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A Generalization of Borcherds Algebra and Denominator Formula
JOURNAL OF ALGEBRA ARTICLE NO.
180, 631]651 Ž1996.
0085
A Generalization of Borcherds Algebra and Denominator Formula Masahiko Miyamoto Department of Mathematics, Ehime Uni¨ ersity, Matsuyama 790, Japan Communicated by Walter Feit Received April 7, 1995 DEDICATED TO PROFESSOR TAKESHI KONDO ON HIS
60TH BIRTHDAY
We study generalized Lie superalgebras Žan extension of Kac’s generalized Lie superalgebras . and show that they are transformed forms of L-graded Lie superalgebras for some abelian group L. We then introduce a generalized Lie superalgebra version of the generalized Kac]Moody algebra ŽBorcherds algebra.. Since it is a transformed Borcherds superalgebra, it has several properties similar to those of Borcherds superalgebra. For example, it is defined by similar relations and has a similar denominator formula and character formulas. Q 1996 Academic Press, Inc.
1. INTRODUCTION A notion of Borcherds algebras Žgeneralized Kac]Moody algebras, GKM for short. was introduced by Borcherds wB1x and he has proved Conway and Norton’s moonshine conjecture using this new algebra wB5x. In general, Borcherds algebras have almost all the properties that ordinary Kac]Moody algebras have, and the only major difference is that Borcherds algebras are allowed to have imaginary simple roots. However, their properties are much the same as those of Kac]Moody algebras and so they have a similar denominator formula and a character formula. The most famous example of Borcherds algebra is the monster Lie algebra, which is given by the following: Let Ž V, Y, w, 1. be a vertex algebra, then VrLy1 V becomes a Lie algebra by w a, b x s a0 bŽmod Ly1V ., where Ly1 s wŽ0.. For example, if V s V > m VII1, 1 is the tensor product of the moonshine module vertex operator algebra V > and the vertex algebra VII1, 1 of the even unimodular Lorentzian lattice II1, 1 of dimension 2, then the monster Lie algebra is given as a subalgebra of VrLy1 V. 631 0021-8693r96 $18.00 Copyright Q 1996 by Academic Press, Inc. All rights of reproduction in any form reserved.
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MASAHIKO MIYAMOTO
Recently, Mossberg extended the concept of vertex algebras wMox. Roughly speaking, the original vertex algebra is defined by the commutativity: Y Ž ¨ , z1 . Y Ž u, z 2 . ; Y Ž u, z 2 . Y Ž ¨ , z1 . . Mossberg’s idea is to introduce a P-gradation of V s Ý a g P V a by some finitely generated abelian group P and a Cr2Z-valued inner product ² ? , ? : on P and a map u : P = P ª C* satisfying
u Ž a, b .u Ž b, a . s 1
Ž 1.1.
u Ž a q b, g . s u Ž a, g .u Ž b, g ..
Ž 1.2.
and
Because of Ž1.2., u is a 2-cocycle of L. He then used the anti-commutativity
Ž z1 y z 2 .
²a , b :
Y Ž ¨ , z1 . Y Ž u, z 2 . ; Ž z 2 y z1 .
²a , b :
u Ž a , b . Y Ž u, z 2 . Y Ž ¨ , z1 .
for ¨ g V a , u g V b Žsee wMox for detail.. Stimulated by these results, it is natural to study the algebra VrLy1 V of Mossberg’s vertex operator algebra V, but Ž VrLy1 V, w ¨ , u x s ¨ 0 u. is not a Lie algebra any more, where ¨ denotes the image of ¨ in VrLy1 V. However, if we take the trivial inner product ² ? , ? : s 0 on P and pay attention to the map u , we can find some nice algebraic structure on VrLy1 V. This is a Žnonassociative. algebra that we shall study in this paper. We should note that our argument includes vertex operator superalgebras since it is possible to replace any Z 2-valued inner product ² ? , ? : on P by some 2-cocycle satisfying Ž1.1. and Ž1.2.. It is easy to see that VrLy1 V has also a P-graded structure Ý a g P V arLy1V a. By the similar argument as in the case of vertex operator algebra, we have
w a, b x s u Ž a , b . b, a g V aq brLy1V aq b
Ž 1.3.
s w a, b x , c q u Ž a , b . b, w a, c x
Ž 1.4.
a, b, c
for a g V a , b g V b , c g V, and ¨ denotes the image of ¨ in VrLy1 V. Kac introduced such an algebra at the end of his paper wK1x and called it a generalized Lie superalgebra where P s Z 2n and a map u is given by u ŽŽ a1 , . . . , a n ., Ž b1 , . . . , bn .. s Žy1. Ý a i b i . In the right side, we consider a i , bi g Z Žmod 2Z.. He also proposed the classification of such a simple
GENERALIZATION OF BORCHERDS ALGEBRA
633
algebra with finite dimension wK1x. We still use the same name for any abelian group P with a 2-cocycle u . For example, if R s Ý a g P R a is a P-graded associative algebra, then R becomes a generalized Lie superalgebra by the product:
w ¨ , u x s ¨ u y u Ž a , b . u¨
for ¨ g R a , u g R b .
Conversely, for a generalized Lie superalgebra R, we will define the universal enveloping algebra UŽ R . of R by the above relation as usual Žsee Ž3.2... An example of Mossberg’s vertex operator algebra is a tensor product of vertex superalgebras and it is used to construct an original vertex operator algebra in wMix. We will show that we can construct a Lie superalgebra from generalized Lie superalgebra by the same way as that used in wMix. Namely, using a group extension C*e P of P by C* and its 2-cocycle f given by e a e b s f Ž a, b . e aq b , we can transform any u-generalized Lie superalgebra R s [ag P R a into a u 9-generalized Lie superalgebra [ag P R a m e a, where u 9Ž a , b . is given by u Ž a , b . f Ž a , b . f Ž b , a .y1 . For example, any generalized Lie superalgebras in a sense of Kac’s are able to be transformed into Lie superalgebras. So we can simply answer the question about the classification of Kac’s generalized Lie superalgebras in wK1x. THEOREM 1.1. Let P be a finitely generated abelian group. Then e¨ ery generalized Lie superalgebra is a transformed form of Lie superalgebra R s @ ¨ g P R ¨ with a P-gradation such that @ ¨ g P , u Ž ¨ , ¨ .s1 R ¨ is the e¨ en part of R and @ ¨ g P , u Ž ¨ , ¨ .s1 R ¨ is the odd part of R. Proof. Set P o s ¨ g P < u Ž ¨ , ¨ . s y14 and Le s L y Lo s ¨ g P < u Ž ¨ , ¨ . s 14 . Define c Ž ¨ , w . s y1 for ¨ , w g Lo and 1 otherwise. Then c satisfies Ž1.1. and Ž1.2.. It is clear that a c-generalized Lie superalgebra is just a Lie superalgebra. Decompose P as a direct product of cyclic groups P s @ i g I Ci and let ¨ i be a generator of Ci . Put an order Ž). in I. Define f Ž ¨ i , ¨ j . s u Ž ¨ i , ¨ j . c Ž ¨ i , ¨ j . for i ) j and 1 otherwise and then extend it to P = P ª C* homomorphically. It is easy to see that f is a 2-cocycle such that u Ž ¨ , w . s c Ž ¨ , w . f Ž ¨ , w . f Ž w, ¨ .y1 . Among Lie algebras, the Borcherds algebra Žthe generalized Kac]Moody algebra. plays an important role in many fields. Especially, its character formulas offer many examples of modular forms wB6x. We will introduce a generalized Lie superalgebra’s version of Borcherds algebra and call it a ‘‘generalized Borcherds superalgebra.’’ The main aim of this paper is to show that this new algebra still has properties and character formulas Ž9.6. similar to those of Borcherds algebra.
634
MASAHIKO MIYAMOTO
Another remarkable point is that this new algebra is also defined by a few easy relations and the same relations Žsee Theorem 7.1. ad Ž e i .
1y 2 a i jra i i
ej s 1
as in the definitions of Kac]Moody algebras and Borcherds algebras.
2. NOTATION A a, b ai B bi j C* Dq Doq Deq ei , fi , h i F Ž A, B .
A symmetric real matrix Ž a i j .. Elements in P. A positive simple root. A C*-matrix Ž bi j . with bi j bji s 1. s u Ž a i , a j . g C y 04 . s C y 04 . The set of all positive roots. The set of positive roots a with u Ž a , a . s y1. The set of positive roots a with u Ž a , a . s 1. The generators of F Ž A, B .. The free generalized Lie superalgebra generated by e i , f i , h i Ž i g I .. Ž . G s G A, B s F Ž A, B .rr. ˜Ž A, B . G s F Ž A, B .rr. ˜ H s ² h i : i g I :. H* s HomŽ H, C.. I The set of indices i of positive simple roots a i . I re The set of indices of real positive simple roots. I im The set of indices of imaginary positive simple roots. m A map m : F Ž A, B . ª P with m Ž e i . s ymŽ f i . s a i and m Ž h i . s 0. V The generalized Casimir operator. P A lattice @ i g I Z a i with Ž a i , a j . s a i j and a map u : P = P ª C*. P* s HomŽ P, C.. Pq See Ž9.1.. r˜ The ideal generated by relations Ž4.7.. r s KerŽ?N ? . l Ý i / 0 F Ž A, B . i . ri A reflection by a real simple root a i . r A Weyl vector with Ž r , a i . s 12 a ii . i u i 4 is a basis of HrRadŽ H . and u i 4 is the dual basis of ui, u u i4.
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GENERALIZATION OF BORCHERDS ALGEBRA
n u W Ž?, ? . Ž?N ? .
A map n : ÝZh i ª P with n Ž h i . s a i . u Ž a i , a j . s bi j . The Weyl group ² ri : i g I re :. An inner product on H and P. An invariant bilinear form of F Ž A, B . Žand GŽ A, B ...
3. GENERALIZED LIE SUPERALGEBRAS Throughout this paper, P denotes a lattice with a map u : P = P ª C* satisfying Ž1.1. and Ž1.2.. In particular, u Ž a , a . s "1. We call a g P e¨ en if u Ž a , a . s 1 and odd otherwise. In this section, we introduce notion of generalized Lie superalgebra Žcf. wK1x. and show several properties of this algebra. We shall omit all proofs in this section, since all results come from the straightforward calculations. DEFINITION 1. Let R s [a g P R a be a P-graded vector space. If R has a bilinear product w?, ? x satisfying: Ž1. w R a , R b x : R aq b , Ž2. w ¨ , u x s yu Ž a , b .w u, ¨ x, and Ž3. w ¨ , w u, w xx s ww ¨ , u x, w x q u Ž a , b .w u, w ¨ , w xx for ¨ g R a , u g R b, w g R, then R is called a generalized Lie superalgebra. LEMMA 3.1.
If R is a generalized Lie superalgebra, then
w w, u x , ¨ s w, w u, ¨ x q u Ž b , a . w w, ¨ x , u
Ž 3.1.
for ¨ g R a , u g R b, w g R. LEMMA 3.2. If R is a P-graded associati¨ e algebra, then R becomes a generalized Lie superalgebra by
w ¨ , u x s ¨ u y u Ž a , b . u¨
for ¨ g R a , u g R b .
Conversely, we shall define the universal enveloping algebra of a generalized Lie superalgebra by the above relation. Namely, for a generalized Lie superalgebra G s @ ag P G a, let T Ž G . be the tensor generalized superalgebra over the space G with the induced P-grading and let n be the
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MASAHIKO MIYAMOTO
ideal of T Ž G . generated by the elements of the form:
w ¨ , u x y ¨ u q u Ž a , b . u¨ for ¨ g G a , u g G b. We set U Ž G . s T Ž G . rn
Ž 3.2.
and call it the universal enveloping algebra of G. We should note that the definition of the tensor product is different from the usual one. Let V s @ ag P V a and W s @ b g P W b be generalized superalgebras. Their tensor product V m W is the generalized superalgebra whose space is the tensor product of the spaces of V and W, with the induced P-grading and the operation defined by
Ž ¨ 1 m w1 . Ž ¨ 2 m w 2 . s u Ž b 1 , b 1 . u Ž g 2 , g 2 . Ž ¨ 1¨ 2 m w1w 2 . , for ¨ i g V g i and wi g W b i ; see wK1x. DEFINITION 2. Let R s Ý a g P R a be a P-graded algebra. For t g P, a t-deri¨ ation of R is an endomorphism ft : R ª R such that ft Ž u¨ . s ft Ž u . ¨ q u Ž t , a . uft Ž ¨ .
Ž 3.3.
for u g R a , ¨ g R. A P-derivation is a linear combination Ý i f t i of t i-derivations ft i with t i g P. LEMMA 3.3. The set of all P-deri¨ ations becomes a generalized Lie superalgebra by
w fa , gt x s fa gt y u Ž a , t . gt fa .
Ž 3.4.
For example, the ordinary derivations are 0-derivations. Let R be a semisimple Lie algebra, H be a Cartan subalgebra, and P be the root lattice. For any 2-cocycle u of P satisfying Ž1.1. and Ž1.2., R is a P-graded algebra and the set of all P-derivations becomes a generalized Lie superalgebra which contains R as a subalgebra.
4. GENERALIZED BORCHERDS SUPERALGEBRAS In this section we will define a concept of generalized Borcherds superalgebra. We adopt the most notation from wK2x. Let I be a countable set and assume that Ž a i j . i, j g I is a real symmetric square matrix and Ž bi j . i, j g I is a complex matrix satisfying the following
GENERALIZATION OF BORCHERDS ALGEBRA
637
properties: 1. If i / j then a i j F 0. 2. If a ii ) 0 then 2 a i jra ii are integers for all j. 3. If a i i ) 0 and bi i s y1, then 2 a i jra i i are even for all j. 4. bi j bji s 1 for all i, j g I. DEFINITION 3. i is called a real index if a ii ) 0 and an imaginary index otherwise. I re and I im denote the set of all real indexes and the set of all imaginary indexes, respectively. i is also called even if bii s 1 and odd otherwise. Let h i , e i , f i Ž i g I . be formal elements ŽChevalley generators.. Let P s Ý i g I Z a i be a lattice with the inner product Ž a i , a j . s a i j and define a map u : P = P ª C* by
u Ž a i , a j . s bi j u Ž a q b, g . s u Ž a, g .u Ž b, g . u Ž a, b q g . s u Ž a, g .u Ž a, b .. It is easy to see that u satisfies Ž1.1. and Ž1.2. since bi j bji s 1. We will call elements of P roots. In particular, a i : i g I 4 are called the positive simple roots and ya i : i g I 4 are called the negative simple roots. Define a map m : h i , e i , f i : i g I 4 ª P so that m Ž e i . s ymŽ f i . s a i and m Ž h i . s 0. Let F Ž B . be the free generalized Lie superalgebra on h i , e i , f i Ž i g I . with a bilinear product w , x satisfying
w x, y x s yu Ž a , b . w y, x x
Ž 4.1.
and x, w y, z x s w x, y x , z q u Ž a , b . y, w x, z x ,
Ž 4.2.
for x g F Ž B . a , y g F Ž B . b, and z g F Ž B ., where the P-gradation F Ž B . s @ a g P F Ž B . a is given by a
0
h i g F Ž B . , ei g F Ž B . i , fi g F Ž B . and a
F Ž B. , F Ž B.
b
: F Ž B.
aq b
.
ya i
,
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MASAHIKO MIYAMOTO
Moreover, let m be the ideal of F Ž B . generated by ei , f j s di , j h i , h i , h j s 0,
Ž 4.3.
h i , e j s ai j e j , and h i , f j s ya i j f j
for i , j g I.
Consider the quotient generalized Lie superalgebra F Ž A, B . s F Ž B . rm.
Ž 4.4.
Setting degŽ e i . s ydegŽ f i . s 1, we also introduce a Z-gradation of F Ž A, B .. Clearly, we have: F Ž A, B . 0 s
@ C hi , igI
F Ž A, B . 1 s
@ C ei , igI
Ž 4.5.
and F Ž A, B . y1 s
@ C fi .
Set H s F Ž A, B . 0 s
@ C hi .
Ž 4.6.
igI
H is an abelian Lie algebra ŽCartan subalgebra . and acts on all F Ž A, B . i diagonally. We introduce an inner product Ž?, ? . in H by Ž h i , h j . s a i j for all i, j g I. We note that we can identify C P and H by a i l h i . Let nq and ny be the subalgebras generated by all e i and f i , respectively, that is, nqs @ `is1 F Ž A, B . i and nys @ `is1 F Ž A, B .yi . Clearly, we have F Ž A, B . s ny[ H [ nq. DEFINITION 4. Let ˜ r be the ideal of F Ž A, B . generated by relations: ei , w ei , ei x , fi , w fi , fi x
Ž ad e i .
1y 2 a i jra i i
ej
for all i for a ii ) 0,
and
w ei , e j x , fi , f j s 0 where Žad e . ¨ denotes w e, ¨ x.
for a i j s 0
Ž 4.7.
GENERALIZATION OF BORCHERDS ALGEBRA
639
Later, we will extend the inner product Ž h i , h j . s a i j of H to an invariant bilinear form Ž?N ? . on F Ž A, B . satisfying the properties: Ž1. Ž?N ? . < H s Ž?, ? . Ž2. Ž?N ? . is nondegenerate on F Ž A, B .1 [ F Ž A, B .y1 , and Ž3. Ž F Ž A, B .a < F Ž A, B .b . s 0 except a q b s 0 for a , b g P. It will be easy to see that ˜ r is contained in the kernel of Ž?N ? ., which means that ˜ r has a trivial intersection with F Ž A, B .1 [ H [ F Ž A, B .y1 because of Ž2.. Set ˜Ž A, B . s F Ž A, B . rr. G Ž 4.8. ˜ Using P s Ý i g I Z a i , we have a P-gradation of both F Ž A, B . and ˜Ž A, B . by mŽ e i . s a i , mŽ f i . s ya i and mŽw x, y x. s mŽ x . q mŽ y . g P. G In our proofs, we will often omit the detail, since most arguments are similar to those in wK2x. The main difference between our algebras and Borcherds algebras is that our following invariant bilinear form is not symmetric. THEOREM 4.1. There exists a bilinear C-¨ alued form Ž?N ? . on F Ž A, B . such that: Ža. Ž?N ? . is in¨ ariant, i.e., Žw x, y x< z . s Ž x
˜Ž A, B ., that is, we shall We shall later prove a kind of simplicity of G prove RadŽ?, ? . [ ˜ r s RadŽ?N ? ., but not now, where Ž?, ? . is the inner product on H and RadŽ?N ? . denotes the radical of Ž?N ? .. So, for a while, we use the following notation. DEFINITION 5. Let r be the maximal ideal of F Ž A, B . containing ˜ r such that it has a zero intersection with F Ž A, B .1 [ H [ F Ž A, B .y1. Set G Ž A, B . s F Ž A, B . rr. We shall call GŽ A, B . the ‘‘generalized Borcherds superalgebra.’’
Ž 4.9.
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MASAHIKO MIYAMOTO
It is clear that r s Ž[n/ 0 F Ž A, B . n . l KerŽ?N ? .. We will later show that this algebra is given by some relations Žsee Theorem 7.1.. and has properties similar to those of Borcherds algebras. We should note that GŽ A, B . has also a P-gradation. For, it follows from Žc. in Theorem 4.1 that the radical of Ž?N ? . is also an orthogonal sum of Rad Ž F Ž A, B . a [ F Ž A, B . y a . s Ž r l F Ž A, B . a . [ Ž r l F Ž A, B . y a . .
Ž 4.10. Since r l H s 04 , we have r s Ž nql r. [ Ž nyl r. and so G Ž A, B . s nyr Ž nyl r . [ H [ nqr Ž nql r . .
Ž 4.11.
In particular, the universal enveloping algebra of GŽ A, B . is U Ž G Ž A, B . . s Uy Ž G Ž A, B . . m H m Uq Ž G Ž A, B . . ,
Ž 4.12.
where Uy Ž GŽ A, B .. s UŽ nyrŽ ny l r.. and Uq Ž GŽ A, B .. s UŽ nqr Ž nql r... 5. EXAMPLES In this section, we shall study a few minimal examples. Fix a real simple root i g I re and set f s f i , h s h i , e s e i , and r s a ii . If u Ž i, i . s bii s 1, then C f [ C h [ C e is isomorphic to sl 2 ŽC. and we know its representations. So assume that u Ž i, i . s bi i s y1. Let F be the free generalized Lie superalgebra generated by e, f, h. Namely, we think of F as the generalized Lie superalgebra F Ž A, B . with 1 = 1-matrices A s Ž r . and B s Žy1.. The products of e, f, and h are given by:
w e, f x s w f , e x s h,
w h, e x s rh,
and
w h, f x s yrh. Ž 5.1.
Let V be an F-module on which e and f act as locally nilpotent. Choose 0 / ¨ g V such that f¨ s 0 and h¨ s s¨ for some s g C. Set ¨ 1 s e¨ and ¨ n s eŽ ¨ ny1 . for n ) 1 inductively. Since the action of e on V is locally nilpotent, there is an m g N such that ¨ m s 0. Take m as the minimum subject to the condition. If we set h¨ n s SŽ n. ¨ n , then since S Ž n . ¨ n s h¨ n s he¨ ny1 s w h, e x ¨ ny1 q eh¨ ny1 s r¨ n q S Ž n y 1 . ¨ n ,
Ž 5.2. we have SŽ n. s r q SŽ n y 1. s nr q s.
641
GENERALIZATION OF BORCHERDS ALGEBRA
Setting f¨ n s T Ž n. ¨ ny1 , we have T Ž n . s Ž n y 1. r q s y T Ž n y 1. s Ž n y 1. r q s y Ž n y 2. r y s q T Ž n y 2. s T Ž n y 2. q r
Ž 5.3.
since T Ž n . ¨ ny 1 s f¨ n s fe¨ ny1 s w f , e x ¨ ny1 y ef¨ ny1 s h¨ ny 1 y eT Ž n y 1 . ¨ ny2 s Ž Ž n y 1 . r q s . y T Ž n y 1 . 4 ¨ ny 1 .
Ž 5.4.
For n s 1, 2, we have f¨ 1 s fe¨ s w f, e x¨ y ef¨ s h¨ s s¨ s T Ž1. ¨ and f¨ 2 s fe¨ 1 s w f, e x¨ 1 y ef¨ 1 s h¨ 1 y es¨ s Ž r q s y s . ¨ 1 s r¨ 1. Hence T Ž1. s s, T Ž2. s r, and so we have T Ž1 q 2 n. s nr q s and T Ž2 n. s nr. By applying them to ¨ m s 0, we have:
½
If m s 2 n,
then 0 s f¨ 2 n s T Ž 2 n . ¨ 2 ny1 s nr¨ 2 ny1 and so r s 0.
If m s 2 n q 1,
then 0 s f¨ 2 nq1 s Ž nr q s . ¨ 2 n and so nr q s s 0.
Ž 5.5 . In particular, we have that m s 1 y 2 srr is a positive odd integer and so srr is a nonpositive integer except for the case r s Ž h, h. s 0 and the eigenvalues of h are s, r q s, . . . , yr y s, ys 4 .
Ž 5.6.
On the other hand, if srr s yn is a non-positive integer, then we have f¨ 2 nq1 s 0 without the assumption that e acts as locally nilpotent. We note that w e, e x is also an element, but we have w e, w e, ¨ xx s ww e, e x, ¨ x y w e, w e, ¨ xx and so ww e, e x, ¨ x s 2w e, w e, ¨ xx for ¨ g V. LEMMA 5.1.
Recall the definition of r from Definition 5, then we ha¨ e: ei , w ei , ei x g r
for all i g I.
If a ii s 0,
then w e i , e i x g r.
Proof. If e i is even Ž bii s 1., then w e i , e i x s 0. So assume that e i is odd Ž bii s y1.. Then we have: fi , w ei , ei x s w h i , ei x y w ei , h i x s 2 ai i ei
642
MASAHIKO MIYAMOTO
and fi , ei , w ei , ei x
s 2 a ii w e i , e i x y w e i , 2 a ii e i x s 0.
Hence w e i , w e i , e i xx Žor w e i , e i x if a i i s 0. are all highest weight vectors and so Ž Vy3 h i
w e, e x , w f , f x s w e, e x , f , f q f , w e, e x , f s w y2 re, f x q w f , y2 re x s y4rh, and w h, w e, e xx s 2 r w e, e x, we have the following lemma. LEMMA 5.2.
If r s a ii / 0 and u Ž m Ž e ., m Ž e .. s y1, then
¦
y
PROPOSITION 5.3.
1 2r
1
1
r
2r
w f , f x , h,
w e, e x ( sl 2 Ž C . .
;
If A s Ž r . / Ž0. and B s Žy1., then
G Ž A, B . s C w f , f x [ C f [ C h [ C e [ C w e, e x is a simple generalized Lie superalgebra Ž super Lie algebra.. If A s Ž0. and B s Žy1., then G Ž A, B . s C f [ C h [ C e is a Lie superalgebra with the center C h and w GŽ A, B ., GŽ A, B .x s C h. As an application of the above arguments, we have a reflection ri on P for every real root i g I re . From the construction, H acts on GŽ A, B . and ˜Ž A, B . diagonally and so GŽ A, B . and G˜Ž A, B . have H *-gradations and G P-gradations, where H * s HomŽ H, C.. If GŽ A, B .a / 0 for a g H * Žor a g P ., then a is called a root of GŽ A, B .. There is a natural homomorphism from P to H * given by a Ž h. s Ž ny1 Ž a ., h.. We will abuse these notations. We note that by the definition, all adŽ e i . and adŽ f i . are locally nilpotent if i g I re . Hence, LEMMA 5.4.
Let a i be a real simple root and a be a root, then ri Ž a . s a y
is also a root.
2Ž a , ai . a ii
ai g H *
Ž or g P .
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GENERALIZATION OF BORCHERDS ALGEBRA
6. GENERALIZED CASIMIR OPERATOR Let r g P* s HomŽ P, C. be a Weyl vector, i.e., r Ž a i . s 12 a i i . We shall abuse the notation Ž c < a i . to denote c Ž a i . for c g HomŽ P, C.. We will define a generalized Casimir operator V of GŽ A, B . as follows: First set a local Casimir operator V0 s 2
Ý Ý eyi a eai .
agD q
i
i 4 Here eai 4 is a basis of GŽ A, B .a for a positive root a and ey a is the dual i j basis of GŽ A, B .ya , that is, Ž ea < ey b . s da , b d i, j . In particular, for a simple root i s a , we take ea s e i and ey a s f i . i We have that for x s Ý x i e ai and y s Ý yi eya , i .Ž e ai < y . s Ý x i yi s Ž x < y . . Ý Ž x < eya
Ž 6.1.
i i . We note that Ž6.1. is well-defined, since for any x, Ž x < ey1 s 0 for all but a finite number of i g I.
LEMMA 6.1 ŽLemma 2.4 in wK2x.. i m Ý eya
For z g G by a ,
e ai , z s
Ý
i z, eyb m e bi .
Ž 6.2.
Proof. For e g Ua , f g Uyb , we have the same inner products: s m Ý Ž eya
e m f. s
e as , z
s .Ž Ý Ž e < eya
e as , z N f .
s s . Ž e as N w z, f x . s Ž e N w z, f x . . Ý Ž e < eya s e m f . s Ý Ž e N z, eyb . Ž e bs N f .
s
ÝŽ
s z, eyb m e bs
s
s Ž w e, z x N f . s Ž e N w z, f x . . COROLLARY 6.2.
In the notation of Lemma 6.1, we ha¨ e i Ý eya
e ai , z s
Ý
i z, eyb e bi .
Ž 6.2.
Proof. Apply to the above lemma the linear map of GŽ A, B . m GŽ A, B . to UŽ GŽ A, B .. given by x m y ª xy. A GŽ A, B .-module V is called restricted if for every ¨ g V, we have GŽ A, B .a ¨ s 0 for all but a finite number of positive roots a .
644
MASAHIKO MIYAMOTO
If V is a restricted GŽ A, B .-module and u g UŽ GŽ A, B ..a ,
THEOREM 6.3. then:
w V 0 , u x s yu Ž 2 Ž r < a . q Ž a < a . q 2 ny1 Ž a . .
Ž 6.3.
Proof. By the same argument as in the proof of Theorem 2.6 of wK2x, it is sufficient to check the cases u s e t and u s eyt .
w V 0 , e t x s 2 Ý Ý eys a eas e t y e t eys a eas a
s
s 2Ý
Ý Ž y yu Ž a t , ya . eys a et q et eys a 4 eas s s s qey a Ž ea e t y u Ž a t , ya . ea e t . .
s 2Ý
Ý Žy
s s s s e t , ey a ea q ey a ea , e t
s y2 w e t , eyt x e t y 2
Ý a gD qy t 4
.
s s e t , ey a ea
žÝ s
s s y Ý ey l qt ea yt , e t
/
s y2 w e t , eyt x e t s y2 h t e t s y2 w h t , e t x y 2 e t h t s y2 a t t e t y 2 e t h t s ye t Ž 2 Ž h t , h t . q 2 h t . s ye t Ž Ž r , h t . q Ž h t , h t . q 2 h t . .
w V 0 , eyt x s 2 Ý Ý eys a eas eyt y eyt eys a eas a
s
s 2Ý
Ý Ž eys a
s s eas , e t y eyt , ey a ea .
s 2 eyt w e t , eyt x q 2
Ý a gD qy t 4
s ya
ž Ýe
s , ey a , et
s
y Ý eyt , eas yt eayt
/
s 2 eyt h t s yeyt Ž Ž r , yh t . q Ž yh t , yh t . q 2 Ž yh t . . Let u i 4 be a basis of HrRadŽ H . and u i 4 be the dual basis of u i 4 . LEMMA 6.4.
Ý ui ui , x i
s x Ž Ž a < a . q 2 ny1 Ž a . .
for x g G Ž A, B . a . Ž 6.4.
GENERALIZATION OF BORCHERDS ALGEBRA
645
Proof.
Ý ui ui , x
s
i
Ý u i u i x y xu i u i i
s
Ý u i xu i y xu i u i q u i u i x y u i xu i i
s
Ý w ui , x x ui q ui w ui , x x i
s
Ý Ž u i , a . xu i q u i Ž u i , a . x i
s
Ý Ž u i , a . xu i q Ž u i , a . w u i , x x q Ž u i , a . xu i i
s
Ý Ž ui , a . Ž ui , a . x q x Ž Ž ui , a . ui q Ž ui , a . ui .
s x Ž Ž a < a . q 2 ny1 Ž a . . We define the generalized Casimir operator by V s 2r q
Ý ui ui q V 0 .
Ž 6.5.
We note that since we use r g P* s HomŽ P, C., V can acts on only P-graded GŽ A, B .-modules. The next results follow from the definition of V and the previous lemmas. THEOREM 6.5. If V is a restricted P-graded GŽ A, B .-module, then V commutes with the action of UŽ GŽ A, B .. on V. COROLLARY 6.6. Let V be a P-graded GŽ A, B .-module and ¨ be a lowest weight ¨ ector in V, that is, e i ¨ s 0 for i g I, and hŽ ¨ . s Ž L < h. ¨ for some L g H *, then VŽ ¨ . s Ž L q 2 r
Ž 6.6.
If, furthermore, UŽ GŽ A, B .. ¨ s V, then V s Ž < L q r < 2 y < r < 2 . IV .
Ž 6.7.
7. RELATIONS We have obtained all necessary results to get the following result similar to Theorem 9.11 in wK2x.
646
MASAHIKO MIYAMOTO
THEOREM 7.1. Recall the definition of r from Definition 5. r is the ideal of F Ž A, B . generated by ei , w ei , ei x , fi , w fi , fi x
Ž ad e i .
1y 2 a i jra i i
for all i for i g I re
ej
Ž 7.1.
and
w ei , e j x , fi , f j
for a i j s 0
˜Ž A, B . s GŽ A, B .. Namely, we ha¨ e G Proof. We use the same notation and arguments as in the proof of Proposition 9.11 in wK2x. We have a GŽ A, B .-homomorphism:
l : rqr w rq , rq x ª mis1 M Ž ya i . . n
Ž 7.2.
n We note that rqrwrq, rq x and [is1 M Žya i . have P-gradations and these are invariant under l, since GŽ A, B . and r have P-gradations. In particular, the generalized Casimir operator V acts on both the GŽ A, B .-modules. Using the degree htŽ e i . s 1, rq has a Z-gradation
rqs
`
Ž 7.3.
Ý Ž rq . i isk
such that Žrq . k / 0. Since the kernel of l is contained in Žrq .Žny .UŽny ., we have lŽr k . / 0. Hence, there is a root a such that Žr k .a / 0. Set
as
k i Ž ya i . q
Ý igJ q
Ý
k j Ž ya j . ,
Ž 7.4.
jgJ y
where a i Ž i g Jq . are positive, a j Ž j g Jy . are negative simple roots, and k i Ž i g J s Jqj Jy . are all positive integers. Since ht ri Ž a . G ht a for any reflection ri , we have Ž a < a i . F 0 for all positive simple roots i and so Ž a < a . F 0. On the other hand, since l is commutative with V, we have
Ž ya q 2 r < y a . s Ž ya i q 2 r < y a i . s 0
Ž 7.5.
for some i and so
Ž a < a . s 2Ž r < a .
Ž 7.6.
647
GENERALIZATION OF BORCHERDS ALGEBRA
since Ž r < a i . s 12 Ž a i < a i .. Using Ž7.4., we have 0 s Ž2 r < a . y Ž a < a . s
Ý ki Ž ai , ai . y ki Ž ai < a . 4 igJ
s
Ý ki Ž ai < ai y a . .
Ž 7.5.
igJ
We will show Ž a i < a i y a . G 0 for all i g J. If i g I re , then Ž a i , a i y a . ) 0, while if i g I im , then
Ž a i , a i y a . s y Ý k j Ž a i , a j . q Ž 1 y k i . Ž a i , l i . G 0.
Ž 7.7.
j/i
The equality holds only if Ž a i < a j . s 0 for all j / i. So we have that there are no real roots in J and all imaginary roots are mutually commutative. This happens only when a is a simple imaginary root, which contradicts r l GŽ A, B .1 s 0. 8. SUPERSYMMETRY A Borcherds algebra has an automorphism exchanging e i and f i and acting on H as y1. However, this is not an automorphism of the generalized Borcherds superalgebra any more. It satisfies the following relation: w Ž w x, y x . s y w Ž y . , w Ž x . . Ž 8.1. However, this is sufficient to have results similar to those of Kac]Moody algebras. We note a
Ž w Ž x . < w Ž y . . s u Ž a , b . Ž x, y .
b
for x g G Ž A, B . , y g G Ž A, B . . Ž 8.2.
Proof. We will prove this by induction. For simple roots,
žw Ž e . w Ž f . / s Ž f N e . s u Ž i , j . Ž e N f . . i
j
i
j
j
i
Ž 8.3.
Also, for x g GŽ A, B . a , y g GŽ A, B . b with i q a q b s 0, we have:
Žw Ž w ei , x x . w Ž y . . s Žy w Ž x . , w Ž ei . w Ž y . . s y Žw Ž x . w Ž ei . , w Ž y . . s Ž w Ž x . w Ž w y, e i x . s u Ž a , b q i . Ž x N w y, e i x . by induction, s u Ž a , b . u Ž a , i . u Ž b , i . u Ž a , i . Ž w ei , x x N y . s u Ž a , a . u Ž i , i . Ž w ei , x x N y . s u Ž a q i , a q i . Ž w ei , x x N y . s u Ž a q i , b . Ž w ei , x x N y . .
Ž 8.4.
648
MASAHIKO MIYAMOTO
9. CHARACTER FORMULAS In this section, we calculate the character formulas of GŽ A, B .-modules with highest weight L on which the actions of real simple roots e i and f i are locally nilpotent. This condition is equivalent to L g Pq, where
¡ ŽlNh . gZ ~ P s l g H * Ž l N h . g 2Z ¢ ŽlNh . G0
G0
i
q
i
G0
i
¦ for real odd simple roots i ¥ for imaginary simple roots i§ for real simple roots i
Ž 9.1.
is the set of dominant roots. We note that the results in this section are independent of deformations. We first calculate the denominator formula of the Verma module M Ž L .. Let b be a positive root. Since ww e, e x, ¨ x s 2w e, w e, ¨ xx for e g Gb if u Ž b , b . s y1, adŽ e . 2 is replaced by 12 adŽ e . in UŽ GŽ A, B ... Hence, the difference between the parts afforded by b and 2 b is
Ž 1 q e Ž yb . .
mult b
s Ž 1r Ž 1 y e Ž yb . .
mu lt b
Ž 1 y e Ž y2 b . .
mult b
.
Hence we easily have the following denominator formula of M Ž l.. THEOREM 9.1. ch M Ž L . s e Ž L .
Łb g Doq Ž 1 q e Ž yb .
mu lt b
Ł a g Deq Ž 1 y e Ž ya . .
mu lt a
,
Ž 9.2.
where Deq denotes the set of all positi¨ e roots a with u Ž a , a . s 1 and Doq denotes the set of all positi¨ e roots b with u Ž b , b . s y1. Set Rs
Ł a g Deq Ž 1 y e Ž ya . . Łb g Doq Ž 1 q e Ž yb .
mult a
mu lt b
Ž 9.3.
and call it the denominator formula of GŽ A, B .. We will also have a character formula for irreducible GŽ A, B .-module LŽ L . with highest weight L g Pq, which is similar to one for an irreducible module of Borcherds algebra. In order to get a character formula, we first prove the following lemma.
GENERALIZATION OF BORCHERDS ALGEBRA
649
LEMMA 9.2. w Ž eŽ r . R. s e Ž w . eŽ r . R
for w g W.
Ž 9.4.
Proof. Let a i be a real simple positive root. If u Ž h i , h i . s bii s 1, Dqy a i 4 is ri-invariant and so we have: ri Ž e Ž r . R . s e Ž r y a i . ri Ž 1 y e Ž ya i . . ri R9 s e Ž r . e Ž ya i . Ž 1 y e Ž a i . . R9 s ye Ž r . R,
Ž 9.5.
where R9 s RrŽ1 y eŽya i ... If bii s y1 and w e i , e i x is not zero, then R has the factor Ž1 y eŽy2 a i ..rŽ1 q eŽya i .. s 1 y eŽya i . and the remaining arguments are the same as the above. If bii s y1 and w e i , e i x s 0, then Ž a i , a i . s 0 and there is no reflection. DEFINITION 6. For L g Pq, let SL s e Ž L q r . Ý e Ž b . e Ž yb . , b
where b runs over all elements of P of the form b s a i1 q ??? qa i m Ž m s 0 if b s 0. such that 1. i1 , . . . , i m are distinct elements of I im , 2. Ž a i k, a i l . s 0 if k / l, and 3. Ž L, a i k . s 0 for 1 F k F m. For such a b , e Ž b . denotes Žy1. m . THEOREM 9.3.
For L g Pq, ch L Ž L . s
1 eŽ r . R
Ý e Ž w . w Ž SL . .
Ž 9.6.
wgW
Proof. This is the same as the proof of Theorem 10.4 in wK2x and let us follow it. By the generalized Casimir operator and the denominator formula, we have: e Ž r . R ch L Ž L . s
Ý
lFL < l q r < 2s < Lq r < 2
cl e Ž l q r .
Ž 9.7.
by the same argument as that in Proposition 9.8 of wK2x. Since eŽ r . R ch LŽ L . is W-skew-invariant Ž9.4., we have cl s e Ž w . c wŽ lq r .y r .
Ž 9.8.
650
MASAHIKO MIYAMOTO
Set B0 s B0 Ž L . s l < cl / 04 . For l g B0 , choose w g W such that htŽ L y s . is minimal, where s s w Ž l q r . y r . The minimality of htŽ L y s . forces Ž s q r , a i . G 0. If Ž s q r , a i . s 0 for some i g I re , then wi Ž s q r . s s q r and so cs s e Ž wi . c w i Ž sq r .y r s ycs . This contradicts cl / 0. Set C s m g P*: Ž m , a i . ) 0 for all i g I re 4 . So the above argument shows s q r g C and such a w is uniquely determined by l. Hence, if we set SL s Ý lq r g C cl eŽ l q r ., then e Ž r . R ch L Ž L . s
Ý e Ž w . w Ž SL . .
Ž 9.9.
wgW
We next show that if l q r g C and cl / 0, then
l s L y Ž a i1 q ??? qa i m .
Ž 9.10.
such that a i1, . . . , a i m satisfy the three conditions of Definition 6 and cl s Žy1. m . Set J s suppŽ L y l. and write L y l s Ý i g J k i a i . By the choice of l, k i are all positive integers, and
Ý ki Ž L , ai . q Ý ki Ž l q 2 r , ai . s Ž L q l q 2 r , L y l . s Ž L q 2 r , L . y Ž l q 2 r , l . s 0. Ž 9.11. By our hypothesis, Ž L, a i . G 0 for all i g I. If i g I re , then
Ž l q 2 r , a i . s Ž l q r , a i . q Ž r , a i . ) 0.
Ž 9.12.
If i g I im , then
Ž l q 2 r , ai . s Ž l, ai . q Ž ai , ai . s Ž L , a i . y Ž k i y 1. Ž a i , a i . y
Ý k j Ž a j , a i . G 0. Ž 9.13. j/i
Hence, we conclude that J : I im , Ž a i , a j . s 0 for all i, j g J with i / j, and Ž L, a i . s 0 for i g J. It remains to show that k i s 1 for i g J. Let b be an element of P j 04 such that the terms eŽyb . and eŽ b q l. occur in R and ch LŽ L ., respectively. Then b and L y b y l are both nonnegative and suppŽ L y l y b . : suppŽ L y l. s J. Since Ž L, a i . s 0 for all i g J as we have mentioned, there are no LŽ L .Ly ŽLy ly b . s LŽ L .bq l for L y l y b / 0 and so we have b s L y l. In particular, the term eŽyb . in R comes only from
Ł Ž 1 y e Ž ya i . .
igI im
Ž 9.14.
GENERALIZATION OF BORCHERDS ALGEBRA
651
and so k i s 1 for all i g J and the coefficient of eŽyb . is Žy1. < J < . Since the coefficient of eŽ L . in ch LŽ L . is 1, we obtain cl s Žy1. < J < . Conversely, if l is an element of P* satisfying three conditions of Definition 6, then it is easy to see that eŽ l. occurs in R ch LŽ L . and l q r g C. This completes the proof of Theorem 9.3. Now set L s 0 in the above Eq. Ž9.6. and we deduce the following ‘‘denominator identity.’’ eŽ r .
Ł a g Deq Ž 1 y e Ž ya . .
mu lt a
Łb g Doq Ž 1 q e Ž yb . .
mult b
s
Ý e Ž w . e Ž w Ž S0 . . . Ž 9.15. wgW
REFERENCES wB1x wB2x wB3x wB4x wB5x wB6x wHMYx wK1x wK2x wKWx wMix wMox
R. E. Borcherds, Vertex algebras, Kac]Moody algebras, and the monster, Proc. Natl. Acad. Sci. USA 83 Ž1986., 3068]3071. R. E. Borcherds, Generalized Kac]Moody algebra, J. Algebra 115 Ž1988., 501]512. R. E. Borcherds, The monster Lie algebras, Ad¨ . in Math. 83 Ž1990., 30]47. R. E. Borcherds, Central extensions of generalized Kac]Moody algebras, J. Algebra 140 Ž1991., 330]335. R. E. Borcherds, Monstrous moonshine and monstrous Lie superalgebras, In¨ ent. Math. 109 Ž1992., 405]444. R. E. Borcherds, Automorphic forms on Osq 2, s Ž< R . and infinite products, preprint. K. Harada, M. Miyamoto, and H. Yamada, A generalization of Kac-Moody algebra, preprint. V. G. Kac, Lie superalgebra, Ad¨ . in Math., 26 Ž1977., 8]96. V. G. Kac, ‘‘Infinite Dimensional Lie Algebras,’’ 3rd ed., Cambridge Univ. Press, Cambridge, 1990. V. G. Kac and W. Wang, Vertex operator superalgebras and their representations, preprint. M. Miyamoto, Binary codes and vertex operator Žsuper.algebras, preprint. G. Mossberg, Axiomatic vertex algebras and the Jacobi identity, J. Algebra 170 Ž1994., 956]1010.
180, 631]651 Ž1996.
0085
A Generalization of Borcherds Algebra and Denominator Formula Masahiko Miyamoto Department of Mathematics, Ehime Uni¨ ersity, Matsuyama 790, Japan Communicated by Walter Feit Received April 7, 1995 DEDICATED TO PROFESSOR TAKESHI KONDO ON HIS
60TH BIRTHDAY
We study generalized Lie superalgebras Žan extension of Kac’s generalized Lie superalgebras . and show that they are transformed forms of L-graded Lie superalgebras for some abelian group L. We then introduce a generalized Lie superalgebra version of the generalized Kac]Moody algebra ŽBorcherds algebra.. Since it is a transformed Borcherds superalgebra, it has several properties similar to those of Borcherds superalgebra. For example, it is defined by similar relations and has a similar denominator formula and character formulas. Q 1996 Academic Press, Inc.
1. INTRODUCTION A notion of Borcherds algebras Žgeneralized Kac]Moody algebras, GKM for short. was introduced by Borcherds wB1x and he has proved Conway and Norton’s moonshine conjecture using this new algebra wB5x. In general, Borcherds algebras have almost all the properties that ordinary Kac]Moody algebras have, and the only major difference is that Borcherds algebras are allowed to have imaginary simple roots. However, their properties are much the same as those of Kac]Moody algebras and so they have a similar denominator formula and a character formula. The most famous example of Borcherds algebra is the monster Lie algebra, which is given by the following: Let Ž V, Y, w, 1. be a vertex algebra, then VrLy1 V becomes a Lie algebra by w a, b x s a0 bŽmod Ly1V ., where Ly1 s wŽ0.. For example, if V s V > m VII1, 1 is the tensor product of the moonshine module vertex operator algebra V > and the vertex algebra VII1, 1 of the even unimodular Lorentzian lattice II1, 1 of dimension 2, then the monster Lie algebra is given as a subalgebra of VrLy1 V. 631 0021-8693r96 $18.00 Copyright Q 1996 by Academic Press, Inc. All rights of reproduction in any form reserved.
632
MASAHIKO MIYAMOTO
Recently, Mossberg extended the concept of vertex algebras wMox. Roughly speaking, the original vertex algebra is defined by the commutativity: Y Ž ¨ , z1 . Y Ž u, z 2 . ; Y Ž u, z 2 . Y Ž ¨ , z1 . . Mossberg’s idea is to introduce a P-gradation of V s Ý a g P V a by some finitely generated abelian group P and a Cr2Z-valued inner product ² ? , ? : on P and a map u : P = P ª C* satisfying
u Ž a, b .u Ž b, a . s 1
Ž 1.1.
u Ž a q b, g . s u Ž a, g .u Ž b, g ..
Ž 1.2.
and
Because of Ž1.2., u is a 2-cocycle of L. He then used the anti-commutativity
Ž z1 y z 2 .
²a , b :
Y Ž ¨ , z1 . Y Ž u, z 2 . ; Ž z 2 y z1 .
²a , b :
u Ž a , b . Y Ž u, z 2 . Y Ž ¨ , z1 .
for ¨ g V a , u g V b Žsee wMox for detail.. Stimulated by these results, it is natural to study the algebra VrLy1 V of Mossberg’s vertex operator algebra V, but Ž VrLy1 V, w ¨ , u x s ¨ 0 u. is not a Lie algebra any more, where ¨ denotes the image of ¨ in VrLy1 V. However, if we take the trivial inner product ² ? , ? : s 0 on P and pay attention to the map u , we can find some nice algebraic structure on VrLy1 V. This is a Žnonassociative. algebra that we shall study in this paper. We should note that our argument includes vertex operator superalgebras since it is possible to replace any Z 2-valued inner product ² ? , ? : on P by some 2-cocycle satisfying Ž1.1. and Ž1.2.. It is easy to see that VrLy1 V has also a P-graded structure Ý a g P V arLy1V a. By the similar argument as in the case of vertex operator algebra, we have
w a, b x s u Ž a , b . b, a g V aq brLy1V aq b
Ž 1.3.
s w a, b x , c q u Ž a , b . b, w a, c x
Ž 1.4.
a, b, c
for a g V a , b g V b , c g V, and ¨ denotes the image of ¨ in VrLy1 V. Kac introduced such an algebra at the end of his paper wK1x and called it a generalized Lie superalgebra where P s Z 2n and a map u is given by u ŽŽ a1 , . . . , a n ., Ž b1 , . . . , bn .. s Žy1. Ý a i b i . In the right side, we consider a i , bi g Z Žmod 2Z.. He also proposed the classification of such a simple
GENERALIZATION OF BORCHERDS ALGEBRA
633
algebra with finite dimension wK1x. We still use the same name for any abelian group P with a 2-cocycle u . For example, if R s Ý a g P R a is a P-graded associative algebra, then R becomes a generalized Lie superalgebra by the product:
w ¨ , u x s ¨ u y u Ž a , b . u¨
for ¨ g R a , u g R b .
Conversely, for a generalized Lie superalgebra R, we will define the universal enveloping algebra UŽ R . of R by the above relation as usual Žsee Ž3.2... An example of Mossberg’s vertex operator algebra is a tensor product of vertex superalgebras and it is used to construct an original vertex operator algebra in wMix. We will show that we can construct a Lie superalgebra from generalized Lie superalgebra by the same way as that used in wMix. Namely, using a group extension C*e P of P by C* and its 2-cocycle f given by e a e b s f Ž a, b . e aq b , we can transform any u-generalized Lie superalgebra R s [ag P R a into a u 9-generalized Lie superalgebra [ag P R a m e a, where u 9Ž a , b . is given by u Ž a , b . f Ž a , b . f Ž b , a .y1 . For example, any generalized Lie superalgebras in a sense of Kac’s are able to be transformed into Lie superalgebras. So we can simply answer the question about the classification of Kac’s generalized Lie superalgebras in wK1x. THEOREM 1.1. Let P be a finitely generated abelian group. Then e¨ ery generalized Lie superalgebra is a transformed form of Lie superalgebra R s @ ¨ g P R ¨ with a P-gradation such that @ ¨ g P , u Ž ¨ , ¨ .s1 R ¨ is the e¨ en part of R and @ ¨ g P , u Ž ¨ , ¨ .s1 R ¨ is the odd part of R. Proof. Set P o s ¨ g P < u Ž ¨ , ¨ . s y14 and Le s L y Lo s ¨ g P < u Ž ¨ , ¨ . s 14 . Define c Ž ¨ , w . s y1 for ¨ , w g Lo and 1 otherwise. Then c satisfies Ž1.1. and Ž1.2.. It is clear that a c-generalized Lie superalgebra is just a Lie superalgebra. Decompose P as a direct product of cyclic groups P s @ i g I Ci and let ¨ i be a generator of Ci . Put an order Ž). in I. Define f Ž ¨ i , ¨ j . s u Ž ¨ i , ¨ j . c Ž ¨ i , ¨ j . for i ) j and 1 otherwise and then extend it to P = P ª C* homomorphically. It is easy to see that f is a 2-cocycle such that u Ž ¨ , w . s c Ž ¨ , w . f Ž ¨ , w . f Ž w, ¨ .y1 . Among Lie algebras, the Borcherds algebra Žthe generalized Kac]Moody algebra. plays an important role in many fields. Especially, its character formulas offer many examples of modular forms wB6x. We will introduce a generalized Lie superalgebra’s version of Borcherds algebra and call it a ‘‘generalized Borcherds superalgebra.’’ The main aim of this paper is to show that this new algebra still has properties and character formulas Ž9.6. similar to those of Borcherds algebra.
634
MASAHIKO MIYAMOTO
Another remarkable point is that this new algebra is also defined by a few easy relations and the same relations Žsee Theorem 7.1. ad Ž e i .
1y 2 a i jra i i
ej s 1
as in the definitions of Kac]Moody algebras and Borcherds algebras.
2. NOTATION A a, b ai B bi j C* Dq Doq Deq ei , fi , h i F Ž A, B .
A symmetric real matrix Ž a i j .. Elements in P. A positive simple root. A C*-matrix Ž bi j . with bi j bji s 1. s u Ž a i , a j . g C y 04 . s C y 04 . The set of all positive roots. The set of positive roots a with u Ž a , a . s y1. The set of positive roots a with u Ž a , a . s 1. The generators of F Ž A, B .. The free generalized Lie superalgebra generated by e i , f i , h i Ž i g I .. Ž . G s G A, B s F Ž A, B .rr. ˜Ž A, B . G s F Ž A, B .rr. ˜ H s ² h i : i g I :. H* s HomŽ H, C.. I The set of indices i of positive simple roots a i . I re The set of indices of real positive simple roots. I im The set of indices of imaginary positive simple roots. m A map m : F Ž A, B . ª P with m Ž e i . s ymŽ f i . s a i and m Ž h i . s 0. V The generalized Casimir operator. P A lattice @ i g I Z a i with Ž a i , a j . s a i j and a map u : P = P ª C*. P* s HomŽ P, C.. Pq See Ž9.1.. r˜ The ideal generated by relations Ž4.7.. r s KerŽ?N ? . l Ý i / 0 F Ž A, B . i . ri A reflection by a real simple root a i . r A Weyl vector with Ž r , a i . s 12 a ii . i u i 4 is a basis of HrRadŽ H . and u i 4 is the dual basis of ui, u u i4.
635
GENERALIZATION OF BORCHERDS ALGEBRA
n u W Ž?, ? . Ž?N ? .
A map n : ÝZh i ª P with n Ž h i . s a i . u Ž a i , a j . s bi j . The Weyl group ² ri : i g I re :. An inner product on H and P. An invariant bilinear form of F Ž A, B . Žand GŽ A, B ...
3. GENERALIZED LIE SUPERALGEBRAS Throughout this paper, P denotes a lattice with a map u : P = P ª C* satisfying Ž1.1. and Ž1.2.. In particular, u Ž a , a . s "1. We call a g P e¨ en if u Ž a , a . s 1 and odd otherwise. In this section, we introduce notion of generalized Lie superalgebra Žcf. wK1x. and show several properties of this algebra. We shall omit all proofs in this section, since all results come from the straightforward calculations. DEFINITION 1. Let R s [a g P R a be a P-graded vector space. If R has a bilinear product w?, ? x satisfying: Ž1. w R a , R b x : R aq b , Ž2. w ¨ , u x s yu Ž a , b .w u, ¨ x, and Ž3. w ¨ , w u, w xx s ww ¨ , u x, w x q u Ž a , b .w u, w ¨ , w xx for ¨ g R a , u g R b, w g R, then R is called a generalized Lie superalgebra. LEMMA 3.1.
If R is a generalized Lie superalgebra, then
w w, u x , ¨ s w, w u, ¨ x q u Ž b , a . w w, ¨ x , u
Ž 3.1.
for ¨ g R a , u g R b, w g R. LEMMA 3.2. If R is a P-graded associati¨ e algebra, then R becomes a generalized Lie superalgebra by
w ¨ , u x s ¨ u y u Ž a , b . u¨
for ¨ g R a , u g R b .
Conversely, we shall define the universal enveloping algebra of a generalized Lie superalgebra by the above relation. Namely, for a generalized Lie superalgebra G s @ ag P G a, let T Ž G . be the tensor generalized superalgebra over the space G with the induced P-grading and let n be the
636
MASAHIKO MIYAMOTO
ideal of T Ž G . generated by the elements of the form:
w ¨ , u x y ¨ u q u Ž a , b . u¨ for ¨ g G a , u g G b. We set U Ž G . s T Ž G . rn
Ž 3.2.
and call it the universal enveloping algebra of G. We should note that the definition of the tensor product is different from the usual one. Let V s @ ag P V a and W s @ b g P W b be generalized superalgebras. Their tensor product V m W is the generalized superalgebra whose space is the tensor product of the spaces of V and W, with the induced P-grading and the operation defined by
Ž ¨ 1 m w1 . Ž ¨ 2 m w 2 . s u Ž b 1 , b 1 . u Ž g 2 , g 2 . Ž ¨ 1¨ 2 m w1w 2 . , for ¨ i g V g i and wi g W b i ; see wK1x. DEFINITION 2. Let R s Ý a g P R a be a P-graded algebra. For t g P, a t-deri¨ ation of R is an endomorphism ft : R ª R such that ft Ž u¨ . s ft Ž u . ¨ q u Ž t , a . uft Ž ¨ .
Ž 3.3.
for u g R a , ¨ g R. A P-derivation is a linear combination Ý i f t i of t i-derivations ft i with t i g P. LEMMA 3.3. The set of all P-deri¨ ations becomes a generalized Lie superalgebra by
w fa , gt x s fa gt y u Ž a , t . gt fa .
Ž 3.4.
For example, the ordinary derivations are 0-derivations. Let R be a semisimple Lie algebra, H be a Cartan subalgebra, and P be the root lattice. For any 2-cocycle u of P satisfying Ž1.1. and Ž1.2., R is a P-graded algebra and the set of all P-derivations becomes a generalized Lie superalgebra which contains R as a subalgebra.
4. GENERALIZED BORCHERDS SUPERALGEBRAS In this section we will define a concept of generalized Borcherds superalgebra. We adopt the most notation from wK2x. Let I be a countable set and assume that Ž a i j . i, j g I is a real symmetric square matrix and Ž bi j . i, j g I is a complex matrix satisfying the following
GENERALIZATION OF BORCHERDS ALGEBRA
637
properties: 1. If i / j then a i j F 0. 2. If a ii ) 0 then 2 a i jra ii are integers for all j. 3. If a i i ) 0 and bi i s y1, then 2 a i jra i i are even for all j. 4. bi j bji s 1 for all i, j g I. DEFINITION 3. i is called a real index if a ii ) 0 and an imaginary index otherwise. I re and I im denote the set of all real indexes and the set of all imaginary indexes, respectively. i is also called even if bii s 1 and odd otherwise. Let h i , e i , f i Ž i g I . be formal elements ŽChevalley generators.. Let P s Ý i g I Z a i be a lattice with the inner product Ž a i , a j . s a i j and define a map u : P = P ª C* by
u Ž a i , a j . s bi j u Ž a q b, g . s u Ž a, g .u Ž b, g . u Ž a, b q g . s u Ž a, g .u Ž a, b .. It is easy to see that u satisfies Ž1.1. and Ž1.2. since bi j bji s 1. We will call elements of P roots. In particular, a i : i g I 4 are called the positive simple roots and ya i : i g I 4 are called the negative simple roots. Define a map m : h i , e i , f i : i g I 4 ª P so that m Ž e i . s ymŽ f i . s a i and m Ž h i . s 0. Let F Ž B . be the free generalized Lie superalgebra on h i , e i , f i Ž i g I . with a bilinear product w , x satisfying
w x, y x s yu Ž a , b . w y, x x
Ž 4.1.
and x, w y, z x s w x, y x , z q u Ž a , b . y, w x, z x ,
Ž 4.2.
for x g F Ž B . a , y g F Ž B . b, and z g F Ž B ., where the P-gradation F Ž B . s @ a g P F Ž B . a is given by a
0
h i g F Ž B . , ei g F Ž B . i , fi g F Ž B . and a
F Ž B. , F Ž B.
b
: F Ž B.
aq b
.
ya i
,
638
MASAHIKO MIYAMOTO
Moreover, let m be the ideal of F Ž B . generated by ei , f j s di , j h i , h i , h j s 0,
Ž 4.3.
h i , e j s ai j e j , and h i , f j s ya i j f j
for i , j g I.
Consider the quotient generalized Lie superalgebra F Ž A, B . s F Ž B . rm.
Ž 4.4.
Setting degŽ e i . s ydegŽ f i . s 1, we also introduce a Z-gradation of F Ž A, B .. Clearly, we have: F Ž A, B . 0 s
@ C hi , igI
F Ž A, B . 1 s
@ C ei , igI
Ž 4.5.
and F Ž A, B . y1 s
@ C fi .
Set H s F Ž A, B . 0 s
@ C hi .
Ž 4.6.
igI
H is an abelian Lie algebra ŽCartan subalgebra . and acts on all F Ž A, B . i diagonally. We introduce an inner product Ž?, ? . in H by Ž h i , h j . s a i j for all i, j g I. We note that we can identify C P and H by a i l h i . Let nq and ny be the subalgebras generated by all e i and f i , respectively, that is, nqs @ `is1 F Ž A, B . i and nys @ `is1 F Ž A, B .yi . Clearly, we have F Ž A, B . s ny[ H [ nq. DEFINITION 4. Let ˜ r be the ideal of F Ž A, B . generated by relations: ei , w ei , ei x , fi , w fi , fi x
Ž ad e i .
1y 2 a i jra i i
ej
for all i for a ii ) 0,
and
w ei , e j x , fi , f j s 0 where Žad e . ¨ denotes w e, ¨ x.
for a i j s 0
Ž 4.7.
GENERALIZATION OF BORCHERDS ALGEBRA
639
Later, we will extend the inner product Ž h i , h j . s a i j of H to an invariant bilinear form Ž?N ? . on F Ž A, B . satisfying the properties: Ž1. Ž?N ? . < H s Ž?, ? . Ž2. Ž?N ? . is nondegenerate on F Ž A, B .1 [ F Ž A, B .y1 , and Ž3. Ž F Ž A, B .a < F Ž A, B .b . s 0 except a q b s 0 for a , b g P. It will be easy to see that ˜ r is contained in the kernel of Ž?N ? ., which means that ˜ r has a trivial intersection with F Ž A, B .1 [ H [ F Ž A, B .y1 because of Ž2.. Set ˜Ž A, B . s F Ž A, B . rr. G Ž 4.8. ˜ Using P s Ý i g I Z a i , we have a P-gradation of both F Ž A, B . and ˜Ž A, B . by mŽ e i . s a i , mŽ f i . s ya i and mŽw x, y x. s mŽ x . q mŽ y . g P. G In our proofs, we will often omit the detail, since most arguments are similar to those in wK2x. The main difference between our algebras and Borcherds algebras is that our following invariant bilinear form is not symmetric. THEOREM 4.1. There exists a bilinear C-¨ alued form Ž?N ? . on F Ž A, B . such that: Ža. Ž?N ? . is in¨ ariant, i.e., Žw x, y x< z . s Ž x
˜Ž A, B ., that is, we shall We shall later prove a kind of simplicity of G prove RadŽ?, ? . [ ˜ r s RadŽ?N ? ., but not now, where Ž?, ? . is the inner product on H and RadŽ?N ? . denotes the radical of Ž?N ? .. So, for a while, we use the following notation. DEFINITION 5. Let r be the maximal ideal of F Ž A, B . containing ˜ r such that it has a zero intersection with F Ž A, B .1 [ H [ F Ž A, B .y1. Set G Ž A, B . s F Ž A, B . rr. We shall call GŽ A, B . the ‘‘generalized Borcherds superalgebra.’’
Ž 4.9.
640
MASAHIKO MIYAMOTO
It is clear that r s Ž[n/ 0 F Ž A, B . n . l KerŽ?N ? .. We will later show that this algebra is given by some relations Žsee Theorem 7.1.. and has properties similar to those of Borcherds algebras. We should note that GŽ A, B . has also a P-gradation. For, it follows from Žc. in Theorem 4.1 that the radical of Ž?N ? . is also an orthogonal sum of Rad Ž F Ž A, B . a [ F Ž A, B . y a . s Ž r l F Ž A, B . a . [ Ž r l F Ž A, B . y a . .
Ž 4.10. Since r l H s 04 , we have r s Ž nql r. [ Ž nyl r. and so G Ž A, B . s nyr Ž nyl r . [ H [ nqr Ž nql r . .
Ž 4.11.
In particular, the universal enveloping algebra of GŽ A, B . is U Ž G Ž A, B . . s Uy Ž G Ž A, B . . m H m Uq Ž G Ž A, B . . ,
Ž 4.12.
where Uy Ž GŽ A, B .. s UŽ nyrŽ ny l r.. and Uq Ž GŽ A, B .. s UŽ nqr Ž nql r... 5. EXAMPLES In this section, we shall study a few minimal examples. Fix a real simple root i g I re and set f s f i , h s h i , e s e i , and r s a ii . If u Ž i, i . s bii s 1, then C f [ C h [ C e is isomorphic to sl 2 ŽC. and we know its representations. So assume that u Ž i, i . s bi i s y1. Let F be the free generalized Lie superalgebra generated by e, f, h. Namely, we think of F as the generalized Lie superalgebra F Ž A, B . with 1 = 1-matrices A s Ž r . and B s Žy1.. The products of e, f, and h are given by:
w e, f x s w f , e x s h,
w h, e x s rh,
and
w h, f x s yrh. Ž 5.1.
Let V be an F-module on which e and f act as locally nilpotent. Choose 0 / ¨ g V such that f¨ s 0 and h¨ s s¨ for some s g C. Set ¨ 1 s e¨ and ¨ n s eŽ ¨ ny1 . for n ) 1 inductively. Since the action of e on V is locally nilpotent, there is an m g N such that ¨ m s 0. Take m as the minimum subject to the condition. If we set h¨ n s SŽ n. ¨ n , then since S Ž n . ¨ n s h¨ n s he¨ ny1 s w h, e x ¨ ny1 q eh¨ ny1 s r¨ n q S Ž n y 1 . ¨ n ,
Ž 5.2. we have SŽ n. s r q SŽ n y 1. s nr q s.
641
GENERALIZATION OF BORCHERDS ALGEBRA
Setting f¨ n s T Ž n. ¨ ny1 , we have T Ž n . s Ž n y 1. r q s y T Ž n y 1. s Ž n y 1. r q s y Ž n y 2. r y s q T Ž n y 2. s T Ž n y 2. q r
Ž 5.3.
since T Ž n . ¨ ny 1 s f¨ n s fe¨ ny1 s w f , e x ¨ ny1 y ef¨ ny1 s h¨ ny 1 y eT Ž n y 1 . ¨ ny2 s Ž Ž n y 1 . r q s . y T Ž n y 1 . 4 ¨ ny 1 .
Ž 5.4.
For n s 1, 2, we have f¨ 1 s fe¨ s w f, e x¨ y ef¨ s h¨ s s¨ s T Ž1. ¨ and f¨ 2 s fe¨ 1 s w f, e x¨ 1 y ef¨ 1 s h¨ 1 y es¨ s Ž r q s y s . ¨ 1 s r¨ 1. Hence T Ž1. s s, T Ž2. s r, and so we have T Ž1 q 2 n. s nr q s and T Ž2 n. s nr. By applying them to ¨ m s 0, we have:
½
If m s 2 n,
then 0 s f¨ 2 n s T Ž 2 n . ¨ 2 ny1 s nr¨ 2 ny1 and so r s 0.
If m s 2 n q 1,
then 0 s f¨ 2 nq1 s Ž nr q s . ¨ 2 n and so nr q s s 0.
Ž 5.5 . In particular, we have that m s 1 y 2 srr is a positive odd integer and so srr is a nonpositive integer except for the case r s Ž h, h. s 0 and the eigenvalues of h are s, r q s, . . . , yr y s, ys 4 .
Ž 5.6.
On the other hand, if srr s yn is a non-positive integer, then we have f¨ 2 nq1 s 0 without the assumption that e acts as locally nilpotent. We note that w e, e x is also an element, but we have w e, w e, ¨ xx s ww e, e x, ¨ x y w e, w e, ¨ xx and so ww e, e x, ¨ x s 2w e, w e, ¨ xx for ¨ g V. LEMMA 5.1.
Recall the definition of r from Definition 5, then we ha¨ e: ei , w ei , ei x g r
for all i g I.
If a ii s 0,
then w e i , e i x g r.
Proof. If e i is even Ž bii s 1., then w e i , e i x s 0. So assume that e i is odd Ž bii s y1.. Then we have: fi , w ei , ei x s w h i , ei x y w ei , h i x s 2 ai i ei
642
MASAHIKO MIYAMOTO
and fi , ei , w ei , ei x
s 2 a ii w e i , e i x y w e i , 2 a ii e i x s 0.
Hence w e i , w e i , e i xx Žor w e i , e i x if a i i s 0. are all highest weight vectors and so Ž Vy3 h i
w e, e x , w f , f x s w e, e x , f , f q f , w e, e x , f s w y2 re, f x q w f , y2 re x s y4rh, and w h, w e, e xx s 2 r w e, e x, we have the following lemma. LEMMA 5.2.
If r s a ii / 0 and u Ž m Ž e ., m Ž e .. s y1, then
¦
y
PROPOSITION 5.3.
1 2r
1
1
r
2r
w f , f x , h,
w e, e x ( sl 2 Ž C . .
;
If A s Ž r . / Ž0. and B s Žy1., then
G Ž A, B . s C w f , f x [ C f [ C h [ C e [ C w e, e x is a simple generalized Lie superalgebra Ž super Lie algebra.. If A s Ž0. and B s Žy1., then G Ž A, B . s C f [ C h [ C e is a Lie superalgebra with the center C h and w GŽ A, B ., GŽ A, B .x s C h. As an application of the above arguments, we have a reflection ri on P for every real root i g I re . From the construction, H acts on GŽ A, B . and ˜Ž A, B . diagonally and so GŽ A, B . and G˜Ž A, B . have H *-gradations and G P-gradations, where H * s HomŽ H, C.. If GŽ A, B .a / 0 for a g H * Žor a g P ., then a is called a root of GŽ A, B .. There is a natural homomorphism from P to H * given by a Ž h. s Ž ny1 Ž a ., h.. We will abuse these notations. We note that by the definition, all adŽ e i . and adŽ f i . are locally nilpotent if i g I re . Hence, LEMMA 5.4.
Let a i be a real simple root and a be a root, then ri Ž a . s a y
is also a root.
2Ž a , ai . a ii
ai g H *
Ž or g P .
643
GENERALIZATION OF BORCHERDS ALGEBRA
6. GENERALIZED CASIMIR OPERATOR Let r g P* s HomŽ P, C. be a Weyl vector, i.e., r Ž a i . s 12 a i i . We shall abuse the notation Ž c < a i . to denote c Ž a i . for c g HomŽ P, C.. We will define a generalized Casimir operator V of GŽ A, B . as follows: First set a local Casimir operator V0 s 2
Ý Ý eyi a eai .
agD q
i
i 4 Here eai 4 is a basis of GŽ A, B .a for a positive root a and ey a is the dual i j basis of GŽ A, B .ya , that is, Ž ea < ey b . s da , b d i, j . In particular, for a simple root i s a , we take ea s e i and ey a s f i . i We have that for x s Ý x i e ai and y s Ý yi eya , i .Ž e ai < y . s Ý x i yi s Ž x < y . . Ý Ž x < eya
Ž 6.1.
i i . We note that Ž6.1. is well-defined, since for any x, Ž x < ey1 s 0 for all but a finite number of i g I.
LEMMA 6.1 ŽLemma 2.4 in wK2x.. i m Ý eya
For z g G by a ,
e ai , z s
Ý
i z, eyb m e bi .
Ž 6.2.
Proof. For e g Ua , f g Uyb , we have the same inner products: s m Ý Ž eya
e m f. s
e as , z
s .Ž Ý Ž e < eya
e as , z N f .
s s . Ž e as N w z, f x . s Ž e N w z, f x . . Ý Ž e < eya s e m f . s Ý Ž e N z, eyb . Ž e bs N f .
s
ÝŽ
s z, eyb m e bs
s
s Ž w e, z x N f . s Ž e N w z, f x . . COROLLARY 6.2.
In the notation of Lemma 6.1, we ha¨ e i Ý eya
e ai , z s
Ý
i z, eyb e bi .
Ž 6.2.
Proof. Apply to the above lemma the linear map of GŽ A, B . m GŽ A, B . to UŽ GŽ A, B .. given by x m y ª xy. A GŽ A, B .-module V is called restricted if for every ¨ g V, we have GŽ A, B .a ¨ s 0 for all but a finite number of positive roots a .
644
MASAHIKO MIYAMOTO
If V is a restricted GŽ A, B .-module and u g UŽ GŽ A, B ..a ,
THEOREM 6.3. then:
w V 0 , u x s yu Ž 2 Ž r < a . q Ž a < a . q 2 ny1 Ž a . .
Ž 6.3.
Proof. By the same argument as in the proof of Theorem 2.6 of wK2x, it is sufficient to check the cases u s e t and u s eyt .
w V 0 , e t x s 2 Ý Ý eys a eas e t y e t eys a eas a
s
s 2Ý
Ý Ž y yu Ž a t , ya . eys a et q et eys a 4 eas s s s qey a Ž ea e t y u Ž a t , ya . ea e t . .
s 2Ý
Ý Žy
s s s s e t , ey a ea q ey a ea , e t
s y2 w e t , eyt x e t y 2
Ý a gD qy t 4
.
s s e t , ey a ea
žÝ s
s s y Ý ey l qt ea yt , e t
/
s y2 w e t , eyt x e t s y2 h t e t s y2 w h t , e t x y 2 e t h t s y2 a t t e t y 2 e t h t s ye t Ž 2 Ž h t , h t . q 2 h t . s ye t Ž Ž r , h t . q Ž h t , h t . q 2 h t . .
w V 0 , eyt x s 2 Ý Ý eys a eas eyt y eyt eys a eas a
s
s 2Ý
Ý Ž eys a
s s eas , e t y eyt , ey a ea .
s 2 eyt w e t , eyt x q 2
Ý a gD qy t 4
s ya
ž Ýe
s , ey a , et
s
y Ý eyt , eas yt eayt
/
s 2 eyt h t s yeyt Ž Ž r , yh t . q Ž yh t , yh t . q 2 Ž yh t . . Let u i 4 be a basis of HrRadŽ H . and u i 4 be the dual basis of u i 4 . LEMMA 6.4.
Ý ui ui , x i
s x Ž Ž a < a . q 2 ny1 Ž a . .
for x g G Ž A, B . a . Ž 6.4.
GENERALIZATION OF BORCHERDS ALGEBRA
645
Proof.
Ý ui ui , x
s
i
Ý u i u i x y xu i u i i
s
Ý u i xu i y xu i u i q u i u i x y u i xu i i
s
Ý w ui , x x ui q ui w ui , x x i
s
Ý Ž u i , a . xu i q u i Ž u i , a . x i
s
Ý Ž u i , a . xu i q Ž u i , a . w u i , x x q Ž u i , a . xu i i
s
Ý Ž ui , a . Ž ui , a . x q x Ž Ž ui , a . ui q Ž ui , a . ui .
s x Ž Ž a < a . q 2 ny1 Ž a . . We define the generalized Casimir operator by V s 2r q
Ý ui ui q V 0 .
Ž 6.5.
We note that since we use r g P* s HomŽ P, C., V can acts on only P-graded GŽ A, B .-modules. The next results follow from the definition of V and the previous lemmas. THEOREM 6.5. If V is a restricted P-graded GŽ A, B .-module, then V commutes with the action of UŽ GŽ A, B .. on V. COROLLARY 6.6. Let V be a P-graded GŽ A, B .-module and ¨ be a lowest weight ¨ ector in V, that is, e i ¨ s 0 for i g I, and hŽ ¨ . s Ž L < h. ¨ for some L g H *, then VŽ ¨ . s Ž L q 2 r
Ž 6.6.
If, furthermore, UŽ GŽ A, B .. ¨ s V, then V s Ž < L q r < 2 y < r < 2 . IV .
Ž 6.7.
7. RELATIONS We have obtained all necessary results to get the following result similar to Theorem 9.11 in wK2x.
646
MASAHIKO MIYAMOTO
THEOREM 7.1. Recall the definition of r from Definition 5. r is the ideal of F Ž A, B . generated by ei , w ei , ei x , fi , w fi , fi x
Ž ad e i .
1y 2 a i jra i i
for all i for i g I re
ej
Ž 7.1.
and
w ei , e j x , fi , f j
for a i j s 0
˜Ž A, B . s GŽ A, B .. Namely, we ha¨ e G Proof. We use the same notation and arguments as in the proof of Proposition 9.11 in wK2x. We have a GŽ A, B .-homomorphism:
l : rqr w rq , rq x ª mis1 M Ž ya i . . n
Ž 7.2.
n We note that rqrwrq, rq x and [is1 M Žya i . have P-gradations and these are invariant under l, since GŽ A, B . and r have P-gradations. In particular, the generalized Casimir operator V acts on both the GŽ A, B .-modules. Using the degree htŽ e i . s 1, rq has a Z-gradation
rqs
`
Ž 7.3.
Ý Ž rq . i isk
such that Žrq . k / 0. Since the kernel of l is contained in Žrq .Žny .UŽny ., we have lŽr k . / 0. Hence, there is a root a such that Žr k .a / 0. Set
as
k i Ž ya i . q
Ý igJ q
Ý
k j Ž ya j . ,
Ž 7.4.
jgJ y
where a i Ž i g Jq . are positive, a j Ž j g Jy . are negative simple roots, and k i Ž i g J s Jqj Jy . are all positive integers. Since ht ri Ž a . G ht a for any reflection ri , we have Ž a < a i . F 0 for all positive simple roots i and so Ž a < a . F 0. On the other hand, since l is commutative with V, we have
Ž ya q 2 r < y a . s Ž ya i q 2 r < y a i . s 0
Ž 7.5.
for some i and so
Ž a < a . s 2Ž r < a .
Ž 7.6.
647
GENERALIZATION OF BORCHERDS ALGEBRA
since Ž r < a i . s 12 Ž a i < a i .. Using Ž7.4., we have 0 s Ž2 r < a . y Ž a < a . s
Ý ki Ž ai , ai . y ki Ž ai < a . 4 igJ
s
Ý ki Ž ai < ai y a . .
Ž 7.5.
igJ
We will show Ž a i < a i y a . G 0 for all i g J. If i g I re , then Ž a i , a i y a . ) 0, while if i g I im , then
Ž a i , a i y a . s y Ý k j Ž a i , a j . q Ž 1 y k i . Ž a i , l i . G 0.
Ž 7.7.
j/i
The equality holds only if Ž a i < a j . s 0 for all j / i. So we have that there are no real roots in J and all imaginary roots are mutually commutative. This happens only when a is a simple imaginary root, which contradicts r l GŽ A, B .1 s 0. 8. SUPERSYMMETRY A Borcherds algebra has an automorphism exchanging e i and f i and acting on H as y1. However, this is not an automorphism of the generalized Borcherds superalgebra any more. It satisfies the following relation: w Ž w x, y x . s y w Ž y . , w Ž x . . Ž 8.1. However, this is sufficient to have results similar to those of Kac]Moody algebras. We note a
Ž w Ž x . < w Ž y . . s u Ž a , b . Ž x, y .
b
for x g G Ž A, B . , y g G Ž A, B . . Ž 8.2.
Proof. We will prove this by induction. For simple roots,
žw Ž e . w Ž f . / s Ž f N e . s u Ž i , j . Ž e N f . . i
j
i
j
j
i
Ž 8.3.
Also, for x g GŽ A, B . a , y g GŽ A, B . b with i q a q b s 0, we have:
Žw Ž w ei , x x . w Ž y . . s Žy w Ž x . , w Ž ei . w Ž y . . s y Žw Ž x . w Ž ei . , w Ž y . . s Ž w Ž x . w Ž w y, e i x . s u Ž a , b q i . Ž x N w y, e i x . by induction, s u Ž a , b . u Ž a , i . u Ž b , i . u Ž a , i . Ž w ei , x x N y . s u Ž a , a . u Ž i , i . Ž w ei , x x N y . s u Ž a q i , a q i . Ž w ei , x x N y . s u Ž a q i , b . Ž w ei , x x N y . .
Ž 8.4.
648
MASAHIKO MIYAMOTO
9. CHARACTER FORMULAS In this section, we calculate the character formulas of GŽ A, B .-modules with highest weight L on which the actions of real simple roots e i and f i are locally nilpotent. This condition is equivalent to L g Pq, where
¡ ŽlNh . gZ ~ P s l g H * Ž l N h . g 2Z ¢ ŽlNh . G0
G0
i
q
i
G0
i
¦ for real odd simple roots i ¥ for imaginary simple roots i§ for real simple roots i
Ž 9.1.
is the set of dominant roots. We note that the results in this section are independent of deformations. We first calculate the denominator formula of the Verma module M Ž L .. Let b be a positive root. Since ww e, e x, ¨ x s 2w e, w e, ¨ xx for e g Gb if u Ž b , b . s y1, adŽ e . 2 is replaced by 12 adŽ e . in UŽ GŽ A, B ... Hence, the difference between the parts afforded by b and 2 b is
Ž 1 q e Ž yb . .
mult b
s Ž 1r Ž 1 y e Ž yb . .
mu lt b
Ž 1 y e Ž y2 b . .
mult b
.
Hence we easily have the following denominator formula of M Ž l.. THEOREM 9.1. ch M Ž L . s e Ž L .
Łb g Doq Ž 1 q e Ž yb .
mu lt b
Ł a g Deq Ž 1 y e Ž ya . .
mu lt a
,
Ž 9.2.
where Deq denotes the set of all positi¨ e roots a with u Ž a , a . s 1 and Doq denotes the set of all positi¨ e roots b with u Ž b , b . s y1. Set Rs
Ł a g Deq Ž 1 y e Ž ya . . Łb g Doq Ž 1 q e Ž yb .
mult a
mu lt b
Ž 9.3.
and call it the denominator formula of GŽ A, B .. We will also have a character formula for irreducible GŽ A, B .-module LŽ L . with highest weight L g Pq, which is similar to one for an irreducible module of Borcherds algebra. In order to get a character formula, we first prove the following lemma.
GENERALIZATION OF BORCHERDS ALGEBRA
649
LEMMA 9.2. w Ž eŽ r . R. s e Ž w . eŽ r . R
for w g W.
Ž 9.4.
Proof. Let a i be a real simple positive root. If u Ž h i , h i . s bii s 1, Dqy a i 4 is ri-invariant and so we have: ri Ž e Ž r . R . s e Ž r y a i . ri Ž 1 y e Ž ya i . . ri R9 s e Ž r . e Ž ya i . Ž 1 y e Ž a i . . R9 s ye Ž r . R,
Ž 9.5.
where R9 s RrŽ1 y eŽya i ... If bii s y1 and w e i , e i x is not zero, then R has the factor Ž1 y eŽy2 a i ..rŽ1 q eŽya i .. s 1 y eŽya i . and the remaining arguments are the same as the above. If bii s y1 and w e i , e i x s 0, then Ž a i , a i . s 0 and there is no reflection. DEFINITION 6. For L g Pq, let SL s e Ž L q r . Ý e Ž b . e Ž yb . , b
where b runs over all elements of P of the form b s a i1 q ??? qa i m Ž m s 0 if b s 0. such that 1. i1 , . . . , i m are distinct elements of I im , 2. Ž a i k, a i l . s 0 if k / l, and 3. Ž L, a i k . s 0 for 1 F k F m. For such a b , e Ž b . denotes Žy1. m . THEOREM 9.3.
For L g Pq, ch L Ž L . s
1 eŽ r . R
Ý e Ž w . w Ž SL . .
Ž 9.6.
wgW
Proof. This is the same as the proof of Theorem 10.4 in wK2x and let us follow it. By the generalized Casimir operator and the denominator formula, we have: e Ž r . R ch L Ž L . s
Ý
lFL < l q r < 2s < Lq r < 2
cl e Ž l q r .
Ž 9.7.
by the same argument as that in Proposition 9.8 of wK2x. Since eŽ r . R ch LŽ L . is W-skew-invariant Ž9.4., we have cl s e Ž w . c wŽ lq r .y r .
Ž 9.8.
650
MASAHIKO MIYAMOTO
Set B0 s B0 Ž L . s l < cl / 04 . For l g B0 , choose w g W such that htŽ L y s . is minimal, where s s w Ž l q r . y r . The minimality of htŽ L y s . forces Ž s q r , a i . G 0. If Ž s q r , a i . s 0 for some i g I re , then wi Ž s q r . s s q r and so cs s e Ž wi . c w i Ž sq r .y r s ycs . This contradicts cl / 0. Set C s m g P*: Ž m , a i . ) 0 for all i g I re 4 . So the above argument shows s q r g C and such a w is uniquely determined by l. Hence, if we set SL s Ý lq r g C cl eŽ l q r ., then e Ž r . R ch L Ž L . s
Ý e Ž w . w Ž SL . .
Ž 9.9.
wgW
We next show that if l q r g C and cl / 0, then
l s L y Ž a i1 q ??? qa i m .
Ž 9.10.
such that a i1, . . . , a i m satisfy the three conditions of Definition 6 and cl s Žy1. m . Set J s suppŽ L y l. and write L y l s Ý i g J k i a i . By the choice of l, k i are all positive integers, and
Ý ki Ž L , ai . q Ý ki Ž l q 2 r , ai . s Ž L q l q 2 r , L y l . s Ž L q 2 r , L . y Ž l q 2 r , l . s 0. Ž 9.11. By our hypothesis, Ž L, a i . G 0 for all i g I. If i g I re , then
Ž l q 2 r , a i . s Ž l q r , a i . q Ž r , a i . ) 0.
Ž 9.12.
If i g I im , then
Ž l q 2 r , ai . s Ž l, ai . q Ž ai , ai . s Ž L , a i . y Ž k i y 1. Ž a i , a i . y
Ý k j Ž a j , a i . G 0. Ž 9.13. j/i
Hence, we conclude that J : I im , Ž a i , a j . s 0 for all i, j g J with i / j, and Ž L, a i . s 0 for i g J. It remains to show that k i s 1 for i g J. Let b be an element of P j 04 such that the terms eŽyb . and eŽ b q l. occur in R and ch LŽ L ., respectively. Then b and L y b y l are both nonnegative and suppŽ L y l y b . : suppŽ L y l. s J. Since Ž L, a i . s 0 for all i g J as we have mentioned, there are no LŽ L .Ly ŽLy ly b . s LŽ L .bq l for L y l y b / 0 and so we have b s L y l. In particular, the term eŽyb . in R comes only from
Ł Ž 1 y e Ž ya i . .
igI im
Ž 9.14.
GENERALIZATION OF BORCHERDS ALGEBRA
651
and so k i s 1 for all i g J and the coefficient of eŽyb . is Žy1. < J < . Since the coefficient of eŽ L . in ch LŽ L . is 1, we obtain cl s Žy1. < J < . Conversely, if l is an element of P* satisfying three conditions of Definition 6, then it is easy to see that eŽ l. occurs in R ch LŽ L . and l q r g C. This completes the proof of Theorem 9.3. Now set L s 0 in the above Eq. Ž9.6. and we deduce the following ‘‘denominator identity.’’ eŽ r .
Ł a g Deq Ž 1 y e Ž ya . .
mu lt a
Łb g Doq Ž 1 q e Ž yb . .
mult b
s
Ý e Ž w . e Ž w Ž S0 . . . Ž 9.15. wgW
REFERENCES wB1x wB2x wB3x wB4x wB5x wB6x wHMYx wK1x wK2x wKWx wMix wMox
R. E. Borcherds, Vertex algebras, Kac]Moody algebras, and the monster, Proc. Natl. Acad. Sci. USA 83 Ž1986., 3068]3071. R. E. Borcherds, Generalized Kac]Moody algebra, J. Algebra 115 Ž1988., 501]512. R. E. Borcherds, The monster Lie algebras, Ad¨ . in Math. 83 Ž1990., 30]47. R. E. Borcherds, Central extensions of generalized Kac]Moody algebras, J. Algebra 140 Ž1991., 330]335. R. E. Borcherds, Monstrous moonshine and monstrous Lie superalgebras, In¨ ent. Math. 109 Ž1992., 405]444. R. E. Borcherds, Automorphic forms on Osq 2, s Ž< R . and infinite products, preprint. K. Harada, M. Miyamoto, and H. Yamada, A generalization of Kac-Moody algebra, preprint. V. G. Kac, Lie superalgebra, Ad¨ . in Math., 26 Ž1977., 8]96. V. G. Kac, ‘‘Infinite Dimensional Lie Algebras,’’ 3rd ed., Cambridge Univ. Press, Cambridge, 1990. V. G. Kac and W. Wang, Vertex operator superalgebras and their representations, preprint. M. Miyamoto, Binary codes and vertex operator Žsuper.algebras, preprint. G. Mossberg, Axiomatic vertex algebras and the Jacobi identity, J. Algebra 170 Ž1994., 956]1010.