimension formula to various Lie algebras such as free Lie algebra, Kac]Moody algebras, and generalized Kac]Moody algebras were discussed in wKaK3x Žsee also wKaK1x.. On the other hand, since 1970s, the Lie superalgebras and their representation have emerged as fundamental algebraic structure behind several areas in mathematical physics. In wK3x, Kac gave a comprehensive presentation of the mathematical theory of Lie superalgebras, and obtained an important classification theorem for finite dimensional simple Lie superalgebras over algebraically closed fields of characteristic zero Žsee also wSx.. In wK4x, the notion of Kac]Moody superalgebras was introduced, and for the nondegenerate Kac]Moody superalgebras, Kac proved the Weyl]Kac character formula for the irreducible highest weight modules with dominant integral highest weights, which yields the denominator identity when applied to the 1-dimensional trivial representation. By specializing to various classes of affine Kac]Moody superalgebras and irreducible highest weight modules, he derived a lot of interesting combinatorial identities that are closely related to number theory. Further developments of the

599

GRADED LIE SUPERALGEBRAS

theory of Lie superalgebras along this line can be found in wKW1, KW2x, where Kac and Wakimoto presented some interesting applications of affine Kac]Moody superalgebras to number theory. In wMi, Rx, the representation theories of nondegenerate Kac]Moody superalgebras and generalized Kac]Moody algebras were combined to give rise to the representation theory of generalized Kac]Moody superalgebras. Following the outline of wK4, K5x, Miyamoto and Ray independently developed the representation theory of generalized Kac]Moody superalgebras. In particular, they proved the Weyl]Kac]Borcherds character formula for the irreducible highest weight modules over generalized Kac]Moody superalgebras with dominant integral highest weights, and hence obtained the denominator identity. In wKaK2x, using the Weyl]Kac]Borcherds formula and the denominator identity, we obtain a closed form root multiplicity formula for all generalized Kac]Moody superalgebras, and discussed its applications to several special cases. In this paper, we consider a very general setting. Let G be a countable Žusually infinite . abelian semigroup and A be a countable Žusually finite. abelian group satisfying a certain finiteness condition. Consider the class of Ž G = A .-graded Lie superalgebras L s [Ž a , a.g G=AA L Ž a , a. with dim L Ž a , a. - ` for all Ž a , a. g G = A. There are many interesting and important Lie superalgebras belonging to this class of Lie superalgebras. For example, the free Lie superalgebras generated by graded superspaces with finite dimensional homogeneous subspaces are of this kind, and so are the positive or negative parts of finite dimensional simple Lie superalgebras, Kac]Moody superalgebras, and generalized Kac]Moody superalgebras. In this work, we propose a general method for investigating the structure of graded Lie superalgebras in a unified way, and discuss its applications to various classes of Lie superalgebras such as free Lie superalgebras, generalized Kac]Moody superalgebras, and Monstrous Lie superalgebras. In Section 1, we recall the basic theory of Lie superalgebras, and show that, if we are given the denominator identity

Ł

Ž a , a .gG= A

Ž 1 y E Ž a , a. .

Dim L Ž a , a.

s1y

Ý

t Ž a , a . E Ž a , a. ,

Ž a , a .gG= A

we can derive a closed form superdimension formula for the homogeneous subspaces of graded Lie superalgebras ŽTheorem 1.2.. Our superdimension formula will be expressed in terms of the Witt partition functions associated with the partitions of the elements in G = A. We believe that the most natural way to derive the denominator identity is to use the Euler]Poincare ´ principle for the homology of graded Lie superalgebras. However, in this work, we will not use the Euler]Poincare ´ principle to derive the

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SEOK-JIN KANG

denominator identity, for not very much has been known about the homology of graded Lie superalgebras. For example, Kostant-type formulas for Kac]Moody superalgebras or generalized Kac]Moody superalgebras are not yet available. Therefore, instead of making use of the Euler]Poincare Theo´ principle, we will use the Poincare]Birkhoff]Witt ´ rem for free Lie superalgebras, and the Weyl]Kac]Borcherds formula for generalized Kac]Moody superalgebras, respectively, to derive the denominator identities for the corresponding Lie superalgebras. In Section 2, we discuss the applications of our superdimension formula to the free Lie superalgebras L s [Ž a , a.g G=AA L Ž a , a. generated by graded superspaces V s [Ž a , a.g G=AA VŽ a , a. with dim VŽ a , a. - ` for all a g G, a g A. When applied to free Lie superalgebras, our superdimension formula will be called the generalized Witt formula ŽTheorem 2.1; see also wKa5x.. We also compare the structure of free Lie algebras and free Lie superalgebras generated by the same vector spaces, and discuss the application of the generalized Witt formula to the product identities for normalized formal power series. In particular, any such product identity can be interpreted as the denominator identity for a suitably defined free Lie superalgebra, and we obtain a number of interesting combinatorial identities arising from the identity Ž2.36.. For example, by applying the above idea to the automorphic forms with infinite product expansions given in wB4x, we obtain some interesting relations for the Fourier coefficients of the corresponding modular functions. In Section 3, we apply our superdimension formula to generalized Kac]Moody superalgebra to derive a closed form root multiplicity formula ŽTheorem 3.4.. The generalized Kac]Moody superalgebras arise naturally in the context of Monstrous Moonshine wB2, B3x, automorphic forms with infinite product expansions wB4, GN1]GN3x, and the string theory wHMx. Our root multiplicity formula enables us to study the structure of a generalized Kac]Moody superalgebra g as a representation of a Kac] Moody algebra or a Kac]Moody superalgebra g 0 contained in g. As an application of this idea, some generalized Kac]Moody algebras will be shown to be the maximal graded Lie algebras with local part V [ g 0 [ V *, where g 0 is a Kac]Moody superalgebra contained in g, V is the direct sum of irreducible highest weight modules over g 0 with highest weight ya i Ž a i runs over all imaginary simple roots counted with multiplicities., and V * is the contragredient module of V ŽProposition 3.6, see also wK3x; for Kac]Moody algebras and generalized Kac]Moody algebras, see wBKM, Ju2, JW, K1x.. The choice of g 0 in our formula gives rise to various expressions of the root multiplicities of g, which would yield combinatorial identities. We will discuss the applications of this idea elsewhere. Finally, in Section 4, we define the notion of Monstrous Lie superalgebras as generalizations of those given in wB3x. The monstrous Lie superalgebras

GRADED LIE SUPERALGEBRAS

601

form a special class of generalized Kac]Moody superalgebras associated with normalized q-series F Ž q . s Ý`ns y1 f Ž n. q n such that f Žy1. s 1, f Ž0. s 0, and f Ž n. g Z for all n G 1. We apply our superdimension formula to obtain a closed form root multiplicity formula for Monstrous Lie superalgebras. We also give an interesting characterization of replicable functions in terms of product identities, and determine the root multiplicities of the Monstrous Lie superalgebras associated with the replicable functions. In wKKKx, we will generalize our superdimension formula to the supertrace formula for graded Lie superalgebras with group actions, and discuss many interesting applications to various graded Lie superalgebras.

1. THE SUPERDIMENSION FORMULA 1.1. Lie Superalgebras We begin with the basic theory of colored Lie superalgebras Žcf. wBMPZ, K3, Sx.. Let A be an abelian group and suppose we have a bimultiplicative map u : A = A ª C= satisfying

u Ž a q b, c . s u Ž a, c . u Ž b, c . , u Ž a, b q c . s u Ž a, b . u Ž a, c . , u Ž a, b . u Ž b, a . s 1

Ž 1.1.

for all a, b, c g A .

The map u : A = A ª C= is called a coloring map on A. Note that we have u Ž a, a. s "1 for all a g A. We will write c Ž a. s u Ž a, a. and call it the sign of a g A. Let A0 s  a g A < c Ž a. s 14 and A1 s  a g A < c Ž a. s y14 , which yields a decomposition A s A0 j A1. The elements of A in A0 Žresp. A1 . are called e¨ en Žresp. odd .. A u-colored superspace is a pair Ž V, u ., where V s [ag A Va is an A-graded vector space and u : A = A ª C= is a coloring map on A. The elements of Va are called e¨ en Žresp. odd . if c Ž a. s 1 Žresp. c Ž a. s y1.. For each a g A, we define the superdimension of Va to be Dim Va s c Ž a . dim Va .

Ž 1.2.

Similarly, we define a u-colored superalgebra to be a pair ŽU, u ., where U s [a g A Ua is an A-graded associative algebra Ži.e., UaUb ; Uaqb for all a, b g A . and u : A = A ª C= is a coloring map on A. The direct sum of A-graded superalgebras is defined in the usual way, but, for u-colored superalgebras U s [ag A Ua and U9 s [a9g A UaX , we define the tensor

602

SEOK-JIN KANG

product of U and U9 to be the u-colored superspace U m U9 with the natural A-gradation u m u9 g ŽU m U9.aq a9 for u g Ua , u9 g Ua9X , and the multiplication given by

Ž u m u9 . Ž ¨ m ¨ 9 . s u Ž a9, b . Ž u¨ m u9¨ 9 . for u g Ua , ¨ g Ub , u9 g Ua9X , ¨ 9 g Ub9X , a, a9, b, b9 g A. DEFINITION 1.1. A u-colored Lie superalgebra is a u-colored superspace L s [ag A L a together with a bilinear operation w , x: L = L ª L satisfying

w L a , L b x ; L aqb , w x, y x s yu Ž a, b . w y, x x ,

Ž 1.3.

x, w y, z x s w x, y x , z q u Ž a, b . y, w x, z x for all x g L a , y g L b , z g L, and a, b g A. Let L Ž0. s [ag A0 L a and L Ž1. s [b g A1 L b . Then we have a decomposition L s L Ž0. [ L Ž1. , and the homogeneous elements of L Ž0. Žresp. L Ž1. . are called e¨ en Žresp. odd .. The uni¨ ersal en¨ eloping algebra of a u-colored Lie superalgebra L s [ag A L a is the pair ŽUŽ L ., i ., where UŽ L . is a u-colored superalgebra and i : L ª UŽ L . is a linear mapping satisfying

i Ž w x, y x . s i Ž x . i Ž y . y u Ž a, b . i Ž y . i Ž x .

for x g L a , y g L b

such that for any u-colored superalgebra U s [ag A Ua and a linear mapping j: L ª U satisfying j Ž w x, y x . s j Ž x . j Ž y . y u Ž a, b . j Ž y . j Ž x .

for x g L a , y g L b ,

there exists a unique homomorphism c : UŽ L . ª U of u-colored superalgebras satisfying c ( i s j. Let X s  xa < a g L4 Žresp. Y s  yb < b g V 4. be a homogeneous basis of the subspace L Ž0. s [ag A0 L a Žresp. L Ž1. s [b g A1 L b .. Then, by the Poincare]Birkhoff]Witt Theorem, the elements of the form ´ xa 1 xa 2 ??? xa k yb 1 yb 2 ??? yb l

with a 1 F ??? F a k , b 1 - ??? - b l Ž 1.4.

together with 1 form a basis of the universal enveloping algebra UŽ L . of L Žcf. wBMPZ, K3, Sx..

GRADED LIE SUPERALGEBRAS

603

1.2. Supercharacters and Denominator Identity Let A be a countable Žusually finite. abelian group with a coloring map u : A = A ª C=, and let G be a countable Žusually infinite . abelian semigroup satisfying the following condition: every element Ž a , a . g G = A can be written as a sum of elements in G = A in only finitely many ways.

Ž 1.5.

For a Ž G = A .-graded u-colored superspace V s [Ž a , a.g G=AA VŽ a , a. , we define the Ž G = A .-graded character of V to be ch G= A V s

Ý

Ž a , a .gG= A

dim VŽ a , a. e Ž a , a. ,

where the e Ž a , a. are the basis elements of the semigroup algebra Cw G = A x with the multiplication e Ž a , a. e Ž b , b. s e Ž aq b , aqb. for all a , b g G, a, b g A. ŽActually, ch G= A V itself is usually an element of Cww G = A xx, the completion of Cw G = A x.. We define the superdimension of the homogeneous subspace VŽ a , a. by Dim VŽ a , a. s c Ž a . dim VŽ a , a.

for Ž a , a . g G = A .

Ž 1.6.

Also, we introduce another basis element of the semigroup algebra Cw G = A x by setting E Ž a , a. s c Ž a . e Ž a , a.

for Ž a , a . g G = A .

Then it is easy to verify that E Ž a , a. E Ž b , b. s E Ž aq b , aqb. for all a , b g G, a,b g A. With these notations, we define the Ž G = A .-graded supercharacter of V by Ch G= A V s

Ý

Ž a , a .gG= A

Ž Dim VŽ a , a. . E Ž a , a. .

Ž 1.7.

Note that the supercharacter Ch G= A V is obtained from the character ch G= A V by replacing dim VŽ a , a. s c Ž a.Dim VŽ a , a. and e Ž a , a. s c Ž a. E Ž a , a. for a g G, a g A. Since c Ž a. 2 s 1, we have Ch G= A V s ch G=AA V. The only Žbut important. difference is that, in the supercharacter Ch G= A V, we allow the negative coefficients. This implies that if we are given a formal power

604

SEOK-JIN KANG

series T G= A s

Ý

t Ž a , a . E Ž a , a.

Ž a , a .gG= A

with t Ž a , a . g Z for all a g G, a g A ,

Ž 1.8.

we can interpret the series T G= A as the Ž G = A .-graded supercharacter of a Ž G = A .-graded superspace V s [Ž a , a.g G=AA VŽ a , a. such that Dim VŽ a , a. s tŽ a , a. for all a g G, a g A. For a Ž G = A .-graded superspace V s [Ž a , a.g G=AA VŽ a , a. , let Va s [ag A VŽ a , a. , and define Dim Va s

Ý

Dim VŽ a , a. s

ag A

Ý

dim VŽ a , a. y

ag A0

Ý

dim VŽ a , a. . Ž 1.9.

ag A1

ŽWe assume that Dim Va is well-defined for all a g G. That is,
Ý Ž Dim Va . E a .

Ž 1.10.

agG

Again, this implies that if we are given a formal power series TG s

Ý t Ž a . Ea

with t Ž a . g Z for all a g G,

Ž 1.11.

agG

we can interpret the series T G as the G-graded supercharacter of a G-graded superspace [a g G Va such that Dim Va s t Ž a . for all a g G. We now consider the Ž G = A .-graded u-colored Lie superalgebras Ls

[

Ž a , a .gG= A

L Ž a , a.

with dim L Ž a , a. - ` for all a g G, a g A .

Let L Ž0. s [Ž a , a.g G=AA0 L Ž a , a. and L Ž1. s [Ž b , b.g G=AA1 L Ž b , b. . Then, by the Poincare]Birkhoff]Witt Theorem, we have ´

ch G= A U Ž L . s

Ł Ž b , b.g G=AA1Ž 1 q e Ž b , b. .

dim L Ž b , b.

Ł Ž a , a.g G=AA0Ž 1 y e Ž a , a. .

dim L Ž a , a.

.

605

GRADED LIE SUPERALGEBRAS

Hence the Ž G = A .-graded supercharacter of the universal enveloping algebra UŽ L . of L is equal to Ch G= A U Ž L . s

Ł

Ž a , a .gG= A

Ž 1 y E Ž a , a. .

yD im L Ž a , a.

.

Ž 1.12.

Let DŽ L . s

Ł

Ž a , a .gG= A

Ž 1 y E Ž a , a. .

Dim L Ž a , a.

.

Ž 1.13.

We will call D Ž L . the denominator function of the Ž G = A .-graded u-colored Lie superalgebra L s [Ž a , a.g G=AA L Ž a , a. . Suppose we have a product identity for the denominator function D Ž L . of the form

Ł

Ž a , a .gG= A

Ž 1 y E Ž a , a. .

Dim L Ž a , a.

s 1 y T G= A

Ž 1.14.

for some formal power series T G= A s ÝŽ a , a.g G=AA t Ž a , a. E Ž a , a. with t Ž a , a. g Z for all a g G, a g A. Then the identity Ž1.14. will be called the denominator identity for the Ž G = A .-graded Lie superalgebra L s [Ž a , a.g G=AA L Ž a , a. . 1.3. The Superdimension Formula and Its Specialization Let P Ž G = A . s Ž a , a. g G = A < t Ž a , a. / 04 , and let Žt i , bj . g G = A < i, j s 1, 2, 3, . . . 4 be an enumeration of the set P Ž G = A .. For Žt , b . g G = A, we define `

T Ž t , b . s s s Ž si j . i , js1 si j g Z G 0 ,

½

Ý si j Ž t i , bj . s Ž t , b . 5 , Ž 1.15.

which is the set of all partitions of Žt , b . into a sum of Žt i , bj .’s. By our finiteness condition Ž1.5. on G = A, the set T Žt , b . must be finite. For a partition s s Ž si j . g T Žt , b ., we will use the notation < s < s Ýsi j and s!s ŁŽ si j!. Žcf. wBox.. For Žt , b . g G = A, we define a function W Žt , b. s

Ž < s < y 1. !

Ý

sgT Žt , b .

s!

Ł t Ž t i , bj .

si j

.

Ž 1.16.

We will call W Žt , b . the Witt partition function. In the next theorem, using the denominator identity Ž1.14., we derive a closed form formula for the superdimensions Dim L Ž a , a. Ž a g G, a g A . in terms of Witt partition functions.

606

SEOK-JIN KANG

THEOREM 1.2. Let A be a countable abelian group with a coloring map u : A = A ª C= and G be a countable abelian semigroup satisfying the finiteness condition Ž1.5.. Let L s [Ž a , a.g G=AA L Ž a , a. be a Ž G = A .-graded u-colored Lie superalgebra with dim L Ž a , a. - ` for all a g G, a g A. Suppose we ha¨ e a denominator identity

Ł

Ž a , a .gG= A

Ž 1 y E Ž a , a. .

Dim L Ž a , a.

s 1 y T G= A

for some formal power series T G= A s ÝŽ a , a.g G=AA t Ž a , a. E Ž a , a. with t Ž a , a. g Z for all Ž a , a. g G = A. Then for any Ž a , a. g G = A, we ha¨ e 1

Dim L Ž a , a. s

Ý d)0 Ž a , a .sd Žt , b .

d

mŽ d. W Žt , b. ,

Ž 1.17.

where m is the classical Mobius function. ¨ Proof. By the denominator identity Ž1.14., we have

Ł

Ž a , a .gG= A

Ž 1 y E Ž a , a. .

yD im L Ž a , a.

s s s

1 1 y T G= A 1 1 y ÝŽ a , a.g G=AA t Ž a , a . E Ž a , a. 1 1y

Ý`i , js1 t

Ž t i , bj . E Ž t , b . i

j

.

1 Using the formal power series logŽ1 y t . s yÝ`ks 1 k t k , we obtain from the left-hand side

log

ž

Ł

Ž a , a .gG= A

sy

Ž 1 y E Ž a , a. .

Ý

Ž a , a .gG= A

s

Ý

Ž a , a .gG= A

s

/

Dim L Ž a , a. log Ž 1 y E Ž a , a. .

Dim L Ž a , a.

`

Ý ks1

`

Ý

yD im L Ž a , a.

Ý

Ž a , a .gG= A ks1

1 k

1 k

E kŽ a , a.

Dim L Ž a , a. E kŽ a , a. .

607

GRADED LIE SUPERALGEBRAS

On the other hand, the right-hand side yields

log

ž

1 1 y T G= A

s log

/ ž

1 1y

Ý`i , js1 t

s s

Ý ks1

k

`

1

Ý ks1

s

k

i

j

i , js1

žÝ

/ /

k

`

t Ž t i , bj . E

Žt i , b j .

i , js1

Ý ss Ž s i j . si jgZ G 0 Ýs i jsk

/

Ž Ýsi j . ! s t Ž t i , bj . Ł ž Ł Ž si j ! . Ž < s < y 1. !

Ý

ž

Ý

W Ž t , b . E Žt , b. .

Žt , b .gG= A

s

j

`

ž

1

i

Ý t Ž t i , bj . E Ž t , b .

s ylog 1 y `

Ž t i , bj . E Ž t , b .

Ý

s!

sgT Žt , b .

ij

/E

Ý s i j Žt i , b j .

Ł t Ž t i , bj .

si j

/

E Žt , b.

Žt , b .gG= A

Therefore, we have W Žt , b. s

1

Ý k)0 Žt , b .sk Ž a , a .

k

Dim L Ž a , a. .

Hence, by Mobius inversion, we obtain ¨ Dim L Ž a , a. s

1

Ý d)0 Ž a , a .sd Žt , b .

d

mŽ d. W Žt , b. .

Remark. For a fixed d g Z ) 0 , there may exist more than one Žt , b . g G = A satisfying Ž a , a. s dŽt , b .. In this case, we need to take the sum over all of those Žt , b .’s. As a special case of Theorem 1.2, we obtain a superdimension formula for ordinary Ž G = Z 2 .-graded Lie superalgebras.

608

SEOK-JIN KANG

COROLLARY 1.3. Let L s [Ž a , i.g G=Z 2 L Ž a , i. be a Ž G = Z 2 .-graded Lie superalgebra with dim L Ž a , i. - ` for all a g G, i g Z 2 . Suppose we ha¨ e a denominator identity

Ł

Ž a , i .gG=Z 2

Ž 1 y E Ž a , i. .

Dim L Ž a , i.

s 1 y T G= Z 2

for some formal power series T G= Z 2 s ÝŽ a , i.g G=Z 2 t Ž a , i . E Ž a , i. with t Ž a , i . g Z for all Ž a , i . g G = Z 2 . Then we ha¨ e Dim L Ž a , 0. s

1

Ý d)0 a sdt

d

1

m Ž d . W Ž t , 0. q

Ý d)0, even a sdt

Dim L Ž a , 1. s

1

Ý d)0, odd a sdt

d

d

m Ž d . W Ž t , 1 . , Ž 1.18.

m Ž d . W Ž t , 1. .

Ž 1.19.

Proof. By Theorem 1.2, we have 1

Dim L Ž a , i. s

Ý d)0 Ž a , i .sd Žt , k .

d

mŽ d.W Žt , k . ,

for a , t g G and i, k g Z 2 . If Ž a , 0. s dŽt , k . for some d g Z ) 0 , then a s dt and dk ' 0 Žmod 2.. Hence, if k s 0, then d can be any positive integer dividing a , and if k s 1, then d must be even, dividing a , which yields Ž1.18.. Similarly, if Ž a , 1. s dŽt , k ., then k s 1 and d must be odd, dividing a . Hence we obtain Ž1.19.. Suppose we have the denominator identity Ž1.14. for a Ž G = A .-graded u-colored Lie superalgebra L s [Ž a , a.g G=AA L Ž a , a. . For each a g G, let L a s Ý ag A L Ž a , a. , and assume that the superdimension Dim L a s Ý ag A Dim L Ž a , a. is well-defined. Then we get a G-graded Lie superalgebra L s [a g G L a with dim L a - ` for all a g G. Furthermore, assume that def

t Ž a . s Ý ag A t Ž a , a. is also well-defined. Then, by specializing E Ž a , a. s E a for all a g A, we obtain a formal power series T G s Ý a g G t Ž a . E a with t Ž a . g Z for all a g G, and the denominator identity Ž1.14. yields a product identity

Ł Ž 1 y E a . DIm L

agG

a

s 1 y TG s 1 y

Ý t Ž a . Ea.

a gG

Ž 1.20.

609

GRADED LIE SUPERALGEBRAS

The identity Ž1.20. will also be called the denominator identity for the G-graded Lie superalgebra L s [a g G L a . Let P Ž G . s  a g G N t Ž a . / 04 , and let t i N i s 1, 2, 3, . . . 4 be an enumeration of the set P Ž G .. For t g G, define `

T Ž t . s s s Ž si . is1 N si g Z G 0 ,

½

Ý sit i s t 5 ,

Ž 1.21.

which is the set of all partitions of t into a sum of t i ’s. Again, by our finiteness condition Ž1.5., the set T Žt . must be finite. For a partition s g T Žt ., we will also use the notation < s < s Ýsi and s!s ŁŽ si!., and for t g G, we define the Witt partition function W Žt . by

Ž < s < y 1. !

W Žt . s

Ý

s!

sgT Žt .

Ł t Žti .

si

.

Ž 1.22.

Then, by Theorem 1.2, we obtain a closed form formula for the superdimensions Dim L a for all a g G. PROPOSITION 1.4. Let L s [a g G L a be a G-graded Lie superalgebra with dim L a - ` for all a g G, and suppose we ha¨ e a denominator identity

Ł Ž 1 y E a . Dim L

agG

a

s 1 y TG

for some formal power series T G s Ý a g G t Ž a . E a with t Ž a . g Z for all a g G. Then, for any a g G, we ha¨ e Dim L a s

1

Ý d)0 a sdt

d

mŽ d.W Žt . ,

Ž 1.23.

where m is the classical Mobius function. ¨ Remark. As in Theorem 1.2, for a fixed d g Z ) 0 , there may exist more than one t g G satisfying a s dt . In this case, we need to take the sum over all of those t ’s.

2. FREE LIE SUPERALGEBRAS 2.1. Generalized Witt Formula Let A be a countable abelian group with a coloring map u : A = A ª C= and G be a countable abelian semigroup satisfying the finiteness condition Ž1.5.. Let V s [Ž a , a.g G=AA VŽ a , a. be a Ž G = A .-graded u-colored super-

610

SEOK-JIN KANG

space with finite dimensional homogeneous subspaes. We denote the superdimension of VŽ a , a. by t Ž a , a. s Dim VŽ a , a. s c Ž a.dim VŽ a , a. . Then the Ž G = A .-graded supercharacter of V can be written as Ch G= A V s

Ý

t Ž a , a . E Ž a , a. .

Ž 2.1.

Ž a , a.gG= A

Let L be the free Lie superalgebra generated by V. Then L has a Ž G = A .-gradation L s [Ž a , a.g G=AA L Ž a , a. induced by V, and as we have seen in Section 1, the Poincare]Birkhoff]Witt Theorem yields ´ Ch G= A U Ž L . s

Ł

Ž a , a.gG= A

Ž 1 y EŽ a , . .

yD im L Ž a , a.

.

Ž 2.2.

On the other hand, since L is free on V, the universal enveloping algebra UŽ L . of L is the tensor algebra T Ž V . s C [ V [ V m2 [ ??? of V. It follows that 2

Ch G= A U Ž L . s Ch G=AA T Ž V . s 1 q Ch G=AA V q Ž Ch G=AA V . q ??? s

1 1 y Ch G= A V

s

1 1 y ÝŽ a , a.g G=AA t Ž a , a . E Ž a , a.

. Ž 2.3.

Hence we obtain a denominator identity for the Ž G = A .-graded free Lie superalgebra L s [Ž a , a.g G=AA L Ž a , a. :

Ł

Ž a , a.gG= A

Ž 1 y E Ž a , a. .

Dim L Ž a , a.

s 1 y Ch G= A V s1y

Ý

t Ž a , a . E Ž a , a. . Ž 2.4.

Ž a , a .gG= A

Let P Ž V, G = A . s Ž a , a. g G = A < dim VŽ a , a. / 04 and let Žt i , bj . < i, j s 1, 2, 3, . . . 4 be an enumeration of the set P Ž V, G = A .. For each Žt , b . g G = A, we denote by T Žt , b . the set of all partitions of Žt , b . into a sum of Žt i , bj .’s as defined in Ž1.15., and let W Žt , b . be the Witt partition function as defined in Ž1.16.. Then our superdimension formula Ž1.17. yields the following generalized version of the Witt formula for the free Lie superalgebras generated by Ž G = A .-graded superspaces: THEOREM 2.1 Žcf. wKa5x.. Let A be a countable abelian group with a coloring map u : A = A ª C= and G be a countable abelian semigroup satisfying the finiteness condition Ž1.5.. Let V s [Ž a , a.g G=AA VŽ a , a. be a Ž G = A .-graded u-colored superspace with finite dimensional homogeneous subspaces, and let L s [Ž a , a.g G=AA L Ž a , a. be the free Lie superalgebra generated by V.

611

GRADED LIE SUPERALGEBRAS

Then for any Ž a , a. g G = A, we ha¨ e 1

Dim L Ž a , a. s

Ý d)0 Ž a , a .sd Žt , b .

d

mŽ d. W Žt , b. ,

Ž 2.5.

where m is the classical Mobius function. ¨ COROLLARY 2.2. Let V s [Ž a , i.g G=Z 2 VŽ a , i. be a Ž G = Z 2 .-graded superspace with finite dimensional homogeneous subspaces and let L s [Ž a , i.g G=Z 2 L Ž a , i. be the free Lie superalgebra generated by V. Then for any a g G, we ha¨ e Dim L Ž a , 0. s

1

Ý d)0 a sdt

d

m Ž d . W Ž t , 0. q

1

Ý d)0, even a sdt

1

Dim L Ž a , 1. s

Ý d)0, odd a sdt

d

d

m Ž d . W Ž t , 1 . , Ž 2.6.

m Ž d . W Ž t , 1. ,

Ž 2.7.

where m is the classical Mobius function. ¨ Furthermore, let Va s [ag A VŽ a , a. and L a s [ag A L Ž a , a. . Then V becomes a G-graded superspace, and the free Lie superalgebra L on V def has a G-gradation L s [a g G L a induced by V. If t Ž a . s Ý ag A t Ž a , a. is well-defined, then Dim Va and Dim L a are also well-defined, and by specializing E Ž a , a. s E a for all a g A in Ž2.4., we obtain a denominator identity for the free Lie superalgebra L s [a g G L a :

Ł Ž 1 y E a . Dim L

agG

a

s 1 y Ch G V s 1 y

Ý t Ž a . Ea.

Ž 2.8.

a gG

As in Section 1, let P Ž V, G . s  a g G < t Ž a . / 04 and let t i < i s 1, 2, 3, ??? 4 be an enumeration of P Ž V, G .. For t g G, define the set T Žt . of all partitions of t into a sum of t i ’s as in Ž1.21. and define the Witt partition function W Žt . by Ž1.22.. Then, by Theorem 2.1, we obtain: PROPOSITION 2.3. Let V s [a g G Va be a G-graded superspace with finite dimensional homogeneous subspaces and let L s [a g G L a be the free Lie superalgebra generated by V. Then for any a g G, we ha¨ e Dim L a s

1

Ý d)0 a sdt

d

where m is the classical Mobius function. ¨

mŽ d.W Žt . ,

Ž 2.9.

612

SEOK-JIN KANG

Remark. The formulas Ž2.5. ] Ž2.7. and Ž2.9. will be called the generalized Witt formulas for free Lie superalgebras. EXAMPLE 2.4. In this example, we will consider the simplest generalization of the classical Witt formula for free Lie algebras to free Lie superalgebras generated by finite dimensional vector spaces. Consider the superspace V s VŽ0. [ VŽ1. with dim VŽ0. s r and dim VŽ1. s s for some r, s g Z ) 0 , and let L be the free Lie superalgebra generated by V. Then, by setting deg ¨ s 1 for all ¨ g V, the free Lie superalgebra L has a ` Z ) 0-gradation L s [ns1 L n induced by V. Since P Ž V, Z ) 0 . s  14 and Dim V s r y s, the denominator identity for L is equal to `

Ł Ž 1 y q n . Dim L

n

s 1 y Ž r y s . q,

ns1

and the Witt partition function W Ž n. is given by

Ž n y 1. !

W Ž n. s

n!

n

Ž r y s. s

1 n

n

Ž r y s. .

Hence, by the generalized Witt formula Ž2.9., we have Dim L n s

1 n

Ý m Ž d . Ž r y s . nrd .

Ž 2.10.

d
Therefore the denominator identity for L yields the following product identity: 1

`

Ł Ž1 y q . n

n

Ý d< n mŽ d.r n r d

s 1 y rq

for all r g Z.

Ž 2.11.

ns1

On the other hand, the superspace V s VŽ0. [ VŽ1. can be regarded as the ŽZ ) 0 = Z 2 .-graded superspace V s VŽ1, 0. [ VŽ1, 1. with Dim VŽ1, 0. s r, Dim VŽ1, 1. s ys. Hence the free Lie superalgebra L generated by V has a ŽZ ) 0 = Z 2 .-gradation L s [Ž n, i.g Z =Z L Ž n, i. induced by V, and by )0 2 letting Q s E Ž1, 0., z s E Ž0, 1., we obtain the denominator identity for the ŽZ ) 0 = Z 2 .-graded free Lie superalgebra L s [Ž n, i.g Z =Z L Ž n, i. : )0 2 `

Ł Ž1 y qn.

ns1

where z 2 s 1.

Dim L Ž n , 0.

`

Ł Ž 1 y q n z . Dim L

ns1

Ž n , 1.

s 1 y q Ž r y sz . ,

613

GRADED LIE SUPERALGEBRAS

Therefore, by Ž2.6. and Ž2.7., we obtain n

Dim L Ž n , 0. s

1 n

2d

Ý mŽ d.

d
Ý js0

n

n

0

d rd 2j n

y

s

2d

1

Ý

n

mŽ d.

d
Ý js0

y2 j

s2 j

n

n

 0

d rd 2jq1

y2 jy1

s 2 jq1

n

1

d mŽ d. Ž r y s. Ý n

d
n

q

2d

1

Ý

n

mŽ d.

d
Ý js0 n

Dim L Ž n , 1. s y

2d

1

Ý

n

mŽ d.

d
Ý js0

n

n

 0  0

d rd 2jq1 n

n

d rd 2jq1

y2 jy1

y2 jy1

s 2 jq1 ,

s 2 jq1 .

Moreover, combining the denominator identity with Ž2.11., we obtain a product identity `

Ł

ns1

ž

1

1 y qn 1 y qnz

s

/

n

nrd n r2 d x Ý d < n , d : odd m Ž d . Ý wjs 0 2 jq1

1 y q Ž r y sz . 1 y Ž r y s. q

Ž

.r

Ž n r d .y 2 jy 1 2 jq 1

s

,

Ž 2.12.

where z 2 s 1, r, s g Z ) 0 . Remark. Since the identity Ž2.11. holds for all r g Z, we have `

Ł

1

Ž1 y a q . n n

n

Ý d< n mŽ d.r n r d

s 1 y raq

ns1

s

1

`

Ł Ž1 y q . n

n

Ý d < n mŽ d. an r dr n r d

ns1

for all a, r g Z, which was called the exchange principle in wKaK1x.

Ž 2.13.

614

SEOK-JIN KANG

2.2. Free Lie Algebras and Free Lie Superalgebras In this subsection, we will discuss the relation of the free Lie algebra and free Lie superalgebra generated by the same vector space. Let A be a countable abelian group with a coloring map u : A = A ª C= and G be a countable abelian semigroup satisfying the finiteness condition Ž1.5.. Let V s [Ž a , a.g G=AA VŽ a , a. be a Ž G = A .-graded u-colored superspace with finite dimensional homogeneous subspaces. First, by neglecting the coloring map u on A, consider V as a Ž G = A .-graded vector space Žnot a superspace., and let L s [Ž a , a.g G=AA LŽ a , a. be the free Lie algebra generated by V. Hence the denominator identity for L is equal to

Ł

Ž a , a.gG= A

Ž 1 y e Ž a , a. .

dim L Ž a , a.

s1y

Ý

Ž a , a .gG= A

dim VŽ a , a. e Ž a , a. . Ž 2.14.

On the other hand, by taking the coloring map u into account, consider V as a Ž G = A .-graded u-colored superspace, and let L s [Ž a , a.g G=AA L Ž a , a. be the free Lie superalgebra generated by V. Then the denominator identity for the free Lie superalgebra L is the same as

Ł

Ž a , a .gG= A

Ž 1 y E Ž a , a. .

dim L Ž a , a.

s1y

t Ž a , a . E Ž a , a. , Ž 2.15.

Ý

Ž a , a .gG= A

where t Ž a , a. s Dim VŽ a , a. s c Ž a.dim VŽ a , a. and E Ž a , a. s c Ž a. e Ž a , a.. Note that the right-hand sides of the identities Ž2.14. and Ž2.15. are the same. Let C Ž a , a. s c Ž a.dim LŽ a , a. . Then the left-hand side of Ž2.14. is equal to

Ł

Ž a , a .gG= A0

s s

Ž 1 y E Ž a , a. .

C Ž a , a.

Ł

Ž b , b .gG= A1

Ł

Ž 1 y E Ž a , a. .

Ł

Ž 1yE Ž a , a. .

Ž a , a .gG= A

Ž a , a.gG= A

C Ž a , a.

Ž 1 q E Ž b , b. .

Ł

Ž b , b .gG= A1

c Ž a .dim L Ž a , a.

yC Ž b , b .

Ž 1 y E 2Ž b , b. .

Ł

Ž a , a.s2 Ž b , b . Ž b , b .gG= A1

yC Ž b , b .

Ž 1yE Ž a , a. .

dim L Ž b , b.

.

Hence the denominator identity for the free Lie superalgebra L is equal to

Ł

Ž a , a .gG= A

Ž 1 y E Ž a , a. .

s1y

Ý

Ž a , a .gG= A

c Ž a .dim L Ž a , a.

Ł

Ž a , a .s2 Ž b , b . Ž b , b .gG= A1

t Ž a , a . E Ž a , a. .

Ž 1 y E Ž a , a. .

dim L Ž b , b.

Ž 2.16.

615

GRADED LIE SUPERALGEBRAS

Therefore, we obtain: PROPOSITION 2.5. Let A be a countable abelian group with a coloring map u : A = A ª C= and G be a countable abelian semigroup satisfying the finiteness condition Ž1.5.. For a Ž G = A .-graded ¨ ector space V s [Ž a , a.g G=AA VŽ a , a. with finite dimensional homogeneous subspaces, let L s [Ž a , a.g G=AA LŽ a , a. be the free Lie algebra generated by V, and let L s [Ž a , a.g G=AA L Ž a , a. be the u-colored free Lie superalgebra generated by V. Then, for any Ž a , a. g G = A, we ha¨ e Dim L Ž a , a. s c Ž a . dim LŽ a , a. q

Ý

Ž a , a.s2 Ž b , b . Ž b , b .gG= A1

dim LŽ b , b. .

Ž 2.17.

EXAMPLE 2.6. Let A be a countable abelian group with a coloring map u and V s [ag A Va be a u-colored superspace. Suppose dim VŽ0. s r and dim VŽ1. s s, where VŽ0. s [ag A0 Va , VŽ1. s [b g A1 Vb . Let  x 1 , . . . , x r 4 be a basis of VŽ0. and  x rq1 , . . . , x rqs 4 be a basis of VŽ1. . Consider V as a vector space and let L be the free Lie algebra rqs  4 Ž . generated by V. Let G s Z G 0 R 0 . By defining deg x i s 0, . . . , 1, . . . , 0 with 1 in the ith place, L becomes a G-graded Lie algebra. For each a s Ž a 1 , . . . , a rqs . g G, set < a < s Ý a i and a !s Ł a i!. Since P Ž V, G . s Ž1, 0, . . . , 0., . . . , Ž0, . . . , 0, 1.4 , we have T Ž a . s Ž a 1 , . . . , a rqs , 0, . . . .4 and Ž < a < y 1. ! WŽ a . s . Hence the generalized Witt formula Ž2.9. yields a!

dim La s

1

Ý

d
d

mŽ d.W

a

ž / d

s

1
< ard < !

Ý m Ž d . Ž ard . !

d
Žcf. wBMPZ, Chap. 2, Theorem 1.16; Kanx.. On the other hand, consider V as a u-colored superspace and let L be the u-colored free Lie superalgebra generated by V. The Lie superalgebra L has a G-gradation L s [a g G L a induced by our definition of the degrees of x i ’s. Note that our choice of degrees of x i ’s defines a coloring map on G. That is, a s Ž a 1 , . . . , a rqs . g G is even Žresp. odd. if and only if a rq1 q ??? qa rqs is even Žresp. odd.. We denote by G0 Žresp. G1 . the set of even Žresp. odd. elements of G. Then, by Proposition 2.5, we obtain Dim L a s

½

dim L a q dim La r2

if a s 2 b for some b g G1 ,

c Ž a . dim La

otherwise.

616

SEOK-JIN KANG

If a s 2 b for some b g G1 , then all a i are even and 12 Ž a rq1 q ??? qa rqs . rqs  4 is odd. Therefore, for a s Ž a 1 , . . . , a rqs . g G s Z G 0 R 0 , we have

¡dim L q dim L ~ s ¢c Ž a . dim L a

Dim L a

a r2

if all a i are even and

1 2

s a rqk . Ž Ý ks1

is odd, otherwise,

a

which recovers the formula in wBMPZ, Chap. 2, Corollary 2.8x. ŽThere is a minor sign error in their formula.. 2.3. 1-Dimensional Generalization In this subsection, we will consider the 1-dimensional generalization of the classical Witt formula for free Lie algebras to free Lie superalgebras. That is, we will discuss the applications of the generalized Witt formula to the Z ) 0-graded free Lie superalgebras. Let TZ ) 0Ž q . s Ý`is1 t Ž i . q i be a formal power series with t Ž i . g Z for all i G 1. As we have seen in Section 1, the series TZ ) 0Ž q . can be interpreted ` as the supercharacter of a Z ) 0-graded superspace V s [is1 Vi with Dim Vi s t Ž i . g Z Ž i G 1.. More precisely, take a Z-graded vector space ` V s [is1 Vi with dim Vi s < t Ž i .<, and let VŽ0. s [tŽ i.) 0 Vi , VŽ1. s [tŽ i.- 0 Vi . Then V becomes a superspace with Dim Vi s t Ž i . for all i G 1. Let L be the free Lie superalgebra generated by V. We will apply the generalized Witt formulas Ž2.5. ] Ž2.7. and Ž2.9. to this setting. First, we consider the Z ) 0-gradation on L. Note that P Ž V, Z ) 0 . s Z ) 0 s  1, 2, 3, . . . 4 , and for n G 1, we have T Ž n . s s s Ž si . G1 si g Z G 0 ,

½

Ý isi s n 5 ,

Ž 2.18.

the set of all partitions of n into a sum of positive integers. Thus the Witt partition function W Ž n. is given by W Ž n. s

Ž < s < y 1. !

Ý sgT Ž n .

s!

Ł tŽ i.

si

.

Therefore, by Proposition 2.3, we obtain the following 1-dimensional generalization of the Witt formula: ` PROPOSITION 2.7. Let V s [is1 Vi be a Z ) 0-graded superspace o¨ er C ` with Dim Vi s t Ž i . g Z for all i G 1, and let L s [ns1 L n be the free Lie superalgebra generated by V with Z ) 0-gradation induced by V.

617

GRADED LIE SUPERALGEBRAS

Then we ha¨ e

Dim L n s

Ž < s < y 1. !

1

Ý d mŽ d. Ý

d
sgT

s!

n

Ł tŽ i.

si

.

Ž 2.19.

ž / d

Next, we consider the ŽZ ) 0 = Z 2 .-gradation on L. By defining « Ž i . s 0 if t Ž i . ) 0 and « Ž i . s 1 if t Ž i . - 0, we obtain a ŽZ ) 0 = Z 2 .-gradation V s [Ž i, k .g Z ) 0=Z 2 VŽ i, k . such that P Ž V, Z ) 0 = Z 2 . s Ž i, « Ž i .. < i s 1, 2, 3, . . . 4 and Dim VŽ i, k . s t Ž i . for all i G 1, k g Z 2 . Then the free Lie superalgebra L on V has a ŽZ ) 0 = Z 2 .-gradation induced by V, and for each n g Z ) 0 , we have T Ž n, 0 . s s s Ž si . iG1 si g Z G 0 ,

½ s ½s s Ž s .

i iG1

½

s s g T Ž n.

si g Z G 0 ,

Ý

Ý si Ž i , « Ž i . . s Ž n, 0. 5 Ý isi s n, Ý si « Ž i . s 0 mod 2. 5

5

si is even .

« Ž i .s1

Similarly, T Ž n, 1. s  s g T Ž n. < Ý« Ž i.s1 si is odd4 . For s s Ž si . g T Ž n., set < s
Wq Ž n . s W Ž n, 0 . s

Ž < s < y 1. !

Ý sgT Ž n . < s
Wy Ž n . s W Ž n, 1 . s

s!

Ž < s < y 1. !

Ý sgT Ž n . < s
s!

Ł tŽ i. Ł tŽ i.

si

si

,

.

Note that WqŽ n. q WyŽ n. s W Ž n.. By Corollary 2.2, we obtain the following twisted 1-dimensional generalization of the Witt formula: PROPOSITION 2.8. Let V s [Ž i, k .g Z ) 0=Z 2 VŽ i, k . be a ŽZ ) 0 = Z 2 .-graded superspace with Dim VŽ i, k . s t Ž i . g Z for all i G 1, and let L s [Ž n, k .g Z ) 0=Z 2 L Ž n, k . be the free Lie superalgebra generated by V with ŽZ ) 0 = Z 2 . gradation induced by V.

618

SEOK-JIN KANG

Then we ha¨ e Dim L Ž n , 0. s

s

1

Ý d m Ž d . Wq

d
n

ž / d

d
sgT

Ý

d
d

Ž < s < y 1. !

1

Ý d mŽ d. Ý

1

q

Ł tŽ i.

s!

n

m Ž d . Wy

n

ž / d

si

ž /

d < s
q

1

Ý

d

d
Dim L Ž n , 1. s

s

d
d
s!

n

Ł tŽ i.

si

, Ž 2.20.

ž /

d < s
d 1

Ý

Ý sgT

1

Ý

Ž < s < y 1. !

mŽ d.

d

m Ž d . Wy

mŽ d.

n

ž / d

Ž < s < y 1. !

Ý sgT

s!

n

Ł tŽ i.

si

.

Ž 2.21.

ž /

d < s
2.4. 2-Dimensional Generalization In this subsection, we will discuss the applications of the generalized Witt formula to the Z 2-graded free Lie superalgebras. Let TZ 2 Ž p, q . s Ý`i, js1 t Ž i, j . p iq j be a formal power series with t Ž i, j . g Z for all i, j G 1, and let V s [`i, js1 VŽ i, j. be a Z 2-graded superspace with Dim VŽ i, j. s t Ž i, j .. Thus an element Ž i, j . g Z 2 is even Žresp. odd. if t Ž i, j . ) 0 Žresp. t Ž i, j . - 0.. Let L be the free Lie superalgebra generated by V. We will apply the generalized Witt formulas Ž2.5. ] Ž2.7. and Ž2.9. to this setting. First, we consider the Z 2-gradation on L . Note that P Ž V , Z 2 . s Z ) 0 = Z ) 0 s  Ž i , j . i , j s 1, 2, 3, . . . 4 , and for m, n G 1, we have T Ž m, n . s s s Ž si j . i , j)1 si , j g Z G 0 ,

½

Ý si j Ž i , j . s Ž m, n . 5 , Ž 2.22.

the set of all partitions of Ž m, n. into a sum of ordered pairs of positive integers. Thus the Witt partition function W Ž m, n. is equal to W Ž m, n . s

Ž < s < y 1. !

Ý sgT Ž m , n .

s!

Ł tŽ i, j.

si j

,

619

GRADED LIE SUPERALGEBRAS

where < s < s Ýsi j and s!s Ł si j!. Therefore, by Proposition 2.3, we obtain the following 2-dimensional generalization of the Witt formula: 2 PROPOSITION 2.9. Let V s [`i, js1 VŽ i, j. be a Z ) 0 -graded superspace o¨ er ` C with Dim VŽ i, j. s t Ž i, j . g Z for all i, j G 1, and let L s [m, ns1 L Ž m, n. 2 be the free Lie superalgebra generated by V with Z -gradation induced by V with Z 2-gradation induced by . Then we ha¨ e

Dim L Ž m , n. s

1

Ý

d <Ž m , n .

d

Ž < s < y 1. !

mŽ d.

Ý

m n sgT , d d

ž

s!

Ł tŽ i, j.

si j

. Ž 2.23.

/

2 . Ž . Next, we consider the ŽZ ) 0 = Z 2 -gradation on L. By defining « i, j s 2 Ž . Ž . Ž . Ž 0 if t i, j ) 0 and « i, j s 1 if t i, j - 0, we obtain a Z ) 0 = Z 2 .gradation on V s [Ž i, j, k .g Z 2=Z 2 VŽ i, j, k . such that 2 P ŽV , Z) 0 = Z 2 . s  Ž i , j, « Ž i , j . . i , j s 1, 2, 3, . . . 4 2 and Dim VŽ i, j, k . s t Ž i, j . for all i, j G 1, k g Z 2 , which induces a ŽZ ) 0= Z 2 .-gradation on L. For s s Ž si j . g T Ž m, n., set < s
T Ž m, n, 0 . s  s g T Ž m, n . < s
Ž < s < y 1. !

Ý sgT Ž m , n . < s
Wy Ž m, n . s W Ž m, n, 1 . s

s!

Ž < s < y 1. !

Ý sgT Ž m , n . < s
s!

Ł tŽ i, j. Ł tŽ i, j.

si j

si j

,

.

Therefore, by Corollary 2.2, we obtain the following twisted 2-dimensional generalization of the Witt formula: 2 . PROPOSITION 2.10. Let V s [Ž i, j, k .g Z 2) 0=Z 2 VŽ i, j, k . be a ŽZ ) 0 = Z2 graded superspace with Dim VŽ i, j, k . s t Ž i, j . g Z for all i, j G 1, and let L 2 . be the free Lie superalgebra generated by V with ŽZ ) 0 = Z 2 -gradation L s [Ž m, n, k .g Z 2) 0=Z 2 L Ž m, n, k . induced by V.

620

SEOK-JIN KANG

Then we ha¨ e Dim L Ž m , n , 0. s

1

Ý

d <Ž m , n .

d 1

s

Ý

d <Ž m , n .

q

d

m Ž d . Wq

ž

m n , q d d

/

Ý

m n , d d < s
sgT

1 d <Ž m , n . d: even

d

ž

d

d <Ž m , n . d : even

Ž < s < y 1. !

mŽ d.

Ý

1

Ý

s!

Ł tŽ i, j.

Ý

m n , d d < s
sgT

ž

m n , d d

/

si j

/ Ž < s < y 1. !

mŽ d.

m Ž d . Wy

ž

s!

Ł tŽ i, j.

si j

,

/ Ž 2.24.

Dim L Ž m , n , 1. s

s

1

Ý

d <Ž m , n . d : odd

d 1

Ý

d <Ž m , n . d: odd

d

m n , d d

m Ž d . Wy

ž

mŽ d.

Ý

/ Ž < s < y 1. !

m n sgT , d d < s
ž

s!

Ł tŽ i, j.

si j

. Ž 2.25.

/

Remark. The above discussion can be generalized to the free Lie n superalgebras generated by Z ) 0 -graded superspaces Vs

[

n Ž i 1 , . . . , i n .gZ ) 0

VŽ i1 , . . . , i n .

such that Dim VŽ i1 , . . . , i n . s t Ž i1 , . . . , i n . g Z

n for all Ž i1 , . . . , i n . g Z ) 0.

For example, the exchange principle Ž2.13. can be generalized to the n-dimensional exchange principle. More precisely, for k 1 , . . . , k n g Z ) 0 , let T Ž k 1 , . . . , k n . s  s s Ž si1 , . . . , i n . si1 , . . . , i n g Z G 0 ,

Ý si , . . . , i Ž i1 , . . . , i n . s Ž k 1 , . . . , k n . 4 , 1

n

621

GRADED LIE SUPERALGEBRAS

the set of all partitions of Ž k 1 , . . . , k n . into a sum of ordered n-tuples of positive integers, and let

Ž < s < y 1. !

Sr Ž k 1 , . . . , k n . s

Ý

s!

sgT Ž k 1 , . . . , k n .

r < s< ,

where < s < s Ýsi1 , . . . , i n and s!s Ł si1 , . . . , i n!. Then, for any r, a1 , . . . , a n g Z, we have `

Ł

1

k 1 , . . . , k n s1

s

Ž 1 y a1k `

Ł

k 1 , . . . , k n s1

1

??? a nk n q1k 1 ??? qnk n .

Ý d <Ž k 1 , . . . , k n .

1

Ž 1 y q1k

1

??? qnk n .

Ý d <Ž k 1 , . . . , k n .

d

d

mŽ d.Sr

ž

k1 d

,...,

kn d

m Ž d . a1k 1 r d ??? a nk n r d S r Ž

/ k1 d

,???,

kn d

.

.

2.5. Product Identities The purpose of this subsection is to investigate the relation of graded Lie superalgebras and product identities for normalized formal power series. Let us begin with the binomial expansion r

r

Ž1 y q. s

Ý Ž y1. k ks0

r k q . k

ž/

Ž 2.27.

Let L s C x 1 [ ??? [ C x r be the r-dimensional abelian Lie algebra with basis  x 1 , . . . , x r 4 . Then we have Hk Ž L . s Lk Ž L . s Span x i1 n ??? n x i k 1 F i1 - ??? - i k F r ,

½

5

and hence the identity Ž2.27. can be interpreted as the Euler]Poincare ´ principle for the abelian Lie algebra L. Similarly, consider the product expansion

Ž1 y q.

yr

s

`

Ý ks0

ž

rqky1 k q , k

/

Ž 2.28.

and let L s C y 1 [ ??? [ C yr be the abelian Lie superalgebra generated by the odd elements  y 1 , . . . , yr 4 . Then we have Hk Ž L . s Ck Ž L . s S k Ž L . s Span  yi1 yi 2 ??? yi k < 1 F i1 F i 2 F ??? F i k F r 4 , and therefore, the identity Ž2.28. can be interpreted as the Euler]Poincare ´ principle for the abelian Lie superalgebra L.

622

SEOK-JIN KANG

In wK2x, Jacobi’s triple product identity `

k Ž ky1 .

Ł Ž1 y p

q . Ž1 y p

n n

ny1 n

q

.Ž 1 y p

n ny1

q

ns1

. s Ý Ž y1.

k

p

2

k Ž kq1 .

q

2

kgZ

Ž 2.29. was interpreted as the denominator identity for affine Kac]Moody algebra Žcf. wK4x.. In fact, Kac showed that all the Macdonald identities of type AŽ1. 1 wMx are equivalent to the denominator identities for affine Kac]Moody algebras wK2x. Furthermore, in wKW1, KW2x, Kac and Wakimoto investigated the relation of affine Lie superalgebras and many interesting product identities arising from number theory. Recently, Borcherds completed the proof of the Moonshine Conjecture by constructing an infinite dimensional Lie algebra, called the Monster Lie algebra wB3x. One of the main ingredients of his proof is the following product identity py1

Ž . Ł Ž 1 y p mq n . c m n s j Ž p . y j Ž q . ,

Ž 2.30.

m)0 ngZ

where the cŽ n. are the coefficients of the elliptic modular function J Ž q . s j Ž q . y 744 s

`

Ý c Ž n. q n nsy1

y1

sq

q 196884 q q 21493760 q 2 q ??? .

Ž 2.31.

It was shown in wB3x that the Monster Lie algebra is a generalized Kac]Moody algebra Žcf. wB2x. and the identity Ž2.30. was interpreted as the denominator identity for the Monster Lie algebra. More generally, let G be a countable abelian semigroup satisfying the finiteness condition Ž1.5., and consider a normalized formal power series 1 y TG s 1 y

Ý t Ž a . Ea

with t Ž a . g Z for all a g G. Ž 2.32.

agG

Suppose we have a product identity for the above formal power series,

Ł Ž1 y Ea . C a

Ž .

agG

s1y

Ý t Ž a . Ea

Ž 2.33.

a gG

with C Ž a . g Z for all a g G. If we could construct a ‘‘natural’’ G-graded Lie superalgebra L s [a g G L a such that the supercharacter of the homology superspace H Ž L . s Ý`ks 1Žy1. kq 1 Hk Ž L . is given by Ch H Ž L . s

Ý

agG

Dim H Ž L . a E a s

Ý t Ž a . Ea,

a gG

Ž 2.34.

623

GRADED LIE SUPERALGEBRAS

then the product identity Ž2.33. can be interpreted as the Euler]Poincare ´ principle, i.e., the denominator identity for the graded Lie superalgebra L s [a g G L a . In particular, we would have Dim L a s C Ž a .

for all a g G.

Ž 2.35.

In general, it is quite complicated and difficult to construct a ‘‘natural’’ graded Lie superalgebra corresponding to the product identity Ž2.33.. Still, we can always interpret the identity Ž2.33. as the denominator identity for a suitably defined free Lie superalgebra. More precisely, consider a G-graded vector space V s [a g G Va over C with dim Va s < t Ž a .< for all a g G. Let VŽ0. s [a : tŽ a .) 0 Va and VŽ1. s [a : tŽ a .- 0 Va . Then V s VŽ0. [ VŽ1. s [Ž a , i.g G=Z 2 VŽ a , i. becomes a Ž G = Z 2 .-graded superspace with Dim VŽ a , i. s t Ž a . for all a g G, i g Z 2 . Thus Dim Va s t Ž a . for all a g G and the right-handed side of Ž2.33. can be interpreted as 1 y Ch V. Let L be the free Lie superalgebra generated by V. Then the free Lie superalgebra L has a G-gradation induced by V, and the denominator identity for L s [a g G L a is the same as the product identity Ž2.33.. Hence we have Dim L a s C Ž a .

for all a g G.

On the other hand, let P Ž V, G . s  a g G < t Ž a . / 04 and let t 1 , t 2 , t 3 , . . . 4 be an enumeration of P Ž V, G .. Then, by the generalized Witt formula Ž2.9., we have Dim L a s

1

Ý d)0 a sdt

d

Ž < s < y 1. !

mŽ d.

Ý

sgT Žt .

s!

Ł t Žti .

si

,

which yields a combinatorial identity CŽ a . s

1

Ý d)0 a sdt

d

mŽ d.

Ž < s < y 1. !

Ý

sgT Žt .

s!

Ł t Žti .

si

.

Ž 2.36.

EXAMPLE 2.11. Ža. Consider the generating function for the partition function pŽ n.: `

`

`

ns1

is0

is1

Ł Ž 1 y q n . y1 s Ý p Ž i . q i s 1 y Ý Ž yp Ž i . . q i .

Ž 2.37.

Let V s [`is1 Vi be a Z ) 0-graded superspace with Dim Vi s ypŽ i . for all ` i G 1 Žthus VŽ0. s 0, VŽ1. s V ., and let L s [ns1 L n be the free Lie superalgebra generated by V. Then Ž2.37. can be interpreted as the

624

SEOK-JIN KANG

denominator identity for the free Lie superalgebra L, and hence we have Dim L n s y1

for all n G 1.

Therefore, by Ž2.36., we obtain

Ž < s < y 1. !

1

Ý d mŽ d. Ý

d
sgT

Ž y1.

s!

n

< s<

Ł pŽ i .

si

s y1.

Ž 2.38.

ž / d

Žb. Recall the definition of the Ramanujan tau-function: `

DŽ q. s q Ł Ž1 y q n .

24

s

ns1

`

Ý t Ž i. qi is1

s q y 24 q 2 q 252 q 3 y 1472 q 4 y ??? .

Ž 2.39.

We can rewrite it as `

`

ns1

is1

Ł Ž 1 y q n . 24 s 1 y Ý Ž yt Ž i q 1. . q i .

Ž 2.40.

Let V s [`is1 Vi be a Z ) 0-graded superspace with Dim Vi s yt Ž i q 1. for all i G 1 Žthus VŽ0. s [i: t Ž iq1.- 0 Vi , VŽ1. s [i: t Ž iq1.) 0 Vi ., and let ` L s [ns1 L n be the free Lie superalgebra generated by V. Then Ž2.40. can be interpreted as the denominator identity for the free Lie superalgebra L, and hence we have Dim L n s 24

for all n G 1.

Therefore, by Ž2.36., we obtain

Ž < s < y 1. !

1

Ý d mŽ d. Ý

d
sgT

n

s!

Ž y1.

< s<

Ł t Ž i q 1.

si

s 24.

Ž 2.41.

ž / d

The relation Ž2.41. allows us to determine the values of the Ramanujan tau-function t Ž n. recursively. Žc. Recall the product identity Ž2.30. for the elliptic modular function J Ž q . s jŽ q . y 744. Observing that cŽ0. s 0 and cŽyk . s 0 for k ) 1, it can be written as `

jŽ p. y jŽ q .

m, ns1

py1 y qy1

Ž . Ł Ž 1 y p mq n . c m n s

s1y

`

Ý c Ž i q j y 1 . p iq j . i , js1

Ž 2.42.

625

GRADED LIE SUPERALGEBRAS

2 Let V s [`i, js1 VŽ i, j. be a Z ) 0 -graded vector space over C with dim VŽ i, j. ` s cŽ i q j y 1. for i, j G 1, and let L s [m, ns1 LŽ m, n. be the free Lie algebra generated by V. Then the product identity Ž2.42. for the elliptic modular function J is the denominator identity for the free Lie algebra L, and we have dim LŽ m, n. s cŽ mn.. 2 Žd. Let V s [i,` js1 VŽ i, j. be a Z ) 0 -graded superspace over C with ` iqj Dim VŽ i, j. s Žy1. cŽ i q j y 1., and let L s [m, ns1 L Ž m, n. be the free Lie superalgebra generated by V. Replacing p and q by yp and yq in Ž2.42. yields

`

Ł Ž 1 y Ž y1.

mqn

m, ns1

p mq n .

cŽ m n.

s1y

`

Ý Ž y1. iqj c Ž i q j y 1. p iq j , i , js1

which can be regarded as the denominator identity for the free Lie superalgebra L. Observe that `

Ł Ž 1 y Ž y1.

mqn

m, ns1

p mq n .

cŽ m n.

`

Ž . Ł Ž 1 y p mq n . y1

s

mq n

cŽ m n.

Ž 1 y p m q n . c m nr4 . Ž

Ł

.

m, n: even mqn'2 Žmod 4 .

m , ns1

Hence the denominator identity for the free Lie superalgebra L s ` [m, ns1 L Ž m, n. is the same as `

Ž . Ł Ž 1 y p mq n . y1

mq n

cŽ m n.

m , ns1

s1y

Ł

Ž 1 y p mq n . c

Ž m n r4.

m, n: even mqn'2 Žmod 4 .

`

Ý Ž y1. iqj c Ž i q j y 1. p iq j ,

Ž 2.43.

i , js1

which implies

Dim L Ž m , n.

¡c Ž mn. s c mn ž4/ s~ ¢Ž y1. c Ž mn. mqn

if m, n are even, m q n ' 2 Ž mod 4 . , otherwise.

Note that this is a special case of Proposition 2.5. In wB4x, Borcherds gave a very important method for constructing automorphic forms on Osq 2, 2 ŽR.q with infinite product expansions. In

626

SEOK-JIN KANG

particular, let F Žt . s

s 1 Ž n . q n s q q 4 q 3 q 6 q 5 q ??? ,

Ý ngZ )0 , odd

2

u Žt . s

Ý

q n s 1 q 2 q q 2 q 4 q ??? ,

ngZ

and define

f 0 Ž t . sF Ž t . u Ž t . s

Ž u Ž t . 4 y2 F Ž t . .Ž u Ž t . 4 y16 F Ž t . . E6 Ž 4t . D Ž 4t .

q56u Ž t .

Ý c0 Ž n . q n s qy3 y 248q q 26752 q 4 y ??? , ngZ

g 0 Ž t . s Ž j Ž 4t . y 876 . u Ž t . y 2 F Žt . u Žt . s

Ý

Ž u Ž t . 4 y 2 F Ž t . .Ž u Ž t . 4 y 16 F Ž t . . E6 Ž 4t . D Ž 4t .

b 0 Ž n . q n s qy4 q 6 q 504 q q 143388q 4 y ??? .

ngZ

Then he proved the following product identities for the elliptic modular function j and the Eisenstein series E4 , E6 , E8 , E10 , and E14 : j Ž t . s qy1 q 744 q 196884 q q ??? s qy1

`

0

ns1

E4 Ž t . s 1 q 240 E6 Ž t . s 1 y 504

`

`

Ý s 3 Ž n . q n s Ł Ž 1 y q n . c Ž n .q8 , ns1

`

`

E10 Ž t . s 1 y 264

0

`

Ý

ns1

s7 Ž n. q n s

`

Ł Ž 1 y q n . 2 c Ž n .q16 , 2

0

ns1

ns1

`

`

Ý s 9 Ž n . q n s Ł Ž 1 y q n . b Ž n .qc Ž n .q8 , 0

ns1

E14 Ž t . s 1 y 24

2

Ý s5 Ž n . q n s Ł Ž 1 y q n . b Ž n . , ns1

E8 Ž t . s 1 q 480

2

0

ns1

`

2

2

0

ns1 `

Ý s13 Ž n . q n s Ł Ž 1 y q n . b Ž n .q2 c Ž n .q16 . 0

ns1

ns1

2

Ł Ž 1 y q n . 3c Žn .,

2

0

2

627

GRADED LIE SUPERALGEBRAS

The product identity for the modular function j can be written as `

2

Ł Ž 1 y q n . 3 c Ž n . s 1 q 744 q q 196884 q 2 q ??? 0

ns1

`

s1y

Ý Žyc1Ž i y 1. . q i ,

Ž 2.44.

is1

where the c1Ž i . are the coefficients of j. ŽNote that c1Ž0. s 744 and c1Ž n. s cŽ n. for all n / 0.. Let V s [`is1 Vi be the Z ) 0-graded superspace with Dim Vi s yc1Ž i y 1. for all i G 1 Žthus VŽ0. s 0, VŽ1. s V ., and ` let L s [ns1 L n be the free Lie superalgebra genreated by V. Then Ž2.44. can be interpreted as the denominator identity for the free Lie superalgebra L, and hence Dim L n s 3c 0 Ž n2 .

for all n G 1.

Therefore, we obtain a combinatorial identity:

3c 0 Ž n2 . s

Ž < s < y 1. !

1

Ý d mŽ d. Ý

d
sgT

s!

n

Ž y1.

< s<

Ł c1 Ž i y 1 .

si

. Ž 2.45.

ž / d

Similarly, the product identities for the Eisenstein series can be interpreted as the denominator identities for the free Lie superalgebras ` ` L s [ns1 L n generated by the Z ) 0-graded superspaces V s [is1 Vi with Dim Vi s y240 s 3 Ž i ., 504s5 Ž n., y480 s 7 Ž i ., 264s 9 Ž n., and 24s 13 Ž n., respectively. Therefore, the superdimensions of the homogeneous subspaces L n are given by Dim L n s c 0 Ž n2 . q 8 Dim L n s b 0 Ž n2 .

Ž corresponding to Ž corresponding to

Dim L n s 2 c 0 Ž n2 . q 16

E4 Ž t . . ,

E6 Ž t . . ,

Ž corresponding to

Dim L n s b 0 Ž n2 . q c 0 Ž n2 . q 8 Dim L n s b 0 Ž n2 . q 2 c 0 Ž n2 . q 16

E8 Ž t . . ,

Ž corresponding to

E10 Ž t . . ,

Ž corresponding to

E14 Ž t . . ,

628

SEOK-JIN KANG

and Ž2.36. yields the following combinatorial identities:

c0 Ž n2 . q 8 s

b 0 Ž n2 . s

Ž < s < y 1. !

1

Ý d mŽ d. Ý

d
sgT

s!

n

2 c0 Ž n2 . q 16 s

Ž < s < y 1. ! s!

n

sgT

s!

n

< s<

Ł s7 Ž i .

si

,

d

Ž < s < y 1. !

sgT

s!

n

s

264 < s < Ł s 9 Ž i . i ,

ž / d

Ž < s < y 1. !

1

Ý d mŽ d. Ý

d
Ž y480.

ž /

Ý d mŽ d. Ý

b 0 Ž n2 . q 2 c 0 Ž n2 . q 16 s

s

504 < s < Ł s5 Ž i . i ,

Ž < s < y 1. !

1

d
,

d

Ý d mŽ d. Ý

b 0 Ž n2 . q c0 Ž n2 . q 8 s

si

ž /

1

d
Ł s3 Ž i .

d

Ý d mŽ d. Ý sgT

< s<

ž /

1

d
Ž y240.

sgT

n

s!

s

24 < s < Ł s 13 Ž i . i .

ž / d

Ž 2.46.

3. GENERALIZED KAC]MOODY SUPERALGEBRAS 3.1. Weyl]Kac]Borcherds Character Formula Let I be a countable Žpossibly infinite . index set. A real square matrix A s Ž a i j . i, j g I is called a Borcherds]Cartan matrix if it satisfies Ži. a ii s 2 or a ii F 0 for all i g I; Žii. a i j F 0 if i / j, and a i j g Z if a ii s 2; Žiii. a i j s 0 implies a ji s 0. We say that an index i is real if a ii s 2 and imaginary if a i i F 0. We denote by I r e s  i g < a ii s 24 , I i m s  i g < a ii F 04 . Let m s Ž m i g Z ) 0 < i g I . be a collection of positive integers such that m i s 1 for all i g I r e. We call m the charge of A. In this paper, we assume that the Borcherds]Cartan matrix A is symmetrizable, i.e., there is a diagonal matrix D s diagŽ si < i g I . with si ) 0 Ž i g I . such that DA is symmetric. Let C s Ž u i j . i, j g I be a complex matrix satisfying u i j u ji s 1 for all i, j g I. Thus we have u ii s "1 for all i g I. We call i g I an e¨ en index if u ii s 1 and an odd index if u ii s y1. We denote by I e ¨ en Žresp. I o d d . the

GRADED LIE SUPERALGEBRAS

629

set of all even Žresp. odd. indices. We say that a Borcherds]Cartan matrix A s Ž a i j . i, j g I is restricted with respect to C if it satisfies the following condition: if a i i s 2 and u i i s y1, then the a i j are even integers for all j g I. Ž 3.1. In this case, the matrix C is called a coloring matrix of A. Let h s Ž[i g I C h i . [ Ž[i g I C d i . be a complex vector space with a basis  h i , d i < i g I 4 , and for each i g I define a linear functional a i g h* by

a i Ž h j . s a ji ,

a i Ž d j . s di j

for all j g I.

Ž 3.2.

The free abelian group Q s [i g I Z a i generated by a i ’s Ž i g I . is called the root lattice associated with A. Since A is assumed to be symmetrizable, there is a symmetric bilinear form Ž < . on Q given by Ž a i < a j . s si a i j s s j a ji for all i, j g I. Let Qqs Ý i g I Z G 0 a i and Qys yQq. There is a partial ordering G on Q given by l G m if and only if l y m g Qq. The coloring matrix C s Ž u i j . i, j g I defines a bimultiplicative map u : Q = Q ª C= on Q by

u Ž ai , a j . s ui j

for all i , j g I,

u Ž a q b, g . s u Ž a, g .u Ž b, g .,

Ž 3.3.

u Ž a, b q g . s u Ž a, b .u Ž a, g . for all a , b , g g Q. Note that, since u i j u ji s 1 for all i, j g I, u satisfies

u Ž a, b .u Ž b, a . s 1

for all a , b g Q.

Ž 3.4.

That is, u is a coloring map on Q. In particular, u Ž a , a . s "1 for all a g Q. We say a g Q is e¨ en if c Ž a . s u Ž a , a . s 1 and odd if c Ž a . s u Ž a , a . s y1. DEFINITION 3.1. The generalized Kac]Moody superalgebra g s g Ž A, m, C . associated with a symmetrizable Borcherds]Cartan matrix A s Ž a i j . i, j g I of charge m s Ž m i < i g I . with a coloring matrix C s Ž u i j . i, j g I is the u-colored Lie superalgebra generated by the elements

630

SEOK-JIN KANG

h i , d i Ž i g I ., e i k , f i k Ž i g I, k s 1, 2, . . . , m i . with the defining relations, h i , h j s h i , d j s d i , d j s 0, h i , e jl s a i j e jl ,

h i , f jl s ya i j f jl ,

d i , e jl s d i j e jl ,

d i , f jl s yd i j f jl ,

Ž 3.5.

e i k , f jl s d i j d k l h i ,

Ž adei k .

1ya i j

1ya if a i i s 2 and i / j, Ž e jl . s Ž adfi k . i j Ž f jl . s 0 w e i k , e jl x s f i k , f jl s 0 if a i j s 0

for i, j g I, k s 1, . . . , m i , l s 1, . . . , m j . The abelian subalgebra h s Ž[i g I C h i . [ Ž[i g I C d i . is called the Cartan subalgebra of g, and the linear functionals a i g h* Ž i g I . defined by Ž3.2. are called the simple roots of g. For each i g I r e, let ri g GLŽ h*. be the simple reflection on h* defined by ri Ž l . s l y lŽ h i . a i

Ž l g h* . .

Ž 3.6.

The subgroup W of GLŽ h*. generated by the ri ’s Ž i g I is called the Weyl group of the generalized Kac]Moody superalgebra g. The generalized Kac]Moody superalgebra g s g Ž A, m, C . has the root space decomposition g s [a g Q g a , where re.

g a s  x g g w h, x x s a Ž h . x for all h g h 4 .

Ž 3.7.

Note that g a i s C e i, 1 [ ??? [ C e i, m i , and gy a i s C f i, 1 [ ??? [ C f i, m i . We say that a g Q= is a root of g if g a / 0. The subspace g a is called the root space of g attached to a . A root a is called real if Ž a < a . ) 0 and imaginary if Ž a < a . F 0. In particular, a simple root a i is real if a ii s 2 Ži.e., i g I r e . and imaginary if a ii F 0 Ži.e., i g I i m .. Note that the imaginary simple roots may have multiplicity ) 1. A root a ) 0 Žresp. a - 0. is called positi¨ e Žresp. negati¨ e .. One can show that all the roots are either positive or negative. We denote by F, Fq, and Fy the set of all roots, positive roots, and negative roots, respectively. We also denote by F 0 Žresp. F 1 . the set of all even Žresp. odd. roots of g. Hence, for example, Fq will denote the set of all positive even roots of g. Define the 0 subalgebras g "s [a g F " g a . Then we have the triangular decomposition of g: g s gy[ h [ gq. A g-module V is called h-diagonalizable if it admits a weight space decomposition V s [m g h* Vm , where Vm s ¨ g V h ? ¨ s m Ž h . ¨ for all h g h 4 .

Ž 3.8.

631

GRADED LIE SUPERALGEBRAS

If Vm / 0, then m is called a weight of V, and Vm is called the m-weight space. We denote by P Ž V . the set of all weights of V. When all the weight spaces are finite dimensional, we define the character of V to be ch V s

Ý Ž dim Vm . e m ,

Ž 3.9.

mgh*

where the e m are the basis elements of the group algebra Cw h*x with the multiplication given by e m e n s e mq n for m , n g h*. An h-diagonalizable g-module V is called a highest weight module with highest weight l g h* if there is a nonzero vector ¨l g V such that Ži. e i k ? ¨l s 0 for all i g I, k s 1, . . . , m i ; Žii. h ? ¨l s lŽ h. ¨l for all h g h; Žiii. V s UŽ g . ? ¨l. The vector ¨l is called a highest weight ¨ ector. For a highest weight module V with highest weight l, we have Ži. V s UŽ gy. ? ¨l; Žii. V s [m F l Vm , Vl s C ¨l ; and Žiii. dim Vm - ` for all m F l. Let bqs h [ gq, and let Cl be the 1-dimensional bq-module defined by h ? 1 s lŽ h.1 for all h g h and gq? 1 s 0. The induced module M Ž l. s UŽ g . mUŽ bq . Cl is called the Verma module over g with highest weight l. Every highest weight g-module with highest weight l is a homomorphic image of M Ž l. and the Verma module M Ž l. contains a unique maximal submodule J Ž l.. Hence the quotient V Ž l. s M Ž l.rJ Ž l. is irreducible. Let P s  l g h* < lŽ h i . g Z for all i g I 4 be the weight lattice of g, and let Pq be the set of all linear functionals l g h* satisfying

¡lŽ h . g Z ~lŽ h . g 2Z ¢lŽ h . G 0 i i i

G0 G0

for all i g I r e , for all i g I r e l I o d d , for all i g I

im

Ž 3.10.

.

The elements of Pq are called the dominant integral weights. Take a C-linear functional r g h* satisfying r Ž h i . s 12 a ii for all i g I, and let R be the set of all imaginary simple roots counted with multiplicities. For l g Pq, let Rl be the set of all elements b g P of the form

b s a i1 q ??? qa i r q pj1 b j1 q ??? qpj s b j s

Ž r s s s 0 if b s 0 . ,

where a i k Žresp. b jl . are distinct even Žresp. odd. imaginary simple roots of R satisfying Ž a i k < a i l . s Ž b jk < b jl . s 0 if k / l, Ž a i k < a jl . s 0 for all k, l, Ž b j < b j . s 0 if pj G 2, and lŽ h i . s lŽ h j . s 0 for all k, l. For such k k k k l s b g Rl, we denote < b < s r q Ý ks1 pjk. Then the character of the irreducible highest weight module V Ž l. with highest weight l g Pq is deter-

632

SEOK-JIN KANG

mined by the Weyl]Kac]Borcherds formula: PROPOSITION 3.2 wMi, Rx. ch V Ž l . s

Ł a g F y1 Ž 1 q e a .

dim g a

a dim g a

Ł a g F y0 Ž 1 y e .

Ý

<

<

Ž y1. l w q b e wŽ lq ry b .y r . Ž .

wgW , b gR l

Ž 3.11. Letting l s 0, we obtain the denominator identity: Ł a g F y0 Ž 1 y e a .

dim g a

a dim g a

Ł a g F y1 Ž 1 q e .

s

Ý

<

<

Ž y1. l w q b e wŽ ry b .y r . Ž 3.12. Ž .

wgW , b gR 0

3.2. Root Multiplicity Formula Let J be a finite subset of I r e, and we denote by FJ s F l ŽÝ j g J Z a j ., s F "l FJ , and F " Ž J . s F "R FJ" . We also denote by Q J s Q l ŽÝ j g J Z a j ., Q J" s Q "l Q J , and Q " Ž J . s Q "R Q J" . Let g Ž0J . s h [ Ž[a g F g a ., and g Ž"J . s [a g F " Ž J . g a . Then we have the triangular J decomposition, FJ"

ŽJ. ŽJ. g s gy [ g Ž0J . [ gq ,

Ž 3.13.

where g Ž0J . is the Kac]Moody superalgebra Žwith an extended Cartan subalgebra . associated with the generalized Cartan matrix A J s Ž a i j . i, j g J ŽJ. and the set of odd indices J o d d s J l I o d d s  j g J < u j j s y14 , and gy Ž J . Žresp. gq . is a direct sum of irreducible highest weight Žresp. lowest weight. modules over g Ž0J . Žcf. wK4x.. Let WJ be the subgroup of W generated by the simple reflections r j with j g J, and let W Ž J . s  w g W < Fw ; FqŽ J .4 , where Fw s  a g Fq < wy1a - 04 . Thus WJ is the Weyl group of the Kac]Moody superalgebra g Ž0J . and W Ž J . is the set of right coset representatives of WJ in W. That is, W s WJ W Ž J .. Let us denote by FJ,"i s FJ l Fi" Ž i s 0, 1. and Fi"Ž J . s Fi" R FJ,"i Ž i s 0, 1.. Then we have the following generalization of the denominator identity for generalized Kac]Moody superalgebras: PROPOSITION 3.3. Then we ha¨ e Ł a g F y0 Ž J . Ž 1 y e a . Ł

a g Fy 1 ŽJ.

Let J be a finite subset of the set of real indices I r e. dim g a

a dim g a

Ž1 q e .

s

Ž . < < Ý Ž y1. l w q b ch VJ Ž w Ž r y b . y r . ,

wgW Ž J . bgR 0

Ž 3.14.

GRADED LIE SUPERALGEBRAS

633

where VJ Ž m . denotes the irreducible highest weight module o¨ er the Kac] Moody superalgebra g Ž0J . with highest weight m. Proof. Let PJq ; Pq be the set of all l g h* such that lŽ h j . g Z G 0 for all j g J and lŽ h j . g 2Z G 0 for all j g J o d d. We first show that w Ž r y b . y r g PJq for all w g W Ž J .. Since w Ž r y b . y r g Qy, it is clear that Ž w Ž r y b . y r .Ž h j . g Z for all j g J, and since the Borcherds]Cartan matrix A is restricted with respect to the coloring matrix C, we have Ž w Ž r y b . y r .Ž h j . g 2Z if j g J o d d. Thus it remains to show that Ž w Ž r y b . y r .Ž h j . G 0 for all j g J. For each j g J, since w g W Ž J ., we have wy1 Ž a j . ) 0. Hence Ž wr < a j . s Ž r < wy1 Ž a j .. ) 0, which implies Ž wr .Ž h j . s 2Ž wr < a j .rŽ a j < a j . ) 0. Therefore, Ž wr y r .Ž h j . s Ž wr .Ž h j . y 1 G 0. Moreover, since J ; I r e, we have b Ž h j . F 0 for all j g J. Hence Ž w Ž r y b . y r .Ž h j . G 0 for all j g J. By the Weyl]Kac character formula for the irreducible highest weight modules over the Kac]Moody superalgebra g Ž0J . with dominant integral highest weights wK4x, we have ch VJ Ž w Ž r y b . y r . s

Ł a g F yJ , 1Ž 1 q e a .

dim g a

a dim g a

Ł a g F yJ , 0Ž 1 y e .

Ž . Ý Ž y1. l w 9 e w 9wŽ ry b .y r .

w9gW J

Therefore, the right-hand side of Ž3.14. yields Ž . < < Ý Ž y1. l w q b ch VJ Ž w Ž r y b . y r .

wgW Ž J . b gR 0

s

Ý Ž y1.

l Ž w .q < b <

wgW Ž J . bgR 0

=

Ł a g F yJ , 1Ž 1 q e a .

dim g a

Ł a g F yJ , 0Ž 1 y e a .

dim g a

Ž . Ý Ž y1. l w 9 e w 9wŽ ry b .y r

w9gW J

s

s

Ł a g F yJ , 1Ž 1 q e a .

dim g a

Ł a g F yJ , 0Ž 1 y e a .

dim g a

Ł a g F yJ , 1Ž 1 q e a .

dim g a

Ł a g F yJ , 0Ž 1 y e a .

dim g a

Ž . Ž . < < Ý Ž y1. l w ql w 9 q b e w 9wŽ ry b .y r

wgW Ž J . w9gW J bgR 0 Ž . < < Ý Ž y1. l w q b e wŽ ry b .y r .

wgW bgR 0

634

SEOK-JIN KANG

By the denominator identity Ž3.12., this is equal to Ł a g F yJ , 1Ž 1qe a .

dim g a

Ł a g F y0 Ž 1ye a .

dim g a

Ł a g F yJ , 0Ž 1ye a .

dim g a

Ł a g F y1 Ž 1qe a .

dim g a

s

Ł a g F y0 Ž J . Ž 1ye a .

dim g a

Ł a g F y1 Ž J . Ž 1 q e a .

dim g a

,

which proves the identity Ž3.14.. Recall that Dim g a s c Ž a .dim g a . Since w Ž r y b . y r is an element of Qy, by setting E a s c Ž a . e a , we can define the supercharacter of the irreducible highest weight g Ž0J .-module VJ Ž w Ž r y b . y r .. Thus, the identity Ž3.14. can be written as

Ł Ž 1 y E a . Dim g

a gF

yŽ

a

J.

s

Ž . < < Ý Ž y1. l w q b Ch VJ Ž w Ž r y b . y r . ,

wgW Ž J . b gR 0

Ž 3.15. which will be called the denominator identity for the u-colored Lie superalŽJ. gebra gy . For each k G 1, let HkŽ J . s

Ý

VJ Ž w Ž r y b . y r . ,

Ž 3.16.

wgW Ž J . b gR 0 l Ž w .q < b
and define the homology superspace HŽJ. s

`

Ý Ž y1. kq 1 HkŽ J . s H1Ž J . ] H2Ž J . [ H3Ž J . ] ??? ,

Ž 3.17.

ks1

an alternating direct sum of superspaces. Then the denominator identity Ž3.15. can be written as

Ł Ž 1 y E a . Dim g

a gF

yŽ

a

s 1 y Ch H Ž J . .

Ž 3.18.

J.

Let P Ž H Ž J . . s t g Qy < Dim HtŽ J . / 04 s t 1 , t 2 , t 3 , . . . 4 , and t Ž i . s Dim HtŽiJ .. For t g Qy, define the set T Ž J . Žt . of all partitions of t into a sum of t i ’s as in Ž1.21. and define the Witt partition function W Ž J . Žt . as in Ž1.22.. Then, our superdimension formula Ž1.23. yields a closed form

635

GRADED LIE SUPERALGEBRAS

root multiplicity formula for all symmetrizable generalized Kac]Moody superalgebras: THEOREM 3.4. ha¨ e

Let J be a finite subset of I r e. Then, for all a g Fy Ž J ., we

Dim g a s s

1

Ý

d
d 1

Ý

d
d

mŽ d.W Ž J.

a

ž / d

Ž < s < y 1. !

mŽ d.

Ý sgT

Ž J .Ž

a rd .

s!

Ł tŽ i.

si

,

Ž 3.19.

where m is the classical Mobius function, and, for a positi¨ e integer d, d < a ¨ denotes a s dt for some t g Qy, in which case ard s t . 3.3. Maximal and Minimal Graded Lie Superalgebras In this subsection, we consider the realization of generalized Kac]Moody superalgebras as the minimal graded Lie superalgebras. We first recall the basic definition introduced by Kac wK1x. Let L s [ng Z L n be a Z-graded Lie superalgebra and let L l o c s Ly1 [ L 0 [ L 1. The subspace L l o c is called the local part of L if it generates the Lie superalgebra L. More generally, a direct sum of superspaces L l o c s Ly1 [ L 0 [ L 1 is called a local Lie superalgebra if there exists a bilinear map w , x: L i = L j ª L iqj , called the superbracket, defined for < i q j < F 1 satisfying the condition Ž1.3. whenever the superbrackets are defined. A Z-graded Lie superalgebra L s [n g Z L n with local part L l o c s Ly1 [ L 0 [ L 1 is called the maximal graded Lie superalgebra Žresp. minimal graded Lie superalgebra. if for any Z-graded Lie superalgebra L9 s X [ng Z LXn with local part LXl o c s Ly1 [ LX0 [ LX1 , every isomorphism beX tween the local parts L l o c and L l o c can be extended to an epimorphism of L onto L9 Žresp. L9 onto L .. In particular, a Z-graded Lie superalgebra L s [ng Z L n is the minimal graded Lie superalgebra with local part L l o c s Ly1 [ L 0 [ L 1 if and only if there is no nonzero Z-graded ideal of L that intersects the local part L l o c trivially. In wK3x, using the same argument in wK1x, Kac proved: PROPOSITION 3.5 wK3x. For any local Lie superalgebra L l o c s Ly1 [ L 0 [ L 1 , there exist unique Ž up to isomorphism. maximal and minimal Z-graded Lie superalgebras whose local parts are isomorphic to L l o c s Ly1 [ L 0 [ L 1. We return to the generalized Kac]Moody superalgebras. Let g s g Ž A, m, C . be a generalized Kac]Moody superalgebra associated with a Borcherds]Cartan matrix A s Ž a i j . i, j g I of charge m s Ž m i g Z ) 0 <

636

SEOK-JIN KANG

i g I . with a coloring matrix C s Ž u i j . i, j g I . Let J be a finite subset of I r e, ŽJ. and consider the corresponding triangular decomposition g s gy [ g Ž0J . ŽJ. ŽJ. [ gq , where the subalgebra g 0 s h [ Ž[a g F J g a . is the Kac]Moody superalgebra Žwith an extended Cartan subalgebra . associated with the generalized Cartan matrix A J s Ž a i j . i, j g J and the set of odd indices J o d d s  j g J < u j j s y14 . For each root a s Ý i g I k i a i g F, we define the generalized height of a to be ht Ž J . Ž a . s Ý i g I R J k i . Note that if J s f , ht Ž J . is the usual height function. For all n G 1, let g ŽnJ . s

[

a gF qŽ J . ht Ž J . Ž a .sn

ga

ŽJ. gyn s

and

[

a gF yŽ J . ht Ž J . Ž a .syn

ga .

Ž 3.20.

Then the generalized Kac]Moody superalgebra g becomes a Z-graded Lie superalgebra g s [ng Z g Ž"J .n generated by the local part g l o c s gy1 [ g 0 [ g 1 , and, by the same argument given in wBKM, Ju2, K1, Ka1x, one can show that there is no nonzero Z-graded ideal of g that intersects the local part g l o c s gy1 [ g 0 [ g 1 trivially. Moreover, as g Ž0J .-modules, we have ŽJ. gy1 (

[ V Ž ya .

igIRJ

J

i

[m i

,

g 1Ž J . (

[V

igIRJ

U J

Ž ya i .

[m i

, Ž 3.21.

where VJ Ž m . Žresp. VJU Ž m .. denotes the irreducible highest weight Žresp. lowest weight. module over the Kac]Moody superalgebra g Ž0J . with highest weight m Žresp. lowest weight ym ., and the m i ’s are the multiplicities of the imaginary simple roots a i ’s. Therefore, the generalized Kac]Moody superalgebra is the minimal Z-graded Lie superalgebra with local part g l o c s gy1 [ g 0 [ g 1 ( V [ g Ž0J . [ V *, where V s [i g I R J VJ Žya i .[m i and V * s [i g I R J VJU Žya i .[m i . Suppose that the Borcherds]Cartan matrix A satisfies Ži. the set I r e is finite, Žii. a i j / 0 for all i, j g I i m . If we take J s I r e, then we have W Ž J . s  14 and every element b g R 0 has the form b s 0 or b s a i for ŽJ. i g I i m . Hence the denominator identity for the Lie superalgebra gy is the same as

Ł Ž 1 y E a . Dim Ž g

a gQ yŽ J .

ŽJ. y .a

s1y

Ý

Ch VJ Ž w Ž r y b . y r .

wgW Ž J . , b gR 0 l Ž w .q < b
s1y

Ý igI im

m i Ch VJ Ž ya i . .

Ž 3.22.

GRADED LIE SUPERALGEBRAS

637

ŽJ. ŽJ. But, since g s gy [ g Ž0J . [ gq is the minimal Z-graded Lie superalgebra ŽJ. with local part g l o c ( V [ g Ž0J . [ V *, the Lie superalgebra gy is a homomorphic image of the free Lie superalgebra F s [a g Qy Ž J . F a generated by the superspace V s [i g I i m VJ Žya i .[m i . By Ž2.8., the denominator identity for the free Lie superalgebra F is equal to

Ł Ž 1 y E a . Dim F

agQ yŽ J .

a

s1y

Ý

m i Ch VJ Ž ya i . .

igI im

Ž J .. yŽ . In particular, Dim F a s DimŽ gy J . Hence the Lie a for all a g Q ŽJ. superalgebra gy is isomorphic to the free Lie superalgebra generated by the superspace V s [i g I i m VJ Žya i .[m i . Therefore, we obtain:

PROPOSITION 3.6. Let A s Ž a i j . i, j g I be a Borcherds]Cartan matrix of charge m s Ž m i g Z ) 0 < i g I . with a coloring matrix C s Ž u i j . i, j g I . Suppose A satisfies Ži. the set I r e is finite, Žii. a i j / 0 for all i, j g I i m . Let J s I r e, and consider the corresponding triangular decomposition of the generalized Kac]Moody superalgebra ŽJ. ŽJ. g s g Ž A, m, C . s gy [ g Ž0J . [ gq . ŽJ. ŽJ. Then the Lie superalgebra gy s [a g F y Ž J . g a Ž resp. gq s [a g F q Ž J . g a . is isomorphic to the free Lie superalgebra generated by V s [i g I i m VJ Žya i .[m i Ž resp. V * s [i g I i m VJU Žya i .[m i ., where VJ Ž m . Ž resp. VJU Ž m .. denotes the irreducible highest weight Ž resp. lowest weight . module o¨ er the Kac]Moody superalgebra g Ž0J . with highest weight m Ž resp. lowest weight ym ..

Remark. Proposition 3.6 shows that, under the above assumptions, the ŽJ. ŽJ. generalized Kac]Moody superalgebra g s gy [ g Ž0J . [ gq is isomorphic to the maximal graded Lie superalgebra with local part g l o c ( V [ g Ž0J . [ V *. 4. MONSTROUS LIE SUPERALGEBRAS 4.1. Monstrous Moonshine The classification theorem of finite simple groups tells that there are exactly 26 sporadic simple groups besides the family of alternating groups on n letters Ž n G 5. and the families of simple groups of Lie type Žsee, for example, wGLSx.. The largest among the sporadic simple groups has order 2 46 ? 3 20 ? 5 9 ? 7 6 ? 112 ? 13 3 ? 17 ? 19 ? 23 ? 29 ? 31 ? 41 ? 47 ? 59 ? 71, and it is called the Monster due to its enormous size.

638

SEOK-JIN KANG

The trivial character degree of the Monster simple group G is, by definition, one, and the smallest nontrivial irreducible character degree of G is 196883 wFLTx. It was noticed by McKay that 1 q 196883 s 196884, which is the first nontrivial coefficient of the elliptic modular function J Ž q . s j Ž q . y 744 s

Ý c Ž n . q n s qy1 q 196884 q q 21493760 q 2 q ??? . Ž 4.1. nGy1

Later, Thompson found that the first few coefficients of the modular function jŽ q . y 744 are simple linear combinations of the irreducible character degrees of G wTx. Motivated by these observations, Conway and Norton conjectured that there exists an infinite dimensional graded representation V h s [n Gy1Vnh of the Monster simple group G with dim Vnh s cŽ n. such that the Thompson series Tg Ž q . s

Ý nGy1

Tr g < Vnh q n s

ž

/

Ý

cg Ž n. q n

Ž 4.2.

nGy1

are the normalized generators of the genus zero function fields arising from certain discrete subgroups of PSLŽ2, R. wCNx. Moreover, they also noticed that the Thompson series seem to satisfy certain functional equation which they call the replication formulae. Their conjecture is referred to as the Moonshine conjecture. The natural graded representation V h s [nGy1 Vnh of the Monster simple group G in the Moonshine conjecture, called the Moonshine module, was constructed by Frenkel, Lepowsky, and Meurman using the theory of vertex Žoperator. algebras wFLMx. They also calculated the Thompson series for some conjugacy classes of the Monster, and verified the Moonshine conjecture for these Thompson series. In wB3x, Borcherds completed the proof of the Moonshine conjecture by constructing a II1, 1-graded Lie algebra L s [Ž m, n.g II1, 1 LŽ m, n. , called the Monster Lie algebra, where II1, 1 is the 2-dimensional even Lorentzian lattice associated with the matrix Ž y10 y10 .. The Monster Lie algebra L is a II1, 1-graded representation of the Monster simple group G such that LŽ m, n. ( Vmh n as G-modules for Ž m, n. / Ž0, 0.. In particular, we have dim LŽ m , n. s dimVmh n s c Ž mn .

for all Ž m, n . / Ž 0, 0 . .

On the other hand, the Monster Lie algebra can be regarded as a generalized Kac]Moody algebra. We take I s  y14 j  1, 2, 3, . . . 4 as the index set, and consider the Borcherds]Cartan matrix A s ŽyŽ i q j .. i, j g I of charge m s Ž cŽ i . < i g I ., where cŽ i . are the coefficients of the elliptic

GRADED LIE SUPERALGEBRAS

639

modular function J Ž q . s jŽ q . y 744 s Ý`ns y1 cŽ n. q n. Then, by using Borcherds’ product identity Ž2.30. for the elliptic modular function J Ž q . s jŽ q . y 744, it can be shown that the Monster Lie algebra is isomorphic to the generalized Kac]Moody algebra g s g Ž A, m.. In wKa3x, we derived a root multiplicity formula for all symmetrizable generalized Kac]Moody algebras, which turns out to be a special case of our formula Ž3.19.. Applying that formula to the Monster Lie algebra, we obtained the following interesting relations for the coefficients cŽ n. of the elliptic modular function J Ž q . s jŽ q . y 744 s Ý`ns y1 cŽ n. q n, 1 Ž < s < y 1. ! s c Ž mn . s Ý mŽ d. Ý Ł c Ž i q j y 1. i j , Ž 4.3. d s! m n d <Ž m , n . sgT

ž

,

d

d

/

where T Ž k, l . denotes the set of all partitions of Ž k, l . as a sum of ordered pairs of positive integers Žsee also wJu1x.. In wJLW, KKx, the relation Ž4.3. was generalized to the relation of the coefficients c g Ž n. of the Thompson series: 1 Ž < s < y 1. ! s c g Ž mn . s Ý mŽ d. Ý Ł c g d Ž i q j y 1. i j . Ž 4.4. d s! m n d <Ž m , n . sgT

ž

d

,

d

/

It was pointed out in wJLWx that these relations completely determine all the coefficients c g Ž n. if the values of c hŽ1., c hŽ2., c hŽ3., and c hŽ5. are known for all h g G. In particular, the relation Ž4.3. is a complete recursive relation determining the coefficients cŽ n. of the elliptic modular function jŽ q . y 744. Moreover in wKKx, we have noticed the ghost functions introduced in wCNx also seem to satisfy the same relations as the Thompson series, which leads us to consider a more general class of modular functions}replicable functions. 4.2. Replicable Functions We recall the definition of replicable functions following the exposition given in wHx Žsee also wACMS, CN, F2, Nx.. Let F Ž q . s Ý`ns y1 f Ž n. q n s qy1 q f Ž1. q q f Ž2. q 2 q ??? be a normalized q-series such that f Žy1. s 1, f Ž0. s 0, and f Ž n. g Z for all n G 1. Observe that, for each m G 1, there exists a unique polynomial PmŽ t . g Zw t x such that Pm Ž F . ' qym

mod qZ w q x .

For example, P1Ž t . s t, P2 Ž t . s t y 2 f Ž1., etc. The polynomials PmŽ t . are uniquely determined by the recursive relation wACMS, Hx 2

my1

Pmq 1 Ž t . q

Ý f Ž i . Pmyi Ž t . q Ž m q 1. f Ž m . s tPm Ž t . . is0

Ž 4.5.

640

SEOK-JIN KANG

DEFINITION 4.1. A normalized q-series F Ž q . s Ý`ns y1 f Ž n. q n s qy1 q f Ž1. q q f Ž2. q 2 q ??? is called replicable if for all m ) 0 and for all a < m, there exist normalized q-series F Ž a. Ž q . s Ý`ns y1 f Ž a. Ž n. q n s qy1 q f Ž a. Ž1. q q f Ž a. Ž2. q 2 q ??? with f Ž a. Žy1. s 1, f Ž a. Ž0. s 0, and f Ž a. Ž n. g Z for all n G 1 such that F Ž1. s F ,

and

F Ž a.

Ý adsm 0Fb-d

ž

az q b d

/

s Pm Ž F . ,

Ž 4.6.

where q s e 2 p i z , Im z ) 0. The normalized q-series F Ž a. is called the ath replicate of F. If all the replicates F Ž a. are also replicable, then F is called completely replicable. EXAMPLE 4.2. Ža. A finite series F Ž q . s qy1 q rq Ž r g Z. is completely replicable with F Ž a. Ž q . s qy1 q r aq for all a G 1. Norton showed N Ž . n that a finite normalized q-series F Ž q . s Ý ns y1 f n q is replicable only if f Ž n. s 0 for all n G 2 wNx. Žb. The elliptic modular function J Ž q . s jŽ q . y 744 is completely replicable with J Ž a. s J for all a G 1 wCN, Se2x. Žc. The Thompson series Tg Ž q . s Ý`nsy1 c g Ž n. q n are completely replicable with TgŽ a. s Tg a for all a G 1 wKo, F2x. Žd. In wF1x, Ferenbaugh classified all genus 0 modular groups between G0 Ž N . and its normalizer in PSLŽ2, R.. The corresponding Hauptmoduls are called n N h-type Hauptmoduls, and it was proved that they are all completely replicable wF2, Kox. This fact played a crucial role in Borcherds’ proof of the Moonshine Conjecture. In wACMSx, it was proposed that we can define the notion of generalized Hecke operators on the replicable functions, and in wF2x, Ferenbaugh verified that for the n < h-type modular functions, it is a reasonable definition with the Galois action incorporated. More generally, for each m G 1, we define the formal Hecke operators Tm on the set of normalized q-series with values in the set of formal Laurent q-series by Tm Ž F . s

1 m

Pm Ž F . .

Ž 4.7.

Hence if F is replicable with replicates F Ž a. for a G 1, we have Tm Ž F . s

1 m

Ý adsm 0Fb-d

F Ž a.

ž

az q b d

/

.

Ž 4.8.

641

GRADED LIE SUPERALGEBRAS

Furthermore, the same argument in wSe2, Chap. VII, Proposition 12x yields 1 m

F Ž a.

Ý adsm 0Fb-d

ž

az q b d

/

s

Ý ngZ



1

Ý a)0 a <Ž m , n .

a

f Ž a.

mn

ž / a2

0

q n.

Hence we obtain: PROPOSITION 4.3. A normalized q-series F Ž q . s Ý`ns y1 f Ž n. q n s qy1 q f Ž1. q q f Ž2. q 2 q ??? is replicable if and only if there exist normalized q-series F Ž a. Ž q . s Ý`ns y1 f Ž a. Ž n. q n s qy1 q f Ž a. Ž1. q q f Ž a. Ž2. q 2 q ??? for all a G 1 satisfying Tm Ž F . s

1 m

Pm Ž F . s

Ý ngZ



1

Ý a)0 a <Ž m , n .

a

f Ž a.

mn

ž / a2

0

qn

Ž 4.9.

for all m G 1. Now, we will characterize the replicable functions in terms of product identities: THEOREM 4.4. The following conditions on a normalized q-series F Ž q . s qy1 q f Ž1. q q f Ž2. q 2 q ??? s Ý`ns y1 f Ž n. q n are equi¨ alent. Ža. F is replicable. Žb. For all k G 1, there exist normalized q-series F Ž k . Ž q . s ` Ý ns y1 f Ž k . Ž n. q n satisfying the product identity py1

Ł exp

m)0 ngZ

Žc .

Ý`ns y1

ž

y

`

Ý ks1

1 k

f Ž k . Ž mn . p k m q k n s F Ž p . y F Ž q . . Ž 4.10.

/

For all k G 1, there exist normalized q-series F Ž k . Ž q . s f n. q n satisfying the product identity Ž k .Ž

`

Ł

m, ns1

exp y

ž

`

Ý ks1

1 k

f Ž k . Ž mn . p k m q k n s 1 y

/

`

Ý f Ž i q j y 1 . p iq j . i , js1

Ž 4.11. Proof. It is easy to prove the equivalence of Žb. and Žc.. Observing that Ž f 0. s 0 and f Žyk . s 0 for k ) 1, the identity Ž4.10. can be rewritten as `

Ł

m, ns1

exp y

ž

`

Ý ks1

1 k

f Ž k . Ž mn . p k m q k n s

/

F Ž p. y F Ž q. py1 y qy1

s1y

`

Ý f Ž i q j y 1 . p iq j , i , js1

which is the identity Ž4.11..

642

SEOK-JIN KANG

We will prove the equivalence of Ža. and Žb.. Suppose F is replicable with replicates F Ž k . Ž q . s Ý`ns y1 f Ž k . Ž n. q n for k G 1. By Proposition 4.3, we have Tm Ž F . s

1 m

Pm Ž F . s



Ý ngZ

1

Ý k)0 k <Ž m , n .

k

f Žk.

mn

ž / k2

0

q n.

Observe that

ž

log

Ł exp

m)0 ngZ

ž

y

`

1

Ý ks1

f Ž k . Ž mn . p k m q k n

k `

sy

1

Ý Ý Ý m)0 ngZ ks1

f Ž k . Ž mn . p k m q k n

k

1

sy

Ý Ý Ý m)0 ngZ `

sy

1

Ý ms1

m

//

k

k)0 k <Ž m , n .

f Žk.

mn

ž / k2

p mq n

Pm Ž F . p m .

So the left-hand side of Ž4.10. is equal to py1 exp y

ž

ms1

y1

sp s

`

Ý

1 m

Pm Ž F . p m

1q y

ž

`

1

Ý ms1

m

/

Pm Ž F . p

m

1

q

/ ž 2!

`

y

Ý ms1

1 m

2

Pm Ž F . p

m

/

q ???

`

Ý

Gm Ž q . p m ,

msy1

where Gy1Ž q . s 1, G 0 Ž q . s yF Ž q ., and Gm Ž q . s

Ž y1.

Ý sgT Ž mq1 .

s!

< s<

Ł

ž

Pi Ž F . i

si

/

Ž m G 1. .

Here, T Ž m q 1. is the set of all partitions of m q 1 into a sum of positive integers. In particular, GmŽ q . is a polynomial in F.

643

GRADED LIE SUPERALGEBRAS

On the other hand, the right-hand side of Ž4.10. can be written as `

`

ms1

msy1

F Ž p . y F Ž q . s py1 y F Ž q . q

Ý f Ž m. p m s Ý

Hm Ž q . p m ,

and each HmŽ q . is obviously a polynomial in F. Thus, in order to prove the identity Ž4.10., we have only to check that the coefficients of q n of GmŽ q . and HmŽ q . are the same for all n F 0. Write GmŽ q . s Ý n g Z g mŽ n. q n and HmŽ q . s Ý ng Z h mŽ n. q n. Then the left-hand side of Ž4.10. is equal to `

Ý

Gm Ž q . p m s

msy1

`

gm Ž n. q n p m s

Ý Ý msy1 ngZ

`

Ý ngZ

ž

Ý

/

gm Ž n. p m q n ,

msy1

and, similarly, the right-hand side of Ž4.10. is equal to `

`

Hm Ž q . p m s

Ý

Ý

msy1

ngZ

ž

Ý

/

hm Ž n. p m q n .

msy1

Suppose Ý`ms y1 g mŽ n. p m s Ý`msy1 h mŽ n. p m for all n F 0. Then g mŽ n. s h mŽ n. for all n F 0, which implies Gm Ž q . s Hm Ž q ., since they are polynomials in F. Hence, to prove Ž4.10., it suffices to check that the coefficients of q n in the left-hand side and the right-hand side of Ž4.10. are the same for all n F 0, which is straightforward. ŽThey are y1 if n s y1, F Ž p . if n s 0, and 0 otherwise.. Conversely, suppose there exist normalized q-series F Ž k . Ž q . s ` Ý ns y1 f Ž k . Ž n. q n for all k G 1 satisfying the identity Ž4.10.. As we have seen in the previous argument, we have

log

ž ž Ł exp

y

m)0 ngZ

sy

`

Ý ks1

1 k

f Ž k . Ž mn . p k m q k n 1

Ý Ý Ý m)0 ngZ

k)0 k <Ž m , n .

k

f Žk.

mn

ž / k2

//

p mq n .

Let Tˆm Ž q . s

Ý ngZ



1

Ý k)0 k <Ž m , n .

k

f Žk.

mn

ž / k2

0

q n.

644

SEOK-JIN KANG

1 We would like to show that TˆmŽ q . s PmŽ F . for all m G 1. The left-hand m side of Ž4.10. is equal to `

py1 exp y

ž

Ý Tˆm p m ms1

`

s py1 1 q y

ž

s

/ Ý

Tˆm p m q

/

ms1

1 2!

ž

`

y

Ý ms1

2

Tˆm p m

/

q ???

`

Ý Gˆm Ž q . p m , msy1

ˆy1Ž q . s 1, Gˆ0 Ž q . s yTˆ1Ž q ., and where G Ž y1.

ˆm Ž q . s G

Ý

< s<

Ł Tˆi Ž q .

s!

sgT Ž mq1 .

si

Ž m G 1. .

Since the right-hand side of Ž4.10. is equal to py1 y F Ž q . q Ý`ms 1 f Ž m. p m , we obtain Tˆ1 Ž q . s F Ž q . ,

Ž y1.

and

Ý

< s<

s!

sgT Ž mq1 .

Ł Tˆi Ž q .

si

s f Ž m.

Ž m G 1 . . Ž 4.12. By induction, it follows from Ž4.12. that TˆmŽ q . is a polynomial in F for all m G 1. It is easy to check that the coefficients of q n in TˆmŽ q . and 1 1 P Ž F . are the same for all n F 0. ŽThey are m if n s ym and 0 m m 1 otherwise.. Hence, TˆmŽ q . s m PmŽ F . for all m G 1, and by Proposition 4.3, F is replicable.

COROLLARY 4.5. If a normalized q-series F Ž q . s Ý`ns y1 f Ž n. q n is completely replicable with replicates F Ž a. Ž q . s Ý`ns y1 f Ž a. Ž n. q n for a G 1, then we ha¨ e f Ž mn . s

1

Ý

d <Ž m , n .

d

mŽ d.

Ž < s < y 1. !

Ý

m n sgT , d d

ž

s!

Ł f Ž d. Ž i q j y 1.

si j

.

/ Ž 4.13.

645

GRADED LIE SUPERALGEBRAS

In particular, the coefficients of a completely replicable function F are determined recursi¨ ely by the coefficients of its replicates F Ž a.. Proof. If F is completely replicable, then all of its replicates F Ž a. are also replicable with Ž F Ž a. .Ž k . s F Ž ak . for all a, k G 1 Žcf. wNx.. Hence, by Ž4.11., we have `

Ł

`

exp y

ž

m, ns1

1

Ý

/

k

ks1

`

f Ž ak . Ž mn . p k m q k n s 1 y

Ý

f Ž a. Ž i q j y 1 . p iq j

i , js1

for all a G 1. By taking the logarithm, we obtain `

log

Ł

exp y

1

ž

ks1

k

`

`

1

m, ns1

sy

`

Ý

Ý Ý m, ns1 ks1

k

f Ž ak . Ž mn . p k m q k n

/

f Ž ak . Ž mn . p k m q k n ,

and `

log 1 y

ž

Ý

f Ž a. Ž i q j y 1 . p iq j

i , js1

sy sy

`

Ý

1

ks1

k

`

1

Ý ks1

k

`

sy

Ý l, ts1

ž

`

žÝ

/ k

f Ž a. Ž i q j y 1 . p iq j

i , js1

Ý ss Ž s i j . si jgZ G 0 Ýs i jsk

/

Ž Ýsi j . ! s f Ž a. Ž i q j y 1 . Ł Ł Ž si j ! . Ž < s < y 1. !

Ý sgT Ž l , t .

s!

ij

p Ýi s i j q Ý js i j

Ł f Ž a. Ž i q j y 1.

si j

/

p lq t .

Therefore, we have 1

Ý

k Ž m , n .s Ž l , t .

k

f Ž ak . Ž mn . s

Ž < s < y 1. !

Ý sgT Ž l , t .

s!

Ł f Ž a. Ž i q j y 1.

si j

.

646

SEOK-JIN KANG

Hence, by Mobius inversion, we obtain ¨

f Ž a. Ž mn . s

1

Ý

d <Ž m , n .

d

Ž < s < y 1. !

mŽ d.

Ý sgT

ž

m n , d d

s!

Ł f Ž ad. Ž i q j y 1.

si j

/

for all a G 1. In particular, if a s 1, we get Ž4.13.. Remark. Note that the relations Ž4.3. and Ž4.4. are special cases of Ž4.13.. 4.3. Monstrous Lie Superalgebras We take I s  y14 j  1, 2, 3, . . . 4 to be the index set, and let A s ŽyŽ i q j .. i, j g I be the Borcherds]Cartan matrix of the Monster Lie algebra. Consider a normalized q-series F Ž q . s Ý`ns y1 f Ž n. q n such that f Žy1. s 1, f Ž0. s 0, and f Ž n. g Z for all n G 1. We define the charge of the matrix A to be m s Ž< f Ž i .<: i g I ., and choose a coloring matrix C s Ž u i j . i, j g I such that u ii s 1 if f Ž i . ) 0 and u ii s y1 if f Ž i . - 0. That is, an index i g I is even if f Ž i . ) 0 and is odd if f Ž i . - 0. DEFINITION 4.6. The generalized Kac]Moody superalgebra L Ž F . s g Ž A, m, C . associated with the above data is called the Monstrous Lie superalgebra associated with the normalized q-series F Ž q . s Ý`ns y1 f Ž n. q n. For example, the Monstrous Lie superalgebra associated with the elliptic modular function J Ž q . s jŽ q . y 744 is the Monster Lie algebra, and the Monstrous Lie superalgebras associated with the Thompson series Tg Ž q . s Ý`nsy1 c g Ž n. q n are the Monstrous Lie superalgebras given in wB3x. By our choice of the charge and the coloring matrix, we see that ay1 is the only real even simple root, a i Ž i G 1. is an imaginary even simple root with multiplicity f Ž i . if f Ž i . ) 0, and a i Ž i G 1. is an imaginary odd simple root with multiplicity yf Ž i . if f Ž i . - 0. We just neglect those a i ’s for which f Ž i . s 0. In short, we have Dim L Ž F .a i s f Ž i . for all i g I. Thus W s  1, ry1 4 and R s  a i , . . . , a i < i G 14 , where each a i is counted < f Ž i .< times. Since Ž a i < a i . s y2 i / 0 for i G 1, b g R 0 if and only if b s 0 or b s a i for some i G 1 Ž< f Ž i .< choices.. Hence, if we take J s f , by Ž3.16., we obtain

H Ž f . s Cya y1 [

`

ž[ is1

`

/ ž[

f Ž i.< C[< ] ya i

is1

f Ž i.< C[< yi a y 1 y a i .

/

Ž 4.14.

647

GRADED LIE SUPERALGEBRAS

Therefore, the denominator identity Ž3.12. for the Lie superalgebra L Ž F .y yields Ł a g F y0 Ž 1 y e a .

dim L Ž F .a

a dim L Ž F .a

Ł a g F y1 Ž 1 q e .

s 1 y eya y1 y

`

Ý f Ž i.

ey a i

is1 `

q Ý f Ž i . eyi ay1 y a i , is1

which is equivalent to Ž . Ł Ž 1 y E a . Dim L F

a

agQ y

s 1 y Ey ay 1 y

`

Ý f Ž i . Ey a

i

is1 `

q Ý f Ž i . Eyi ay1 y a i . is1

We identify the simple roots ay1 with Ž1, y1. g II1, 1 and a i with Ž1, i . g II1, 1 Ž i G 1.. Then the Monstrous Lie superalgebra L Ž F . becomes a II1, 1-graded Lie superalgebra, and we have L Ž F . qs

[LŽ F.

Ž m , n. ,

m)0 ngZ

L Ž F . ys

[LŽ F. m)0 ngZ

Žym , n . .

By letting p s EyŽ1 , 0. and q s EyŽ0 , 1., the denominator identity for the Lie superalgebra L Ž F .y can be written as Ž . Ł Ž 1 y p mq n . Dim L F

Ž m , n.

s1 y pqy1 y

m)0 ngZ

sp py1 q

ž

`

`

is1

is1

Ý f Ž i . pq i q Ý f Ž i . p iq1

`

Ý f Ž i . pi is1

yp qy1 q

/ ž

`

Ý f Ž i. qi is1

/

sp Ž F Ž p . y F Ž q . . , which yields: PROPOSITION 4.7. Let L Ž F . s [Ž m, n.g II1, 1 L Ž F .Ž m, n. be the Monstrous Lie superalgebra associated with a normalized q-series F Ž q . s Ý`ns y1 f Ž n. q n such that f Žy1. s 1, f Ž0. s 0, and f Ž n. g Z for all n G 1. Then the denominator identity for the Lie superalgebra L Ž E .y is gi¨ en by py1

Ž . Ł Ž 1 y p mq n . Dim L F

m)0 ngZ

Ž m , n.

s F Ž p. y F Ž q . .

Ž 4.15.

648

SEOK-JIN KANG

We will apply our root multiplicity formula Ž3.19. to the Monstrous Lie superalgebra L Ž F . s [Ž m, n.g II1, 1 L Ž F .Ž m, n. . Take J s  y14 . Then L Ž F .Ž0J . s ² ey1 , fy1 , hy1 : q h ( sl Ž2, C. q h, and W Ž J . s  14 . By Ž3.16., we obtain `

H Ž J . s H1Ž J . s

[ V Ž ya . is1

J

[ < f Ž i .<

i

,

where VJ Žya i . is the i-dimensional irreducible representation of sl Ž2, C. Žsince ya i Ž hy1 . s i y 1.. Hence the weights of VJ Žya i . are ya i s Žy1, yi ., ya i y ay1 s Žy2, yi q 1., . . . , ya i y Ž i y 1. ay1 s Žyi, y1. with the superdimensions f Ž i . Ž i G 1.. It follows that P Ž H Ž J . . s  Ž yi, yj . i , j g Z ) 0 4 and ŽJ. Dim HŽyi , yj. s f Ž i q j y 1 .

for all i , j G 1.

ŽJ. Hence the denominator identity for the Lie superalgebra L Ž F .y is equal to

`

Ł

Ž 1 y p m q n . Dim L

Ž F . Ž m , n.

s1y

m, ns1

`

Ý f Ž i q j y 1. p iq j . Ž 4.16. i , js1

For k, l ) 0, we have T Ž J . Ž k, l . s T Ž k, l . s s s Ž si j . i , jG1 si j g Z G 0 ,

½

Ý si j Ž i , j . s Ž k , l . 5 ,

the set of all partitions of Ž k, l . into a sum of ordered pairs of positive integers, and the Witt partition function W Ž J . Ž k, l . is given by W Ž J . Ž k, l . s

Ž < s < y 1. !

Ý sgT Ž k , l .

s!

Ł f Ž i q j y 1.

si j

.

Therefore, our root multiplicity formula Ž3.19. yields: THEOREM 4.8. Let L Ž F . s [Ž m, n.g II1, 1 L Ž F .Ž m, n. be the Monstrous Lie superalgebra associated with a normalized q-series F Ž q . s Ý`ns y1 f Ž n. q n such that f Žy1. s 1, f Ž0. s 0, and f Ž n. g Z for all n G 1.

649

GRADED LIE SUPERALGEBRAS

Then, for all m, n g Z ) 0 , we have 1

Dim L Ž F . Ž m , n . s

Ý

d <Ž m , n .

d

Ž < s < y 1. !

mŽ d.

Ý sgT

ž

m n , d d

s!

Ł f Ž i q j y 1.

si j

.

/ Ž 4.17.

Remark. It follows from Proposition 3.6 that the Lie superalgebra ŽJ. L Ž F .y is isomorphic to the free Lie superalgebra generated by `

HŽJ. s

[ V Ž ya . is1

J

[ < f Ž i .<

i

,

2 where H Ž J . is regarded as a Z ) 0 -graded superspace

`

H

ŽJ.

s

[H

i , js1

ŽJ. Žyi , yj.

ŽJ. with Dim HŽyi , yj. s f Ž i q j y 1 . .

Hence the Monstrous Lie superalgebra L Ž F . is isomorphic to the maximal graded Lie superalgebra with local part L Ž F . l o c s H Ž J . [ Ž sl Ž2, C. q h . [ H Ž J .*. Since we have identified the roots of the Monstrous Lie superalgebra L Ž F . with the elements of the lattice II1, 1 , the superdimension Dim L Ž F .Ž m, n. is the difference of the e¨ en dimension and the odd dimension. Still, thanks to Corollary 1.3 and Proposition 2.10, we can recover its even dimension and odd dimension explicitly. We consider the Ž II1, 1 = Z 2 .-gradation on the superspace H Ž J . and the Monstrous Lie superalgebra L Ž F . as follows. For each Ž i, j . g Z ) 0 = Z ) 0 , define « Ž i, j . s 0 if f Ž i q j y 1. ) 0 and « Ž i, j . s 1 if f Ž i q j y 1. - 0. Then we ŽJ. obtain a Ž II1, 1 = Z 2 .-gradation on H Ž J . s [Ž i, j, k .g II1, 1=Z 2 HŽyi, yj, k . such that P Ž H Ž J . , II1, 1 = Z 2 . s  Ž yi, yj, « Ž i , j . . i , j s 1, 2, 3, ??? . 4 and Dim HŽŽi,J .j, k . s f Ž i q j y 1. for all i, j G 1, k g Z 2 , which induces a Ž II1, 1 = Z 2 .-gradation on L Ž F .. For s s Ž si j . g T Ž m, n., set < s
Ž < s < y 1. !

Ý sgT Ž m , n . < s
Wy Ž m, n . s W Ž m, n, 1 . s

s!

Ž < s < y 1. !

Ý sgT Ž m , n . < s
s!

Ł f Ž i q j y 1.

si j

Ł f Ž i q j y 1.

si j

,

.

650

SEOK-JIN KANG

Therefore, by Corollary 1.3 and Proposition 2.10, we recover the even dimensions and the odd dimensions of homogeneous subspaces of the Monstrous Lie superalgebras: PROPOSITION 4.9. Let L Ž F . s [Ž m, n.g II1, 1 L Ž F .Ž m, n. be the Monstrous Lie superalgebra associated with a normalized q-series F Ž q . s Ý`ns y1 f Ž n. q n such that f Žy1. s 1, f Ž0. s 0, and f Ž n. g Z for all n G 1. Then, for each m, n g Z ) 0 , the e¨ en dimension and the odd dimension of the homogeneous subspace L Ž F .Ž m, n. are gi¨ en by Dim L Ž F . Ž m , n , 0. s

1

Ý

d <Ž m , n .

d

m Ž d . Wq 1

q

Ý

d <Ž m , n . d : even

s

1

Ý

d <Ž m , n .

q

d

d

d <Ž m , n . d: even

Dim L Ž m , n , 1. s

ž

m n , d d

m n , d d < s
mŽ d.

ž

d <Ž m , n . d : odd

d <Ž m , n .

d 1 d

Ł f Ž i q j y 1.

si j

/ Ž < s < y 1. !

m n sgT , d d < s
1

Ý

s!

Ý

ž

Ý

/

Ž < s < y 1. ! sgT

d

/

Ý

d: odd

s

m n , d d

m Ž d . Wy

mŽ d.

1

Ý

ž

s!

Ł f Ž i q j y 1.

si j

, Ž 4.18.

/ m n , d d

m Ž d . Wy

ž

mŽ d.

Ý

/ Ž < s < y 1. !

m n , d d < s
sgT

ž

s!

Ł f Ž i q j y 1.

si j

.

/ Ž 4.19.

EXAMPLE 4.10. In this example, we will apply Theorem 4.4, Corollary 4.5, and Theorem 4.8 to the Monstrous Lie superalgebras associated with replicable functions. Ža. If F Ž q . s qy1 , the trivial case, then L Ž F . ( sl Ž 2, C . [ C d ( gl Ž 2, C . .

651

GRADED LIE SUPERALGEBRAS

Žb. If F Ž q . s qy1 q rq Ž r g Z., then P Ž H Ž J . . s  Ž y1, y1 . 4

ŽJ. with Dim HŽy1, y1. s r .

ŽJ. Hence the denominator identity for the Lie superalgebra L Ž F .y is equal to

`

Ł

Ž 1 y p m q n . Dim L

Ž F . Ž m , n.

`

s1y

m, ns1

Ý f Ž i q j y 1. p iq j s 1 y rpq. i , js1

Thus, we have W Žk , l. s which yields Dim L Ž F . Ž m , n .

¡1 ¢0

~kr

if k s l,

k

otherwise,

¡1 mŽ d. r s~ n Ý ¢0

n

if m s n,

d

d
otherwise.

Therefore, the Lie superalgebra [ L Ž F .Žyn, yn. is isomorphic to the free Lie superalgebra generated by a superspace of superdimension r g Z. Žc. If F Ž a. s F for all a G 1 Že.g., F Ž q . s jŽ q . y 744., then by Corollary 4.5 and Theorem 4.8, we have ` ns1

ŽJ. L Ž F .y s

Dim L Ž F . Ž m , n . s

1

Ý

d <Ž m , n .

s

d

d <Ž m , n .

Ý sgT

1

Ý

Ž < s < y 1. !

mŽ d.

d

ž

m n , d d

s!

Ý sgT

ž

m n , d d

si j

/ Ž < s < y 1. !

mŽ d.

Ł f Ž i q j y 1.

s!

Ł f Ž d. Ž i q j y 1.

si j

/

s f Ž mn . . Žd. More generally, if F is replicable, then by Theorem 4.4 and the ŽJ. identity Ž4.16., the denominator identity for the Lie superalgebra L Ž F .y is equal to `

Ž . Ł Ž 1 y p mq n . Dim L F

Ž m , n.

s1y

m, ns1

`

Ý f Ž i q j y 1. p iq j i , js1

s

`

Ł

m, ns1

exp y

ž

`

Ý as1

1 a

/

f Ž a. Ž mn . p am q an .

652

SEOK-JIN KANG

Taking the logarithm, we have `

log

Ž . Ł Ž 1 y p m q n . Dim L F

Ž m , n.

m, ns1

`

sy

`

1

Ý Ý

Dim L Ž F . Ž m , n . p k m q k n

k

m, ns1 ks1 `

sy

1

Ý

Ý

m, ns1

k)0 k <Ž m , n .

k

Dim L Ž F . Ž mrk , nrk . p m q n ,

and `

log

Ł

exp y

m, ns1

sy

` as1

a

`

`

1

Ý Ý m, ns1 as1

sy

1

ž

Ý

a

`

f Ž a. Ž mn . p am q an 1

Ý

Ý

m, ns1

a)0 a <Ž m , n .

/

f Ž a. Ž mn . p am q an

a

f Ž a.

mn

p mq n .

ž / a2

It follows that 1

Ý k)0 k <Ž m , n .

k

Dim L Ž F . Ž mrk , nrk . s

1

Ý a)0 a <Ž m , n .

a

f Ž a.

mn

ž / a2

.

Therefore, by Mobius inversion, we obtain ¨ Dim L Ž F . Ž m , n . s

1

Ý

ad <Ž m , n .

ad

m Ž d . f Ž a.

ž

mn a2 d 2

/

.

ACKNOWLEDGMENTS Part of this work was completed while I was visiting the Hong Kong University of Science and Technology and Research Institute for Mathematical Sciences at Kyoto University at the beginning of 1997. I express my sincere gratitude to Professors Xiaoping Xu, Masaki Kashiwara, Tetsuji Miwa, and Kyoji Saito for their hospitality. I also thank Professor Isaiah L. Kantor for sending me his paper on free Lie superalgebras wKanx. Special thanks should be given to Professor Georgia Benkart and my student Jae-Hoon Kwon for their interest in this work and many valuable discussions.

GRADED LIE SUPERALGEBRAS

653

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654 wHMx wJx wJu1x wJu2x

wJWx wJLWx wK1x wK2x wK3x wK4x wK5x wKKx

wKKKx wKW1x

wKW2x

wKa1x wKa2x wKa3x wKa4x

wKa5x wKaK1x wKaK2x wKaK3x wKanx

SEOK-JIN KANG

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