- Email: [email protected]
Vertex Operator Algebras and Associative Algebras
206, 67]96 Ž1998. JA987425
JOURNAL OF ALGEBRA ARTICLE NO.
Vertex Operator Algebras and Associative Algebras Chongying Dong* Department of Mathematics, Uni¨ ersity of California, Santa Cruz, California 95064 E-mail: [email protected]
Haisheng Li † Department of Mathematical Sciences, Rutgers Uni¨ ersity, Camden, New Jersey 08102 E-mail: [email protected]
and Geoffrey Mason‡ Department of Mathematics, Uni¨ ersity of California, Santa Cruz, California 95064 E-mail: [email protected] Communicated by Walter Feit Received January 16, 1997
Let V be a vertex operator algebra. We construct a sequence of associative algebras A nŽ V . Ž n s 0, 1, 2, . . . . such that A nŽ V . is a quotient of A nq1Ž V . and a pair of functors between the category of A nŽ V .-modules which are not A ny1Ž V .modules and the category of admissible V-modules. These functors exhibit a bijection between the simple modules in each category. We also show that V is rational if and only if all A nŽ V . are finite-dimensional semisimple algebras. Q 1998 Academic Press
* Partially supported by NSF Grant DMS-9303374 and a research grant from the Committee on Research, UC Santa Cruz. † Partially supported by NSF Grant DMS-9616630. ‡ Partially supported by NSF Grant DMS-9401272 and a research grant from the Committee on Research, UC Santa Cruz. 67 0021-8693r98 $25.00 Copyright Q 1998 by Academic Press All rights of reproduction in any form reserved.
68
DONG, LI, AND MASON
1. INTRODUCTION For a vertex operator algebra V Zhu constructed an associative algebra AŽ V . wZx such that there is a one-to-one correspondence between irreducible admissible V-modules and irreducible AŽ V .-modules. In the case that V is rational the admissible V-module category and AŽ V .-module category are in fact equivalent. But if V is not rational, AŽ V . does not carry enough information for the representation of V. In this paper we construct a sequence of associative algebras A nŽ V . Ž n s 0, 1, 2, . . . . such that A 0 Ž V . s AŽ V . and A nŽ V . is an epimorphic image of A nq 1Ž V .. As in wZx, we use A nŽ V . to study representation theory of V. Let M s [k G 0 M Ž k . be an admissible V-module as defined in wDLMx with M Ž0. / 0. Then each M Ž k . for k F n is an A nŽ V .-module. In some sense, A nŽ V . takes care of the first n q 1 homogeneous subspaces of M while AŽ V . concerns the top level M Ž0.. The results of the present paper are modeled on the results in wDLMx and the methods are also similar. However, the situation for constructing admissible V-modules from A nŽ V .-modules turns out to be very complicated. As in wL2, DLMx we extensively use the Lie algebra Vˆ s V m C w t , ty1 x r L Ž y1 . m 1 q 1 m
ž
d dt
/Ž
V m C w t , ty1 x .
to construct admissible V-modules from A nŽ V .-modules. It should be pointed out that the A nŽ V .4 in fact form an inverse system. So it is natural to consider the inverse limit lim A nŽ V . and its representa¤
tions. This problem will be addressed in a separate paper. One of the important motivations for constructing A nŽ V . is to study induced modules from a subalgebra to V as initiated in wDLx. Induced module theory is very important in the representation theory of classical objects such as groups, rings, Lie algebras. The theory of A nŽ V . developed in this paper will definitely play a role in the study of induced modules for vertex operator algebras. In order to see this, we consider a subalgebra U of V and a U-submodule W of M which is an admissible V-module. In general, the top level of W is not necessarily a subspace of the top level of M. In other words, an AŽU .-module can be a subspace of an A nŽ V .module for some n ) 0. One can now see how the A nŽ V . enter the picture of studying the induced module for the pair ŽU, V . along this line. This paper is organized as follows: In Section 2 we introduce the algebra A nŽ V . which is a quotient of V modulo a subspace OnŽ V . consisting of u(n ¨ Žsee Section 2 for the definition. and LŽy1. u q LŽ0. u for u, ¨ g V. In the case n s 0, Ž LŽy1. q LŽ0.. u can be expressed as v (0 u. But in general it is not clear if one can write Ž LŽy1. q LŽ0.. u as a linear
69
VERTEX OPERATOR ALGEBRAS
combination of ¨ (n w’s. On the other hand, the weight zero component of the vertex operator Y ŽŽ LŽy1. q LŽ0.. u, z . is zero on any weak V-module. So we have to put Ž LŽy1. q LŽ0..V artificially in OnŽ V . for general n. We also show in this section how the identity map on V induces an epimorphism of algebras from A nq 1Ž V . to A nŽ V .. In Section 3, we construct a functor V n from the category of weak V-modules to the category of A nŽ V .-modules such that if M s [k G 0 M Ž k . is an admissible V-module n Ž . Ž . Ž . Ž . then [ks 0 M k with M 0 / 0 is contained in V n M and each M k for Ž . k F n is an A n V -submodule. In particular, if M is irreducible then n Ž . Ž . Ž . Ž . [ks 0 M k s V n M and each M k is an irreducible A n V -module. Section 4 is the core of this paper. In this section we construct a functor L n from the category of A nŽ V .-modules which cannot factor through A ny 1Ž V . to the category of admissible V-modules. For any such A nŽ V .module U we first construct a universal admissible V-module MnŽU . which is somehow a ‘‘generalized Verma module.’’ The L nŽ V . is then a suitable quotient of MnŽU .; the proof of this result is technically the most difficult part of this paper. We also show that V nŽ L nŽU ..rV ny1Ž L nŽU .. is isomorphic to U as A nŽ V .-modules. Moreover, V is rational if and only if the A nŽ V . are finite-dimensional semisimple algebras for all n. Section 5 deals with several combinatorial identities used in previous sections. We assume that the reader is familiar with the basic knowledge on vertex operator algebras as presented in wB, FHL, FLMx. We also refer the reader to wDLMx for the definitions of weak modules, admissible modules, and Žordinary. modules.
2. THE ASSOCIATIVE ALGEBRA A nŽ V . Let V s Ž V, Y, 1, v . be a vertex operator algebra. We will construct an associative algebra A nŽ V . for any nonnegative integer n generalizing the Zhu’s algebra AŽ V . which is our A 0 Ž V .. Let OnŽ V . be the linear span of all u(n ¨ and LŽy1. u q LŽ0. u where for homogeneous u g V and ¨ g V, u(n ¨ s Res z Y Ž u, z . ¨
Ž1 q z.
wt uqn
z 2 nq2
.
Ž 2.1.
Define the linear space A nŽ V . to be the quotient VrOnŽ V .. We also define a second product )n on V for u and ¨ as above: n
u)n ¨ s
Ý Ž y1. ms0
wt uqn
m
ž
m q n Res Y Ž u, z . Ž 1 q z . z n z nqmq1
/
¨.
Ž 2.2.
70
DONG, LI, AND MASON
Extend linearly to obtain a bilinear product on V which coincides with that of Zhu wZx if n s 0. We denote the product Ž2.2. by ) in this case. Note that Ž2.2. may be written in the form `
n
u)n ¨ s
Ý Ý Ž y1. ms0 is0
m
ž m qn n / ž wt uiq n / u
iymyny1¨ .
Ž 2.3.
The following lemma generalizes Lemmas 2.1.2 and 2.1.3 of wZx. LEMMA 2.1. Ži. Assume that u g V is homogeneous, ¨ g V, and m G k G 0. Then Res z Y Ž u, z . ¨
Ž1 q z.
wt uqnqk
z 2 nq2qm Žii. Assume that ¨ is also homogeneous. Then
wt ¨ qm y1
n
u)n ¨ y
g On Ž V . .
Ý ms0
ž
m q n Ž y1 . n Res Y Ž ¨ , z . u Ž 1 q z . z n z 1qmqn
g On Ž V .
/
and Žiii. u)n ¨ y ¨ )n u y Res z Y Ž u, z . ¨ Ž1 q z . wt uy1 g OnŽ V .. Proof. The proof of Ži. is similar to that of Lemma 2.1.2 of wZx. As in wZx we use LŽy1. u q LŽ0. u g OnŽ V . to derive the formula yz yw t uywt ¨ Y Ž u, z . ¨ ' Ž 1 q z . Y ¨, u mod On Ž V . . 1qz Thus we have
ž
wt uqn
n
u)n ¨ s
Ž y1.
m
Ý Ž y1.
m
Ý ms0
ž
n q m Res Y Ž u, z . ¨ Ž 1 q z . z n z mqnq1
ž
n q m Res Y ¨ , yz u Ž 1 q z . z n 1qz z mqnq1
/
yw t ¨ qn
n
'
/
ms0
/
ž
/
mod On Ž V . wt ¨ qm y1
n
s
Ý
Ž y1.
n
ms0
ž
n q m Res Y Ž ¨ , z . u Ž 1 q z . z n z mqnq1
/
and Žii. is proved. Using Žii. we have u)n ¨ y ¨ )n u ' Res z Y Ž u, z . ¨ Ž 1 q z .
m
n
=
ž
Ý ms0
wt uy1
ž
nq1
n
y Ž y1 . Ž 1qz . mqn Ž y1 . Ž 1qz . nq mq1 n z
/
m
/
.
71
VERTEX OPERATOR ALGEBRAS
By Proposition 5.2 in the Appendix we know that m
n
Ý ms0
ž
n
nq1
y Ž y1 . Ž 1 q z . m q n Ž y1 . Ž 1 q z . nq mq1 n z
m
/
s 1.
The proof is complete. LEMMA 2.2. OnŽ V . is a 2 sided ideal of V under )n . Proof. First we show that Ž LŽy1. u q LŽ0. u.)n ¨ g OnŽ V . for any homogeneous u g V. From the definition we see that
Ž L Ž y1. u . )n ¨ wt uqnq1
n
s
ž
Ý ms0
¨
/
wt uqnq1
m q n Ž y1 . m Res d Y Ž u, z . ¨ Ž 1 q z . z n dz z nqmq1
n
s
m q n Ž y1 . m Res Y Ž L Ž y1 . u, z . Ž 1 q z . z n z nqmq1
ž / s Ý ž m q n / Ž y1 . n Ý
ms0 n
ž
mq 1
/
Res z Y Ž u, z . ¨
ms0
=
ž
Ž yn y m y 1 . Ž 1 q z .
wt uqnq1
q
z nq mq2
z Ž wtu q n q 1 . Ž 1 q z .
wt uqn
z nqmq2
/
Thus
Ž L Ž y1. u q wt uu . )n ¨ n
s
Ý ms0
ž m qn n / Ž y1.
m
Res z Y Ž u, z . ¨ Ž 1 q z .
wt uqn
mz q n q m q 1 z nqmq2
It is straightforward to show that n
Ý ms0
ž m qn n / Ž y1. n
s
Ý ms0
ž
s Ž y1 .
m
mz q n q m q 1 z nq mq2
m q n Ž y1 . m mz q n z nq mq2
/
n
ž 2 n nq 1 /
2n q 1 z 2 nq2
n
Ý ms0
ž m qnn q 1 / Ž y1.
.
It is clear now that Ž LŽy1. u q LŽ0. u.)n ¨ g OnŽ V ..
m
mq1 z nqmq2
.
.
72
DONG, LI, AND MASON
Second, we show that u)n Ž LŽy1. ¨ q LŽ0. ¨ . g OnŽ V .. Using the result that Ž LŽy1. ¨ q LŽ0. ¨ .)n ¨ g OnŽ V . and Lemma 2.1Žiii. we have u)n Ž L Ž y1 . ¨ q L Ž 0 . ¨ . ' yRes z Ž Y Ž L Ž y1 . ¨ , z . u Ž 1 q z .
wt ¨
q Y Ž L Ž 0. ¨ , z . u Ž 1 q z .
wt ¨ y1
.
mod On Ž V . s Res z Y Ž ¨ , z . u
ž
d dz
Ž1 q z.
wt ¨
y Y Ž L Ž 0. , ¨ , z . u Ž 1 q z .
wt ¨ y1
/
s 0. Third, a similar argument as in wZx using Lemma 2.1Ži. shows that u)n Ž ¨ (n w . g OnŽ V . for u, ¨ , w g V. Finally, use u)n Ž ¨ (n w . g OnŽ V . and Lemma 2.1Žiii. to obtain
Ž ¨ (n w . )n u ' yRes z 1 Res z 2 Y Ž u, z1 . Y Ž ¨ , z 2 . w
Ž 1 q z1 .
wt uy1
Ž 1 q z2 .
wt ¨ qn
z 22 nq2 mod On Ž V .
' yRes z 2 Res z 1yz 2 Y Ž Y Ž u, z1 y z 2 . ¨ , z 2 . w
Ž 1 q z1 .
wt uy1
Ž 1 q z2 .
wt ¨ qn
z 22 nq2 wt uqwt ¨ qn y1yi
sy Ý iG0
ž
Ž 1 q z2 . wt u y 1 Res z 2 Y Ž u i ¨ , z 2 . w i z 22 nq2
/
which belongs to OnŽ V . as wt u i ¨ s wt u q wt ¨ y i y 1. This completes the proof. Our first main result is the following. THEOREM 2.3. Ži. The product ) n induces the structure of an associati¨ e algebra on A nŽ V . with identity 1 q OnŽ V .. Žii. The linear map
f : ¨ ¬ e LŽ1. Ž y1 .
L Ž0 .
¨
induces an anti-isomorphism A nŽ V . ª A nŽ V .. Žiii. v q OnŽ V . is a central element of A nŽ V ..
73
VERTEX OPERATOR ALGEBRAS
Proof. For Ži. we only need to prove that A nŽ V . is associative. Let u, ¨ , w g V be homogeneous. Then
Ž u)n ¨ . )n w n
s
Ý Ý Ž y1.
m1
m1s0 iG0
ž
n
s
Ý
Ý Ž y1.
m1 q n n m 1 qm 2
m1 , m 2s0 iG0
ž
/ž
wt u q n Ž uym 1yny1qi ¨ . )n w i
/
m1 q n n
=Res z 2 Y Ž uym 1yny1qi ¨ , z 2 . w n
m qm Ý Ž y1.
s
1
2
m1 , m 2s0
m1 q n n
ž
=Y Ž Y Ž u, z1 y z 2 . ¨ , z 2 . w n
m qm Ý Ž y1.
s
1
m1 , m 2s0
2
n
m qm Ý Ž y1. 1
2
m1 , m 2s0
ž
/ž
n
Ý
Ý Ž y1.
m 1 qm 2
m1 , m 2s0 iG0
ž
/
/ž
n
Ý
m1 , m 2s0 iG0
2
ž
wt ¨ qn qm 1
/
Ž 1 q z1 .
wt uqn
Ž z1 y z 2 .
Ž 1 q z2 .
wt ¨ qn qm 1
m 1 qnq1 1q m 2 qn z2
/
Ž 1 q z1 .
wt uqn
Ž z1 y z 2 .
/ž
m2 q n n
Ž 1 q z1 .
/ž
Ž 1 q z2 .
m 1 qnq1
wt ¨ qn qm 1
z 21q m 2qn
ym1 y n y 1 i Ž y1. i
wt uqn
/
Ž 1 q z2 .
wt ¨ qn qm 1
z1m 1qnq1qi z 21qm 2qnyi
m1 q n n
=Res z 2 Res z 1Y Ž ¨ , z 2 . Y Ž u, z1 . w
Ž 1 q z2 .
m 1 qnq1 1qm 2 qn z2
m2 q n n
/ž
m1 q n n
Ý Ž y1. m qnq1qi
wt uqn
m2 q n n
=Res z 1 Res z 2 Y Ž u, z1 . Y Ž ¨ , z 2 . w y
/
m2 q n Res z 2 Res z 1yz 2 n
=Res z 2 Res z 1Y Ž ¨ , z 2 . Y Ž u, z1 . w s
wt u q n i
wt uqwt ¨ q2 nqm1yi
Ž z1 y z 2 .
m1 q n n
/ž
z 21qm 2qn
=Res z 1 Res z 2 Y Ž u, z1 . Y Ž ¨ , z 2 . w y
m2 q n n
Ž 1 q z2 .
Ž 1 q z1 .
m1 q n n
ž
/ž
/ž
m2 q n n
Ž 1 q z1 .
wt uqn
/ž
ym1 y n y 1 i
Ž 1 q z2 .
/
wt ¨ qn qm 1
z1m 1qnq1yi z 22qm 1qm 2q2 nqi
.
74
DONG, LI, AND MASON
From Lemma 2.1 we know that Res z 2 Res z 1Y Ž ¨ , z 2 . Y Ž u, z1 . w
Ž 1 q z1 .
wt uqn
Ž 1 q z2 .
wt ¨ qn qm 1
z1m 1qnq1yi z 22qm 1qm 2q2 nqi
lies in OnŽ V .. Also if i ) n y m1 Res z 1 Res z 2 Y Ž u, z1 . Y Ž ¨ , z 2 . w
Ž 1 q z1 .
wt uqn
Ž 1 q z2 .
wt ¨ qn qm 1
z1m 1qnq1qi z 21qm 2qnyi
is in OnŽ V .. Thus n
Ž u)n ¨ . )n w ' u)n Ž ¨ )n w . q
m qm Ý Ž y1. 1
m1 , m 2s0
=Res z 1 Res z 2 Y Ž u, z1 . Y Ž ¨ , z 2 . nym 1
=
ž
Ý Ý is0
jG0
ž
ym1 y n y 1 i
2
ž
m1 q n n
Ž 1 q z1 .
/ž
wt uqn
m2 q n n
Ž 1 q z2 .
/
wt ¨ qn
z1m 1qnq1 z 21qm 2qn z iqj m1 i 2 Ž y1. i y 1 . j z1
/
/ž /
From Proposition 5.3 in the Appendix we know that n
m Ý Ž y1.
1
m1s0
ž
m1 q n n
nym 1
ž
Ý Ý is0
jG0
ž
/
ym1 y n y 1 i
z 2iqj 1 m1 i Ž y1. iqm 1 y m 1 j z1 z1
/ž /
/
s 0.
This implies that the product )n of A nŽ V . is associative. The proof of Žii. is similar to that of Žii. of Theorem 2.4 of wDLMx. We refer the reader to wDLMx for detail. Note that 1)n u s u for any u g V and that u)n 1 y 1)n u ' Res z Y Ž u, z . 1 Ž 1 q z .
wt uy1
s 0.
This shows that 1 q OnŽ V . is the identity of A nŽ V .. Again by Lemma 2.1Žiii.,
v )n u y u)n v s Res z Y Ž v , z . u Ž 1 q z . s L Ž y1 . u q L Ž 0 . u g On Ž V . . So Žiii. is proved.
75
VERTEX OPERATOR ALGEBRAS
PROPOSITION 2.4. The identity map on V induces an onto algebra homomorphism from A nŽ V . to A ny1Ž V .. Proof. First by Lemma 2.1Ži., OnŽ V . ; Ony1Ž V .. It remains to show that u)n ¨ ' u)ny1 ¨ mod Ony1Ž V .. Let u be homogeneous. Then wt uqny1
n
u)n ¨ s
Ý ms0
ž
m q n Ž y1 . m Res Y Ž u, z . ¨ Ž 1 q z . z n z nqm
/
wt uqny1
m q n Ž y1 . m Res Y Ž u, z . ¨ Ž 1 q z . z n z nqmq1
n
q
Ý ms0
Ý ms0
/
wt uqny1
ny1
'
ž
ž
m q n Ž y1 . m Res Y Ž u, z . ¨ Ž 1 q z . z n z nqm
/
wt uqny1
m q n Ž y1 . m Res Y Ž u, z . Ž 1 q z . z n z nqmq1
ny2
q
Ý ms0
ž
/
mod Ony 1 Ž V . sRes z Y Ž u, z . ¨ ? Ž y1 .
ž
m
Ž 1qz .
wt uqny1
ny1
q Ý Res z Y Ž u, z . ¨
zn
ms1
Ž 1qz .
wt uqny1
z nqm
ž m qn n / q Ž y1. ž m qnn y 1 / / mq1
s u)ny 1 ¨ , as desired. From Proposition 2.4 we in fact have an inverse system A nŽ V .4 . Denote by I Ž V . the inverse limit lim A nŽ V .. Then ¤
½
I Ž V . s a s Ž a n q On Ž V . . g
`
Ł A n Ž V . < an y any1 g Ony1Ž ¨ .
ns0
5
.
Ž 2.4. Define i: V ª I Ž V . such that iŽ ¨ . s Ž ¨ q OnŽ V .. for ¨ g V. Then Vrker i is linearly isomorphic to a subspace of I Ž V .. It is easy to see that iŽ V . is not closed under the product. But one can introduce a suitable topology on I Ž V . so that iŽ V . is a dense subspace of I Ž V . under the topology. An interesting problem is to determine the kernel of i. From the definition of OnŽ V . we see immediately that Ž LŽy1. q LŽ0..V is contained in the kernel. It will be proved in Section 3 that if ¨ g OnŽ V . then a wt ay1 s 0 on
76
DONG, LI, AND MASON
n ` Ž . Ž . [ks 0 M n for any admissible V-module [ks0 M k . Thus a g ker i if and only if a wt ay1 s 0 on any admissible V-module. It is proved in wDLMMx that if V is a simple vertex operator algebra satisfying Vk s 0 for all k - 0, and V0 s C1 then the subspace of V consisting of vectors ¨ whose component operators ¨ wt ¨ y1 are 0 on V is essentially Ž LŽ0. q LŽy1..V. We suspect that if V is a rational vertex operator algebra then the kernel of i is exactly Ž LŽ0. q LŽy1..V.
3. THE FUNCTOR V n Consider the quotient space Vˆ s C w t , ty x m VrDC w t , ty x m V ,
Ž 3.1.
where D s dtd m 1 q 1 m LŽy1.. Denote by ¨ Ž m. the image of ¨ m t m in Vˆ for ¨ g V and m g Z. Then Vˆ is Z-graded by defining the degree of ¨ Ž m. to be wt ¨ y m y 1 if ¨ is homogeneous. Denote the homogeneous subspace of degree m by VˆŽ m.. The space Vˆ is, in fact, a Z-graded Lie algebra with bracket aŽ p . , b Ž q . s
`
Ý is0
p i
ž /Ž
ai b . Ž p q q y i .
Ž 3.2.
Žsee wL2, DLMx.. In particular, VˆŽ0. is a Lie subalgebra. By Lemma 2.1Žiii. we have PROPOSITION 3.1. Regarded A nŽ V . as a Lie algebra, the map ¨ Žwt ¨ y 1. ¬ ¨ q OnŽ V . is a well-defined onto Lie algebra homomorphism from VˆŽ0. to A nŽ V .. By Lemmas 5.1 and 5.2 of wDLMx, any weak V-module M is a module for Vˆ under the map aŽ m. ¬ a m and a weak V-module which carries a Zq-grading is an admissible V-module if, and only if, M is a Zq-graded ˆ module for the graded Lie algebra V. For a module W for the Lie algebra Vˆ and a nonnegative m we let V mŽW . denote the space of ‘‘mth lowest weight vectors,’’ that is, V m Ž W . s u g W < VˆŽ yk . u s 0 if k ) m4 .
Ž 3.3.
Then V mŽW . is a module for the Lie algebra VˆŽ0.. THEOREM 3.2. Suppose that M is a weak V-module. Then there is a representation of the associati¨ e algebra A nŽ V . on V nŽ M . induced by the map a ¬ oŽ a. s a wt ay1 for homogeneous a g V.
77
VERTEX OPERATOR ALGEBRAS
Proof. We need to show that oŽ a. s 0 for all a g OnŽ V . and oŽ u)n ¨ . s oŽ u. oŽ ¨ . for u, ¨ g V. Using Y Ž LŽy1. u, z . s dzd Y Ž u, z . we immediately see that oŽ LŽy1. u q LŽ0. u. s 0. From the proof of Lemma 2.1 we know that Ž LŽy1. u q LŽ0. u.)n ¨ s Žy1. n Ž 2nn .Ž2 n q 1. u(n ¨ . It suffices to show that oŽ u)n ¨ . s oŽ u. oŽ ¨ .. Let u, ¨ be homogeneous and 0 F k F n. Note that ¨ wt ¨ qp s u wt uqp s 0 on V nŽ M . if p G n. We assert that the following identity holds on V nŽ M ., wt uqn
k
Ž y1.
Ý
m
ms0
ž
2 n q m y k o Res Y Ž u, z . ¨ Ž 1 q z . z m z 2 nq1ykqm
ž
/
s u wt uynqky1¨ wt ¨ qn yky1
/ Ž 3.4.
which reduces to oŽ u)n ¨ . s oŽ u. oŽ ¨ . if k s n. The proof of Ž3.4. is a straightforward computation involving the Jacobi identity on modules in terms of residues. On V nŽ M . we have wt uqn
k
m Ý Ž y1.
ms0
ž
2 n q m y k o Res Y Ž u, z . ¨ Ž 1 q z . z m z 2 nq1ykqm
/
k
s
Ý Ý Ž y1.
m
ms0 iG0 k
s
Ý Ý Ž y1.
m
ms0 iG0
ž
ž 2 n qmm y k / ž wt uiq n / o Ž u ž 2 n qmm y k / ž wt uiq n /
= Ž u iy2 ny1ymqk¨ . wt uqwt ¨ yiq2 nqmy1yk n
s
Ý Ý Ž y1. ms0 iG0
m
ž 2 n qmm y k / ž wt uiq n /
=Res z 2 Res z 1yz 2 Y Ž Y Ž u, z1 y z 2 . ¨ , z 2 . = Ž z1 y z 2 . k
s
m Ý Ž y1.
ms0
iy2 nymy1qk
z 2wt uqwt ¨ yiq2 nqmy1yk
ž 2 n qmm y k / Res
=Y Ž Y Ž u, z1 y z 2 . ¨ , z 2 . k
s
m Ý Ž y1.
ms0
z2
Res z 1yz 2
z1wt uqn z 2wt ¨ qn qmy1yk
Ž z1 y z 2 .
2 nqmq1yk
ž 2 n qmm y k / Res Res z1
z2
/
iy2 ny1ymqk¨
.
78
DONG, LI, AND MASON
=Y Ž u, z1 . Y Ž ¨ , z 2 . k
m Ý Ž y1.
y
ms0
k
Ž z1 y z 2 .
2 nqmq1yk
ž 2 n qmm y k / Res
=Y Ž ¨ , z 2 . Y Ž u, z1 . s
z1wt uqn z 2wt ¨ qn qm y1 y k
kym
Ý Ý Ž y1. mq i ms0 is0
z2
Res z 1
z1wt uqn z 2wt ¨ qn qmy1yk
Ž z1 y z 2 .
2 nqmq1yk
ž 2 n qmm y k / ž ym y 2 ni y 1 q k /
=u wt uynymy1qkyi ¨ wt ¨ qn qmy1ykqi k
s
k
ž 2 n qmm y k / ž ym yi 2yn my 1 q k / Ž y1.
Ý Ý ms0 ism
i
=u wt uynqkyiy1¨ wt ¨ qn yky1qi k
s
i
Ý Ý is0 ms0
ž 2 n qmm y k / ž ym yi 2yn my 1 q k / Ž y1.
i
=u wt uynqkyiy1¨ wt ¨ qn yky1qi s u wt uynqky1¨ wt ¨ qn yky1 k
i
qÝ
Ý
is1 ms0
ž 2 n qmm y k / ž ym yi 2yn my 1 q k / Ž y1.
=u wt uynqkyiy 1¨ wt ¨ qn yky1qi . It is enough to show that for i s 1, . . . , k. i
Ý ms0
ž 2 n qmm y k / ž ym yi 2yn my 1 q k / s 0,
which follows from an easy calculation: i
Ý ms0
ž 2 n qmm y k / ž ym yi 2yn my 1 q k / s Ý Ž y1 . ž 2 n qmm y k / ž 2 niqyimy k / s Ý Ž y1 . ž 2 n2qn yi yk k / ž mi / i
iym
ms0 i
ms0
s 0. This completes the proof.
iym
i
79
VERTEX OPERATOR ALGEBRAS
Remark 3.3. For homogeneous u, ¨ g V and j g Z we set oj Ž u. s u wt uy1yj and extend to all u g V by linearity. Then o 0 Ž u. s oŽ u.. Using associativity of the vertex operators
Ž z0 q z2 .
wt uqn
Y Ž u, z 0 q z 2 . Y Ž ¨ , z 2 . s Ž z 2 q z 0 .
wt uqn
Y Ž Y Ž u, z 0 . ¨ , z 2 .
on V nŽ M . we have that for i G j with i q j G 0 these exists a unique wu,i, ¨j g V such that oi Ž u. oj Ž ¨ . s oiqj Ž wu,i, ¨j . on V nŽ M .. In fact one can write wu,i, ¨j explicitly in terms of u and ¨ . But for our later purpose it is Ž i G 0. which is given by enough to know the explicit expression of wu,i, yi ¨ wui ,, yi ¨ s
wt uqn
nyi
Ý ms0
Ž y1.
m
ž
n q m q i Res Y Ž u, z . ¨ Ž 1 q z . z m z nq1qiqm
/
in the proof of Theorem 3.2. It is clear that V n is a covariant functor from the category of weak V-modules to the category of A nŽ V .-modules. To be more precise, if f : M ª N is a morphism in the first category we define V nŽ f . to be the ˆ restriction of f to V nŽ M .. Then f induces a morphism of V-modules M ª N by Lemma 5.1 of wDLMx. Moreover V nŽ f . maps V nŽ M . to V nŽ N .. Now Theorem 3.2 implies that V nŽ f . is a morphism of A nŽ V .modules. Let M be such a module. As long as M / 0, then some M Ž m. / 0, and it is no loss to shift the grading so that in fact M Ž0. / 0. If M s 0, let M Ž0. s 0. With these conventions we prove PROPOSITION 3.4. following hold
Suppose that M is an admissible V-module. Then the
n n Ži. V nŽ M . > [is0 M Ž i .. If M is simple then V nŽ M . s [is0 M Ž i .. Žii. Each M Ž p . is an VˆŽ0.-module and M Ž p . and M Ž q . are inequi¨ alent if p / q and both M Ž p . and M Ž q . are nonzero. If M is simple then each M Ž p . is an irreducible VˆŽ0.-module. Žiii. Assume that M is simple. Then each M Ž i . for i s 0, . . . , n is a simple A nŽ V .-module and M Ž i . and M Ž j . are inequi¨ alent A nŽ V .-modules.
Proof. An easy argument shows that V nŽ M . is a graded subspace of M. That is, V nŽ M . s
[i G 0 V n Ž M . l M Ž i . .
Ž 3.5.
Set V nŽ i . s V nŽ M . l M Ž i .. It is clear that M Ž i . ; V nŽ M . if i F n. In order to prove Ži. we must show that V nŽ i . s 0 if i ) n.
80
DONG, LI, AND MASON
By Proposition 2.4 of wDMx or Lemma 6.1.1 of wL2x, M s span u n w < u g V, n g Z4 where w is any nonzero vector in M. If V nŽ i . / 0 for some i ) n we can take 0 / w g V nŽ i .. Since u wt uqp w s 0 for all p G n we see that M s span u wt uqp w < u g V, p g Z, p - n4 . This implies that M Ž0. s 0, a contradiction. ˆ s It is clear that Žiii. follows from Žii.. For Žii., note that M s Vw [p g Z VˆŽ p . w. Thus if 0 / w g M Ž i . then VˆŽ p . w s M Ž i q p .. In particular, VˆŽ0. w s M Ž i ., as required. It was pointed out in wZx that LŽ0. is semisimple on M and M Ž k . s w g M < LŽ0. w s Ž h q k . w4 for some fixed h. The inequivalence follows from the fact that LŽ0. has different eigenvalues on M Ž p . and M Ž q ..
4. THE FUNCTOR L n We show in this section that there is a universal way to construct an admissible V-module from an A nŽ V .-module which cannot factor through A ny 1. ŽIf it can factor through A ny1Ž V . we can consider the same procedure for A ny 1Ž V ... Moreover a certain quotient of the universal object is an admissible V-module L nŽU . and L n defines a functor which is a right inverse to the functor V nrV ny1 , where V nrV ny1 is the quotient functor M ¬ V nŽ M .rV ny1Ž M .. Fix an A nŽ V .-module U which cannot factor through A ny1Ž V .. Then it is a module for A nŽ V .L i e in an obvious way. By Proposition 3.1 we can lift U to a module for the Lie algebra VˆŽ0., and then to one for Pn s [p ) n VˆŽyp . [ VˆŽ0. by letting VˆŽyp . act trivially. Define ˆ
Mn Ž U . s IndVP nŽ U . s U Ž Vˆ . mUŽ P n . U.
Ž 4.1.
If we give U degree n, the Z-gradation of Vˆ lifts to MnŽU . which thus ˆ It is easy to see that MnŽU .Ž i . s becomes a Z-graded module for V. ˆ Ž . U V iynU. We define for ¨ g V, YM n ŽU . Ž ¨ , z . s
Ý ¨ Ž m . zym y1 .
Ž 4.2.
mgZ
As in wDLMx, YM ŽU .Ž ¨ , z . satisfies all conditions of a weak V-module except the associativity which does not hold on MnŽU . in general. We have to divide out by the desired relations. Let W be the subspace of MnŽU . spanned linearly by the coefficients of
Ž z0 q z2 .
wt aqn
Y Ž a, z 0 q z 2 . Y Ž b, z 2 . u y Ž z 2 q z 0 .
wt aqn
Y Ž Y Ž a, z 0 . b, z 2 . u
Ž 4.3.
81
VERTEX OPERATOR ALGEBRAS
for any homogeneous a g V, b g V, u g U. Set Mn Ž U . s Mn Ž U . rU Ž Vˆ . W.
Ž 4.4.
THEOREM 4.1. The space MnŽU . s Ý m G 0 MnŽU .Ž m. is an admissible V-module with MnŽU .Ž0. / 0, MnŽU .Ž n. s U and with the following uni¨ ersal property: for any weak V-module M and any A nŽ V .-morphism f : U ª V nŽ M ., there is a unique morphism f : MnŽU . ª M of weak V-modules which extends f . Proof. By Proposition 6.1 of wDLMx, we know that MnŽU . is a Z-graded weak V-module generated by U q UŽ Vˆ .W. By Proposition 2.4 of wDMx or Lemma 6.1.1 of wL2x, MnŽU . is spanned by
½ a ŽU q U Ž Vˆ . W . < a g V , n g Z 5 . n
Thus MnŽU .Ž m. s VˆŽ m y n.ŽU q UŽ Vˆ .W . for all m g Z. In particular, MnŽU .Ž m. s 0 if m - 0 and MnŽU .Ž n. s A nŽ V .ŽU q UŽ Vˆ .W . which is a quotient module of U. A proof that MnŽU .Ž0. / 0 and MnŽU .Ž n. s U will be given after Proposition 4.7. The universal property of MnŽU . follows from its construction. In the following we let U* s Hom C ŽU, C. and let Us be the subspace of MnŽU .Ž n. spanned by ‘‘length’’ s vectors op 1Ž a1 . ??? op sŽ a s . U, where p1 G ??? G ps , p1 q ??? ps s 0, pi / 0, ps G yn, and a i g V. Then by the PBW theorem MnŽU .Ž n. s Ý s G 0 Us with U0 s U and Us l Ut s 0 if s / t. Recall Remark 3.3. We extend U* to MnŽU .Ž n. inductively so that ² u9, op Ž a1 . ??? op Ž a s . u: s ² u9, op qp Ž wap 1, ,ap 2 . op Ž a3 . ??? op Ž a s . u . , 1 s 1 2 1 2 3 s
Ž 4.5. where oj Ž a. s aŽwt a y 1 y j . for homogeneous a g V. We further extend U* to MnŽU . by letting U* annihilate [i/ n M ŽU .Ž i .. Set J s ¨ g Mn Ž U . <² u9, x¨ : s 0 for all u9 g U*, all x g U Ž Vˆ . 4 . We can now state the second main result of this section. THEOREM 4.2. The space L nŽU . s MnŽU .rJ is an admissible V-module satisfying L nŽU .Ž0. / 0 and V nrV ny1Ž L nŽU .. ( U. Moreo¨ er L n defines a functor from the category of A nŽ V .-modules which cannot factor through A ny 1Ž V . to the category of admissible V-modules such that V nrV ny1 ( L n is naturally equi¨ alent to the identity.
82
DONG, LI, AND MASON
The main point in the proof of the theorem is to show that UŽ Vˆ .W ; J. The next three results are devoted to this goal. PROPOSITION 4.3. The following hold for all homogeneous a g V, b g V, u9 g U*, u g U, j g Zq, ² u9, Ž z 0 q z 2 . wt aqnqj YM s ² u9, Ž z 2 q z 0 .
n ŽU .
Ž a, z 0 q z 2 . YM n ŽU . Ž b, z 2 . u:
wt aqnqj
YM n ŽU . Ž Y Ž a, z 0 . b, z 2 . u: .
Ž 4.6.
In the following we simply write Y for YM n ŽU . , which should cause no confusion. The following is the key lemma. LEMMA 4.4.
For any i, j g Zq,
Res z 0 zy1qi Ž z0 q z2 . 0
wt aqnqj
² u9, Y Ž a, z 0 q z 2 . Y Ž b, z 2 . u:
s Res z 0 zy1qi Ž z2 q z0 . 0
wt aqnqj
² u9, Y Ž Y Ž a, z 0 . b, z 2 . u: .
Proof. Since j G 0 then aŽwt a q n q j . lies in [p ) n VˆŽyp . and hence annihilates u. Then for all i g Zq we get i
Res z 1Ž z1 y z 2 . z1wt aqnqj Y Ž b, z 2 . Y Ž a, z1 . u s 0.
Ž 4.7.
Note that Ž3.2. is equivalent to Y Ž a, z1 . , Y Ž b, z 2 . s Res z 0 zy1 2 d
ž
z1 y z 0 z2
/
Y Ž Y Ž u, z 0 . ¨ , z 2 . . Ž 4.8.
Using Ž4.7. and Ž4.8. we obtain Res z 0 z 0i Ž z 0 q z 2 .
wt aqnqj
Y Ž a, z 0 q z 2 . Y Ž b, z 2 . u
i
s Res z 1Ž z1 y z 2 . z1wt aqnqj Y Ž a, z1 . Y Ž b, z 2 . u i
s Res z 1Ž z1 y z 2 . z1wt aqnqj Y Ž a, z1 . Y Ž b, z 2 . u i
y Res z 1Ž z1 y z 2 . z1wt aqnqj Y Ž b, z 2 . Y Ž a, z1 . u i
s Res z 1Ž z1 y z 2 . z1wt aqnqj Y Ž a, z1 . , Y Ž b, z 2 . u i
s Res z 0 Res z 1Ž z1 y z 2 . z1wt aqnqj zy1 2 d s Res z 0 Res z 1 z 0i z1wt aqnqj zy1 1 d s Res z 0 z 0i Ž z 2 q z 0 .
wt aqnqj
ž
ž
z2 q z0 z1
z1 y z 0 z2
/
/
Y Ž Y Ž a, z 0 . b, z 2 . u
Y Ž Y Ž a, z 0 . b, z 2 . u
Y Ž Y Ž a, z 0 . b, z 2 . u.
Ž 4.9.
83
VERTEX OPERATOR ALGEBRAS
Thus Lemma 4.4 holds if i G 1, and we may now assume i s 0. Next us Ž4.9. to calculate that Res z 0 zy1 0 Ž z0 q z2 . s
`
² u9, Y Ž a, z 0 q z 2 . Y Ž b, z 2 . u:
j wt aqn ² u9, Y Ž a, z 0 qz 2 . Y Ž b, z 2 . u: Res z 0 z 0ky1 z 2jyk Ž z 0 qz 2 . k
ž/ Ýž / Ý
ks0
s
wt aqnqj
`
j wt aqn ² u9, Y Ž Y Ž a, z 0 . b, z 2 . u: Res z 0 z 0ky1 z 2jyk Ž z 2 q z 0 . k
ks1
j qRes z 0 zy1 0 z2 Ž z2 q z0 .
wt aqn
² u9, Y Ž a, z 0 q z 2 . Y Ž b, z 2 . u: .
Ž 4.10.
It reduces to show that Res z 0 zy1 0 Ž z2 q z0 .
wt aqn
² u9, Y Ž a, z 0 q z 2 . Y Ž b, z 2 . u:
s Res z 0 zy1 0 Ž z2 q z0 .
wt aqn
Ž 4.11.
² u9, Y Ž a, z 0 . u, z 2 . u: .
Ž 4.12.
Since ² u9, MnŽU .Ž m.: s 0 if m / n, we see that Res z 0 zy1 0 Ž z2 q z0 .
¦ ¦
s u9,
Ý kgZ q
s u9,
Ý kgZ q
¦ ž
wt aqn
ž ž
z 2wt byn ² u9, Y Ž Y Ž a, z 0 . b, z 2 . u:
wt a q n Ž aky 1 b . Ž wt Ž aky1 b . y 1 . u k
;
/ /
wt a q n o Ž a ky 1 b . u k
s u9, o Res z Y Ž a, z . b
;
Ž1 q z.
wt aqn
/;
u .
z
On the other hand, note that bŽwt b y 1 q p . u s 0 if p ) n. So Res z 0 zy1 0 Ž z0 q z2 .
¦
s u9,
wt aqn
z 2wt byn ² u9, Y Ž a, z 0 q z 2 . Y Ž b, z 2 . u:
Ý aŽ wt a y 2 y i q n . Ý igZ q
b Ž wt b y 1 y m . zynqiqm u 2
;
mGyn
Ž 4.13. n
¦
s u9,
;
Ý aŽ wt a y 1 y i . b Ž wt b y 1 q i . u is0
¦
q u9,
n
Ž 4.14.
;
Ý aŽ wt a y 1 q i . b Ž wt b y 1 y i . u is1
.
Ž 4.15.
84
DONG, LI, AND MASON
Note that the A nŽ V .-module structure on U is equivalent to o Ž a . o Ž b . u s a Ž wt a y 1 . b Ž wt b y 1 . u wt aqn
n
s
Ý
Ž y1.
ms0
m
ž
m q n o Res Y Ž a, z . b Ž 1 q z . z n z mqnq1
ž
/
/
u.
By Ž4.5. with s s 2, a1 s a, a2 s b, p1 s k s yp 2 Ž k ) 0. we see that ² u9, ok Ž a . oyk Ž b . u: s ² u9, a Ž wt a y 1 y k . b Ž wt b y 1 q k . u: wt aqn
nyk
¦
s u9,
Ý
Ž y1.
m
ms0
ž
m q n q k o Res Y Ž a, z . b Ž 1 q z . z m z mqnq1qk
/;
ž
/
u .
Ž 4.16. Thus n
¦
u9,
;
Ý aŽ wt a y 1 y k . b Ž wt b y 1 q k . u ks0 n
¦
s u9,
wt aqn
nyk
Ý Ý Ž y1.
m
ks0 ms0
ž
m q n q k o Res Y Ž a, z . b Ž 1 q z . z m z mqnq1qk
ž
/
/;
u .
Use Lie algebra bracket Ž3.2. to get a Ž wt a y 1 q k . b Ž wt b y 1 y k . s b Ž wt b y 1 y k . a Ž wt a y 1 q k . qÝ iG0
ž
wt a y 1 q k Ž ai b . Ž wt a q wt b y 2 y i . . i
/
By Ž4.16., ² u9, b Ž wt b y 1 y k . a Ž wt a y 1 q k . u: s
¦
wt bqn
nyk
Ý Ž y1.
ms0
m
ž
m q n q k o Res Y Ž b, z . a Ž 1 q z . z m z mqnq1qk
/
ž
/;
u .
A proof similar to that of Lemma 2.1Žii. shows that wt bqn
nyk
Ý Ž y1. ms0
Ý ms0
ž
m q n q k Res Y Ž b, z . a Ž 1 q z . z m z mqnq1qk
/
wt aqmqky1
nyk
y
m
ž
m q n q k Ž y1 . nq k Res Y Ž a, z . b Ž 1 q z . z m z 1qmqnqk
/
g On Ž V . .
85
VERTEX OPERATOR ALGEBRAS
We now have
¦
u9,
n
;
Ý aŽ wt a y 1 q k . b Ž wt b y 1 y k . u ks1 n
s
nyk
ž m qmn q k / Ž y1.
Ý Ý ks1 ms0
¦ ž
= u9, o Res z Y Ž a, z . b n
q
Ý Ý ks1 iG0
n
s
nyk
Ž1 q z.
ks1 ms0
wt aqmqky1
z 1qmqnqk
/; u
wt a y 1 q k ² u9, Ž a i b . Ž wt a q wt b y 2 y i . u: i
ž
/
ž m qmn q k / Ž y1.
Ý Ý
nq k
¦ ž
= u9, o Res z Y Ž a, z . b
nq k
Ž1 q z.
wt aqmqky1
z 1qmqnqk
/; u
n
q
Ý ² u9, o Ž Res z Y Ž a, z . b Ž 1 q z . wt ay1qk . u: . ks1
So it is enough to show the identity n
wt aqn
nyk
Ž y1.
Ý Ý
m
ks0 ms0 n
q
ž
m q n q k Ž y1 . nq k Ž 1 q z . m z 1q mqnqk
Ý Ý
/
wt aqmqky1
nyk
ks1 ms0 n
q
ž
m q n q k Ž1 q z. m z mq nq1qk
/
Ý Ž 1 q z . wt ay1qk
ks1
Ž1 q z.
s
wt aqn
,
z
or equivalently, n
n
nyk
Ý Ý Ž y1.
m
ks0 ms0 n
q
ž
m q n q k Ž1 q z. m z mq nqk
ž
m q n q k Ž y1 . nq k Ž 1 q z . m z mq nqk
mq ky1
nyk
Ý Ý ks1 ms0
/
/
s 1. This identity is proved in Proposition 5.1 in the Appendix.
86
DONG, LI, AND MASON
Proposition 4.3 is a consequence of the next lemma. LEMMA 4.5.
For all m g Z we ha¨ e
Res z 0 z 0m Ž z 0 q z 2 .
wt aqmqj
s Res z 0 z 0m Ž z 2 q z 0 .
² u9, Y Ž a, z 0 q z 2 . Y Ž b, z 2 . u: wt aqmqj
² u9, Y Ž Y Ž a, z 0 . b, z 2 . u: .
Proof. This is true for m G y1 by Lemma 4.4. Let us write m s yk q i with i g Zq and proceed by induction k. Induction yields Res z 0 zyk 0 Ž z0 q z2 .
wt aqmqj
² u9, Y Ž L Ž y1 . a, z 0 q z 2 . Y Ž b, z 2 . u:
s Res z 0 zyk 0 Ž z2 q z0 .
wt aqmqj
² u9, Y Ž Y Ž L Ž y1 . a, z 0 . b, z 2 . u: .
Using the residue property Res z f 9Ž z . g Ž z . q Res z f Ž z . g 9Ž z . s 0 and the LŽy1.-derivation property Y Ž LŽy1. a, z . s dzd Y Ž a, z . we have Res z 0 zyk 0 Ž z0 q z2 . s yRes z 0
ž
wt aq1qmqj
z0
² u9, Y Ž L Ž y1 . a, z 0 q z 2 . Y Ž b, z 2 . u:
zyk 0 Ž z0 q z2 .
s Res z 0 kzyky1 Ž z0 q z2 . 0
wt aq1qmqj
wt aq1qmqj
/
² u9, Y Ž a, z 0 q z 2 . Y Ž b, z 2 . u:
² u9, Y Ž a, z 0 q z 2 . Y Ž b, z 2 . u:
y Res z 0Ž wt a q 1 q m q j . zyk 0 Ž z0 q z2 .
wt aqmqj
= ² u9, Y Ž a, z 0 q z 2 . Y Ž b, z 2 . u: s Res z 0 kzyky1 z2 Ž z0 q z2 . 0 q Res z 0 kzyk 0 Ž z0 q z2 .
wt aqmqj
wt aqmqj
² u9, Y Ž a, z 0 q z 2 . Y Ž b, z 2 . u:
² u9, Y Ž a, z 0 q z 2 . Y Ž b, z 2 . u:
y Res z 0Ž wt a q 1 q m q j . zyk 0 Ž z2 q z0 .
wt aqmqj
= ² u9, Y Ž Y Ž a, z 0 . b, z 2 . u: s Res z 0 kzyky1 z2 Ž z0 q z2 . 0 q Res z 0 kzyk 0 Ž z2 q z0 .
wt aqmqj
wt aqmqj
² u9, Y Ž a, z 0 q z 2 . Y Ž b, z 2 . u:
² u9, Y Ž Y Ž a, z 0 . b, z 2 . u:
y Res z 0Ž wt a q 1 q m q j . zyk 0 Ž z2 q z0 . = ² u9, Y Ž Y Ž a, z 0 . b, z 2 . u: ,
wt aqmqj
87
VERTEX OPERATOR ALGEBRAS
and Res z 0 zyk 0 Ž z2 q z0 . s yRes z 0
ž
wt aq1qmqj
z0
² u9, Y Ž Y Ž L Ž y1 . a, z 0 . b, z 2 . u:
zyk 0 Ž z2 q z0 .
s Res z 0 kzyky1 Ž z2 q z0 . 0
wt aq1qmqj
wt aq1qmqj
/
² u9, Y Ž Y Ž a, z 0 . b, z 2 . u:
² u9, Y Ž Y Ž a, z 0 . b, z 2 . : u:
y Res z 0Ž wt a q 1 q m q j . zyk 0 Ž z2 q z0 .
wt aqmqj
= ² u9, Y Ž Y Ž a, z 0 . b, z 2 . u: s Res z 0 kz 2 zyky1 Ž z2 q z0 . 0 q Res z 0 kzyk 0 Ž z2 q z0 .
wt aqmqj
wt aqmqj
² u9, Y Ž Y Ž a, z 0 . b, z 2 . u:
² u9, Y Ž Y Ž a, z 0 . b, z 2 . u:
y Res z 0Ž wt a q 1 q m q j . zyk 0 Ž z2 q z0 .
wt aqmqj
= ² u9, Y Ž Y Ž a, z 0 . b, z 2 . u: . This yields the identity Res z 0 zyky1 Ž z0 q z2 . 0
wt aqmqj
s Res z 0 zyky1 Ž z2 q z0 . 0
² u9, Y Ž a, z 0 q z 2 . Y Ž b, z 2 . u: wt aqmqj
² u9, Y Ž Y Ž a, z 0 . b, z 2 . u: ,
and the lemma is proved.
ˆ Let us now introduce an arbitrary Z-graded V-module M s [m g Z M Ž m.. As before we extend M Ž n.* to M by letting it annihilate M Ž m. for m / n. The proof of Proposition of 6.1 of wDLMx with ² u9, ? : suitably inserted gives: PROPOSITION 4.6. M Ž n.9 such that Ži. Žii.
Let U be a subspace of M Ž n. and U9 a subspace of
M s UŽ Vˆ .U. For a g V and u g U there is k g Z such that ² u9, Ž z 0 q z 2 . kq n Y Ž a, z 0 q z 2 . Y Ž b, z 2 . u: s ² u9, Ž z 2 q z 0 .
kq n
Y Ž Y Ž a, z 0 . b, z 2 . u:
for any b g V, u9 g U9. Then in fact Ž4.17. holds for any u g M.
Ž 4.17.
88
DONG, LI, AND MASON
PROPOSITION 4.7. Let M be as in Proposition 4.6. Then for any x g UŽ Vˆ ., a g V, u g M, there is an integer k such that ² u9, Ž z 0 q z 2 . kq n x ? Y Ž a, z 0 q z 2 . Y Ž b, z 2 . u: s ² u9, Ž z 2 q z 0 .
kq n
x ? Y Ž Y Ž a, z 0 . b, z 2 . u:
Ž 4.18.
for any b g V, u9 g U9. Proof. Let L be the subspace of UŽ Vˆ . consisting of those x for which Ž4.18. holds. Let x g L, let c be any homogeneous element of V, and let m g Z. Then from Ž4.8. we have ² u9, xc Ž m . Y Ž a, z 0 q z 2 . Y Ž b, z 2 . u: Ž z 0 q z 2 . kq n s
`
Ý is0
q
m i
z0 q z2 .
ž /Ž Ýž / `
is0
kq nqmyi
² u9, xY Ž c Ž i . a, z 0 q z 2 . Y Ž b, z 2 . u:
kq n m my i ² u9, xY Ž a, z 0 q z 2 . Y Ž c Ž i . b, z 2 . u: z2 Ž z0 q z2 . i
qŽ z0 q z2 .
kq n
² u9, xY Ž a, z 0 q z 2 . Y Ž b, z 2 . c Ž m . u: .
Ž 4.19.
The same method that was used in the proof of Proposition 4.6 shows that xcŽ m. g L. Since UŽ Vˆ . is generated by all such cŽ n.’s, and since Ž4.18. holds for x s 1 by Proposition 4.6, we conclude that L s UŽ Vˆ ., as desired. We can now finish the proof of Theorems 4.1 and Theorem 4.2. We can take M s MnŽU . in Proposition 4.7, as we may since MnŽU . certainly satisfies the conditions placed on M prior to Proposition 4.6 and in Proposition 4.6. Then from the definition of W Ž4.3. and Propositions 4.3, 4.6, and 4.7 we conclude that UŽ Vˆ .W ; J. It is clear that LŽU . is a quotient of MnŽU . and hence an admissible V-module. Note that J l U s 0. So LŽU .Ž n. contains U as an A nŽ V .-submodule. This shows that MnŽU .Ž n. ( U as A nŽ V .-modules. If MnŽU .Ž0. s 0 then U will be an A ny 1Ž V .-module, contradicting the assumption on U. This finishes the proof of Theorem 4.1. Theorem 4.2 is now obvious. At this point we have a pair of functors V n , L n defined on appropriate module categories. It is clear that V nrV ny1 ( L n is equivalent to the identity. LEMMA 4.8. Suppose that U is a simple A nŽ V .-module. Then L nŽU . is a simple admissible V-module. Proof. If 0 / W ; L nŽU . is an admissible submodule then, by the definition of L nŽU ., we have W Ž n. s W l L nŽU .Ž n. / 0. As W Ž n. is an
VERTEX OPERATOR ALGEBRAS
89
A nŽ V .-submodule of U s L nŽU .Ž n. by Theorem 3.2 then U s W Ž n., whence W > UŽ Vˆ .W Ž n. s UŽ Vˆ .U s L nŽU .. THEOREM 4.9. L n and V nrOny1 are equi¨ alences when restricted to the full subcategories of completely reducible A nŽ V .-modules whose irreducible components cannot factor through A ny 1Ž V . and completely reducible admissible V-modules, respecti¨ ely. In particular, L n and V nrV ny1 induce mutually in¨ erse bijections on the isomorphism classes of simple objects in the category of A nŽ V .-modules which cannot factor through A ny1Ž V . and admissible V-modules, respecti¨ ely. Proof. We have V nrOny1Ž LŽU .. ( U for any A nŽ V .-module by Theorem 4.2. If M is a completely reducible admissible V-module we must show L nŽ V nrV ny1Ž M .. ( M. For this we may take M simple, whence V nrV ny1Ž M . is simple by Proposition 3.4Žii. and then L nŽ V nrV ny1Ž M .. is simple by Lemma 4.8. Since both M and L nŽ V nrV ny1Ž M .. are simple quotients of the universal object MnŽ V nrV ny1Ž M .. then they are isomorphic by Theorems 4.1 and 4.2. The following theorem is a generalization of Theorem 8.1 of wDLMx. THEOREM 4.10. Suppose that V is a rational ¨ ertex operator algebra. Then the following hold: Ža. A nŽ V . is a finite-dimensional, semisimple associati¨ e algebra. Žb. The functors L n , V nrOny1 are mutually in¨ erse categorical equi¨ alences between the category of A nŽ V .-modules whose irreducible components cannot factor through A ny 1Ž V . and the category of admissible V-modules. Žc. The functors L n , V nrV ny1 induce mutually in¨ erse categorical equi¨ alences between the category of finite-dimensional A nŽ V .-modules whose irreducible components cannot factor through A ny 1Ž V . and the category of ordinary V-modules. Proof. Part Žb. follows from Theorem 4.9 and Ža.. Since V is rational any irreducible admissible V-module is an ordinary module by Theorem 8.1 of wDLMx. Now Žc. follows from Žb.. It remains to prove Ži.. Let W be an A nŽ V .-module. Set U s W [ V Ž n.. Then U is an A nŽ V .module which cannot factor through A ny 1Ž V .. Now L nŽU . is admissible and hence a direct sum of irreducible ordinary V-modules. Thus V nŽ L nŽU ..rV ny1Ž L nŽU .. , U is a direct sum of finite-dimensional irreducible A nŽ V .-modules and so is W. It is believed that if AŽ V . s A 0 Ž V . is semisimple then V is rational. We cannot solve this problem completely in this paper. But we have some partial results which are applications of A nŽ V .-theory.
90 THEOREM 4.11. then V is rational.
DONG, LI, AND MASON
If all A nŽ V . are finite-dimensional semisimple algebras
Proof. Since AŽ V . is semisimple V has only finitely many irreducible admissible modules which are necessarily ordinary V-modules. For any l g C let Ml be the set of irreducible admissible modules whose weights are congruent to l module Z. Then for each W g Ml we have W s [ng Z q Wlqn W qn s [ng Z q W Ž n. where nW g Z and Wlqn W qn s W Ž n.. Since LŽy1.: W Ž n. ª W Ž n q 1. is injective if n is large Žsee wL1x. there exists an ml g N such that the weight space Wlqm / 0 for any W g Wl and m G ml. Consider any admissible module M whose weights are in l q Z and whose homogeneous subspace Mlqm with some m G ml is 0. Let U be an irreducible AŽ V .-submodule of M Ž0.. Then L0 ŽU . s LŽU . is an irreducible V-module such that LŽU .Ž0. s U and LŽU .lqm s 0. Thus LŽU . s 0 and U s 0. This implies that M s 0. Now take an admissible module M s [k g Z q M Ž k .. Then M Ž0. is a direct sum of simple AŽ V .-modules as AŽ V . is semisimple. Let U be an AŽ V .-submodule of M Ž0. isomorphic to W Ž0. s Wlqn W for some W g Ml . We assert that the submodule N of M generated by U is irreducible and necessarily isomorphic to W. First note that N has an irreducible quotient isomorphic to W. Take n g N such that n q nW G ml . Observe that MnŽW Ž n..rJ s L nŽW Ž n.. is isomorphic to W where J is a maximal submodule of MnŽW Ž n.. such that J l W Ž n. s 0. Since Jlqn W qn s 0 we see that J s 0 and MnŽW Ž n.. s L nŽW Ž n.. , W. Write N Ž n. as a direct sum of W Ž n. and another A nŽ V .-submodule N Ž n.9 of N Ž n. as A nŽ V . is semisimple. Clearly the submodule of N Ž n. generated by W Ž n. is isomorphic to W. This shows that N must be isomorphic to W, as claimed. It is obvious now that the submodule UŽ Vˆ . M Ž0. generated by M Ž0. is completely reducible. Using the semisimplicity of A1Ž V . we can decompose M Ž1. into a direct sum of A1Ž V .-modules ŽUŽ Vˆ . M Ž0..Ž1. [ M Ž1.9. The same argument shows that UŽ Vˆ . M Ž1.9 is a completely reducible submodule of M. Continuing in this way proves that M is completely reducible. Remark 4.12. From the proof of Theorem 4.11, we see, in fact, that we can weaken the assumption in Theorem 4.11. Namely we only need to assume that A nŽ V . is semisimple if n is large. 5. APPENDIX In this appendix we prove several combinatorial identities which are used in the previous sections.
91
VERTEX OPERATOR ALGEBRAS
For n G 0 define n
AnŽ z . s
n
nyk
Ý Ý Ž y1.
m
ž
ks0 ms0 n
q
Ý Ý
/
mq ky1
nyk
ks1 ms0
m q n q k Ž1 q z. m z mqnqk
ž
m q n q k Ž y1 . nq k Ž 1 q z . m z mq nqk
/
Using the well-known identity i
Ý Ž y1. k ks0
i ny1 n s Ž y1 . k i
ž /
ž
/
we can rewrite A nŽ z . as n
AnŽ z . s
k
Ý Ý Ž y1.
m
ks0 ms0 n
q
Ý Ý
n
s
Ý
Ž y1.
k
ks0
y Ž y1 .
ž
n q k y 1 Ž1 q z. k z nq k
/
Ý ks1
PROPOSITION 5.1.
/
ž
n
/
n
n
n
n q k Ž y1 . nq kqm Ž 1 q z . m z nq k
ky1
ks1 ms0
ž
n q k Ž1 q z. m z nqk
ž
n q k y 1 Ž1 q z. ky1 z nq k
ky 1
/
.
A nŽ z . s 1 for all n G 0.
Proof. Set n
Bn Ž z . s
Ý Ž y1. ks0 n
Cn Ž z . s
Ý ks1
ž
k
ž
n q k y 1 Ž1 q z. k z nqk
/
n q k y 1 Ž1 q z. ky1 z nqk
/
ky 1
.
n
ky 1
.
92
DONG, LI, AND MASON
Then ny1
Bn Ž z . s
k
Ž y1.
Ý ks0
q Ž y1 . s
1qz z
n
/ ž
2 n y 1 Ž1 q z. n z2n
ž
Ý Ž y1.
kq 1
ks0
1qz z
//
z nqk
/
Bny 1 Ž z . q n
n
Ž1 q z.
n
ny2
q Ž y1 . s
žž
nqky2 nqky2 q k ky1
ž
2 n y 1 Ž1 q z. n z2n
ž
n q k y 1 Ž1 q z. k z nqkq1
n
/
n
/
Bny 1 Ž z . y
1 z
Bn Ž z . q Ž y1 .
ny 1
ž
2 n y 2 Ž1 q z. ny1 z2n
/
nq 1
q Ž y1 .
n
ž
2 n y 1 Ž1 q z. n z 2 nq1
/
.
Solving BnŽ z . gives ny 1
Bn Ž z . s Bny1 Ž z . q Ž y1 . q Ž y1 .
n
Ž1 q z. z
n
Ž1 q z. z2n
ny 1
2 n y 1. ny1
žŽ
2 ny1
/
ž 2 n ny 1 / .
Similarly, ny1
Cn Ž z . s
Ý ks1
žž
nqky2 nqky2 q ky1 ky2
/ ž
2 n y 1 Ž1 q z. q ny1 z2n
ž
s
1 z
ny2
s
1 z
Ý ks0
ž
ž
n q k y 1 Ž1 q z. ky1 z nqkq1
k
/
ny 1
/
1qz z
2 n y 2 Ž1 q z. Cn Ž z . q ny1 z2n
ž
n
ž
z nqk
ny 1
2 n y 1 Ž1 q z. ny1 z2n
Cny 1 Ž z . q
y
ky 1
/
Cny 1 Ž z . q
q
//
Ž1 q z.
2 n y 1 Ž1 q z. . ny1 z 2 nq1
/
/
ny 1
n
93
VERTEX OPERATOR ALGEBRAS
Thus ny 1
Ž y1.
nq1
n
Cn Ž z . s Ž y1 . Cny1 Ž z . q Ž y1 .
n
ž
2 n y 2 Ž1 q z. ny1 z 2 ny1
/
n
q Ž y1 .
ny 1
ž
2 n y 1 Ž1 q z. . ny1 z2n
/
Thus A n Ž z . s Bn Ž z . q Ž y1 .
ny 1
Cn Ž z . s A ny1 .
Note that A 0 Ž z . s 1 and the proposition follows. For n G 0 we define m
n
Fn Ž z . s
Ý ms0
PROPOSITION 5.2.
ž
n
nq1
y Ž y1 . Ž 1 q z . m q n Ž y1 . Ž 1 q z . n z nqmq1
/
FnŽ z . s 1 for all n.
Proof. Set nq 1
ž
m q n Ž y1 . m Ž 1 q z . n z nqmq1
ž
m q n Ž1 q z. . n z nqmq1
n
Dn Ž z . s
Ý ms0
m
n
En Ž z . s
Ý ms0
/ /
Then nq 1
Dn Ž z . s Bnq1 Ž z . q Ž y1 . s Bn Ž z . q Ž y1 . q Ž y1 .
nq 1
q Ž y1 .
n
n
n
ž
2 n q 1 Ž1 q z. n z 2 nq2
/
Ž1 q z.
n
ž 2nn /
z 2 nq1
Ž1 q z.
nq 1
z 2 nq2
ž
2n q 1 nq1 nq 1
ž
2 n q 1 Ž1 q z. n z 2 nq2
s Bn Ž z . q Ž y1 .
/
n
2 n Ž1 q z. n z 2 nq1
n
ž /
and n
Ž1 q z. En Ž z . s Cn Ž z . q 2 n . n z 2 nq1
ž /
/
m
.
94
DONG, LI, AND MASON
Thus Fn Ž z . s Dn Ž z . q Ž y1 .
nq 1
En Ž z . s A n Ž z . s 1,
as required. For n G 0 define n
a n Ž w, z . s
m Ý Ž y1.
ms0
ž m qn n /
nym
=
ž
Ý Ý is0 jG0
ž
ym y n y 1 i
w iqj 1 m i y1 . iqm y m . Ž j z z
/
/ž /
. s Žy1. k Ž p q kk y 1 .. We can rewrite Note that if p ) 0, k ) 0 then Ž yp k a nŽ w, z . as n
a n Ž w, z . s
Ý
Ž y1 .
m
ms0
ž
mqn n
nym
/ž Ý
Ý
is0 jG0
ž
mqnqi i
iqj 1 m w y m . iqm j z z
/
/ž /
PROPOSITION 5.3. The a nŽ w, z . s 0 for all n G 0. Proof. Regarding a nŽ w, z . as a polynomial in zy1 , the coefficient of z in a nŽ w, z . Ž0 F p F n. is equal to Žsetting m q i s p . yp
p
m Ý Ž y1.
ms0
ž m qn n / Ý ž pn yq mp / ž mj / w
py mqj
y Ž y1 .
jG0
p
swp
m Ý Ž y1.
ms0
ž m qn n / ž pn yq mp / Ž 1 q 1rw .
m
p
ž
pqn n
y Ž y1 .
p
ž
/ pqn . n
So the coefficient of zyp w 0 in a nŽ w, z . equals 0. If 0 F q F p y 1, the coefficient of zyp w pyq in a nŽ w, z . is equal to p
c n Ž p, q . s
m Ý Ž y1.
ms0
ž m qn n / ž nnqqmp / ž mq /
/
95
VERTEX OPERATOR ALGEBRAS
which is defined for any n, p, q G 0. So we must prove that a nŽ p, q . s 0 1. Ž . for 1 F q q 1 F p F n. Recall Ž kl . s Ž l yk 1 . q Ž kl y y 1 . Then c n p, q is equal to p
m Ý Ž y1.
ms0
ž m qn n / ž ž n qn qp ym 1 / q ž nnqqmp yy11 / / ž mq /
s Ž y1 .
p
ž
pqn n
p q q
/ž /
py1
m Ý Ž y1.
ms0
ž m qn n / ž nnqqmp yy11 / ž mq /
qc n Ž p y 1, q . py1
s
m Ý Ž y1.
ms0
q Ž y1 .
p
ž
žž
mqny1 q mqny1 n ny1
pqn n
/ ž
p q c n Ž p y 1, q . q
/ž /
s c ny 1 Ž p, q . q c n Ž p y 1, q . y Ž y1 . q Ž y1 .
p
ž
pqn n
py1
y
Ý Ž y1. my 1 ms0
=
žž
pqny1 ny1
my1 my1 q q qy1
/ ž
my1s0
// p
ž
pqny1 n
/ ž
//
s c ny 1 Ž p, q . y c n Ž p y 1, q y 1 . q Ž y1 . py 1
ž
p q
/ž /
ž m qnn y 1 / ž nnqqmp yy11 /
my1 my1 q q qy1
q Ž y1 .
p q
/ž /
ž m qnn y 1 / ž nnqqmp yy11 /
Ý Ž y1. my 1
žž
ž
p q
py2
=
p
/ž /
s c ny 1 Ž p, q . q c n Ž p y 1, q . q Ž y1 . y
/ / ž nnqqmp yy11 / ž mq /
py1qn n
/žž
p
ž
pqny1 n
py1 py1 q q qy1
s c ny 1 Ž p, q . y c n Ž p y 1, q y 1 . .
/ ž
//
p q
/ž /
96
DONG, LI, AND MASON
That is, c n Ž p, q . s c ny1 Ž p, q . y c n Ž p y 1, q y 1 . . so by induction it is enough to show that c 0 Ž p, q . s 0 and c nŽ p, 0. s 0 if p ) q. But this is clear from the definition.
REFERENCES wBx
R. E. Borcherds, Vertex algebras, Kac]Moody algebras, and the Monster, Proc. Nat. Acad. Sci. U.S. A. 83 Ž1986., 3068]3071. wDLMx C. Dong, H. Li, and G. Mason, Twisted representations of vertex operator algebras, Math. Ann. 310 Ž1998., 571]600. wDLMMx C. Dong, H. Li, G. Mason, and P. Montague, The radical of a vertex operator algebra, in ‘‘Proc. of the Conference on the Monster and Lie algebras at The Ohio State University, May 1996’’ ŽJ. Ferrar and K. Harada, Eds.., de Gruyter, BerlinrNew York. wDLx C. Dong and Z. Lin, Induced modules for vertex operator algebras, Comm. Math. Phys. 179 Ž1996., 157]184. wDMx C. Dong and G. Mason, On quantum Galois theory, Duke Math. J. 86 Ž1997., 305]321. wFHLx I. Frenkel, Y. Huang, and J. Lepowsky, On axiomatic approaches to vertex operator algebras and modules, Mem. Amer. Math. Soc. 104 Ž1993.. wFLMx I. Frenkel, J. Lepowsky, and A. Meurman, ‘‘Vertex Operator Algebras and the Monster,’’ Pure and Appl. Math., Vol. 134, Academic Press, Boston, 1988. wL1x H. Li, Symmetric invariant bilinear forms on vertex operator algebras, J. Pure Appl. Algebra 96 Ž1994., 279]297. wL2x H. Li, ‘‘Representation Theory and Tensor Product Theory for Vertex Operator Algebras,’’ Ph.D. thesis, Rutgers University, 1994. wZx Y. Zhu, Modular invariance of characters of vertex operator algebras, J. Amer. Math. Soc. 9 Ž1996., 237]302.
JOURNAL OF ALGEBRA ARTICLE NO.
Vertex Operator Algebras and Associative Algebras Chongying Dong* Department of Mathematics, Uni¨ ersity of California, Santa Cruz, California 95064 E-mail: [email protected]
Haisheng Li † Department of Mathematical Sciences, Rutgers Uni¨ ersity, Camden, New Jersey 08102 E-mail: [email protected]
and Geoffrey Mason‡ Department of Mathematics, Uni¨ ersity of California, Santa Cruz, California 95064 E-mail: [email protected] Communicated by Walter Feit Received January 16, 1997
Let V be a vertex operator algebra. We construct a sequence of associative algebras A nŽ V . Ž n s 0, 1, 2, . . . . such that A nŽ V . is a quotient of A nq1Ž V . and a pair of functors between the category of A nŽ V .-modules which are not A ny1Ž V .modules and the category of admissible V-modules. These functors exhibit a bijection between the simple modules in each category. We also show that V is rational if and only if all A nŽ V . are finite-dimensional semisimple algebras. Q 1998 Academic Press
* Partially supported by NSF Grant DMS-9303374 and a research grant from the Committee on Research, UC Santa Cruz. † Partially supported by NSF Grant DMS-9616630. ‡ Partially supported by NSF Grant DMS-9401272 and a research grant from the Committee on Research, UC Santa Cruz. 67 0021-8693r98 $25.00 Copyright Q 1998 by Academic Press All rights of reproduction in any form reserved.
68
DONG, LI, AND MASON
1. INTRODUCTION For a vertex operator algebra V Zhu constructed an associative algebra AŽ V . wZx such that there is a one-to-one correspondence between irreducible admissible V-modules and irreducible AŽ V .-modules. In the case that V is rational the admissible V-module category and AŽ V .-module category are in fact equivalent. But if V is not rational, AŽ V . does not carry enough information for the representation of V. In this paper we construct a sequence of associative algebras A nŽ V . Ž n s 0, 1, 2, . . . . such that A 0 Ž V . s AŽ V . and A nŽ V . is an epimorphic image of A nq 1Ž V .. As in wZx, we use A nŽ V . to study representation theory of V. Let M s [k G 0 M Ž k . be an admissible V-module as defined in wDLMx with M Ž0. / 0. Then each M Ž k . for k F n is an A nŽ V .-module. In some sense, A nŽ V . takes care of the first n q 1 homogeneous subspaces of M while AŽ V . concerns the top level M Ž0.. The results of the present paper are modeled on the results in wDLMx and the methods are also similar. However, the situation for constructing admissible V-modules from A nŽ V .-modules turns out to be very complicated. As in wL2, DLMx we extensively use the Lie algebra Vˆ s V m C w t , ty1 x r L Ž y1 . m 1 q 1 m
ž
d dt
/Ž
V m C w t , ty1 x .
to construct admissible V-modules from A nŽ V .-modules. It should be pointed out that the A nŽ V .4 in fact form an inverse system. So it is natural to consider the inverse limit lim A nŽ V . and its representa¤
tions. This problem will be addressed in a separate paper. One of the important motivations for constructing A nŽ V . is to study induced modules from a subalgebra to V as initiated in wDLx. Induced module theory is very important in the representation theory of classical objects such as groups, rings, Lie algebras. The theory of A nŽ V . developed in this paper will definitely play a role in the study of induced modules for vertex operator algebras. In order to see this, we consider a subalgebra U of V and a U-submodule W of M which is an admissible V-module. In general, the top level of W is not necessarily a subspace of the top level of M. In other words, an AŽU .-module can be a subspace of an A nŽ V .module for some n ) 0. One can now see how the A nŽ V . enter the picture of studying the induced module for the pair ŽU, V . along this line. This paper is organized as follows: In Section 2 we introduce the algebra A nŽ V . which is a quotient of V modulo a subspace OnŽ V . consisting of u(n ¨ Žsee Section 2 for the definition. and LŽy1. u q LŽ0. u for u, ¨ g V. In the case n s 0, Ž LŽy1. q LŽ0.. u can be expressed as v (0 u. But in general it is not clear if one can write Ž LŽy1. q LŽ0.. u as a linear
69
VERTEX OPERATOR ALGEBRAS
combination of ¨ (n w’s. On the other hand, the weight zero component of the vertex operator Y ŽŽ LŽy1. q LŽ0.. u, z . is zero on any weak V-module. So we have to put Ž LŽy1. q LŽ0..V artificially in OnŽ V . for general n. We also show in this section how the identity map on V induces an epimorphism of algebras from A nq 1Ž V . to A nŽ V .. In Section 3, we construct a functor V n from the category of weak V-modules to the category of A nŽ V .-modules such that if M s [k G 0 M Ž k . is an admissible V-module n Ž . Ž . Ž . Ž . then [ks 0 M k with M 0 / 0 is contained in V n M and each M k for Ž . k F n is an A n V -submodule. In particular, if M is irreducible then n Ž . Ž . Ž . Ž . [ks 0 M k s V n M and each M k is an irreducible A n V -module. Section 4 is the core of this paper. In this section we construct a functor L n from the category of A nŽ V .-modules which cannot factor through A ny 1Ž V . to the category of admissible V-modules. For any such A nŽ V .module U we first construct a universal admissible V-module MnŽU . which is somehow a ‘‘generalized Verma module.’’ The L nŽ V . is then a suitable quotient of MnŽU .; the proof of this result is technically the most difficult part of this paper. We also show that V nŽ L nŽU ..rV ny1Ž L nŽU .. is isomorphic to U as A nŽ V .-modules. Moreover, V is rational if and only if the A nŽ V . are finite-dimensional semisimple algebras for all n. Section 5 deals with several combinatorial identities used in previous sections. We assume that the reader is familiar with the basic knowledge on vertex operator algebras as presented in wB, FHL, FLMx. We also refer the reader to wDLMx for the definitions of weak modules, admissible modules, and Žordinary. modules.
2. THE ASSOCIATIVE ALGEBRA A nŽ V . Let V s Ž V, Y, 1, v . be a vertex operator algebra. We will construct an associative algebra A nŽ V . for any nonnegative integer n generalizing the Zhu’s algebra AŽ V . which is our A 0 Ž V .. Let OnŽ V . be the linear span of all u(n ¨ and LŽy1. u q LŽ0. u where for homogeneous u g V and ¨ g V, u(n ¨ s Res z Y Ž u, z . ¨
Ž1 q z.
wt uqn
z 2 nq2
.
Ž 2.1.
Define the linear space A nŽ V . to be the quotient VrOnŽ V .. We also define a second product )n on V for u and ¨ as above: n
u)n ¨ s
Ý Ž y1. ms0
wt uqn
m
ž
m q n Res Y Ž u, z . Ž 1 q z . z n z nqmq1
/
¨.
Ž 2.2.
70
DONG, LI, AND MASON
Extend linearly to obtain a bilinear product on V which coincides with that of Zhu wZx if n s 0. We denote the product Ž2.2. by ) in this case. Note that Ž2.2. may be written in the form `
n
u)n ¨ s
Ý Ý Ž y1. ms0 is0
m
ž m qn n / ž wt uiq n / u
iymyny1¨ .
Ž 2.3.
The following lemma generalizes Lemmas 2.1.2 and 2.1.3 of wZx. LEMMA 2.1. Ži. Assume that u g V is homogeneous, ¨ g V, and m G k G 0. Then Res z Y Ž u, z . ¨
Ž1 q z.
wt uqnqk
z 2 nq2qm Žii. Assume that ¨ is also homogeneous. Then
wt ¨ qm y1
n
u)n ¨ y
g On Ž V . .
Ý ms0
ž
m q n Ž y1 . n Res Y Ž ¨ , z . u Ž 1 q z . z n z 1qmqn
g On Ž V .
/
and Žiii. u)n ¨ y ¨ )n u y Res z Y Ž u, z . ¨ Ž1 q z . wt uy1 g OnŽ V .. Proof. The proof of Ži. is similar to that of Lemma 2.1.2 of wZx. As in wZx we use LŽy1. u q LŽ0. u g OnŽ V . to derive the formula yz yw t uywt ¨ Y Ž u, z . ¨ ' Ž 1 q z . Y ¨, u mod On Ž V . . 1qz Thus we have
ž
wt uqn
n
u)n ¨ s
Ž y1.
m
Ý Ž y1.
m
Ý ms0
ž
n q m Res Y Ž u, z . ¨ Ž 1 q z . z n z mqnq1
ž
n q m Res Y ¨ , yz u Ž 1 q z . z n 1qz z mqnq1
/
yw t ¨ qn
n
'
/
ms0
/
ž
/
mod On Ž V . wt ¨ qm y1
n
s
Ý
Ž y1.
n
ms0
ž
n q m Res Y Ž ¨ , z . u Ž 1 q z . z n z mqnq1
/
and Žii. is proved. Using Žii. we have u)n ¨ y ¨ )n u ' Res z Y Ž u, z . ¨ Ž 1 q z .
m
n
=
ž
Ý ms0
wt uy1
ž
nq1
n
y Ž y1 . Ž 1qz . mqn Ž y1 . Ž 1qz . nq mq1 n z
/
m
/
.
71
VERTEX OPERATOR ALGEBRAS
By Proposition 5.2 in the Appendix we know that m
n
Ý ms0
ž
n
nq1
y Ž y1 . Ž 1 q z . m q n Ž y1 . Ž 1 q z . nq mq1 n z
m
/
s 1.
The proof is complete. LEMMA 2.2. OnŽ V . is a 2 sided ideal of V under )n . Proof. First we show that Ž LŽy1. u q LŽ0. u.)n ¨ g OnŽ V . for any homogeneous u g V. From the definition we see that
Ž L Ž y1. u . )n ¨ wt uqnq1
n
s
ž
Ý ms0
¨
/
wt uqnq1
m q n Ž y1 . m Res d Y Ž u, z . ¨ Ž 1 q z . z n dz z nqmq1
n
s
m q n Ž y1 . m Res Y Ž L Ž y1 . u, z . Ž 1 q z . z n z nqmq1
ž / s Ý ž m q n / Ž y1 . n Ý
ms0 n
ž
mq 1
/
Res z Y Ž u, z . ¨
ms0
=
ž
Ž yn y m y 1 . Ž 1 q z .
wt uqnq1
q
z nq mq2
z Ž wtu q n q 1 . Ž 1 q z .
wt uqn
z nqmq2
/
Thus
Ž L Ž y1. u q wt uu . )n ¨ n
s
Ý ms0
ž m qn n / Ž y1.
m
Res z Y Ž u, z . ¨ Ž 1 q z .
wt uqn
mz q n q m q 1 z nqmq2
It is straightforward to show that n
Ý ms0
ž m qn n / Ž y1. n
s
Ý ms0
ž
s Ž y1 .
m
mz q n q m q 1 z nq mq2
m q n Ž y1 . m mz q n z nq mq2
/
n
ž 2 n nq 1 /
2n q 1 z 2 nq2
n
Ý ms0
ž m qnn q 1 / Ž y1.
.
It is clear now that Ž LŽy1. u q LŽ0. u.)n ¨ g OnŽ V ..
m
mq1 z nqmq2
.
.
72
DONG, LI, AND MASON
Second, we show that u)n Ž LŽy1. ¨ q LŽ0. ¨ . g OnŽ V .. Using the result that Ž LŽy1. ¨ q LŽ0. ¨ .)n ¨ g OnŽ V . and Lemma 2.1Žiii. we have u)n Ž L Ž y1 . ¨ q L Ž 0 . ¨ . ' yRes z Ž Y Ž L Ž y1 . ¨ , z . u Ž 1 q z .
wt ¨
q Y Ž L Ž 0. ¨ , z . u Ž 1 q z .
wt ¨ y1
.
mod On Ž V . s Res z Y Ž ¨ , z . u
ž
d dz
Ž1 q z.
wt ¨
y Y Ž L Ž 0. , ¨ , z . u Ž 1 q z .
wt ¨ y1
/
s 0. Third, a similar argument as in wZx using Lemma 2.1Ži. shows that u)n Ž ¨ (n w . g OnŽ V . for u, ¨ , w g V. Finally, use u)n Ž ¨ (n w . g OnŽ V . and Lemma 2.1Žiii. to obtain
Ž ¨ (n w . )n u ' yRes z 1 Res z 2 Y Ž u, z1 . Y Ž ¨ , z 2 . w
Ž 1 q z1 .
wt uy1
Ž 1 q z2 .
wt ¨ qn
z 22 nq2 mod On Ž V .
' yRes z 2 Res z 1yz 2 Y Ž Y Ž u, z1 y z 2 . ¨ , z 2 . w
Ž 1 q z1 .
wt uy1
Ž 1 q z2 .
wt ¨ qn
z 22 nq2 wt uqwt ¨ qn y1yi
sy Ý iG0
ž
Ž 1 q z2 . wt u y 1 Res z 2 Y Ž u i ¨ , z 2 . w i z 22 nq2
/
which belongs to OnŽ V . as wt u i ¨ s wt u q wt ¨ y i y 1. This completes the proof. Our first main result is the following. THEOREM 2.3. Ži. The product ) n induces the structure of an associati¨ e algebra on A nŽ V . with identity 1 q OnŽ V .. Žii. The linear map
f : ¨ ¬ e LŽ1. Ž y1 .
L Ž0 .
¨
induces an anti-isomorphism A nŽ V . ª A nŽ V .. Žiii. v q OnŽ V . is a central element of A nŽ V ..
73
VERTEX OPERATOR ALGEBRAS
Proof. For Ži. we only need to prove that A nŽ V . is associative. Let u, ¨ , w g V be homogeneous. Then
Ž u)n ¨ . )n w n
s
Ý Ý Ž y1.
m1
m1s0 iG0
ž
n
s
Ý
Ý Ž y1.
m1 q n n m 1 qm 2
m1 , m 2s0 iG0
ž
/ž
wt u q n Ž uym 1yny1qi ¨ . )n w i
/
m1 q n n
=Res z 2 Y Ž uym 1yny1qi ¨ , z 2 . w n
m qm Ý Ž y1.
s
1
2
m1 , m 2s0
m1 q n n
ž
=Y Ž Y Ž u, z1 y z 2 . ¨ , z 2 . w n
m qm Ý Ž y1.
s
1
m1 , m 2s0
2
n
m qm Ý Ž y1. 1
2
m1 , m 2s0
ž
/ž
n
Ý
Ý Ž y1.
m 1 qm 2
m1 , m 2s0 iG0
ž
/
/ž
n
Ý
m1 , m 2s0 iG0
2
ž
wt ¨ qn qm 1
/
Ž 1 q z1 .
wt uqn
Ž z1 y z 2 .
Ž 1 q z2 .
wt ¨ qn qm 1
m 1 qnq1 1q m 2 qn z2
/
Ž 1 q z1 .
wt uqn
Ž z1 y z 2 .
/ž
m2 q n n
Ž 1 q z1 .
/ž
Ž 1 q z2 .
m 1 qnq1
wt ¨ qn qm 1
z 21q m 2qn
ym1 y n y 1 i Ž y1. i
wt uqn
/
Ž 1 q z2 .
wt ¨ qn qm 1
z1m 1qnq1qi z 21qm 2qnyi
m1 q n n
=Res z 2 Res z 1Y Ž ¨ , z 2 . Y Ž u, z1 . w
Ž 1 q z2 .
m 1 qnq1 1qm 2 qn z2
m2 q n n
/ž
m1 q n n
Ý Ž y1. m qnq1qi
wt uqn
m2 q n n
=Res z 1 Res z 2 Y Ž u, z1 . Y Ž ¨ , z 2 . w y
/
m2 q n Res z 2 Res z 1yz 2 n
=Res z 2 Res z 1Y Ž ¨ , z 2 . Y Ž u, z1 . w s
wt u q n i
wt uqwt ¨ q2 nqm1yi
Ž z1 y z 2 .
m1 q n n
/ž
z 21qm 2qn
=Res z 1 Res z 2 Y Ž u, z1 . Y Ž ¨ , z 2 . w y
m2 q n n
Ž 1 q z2 .
Ž 1 q z1 .
m1 q n n
ž
/ž
/ž
m2 q n n
Ž 1 q z1 .
wt uqn
/ž
ym1 y n y 1 i
Ž 1 q z2 .
/
wt ¨ qn qm 1
z1m 1qnq1yi z 22qm 1qm 2q2 nqi
.
74
DONG, LI, AND MASON
From Lemma 2.1 we know that Res z 2 Res z 1Y Ž ¨ , z 2 . Y Ž u, z1 . w
Ž 1 q z1 .
wt uqn
Ž 1 q z2 .
wt ¨ qn qm 1
z1m 1qnq1yi z 22qm 1qm 2q2 nqi
lies in OnŽ V .. Also if i ) n y m1 Res z 1 Res z 2 Y Ž u, z1 . Y Ž ¨ , z 2 . w
Ž 1 q z1 .
wt uqn
Ž 1 q z2 .
wt ¨ qn qm 1
z1m 1qnq1qi z 21qm 2qnyi
is in OnŽ V .. Thus n
Ž u)n ¨ . )n w ' u)n Ž ¨ )n w . q
m qm Ý Ž y1. 1
m1 , m 2s0
=Res z 1 Res z 2 Y Ž u, z1 . Y Ž ¨ , z 2 . nym 1
=
ž
Ý Ý is0
jG0
ž
ym1 y n y 1 i
2
ž
m1 q n n
Ž 1 q z1 .
/ž
wt uqn
m2 q n n
Ž 1 q z2 .
/
wt ¨ qn
z1m 1qnq1 z 21qm 2qn z iqj m1 i 2 Ž y1. i y 1 . j z1
/
/ž /
From Proposition 5.3 in the Appendix we know that n
m Ý Ž y1.
1
m1s0
ž
m1 q n n
nym 1
ž
Ý Ý is0
jG0
ž
/
ym1 y n y 1 i
z 2iqj 1 m1 i Ž y1. iqm 1 y m 1 j z1 z1
/ž /
/
s 0.
This implies that the product )n of A nŽ V . is associative. The proof of Žii. is similar to that of Žii. of Theorem 2.4 of wDLMx. We refer the reader to wDLMx for detail. Note that 1)n u s u for any u g V and that u)n 1 y 1)n u ' Res z Y Ž u, z . 1 Ž 1 q z .
wt uy1
s 0.
This shows that 1 q OnŽ V . is the identity of A nŽ V .. Again by Lemma 2.1Žiii.,
v )n u y u)n v s Res z Y Ž v , z . u Ž 1 q z . s L Ž y1 . u q L Ž 0 . u g On Ž V . . So Žiii. is proved.
75
VERTEX OPERATOR ALGEBRAS
PROPOSITION 2.4. The identity map on V induces an onto algebra homomorphism from A nŽ V . to A ny1Ž V .. Proof. First by Lemma 2.1Ži., OnŽ V . ; Ony1Ž V .. It remains to show that u)n ¨ ' u)ny1 ¨ mod Ony1Ž V .. Let u be homogeneous. Then wt uqny1
n
u)n ¨ s
Ý ms0
ž
m q n Ž y1 . m Res Y Ž u, z . ¨ Ž 1 q z . z n z nqm
/
wt uqny1
m q n Ž y1 . m Res Y Ž u, z . ¨ Ž 1 q z . z n z nqmq1
n
q
Ý ms0
Ý ms0
/
wt uqny1
ny1
'
ž
ž
m q n Ž y1 . m Res Y Ž u, z . ¨ Ž 1 q z . z n z nqm
/
wt uqny1
m q n Ž y1 . m Res Y Ž u, z . Ž 1 q z . z n z nqmq1
ny2
q
Ý ms0
ž
/
mod Ony 1 Ž V . sRes z Y Ž u, z . ¨ ? Ž y1 .
ž
m
Ž 1qz .
wt uqny1
ny1
q Ý Res z Y Ž u, z . ¨
zn
ms1
Ž 1qz .
wt uqny1
z nqm
ž m qn n / q Ž y1. ž m qnn y 1 / / mq1
s u)ny 1 ¨ , as desired. From Proposition 2.4 we in fact have an inverse system A nŽ V .4 . Denote by I Ž V . the inverse limit lim A nŽ V .. Then ¤
½
I Ž V . s a s Ž a n q On Ž V . . g
`
Ł A n Ž V . < an y any1 g Ony1Ž ¨ .
ns0
5
.
Ž 2.4. Define i: V ª I Ž V . such that iŽ ¨ . s Ž ¨ q OnŽ V .. for ¨ g V. Then Vrker i is linearly isomorphic to a subspace of I Ž V .. It is easy to see that iŽ V . is not closed under the product. But one can introduce a suitable topology on I Ž V . so that iŽ V . is a dense subspace of I Ž V . under the topology. An interesting problem is to determine the kernel of i. From the definition of OnŽ V . we see immediately that Ž LŽy1. q LŽ0..V is contained in the kernel. It will be proved in Section 3 that if ¨ g OnŽ V . then a wt ay1 s 0 on
76
DONG, LI, AND MASON
n ` Ž . Ž . [ks 0 M n for any admissible V-module [ks0 M k . Thus a g ker i if and only if a wt ay1 s 0 on any admissible V-module. It is proved in wDLMMx that if V is a simple vertex operator algebra satisfying Vk s 0 for all k - 0, and V0 s C1 then the subspace of V consisting of vectors ¨ whose component operators ¨ wt ¨ y1 are 0 on V is essentially Ž LŽ0. q LŽy1..V. We suspect that if V is a rational vertex operator algebra then the kernel of i is exactly Ž LŽ0. q LŽy1..V.
3. THE FUNCTOR V n Consider the quotient space Vˆ s C w t , ty x m VrDC w t , ty x m V ,
Ž 3.1.
where D s dtd m 1 q 1 m LŽy1.. Denote by ¨ Ž m. the image of ¨ m t m in Vˆ for ¨ g V and m g Z. Then Vˆ is Z-graded by defining the degree of ¨ Ž m. to be wt ¨ y m y 1 if ¨ is homogeneous. Denote the homogeneous subspace of degree m by VˆŽ m.. The space Vˆ is, in fact, a Z-graded Lie algebra with bracket aŽ p . , b Ž q . s
`
Ý is0
p i
ž /Ž
ai b . Ž p q q y i .
Ž 3.2.
Žsee wL2, DLMx.. In particular, VˆŽ0. is a Lie subalgebra. By Lemma 2.1Žiii. we have PROPOSITION 3.1. Regarded A nŽ V . as a Lie algebra, the map ¨ Žwt ¨ y 1. ¬ ¨ q OnŽ V . is a well-defined onto Lie algebra homomorphism from VˆŽ0. to A nŽ V .. By Lemmas 5.1 and 5.2 of wDLMx, any weak V-module M is a module for Vˆ under the map aŽ m. ¬ a m and a weak V-module which carries a Zq-grading is an admissible V-module if, and only if, M is a Zq-graded ˆ module for the graded Lie algebra V. For a module W for the Lie algebra Vˆ and a nonnegative m we let V mŽW . denote the space of ‘‘mth lowest weight vectors,’’ that is, V m Ž W . s u g W < VˆŽ yk . u s 0 if k ) m4 .
Ž 3.3.
Then V mŽW . is a module for the Lie algebra VˆŽ0.. THEOREM 3.2. Suppose that M is a weak V-module. Then there is a representation of the associati¨ e algebra A nŽ V . on V nŽ M . induced by the map a ¬ oŽ a. s a wt ay1 for homogeneous a g V.
77
VERTEX OPERATOR ALGEBRAS
Proof. We need to show that oŽ a. s 0 for all a g OnŽ V . and oŽ u)n ¨ . s oŽ u. oŽ ¨ . for u, ¨ g V. Using Y Ž LŽy1. u, z . s dzd Y Ž u, z . we immediately see that oŽ LŽy1. u q LŽ0. u. s 0. From the proof of Lemma 2.1 we know that Ž LŽy1. u q LŽ0. u.)n ¨ s Žy1. n Ž 2nn .Ž2 n q 1. u(n ¨ . It suffices to show that oŽ u)n ¨ . s oŽ u. oŽ ¨ .. Let u, ¨ be homogeneous and 0 F k F n. Note that ¨ wt ¨ qp s u wt uqp s 0 on V nŽ M . if p G n. We assert that the following identity holds on V nŽ M ., wt uqn
k
Ž y1.
Ý
m
ms0
ž
2 n q m y k o Res Y Ž u, z . ¨ Ž 1 q z . z m z 2 nq1ykqm
ž
/
s u wt uynqky1¨ wt ¨ qn yky1
/ Ž 3.4.
which reduces to oŽ u)n ¨ . s oŽ u. oŽ ¨ . if k s n. The proof of Ž3.4. is a straightforward computation involving the Jacobi identity on modules in terms of residues. On V nŽ M . we have wt uqn
k
m Ý Ž y1.
ms0
ž
2 n q m y k o Res Y Ž u, z . ¨ Ž 1 q z . z m z 2 nq1ykqm
/
k
s
Ý Ý Ž y1.
m
ms0 iG0 k
s
Ý Ý Ž y1.
m
ms0 iG0
ž
ž 2 n qmm y k / ž wt uiq n / o Ž u ž 2 n qmm y k / ž wt uiq n /
= Ž u iy2 ny1ymqk¨ . wt uqwt ¨ yiq2 nqmy1yk n
s
Ý Ý Ž y1. ms0 iG0
m
ž 2 n qmm y k / ž wt uiq n /
=Res z 2 Res z 1yz 2 Y Ž Y Ž u, z1 y z 2 . ¨ , z 2 . = Ž z1 y z 2 . k
s
m Ý Ž y1.
ms0
iy2 nymy1qk
z 2wt uqwt ¨ yiq2 nqmy1yk
ž 2 n qmm y k / Res
=Y Ž Y Ž u, z1 y z 2 . ¨ , z 2 . k
s
m Ý Ž y1.
ms0
z2
Res z 1yz 2
z1wt uqn z 2wt ¨ qn qmy1yk
Ž z1 y z 2 .
2 nqmq1yk
ž 2 n qmm y k / Res Res z1
z2
/
iy2 ny1ymqk¨
.
78
DONG, LI, AND MASON
=Y Ž u, z1 . Y Ž ¨ , z 2 . k
m Ý Ž y1.
y
ms0
k
Ž z1 y z 2 .
2 nqmq1yk
ž 2 n qmm y k / Res
=Y Ž ¨ , z 2 . Y Ž u, z1 . s
z1wt uqn z 2wt ¨ qn qm y1 y k
kym
Ý Ý Ž y1. mq i ms0 is0
z2
Res z 1
z1wt uqn z 2wt ¨ qn qmy1yk
Ž z1 y z 2 .
2 nqmq1yk
ž 2 n qmm y k / ž ym y 2 ni y 1 q k /
=u wt uynymy1qkyi ¨ wt ¨ qn qmy1ykqi k
s
k
ž 2 n qmm y k / ž ym yi 2yn my 1 q k / Ž y1.
Ý Ý ms0 ism
i
=u wt uynqkyiy1¨ wt ¨ qn yky1qi k
s
i
Ý Ý is0 ms0
ž 2 n qmm y k / ž ym yi 2yn my 1 q k / Ž y1.
i
=u wt uynqkyiy1¨ wt ¨ qn yky1qi s u wt uynqky1¨ wt ¨ qn yky1 k
i
qÝ
Ý
is1 ms0
ž 2 n qmm y k / ž ym yi 2yn my 1 q k / Ž y1.
=u wt uynqkyiy 1¨ wt ¨ qn yky1qi . It is enough to show that for i s 1, . . . , k. i
Ý ms0
ž 2 n qmm y k / ž ym yi 2yn my 1 q k / s 0,
which follows from an easy calculation: i
Ý ms0
ž 2 n qmm y k / ž ym yi 2yn my 1 q k / s Ý Ž y1 . ž 2 n qmm y k / ž 2 niqyimy k / s Ý Ž y1 . ž 2 n2qn yi yk k / ž mi / i
iym
ms0 i
ms0
s 0. This completes the proof.
iym
i
79
VERTEX OPERATOR ALGEBRAS
Remark 3.3. For homogeneous u, ¨ g V and j g Z we set oj Ž u. s u wt uy1yj and extend to all u g V by linearity. Then o 0 Ž u. s oŽ u.. Using associativity of the vertex operators
Ž z0 q z2 .
wt uqn
Y Ž u, z 0 q z 2 . Y Ž ¨ , z 2 . s Ž z 2 q z 0 .
wt uqn
Y Ž Y Ž u, z 0 . ¨ , z 2 .
on V nŽ M . we have that for i G j with i q j G 0 these exists a unique wu,i, ¨j g V such that oi Ž u. oj Ž ¨ . s oiqj Ž wu,i, ¨j . on V nŽ M .. In fact one can write wu,i, ¨j explicitly in terms of u and ¨ . But for our later purpose it is Ž i G 0. which is given by enough to know the explicit expression of wu,i, yi ¨ wui ,, yi ¨ s
wt uqn
nyi
Ý ms0
Ž y1.
m
ž
n q m q i Res Y Ž u, z . ¨ Ž 1 q z . z m z nq1qiqm
/
in the proof of Theorem 3.2. It is clear that V n is a covariant functor from the category of weak V-modules to the category of A nŽ V .-modules. To be more precise, if f : M ª N is a morphism in the first category we define V nŽ f . to be the ˆ restriction of f to V nŽ M .. Then f induces a morphism of V-modules M ª N by Lemma 5.1 of wDLMx. Moreover V nŽ f . maps V nŽ M . to V nŽ N .. Now Theorem 3.2 implies that V nŽ f . is a morphism of A nŽ V .modules. Let M be such a module. As long as M / 0, then some M Ž m. / 0, and it is no loss to shift the grading so that in fact M Ž0. / 0. If M s 0, let M Ž0. s 0. With these conventions we prove PROPOSITION 3.4. following hold
Suppose that M is an admissible V-module. Then the
n n Ži. V nŽ M . > [is0 M Ž i .. If M is simple then V nŽ M . s [is0 M Ž i .. Žii. Each M Ž p . is an VˆŽ0.-module and M Ž p . and M Ž q . are inequi¨ alent if p / q and both M Ž p . and M Ž q . are nonzero. If M is simple then each M Ž p . is an irreducible VˆŽ0.-module. Žiii. Assume that M is simple. Then each M Ž i . for i s 0, . . . , n is a simple A nŽ V .-module and M Ž i . and M Ž j . are inequi¨ alent A nŽ V .-modules.
Proof. An easy argument shows that V nŽ M . is a graded subspace of M. That is, V nŽ M . s
[i G 0 V n Ž M . l M Ž i . .
Ž 3.5.
Set V nŽ i . s V nŽ M . l M Ž i .. It is clear that M Ž i . ; V nŽ M . if i F n. In order to prove Ži. we must show that V nŽ i . s 0 if i ) n.
80
DONG, LI, AND MASON
By Proposition 2.4 of wDMx or Lemma 6.1.1 of wL2x, M s span u n w < u g V, n g Z4 where w is any nonzero vector in M. If V nŽ i . / 0 for some i ) n we can take 0 / w g V nŽ i .. Since u wt uqp w s 0 for all p G n we see that M s span u wt uqp w < u g V, p g Z, p - n4 . This implies that M Ž0. s 0, a contradiction. ˆ s It is clear that Žiii. follows from Žii.. For Žii., note that M s Vw [p g Z VˆŽ p . w. Thus if 0 / w g M Ž i . then VˆŽ p . w s M Ž i q p .. In particular, VˆŽ0. w s M Ž i ., as required. It was pointed out in wZx that LŽ0. is semisimple on M and M Ž k . s w g M < LŽ0. w s Ž h q k . w4 for some fixed h. The inequivalence follows from the fact that LŽ0. has different eigenvalues on M Ž p . and M Ž q ..
4. THE FUNCTOR L n We show in this section that there is a universal way to construct an admissible V-module from an A nŽ V .-module which cannot factor through A ny 1. ŽIf it can factor through A ny1Ž V . we can consider the same procedure for A ny 1Ž V ... Moreover a certain quotient of the universal object is an admissible V-module L nŽU . and L n defines a functor which is a right inverse to the functor V nrV ny1 , where V nrV ny1 is the quotient functor M ¬ V nŽ M .rV ny1Ž M .. Fix an A nŽ V .-module U which cannot factor through A ny1Ž V .. Then it is a module for A nŽ V .L i e in an obvious way. By Proposition 3.1 we can lift U to a module for the Lie algebra VˆŽ0., and then to one for Pn s [p ) n VˆŽyp . [ VˆŽ0. by letting VˆŽyp . act trivially. Define ˆ
Mn Ž U . s IndVP nŽ U . s U Ž Vˆ . mUŽ P n . U.
Ž 4.1.
If we give U degree n, the Z-gradation of Vˆ lifts to MnŽU . which thus ˆ It is easy to see that MnŽU .Ž i . s becomes a Z-graded module for V. ˆ Ž . U V iynU. We define for ¨ g V, YM n ŽU . Ž ¨ , z . s
Ý ¨ Ž m . zym y1 .
Ž 4.2.
mgZ
As in wDLMx, YM ŽU .Ž ¨ , z . satisfies all conditions of a weak V-module except the associativity which does not hold on MnŽU . in general. We have to divide out by the desired relations. Let W be the subspace of MnŽU . spanned linearly by the coefficients of
Ž z0 q z2 .
wt aqn
Y Ž a, z 0 q z 2 . Y Ž b, z 2 . u y Ž z 2 q z 0 .
wt aqn
Y Ž Y Ž a, z 0 . b, z 2 . u
Ž 4.3.
81
VERTEX OPERATOR ALGEBRAS
for any homogeneous a g V, b g V, u g U. Set Mn Ž U . s Mn Ž U . rU Ž Vˆ . W.
Ž 4.4.
THEOREM 4.1. The space MnŽU . s Ý m G 0 MnŽU .Ž m. is an admissible V-module with MnŽU .Ž0. / 0, MnŽU .Ž n. s U and with the following uni¨ ersal property: for any weak V-module M and any A nŽ V .-morphism f : U ª V nŽ M ., there is a unique morphism f : MnŽU . ª M of weak V-modules which extends f . Proof. By Proposition 6.1 of wDLMx, we know that MnŽU . is a Z-graded weak V-module generated by U q UŽ Vˆ .W. By Proposition 2.4 of wDMx or Lemma 6.1.1 of wL2x, MnŽU . is spanned by
½ a ŽU q U Ž Vˆ . W . < a g V , n g Z 5 . n
Thus MnŽU .Ž m. s VˆŽ m y n.ŽU q UŽ Vˆ .W . for all m g Z. In particular, MnŽU .Ž m. s 0 if m - 0 and MnŽU .Ž n. s A nŽ V .ŽU q UŽ Vˆ .W . which is a quotient module of U. A proof that MnŽU .Ž0. / 0 and MnŽU .Ž n. s U will be given after Proposition 4.7. The universal property of MnŽU . follows from its construction. In the following we let U* s Hom C ŽU, C. and let Us be the subspace of MnŽU .Ž n. spanned by ‘‘length’’ s vectors op 1Ž a1 . ??? op sŽ a s . U, where p1 G ??? G ps , p1 q ??? ps s 0, pi / 0, ps G yn, and a i g V. Then by the PBW theorem MnŽU .Ž n. s Ý s G 0 Us with U0 s U and Us l Ut s 0 if s / t. Recall Remark 3.3. We extend U* to MnŽU .Ž n. inductively so that ² u9, op Ž a1 . ??? op Ž a s . u: s ² u9, op qp Ž wap 1, ,ap 2 . op Ž a3 . ??? op Ž a s . u . , 1 s 1 2 1 2 3 s
Ž 4.5. where oj Ž a. s aŽwt a y 1 y j . for homogeneous a g V. We further extend U* to MnŽU . by letting U* annihilate [i/ n M ŽU .Ž i .. Set J s ¨ g Mn Ž U . <² u9, x¨ : s 0 for all u9 g U*, all x g U Ž Vˆ . 4 . We can now state the second main result of this section. THEOREM 4.2. The space L nŽU . s MnŽU .rJ is an admissible V-module satisfying L nŽU .Ž0. / 0 and V nrV ny1Ž L nŽU .. ( U. Moreo¨ er L n defines a functor from the category of A nŽ V .-modules which cannot factor through A ny 1Ž V . to the category of admissible V-modules such that V nrV ny1 ( L n is naturally equi¨ alent to the identity.
82
DONG, LI, AND MASON
The main point in the proof of the theorem is to show that UŽ Vˆ .W ; J. The next three results are devoted to this goal. PROPOSITION 4.3. The following hold for all homogeneous a g V, b g V, u9 g U*, u g U, j g Zq, ² u9, Ž z 0 q z 2 . wt aqnqj YM s ² u9, Ž z 2 q z 0 .
n ŽU .
Ž a, z 0 q z 2 . YM n ŽU . Ž b, z 2 . u:
wt aqnqj
YM n ŽU . Ž Y Ž a, z 0 . b, z 2 . u: .
Ž 4.6.
In the following we simply write Y for YM n ŽU . , which should cause no confusion. The following is the key lemma. LEMMA 4.4.
For any i, j g Zq,
Res z 0 zy1qi Ž z0 q z2 . 0
wt aqnqj
² u9, Y Ž a, z 0 q z 2 . Y Ž b, z 2 . u:
s Res z 0 zy1qi Ž z2 q z0 . 0
wt aqnqj
² u9, Y Ž Y Ž a, z 0 . b, z 2 . u: .
Proof. Since j G 0 then aŽwt a q n q j . lies in [p ) n VˆŽyp . and hence annihilates u. Then for all i g Zq we get i
Res z 1Ž z1 y z 2 . z1wt aqnqj Y Ž b, z 2 . Y Ž a, z1 . u s 0.
Ž 4.7.
Note that Ž3.2. is equivalent to Y Ž a, z1 . , Y Ž b, z 2 . s Res z 0 zy1 2 d
ž
z1 y z 0 z2
/
Y Ž Y Ž u, z 0 . ¨ , z 2 . . Ž 4.8.
Using Ž4.7. and Ž4.8. we obtain Res z 0 z 0i Ž z 0 q z 2 .
wt aqnqj
Y Ž a, z 0 q z 2 . Y Ž b, z 2 . u
i
s Res z 1Ž z1 y z 2 . z1wt aqnqj Y Ž a, z1 . Y Ž b, z 2 . u i
s Res z 1Ž z1 y z 2 . z1wt aqnqj Y Ž a, z1 . Y Ž b, z 2 . u i
y Res z 1Ž z1 y z 2 . z1wt aqnqj Y Ž b, z 2 . Y Ž a, z1 . u i
s Res z 1Ž z1 y z 2 . z1wt aqnqj Y Ž a, z1 . , Y Ž b, z 2 . u i
s Res z 0 Res z 1Ž z1 y z 2 . z1wt aqnqj zy1 2 d s Res z 0 Res z 1 z 0i z1wt aqnqj zy1 1 d s Res z 0 z 0i Ž z 2 q z 0 .
wt aqnqj
ž
ž
z2 q z0 z1
z1 y z 0 z2
/
/
Y Ž Y Ž a, z 0 . b, z 2 . u
Y Ž Y Ž a, z 0 . b, z 2 . u
Y Ž Y Ž a, z 0 . b, z 2 . u.
Ž 4.9.
83
VERTEX OPERATOR ALGEBRAS
Thus Lemma 4.4 holds if i G 1, and we may now assume i s 0. Next us Ž4.9. to calculate that Res z 0 zy1 0 Ž z0 q z2 . s
`
² u9, Y Ž a, z 0 q z 2 . Y Ž b, z 2 . u:
j wt aqn ² u9, Y Ž a, z 0 qz 2 . Y Ž b, z 2 . u: Res z 0 z 0ky1 z 2jyk Ž z 0 qz 2 . k
ž/ Ýž / Ý
ks0
s
wt aqnqj
`
j wt aqn ² u9, Y Ž Y Ž a, z 0 . b, z 2 . u: Res z 0 z 0ky1 z 2jyk Ž z 2 q z 0 . k
ks1
j qRes z 0 zy1 0 z2 Ž z2 q z0 .
wt aqn
² u9, Y Ž a, z 0 q z 2 . Y Ž b, z 2 . u: .
Ž 4.10.
It reduces to show that Res z 0 zy1 0 Ž z2 q z0 .
wt aqn
² u9, Y Ž a, z 0 q z 2 . Y Ž b, z 2 . u:
s Res z 0 zy1 0 Ž z2 q z0 .
wt aqn
Ž 4.11.
² u9, Y Ž a, z 0 . u, z 2 . u: .
Ž 4.12.
Since ² u9, MnŽU .Ž m.: s 0 if m / n, we see that Res z 0 zy1 0 Ž z2 q z0 .
¦ ¦
s u9,
Ý kgZ q
s u9,
Ý kgZ q
¦ ž
wt aqn
ž ž
z 2wt byn ² u9, Y Ž Y Ž a, z 0 . b, z 2 . u:
wt a q n Ž aky 1 b . Ž wt Ž aky1 b . y 1 . u k
;
/ /
wt a q n o Ž a ky 1 b . u k
s u9, o Res z Y Ž a, z . b
;
Ž1 q z.
wt aqn
/;
u .
z
On the other hand, note that bŽwt b y 1 q p . u s 0 if p ) n. So Res z 0 zy1 0 Ž z0 q z2 .
¦
s u9,
wt aqn
z 2wt byn ² u9, Y Ž a, z 0 q z 2 . Y Ž b, z 2 . u:
Ý aŽ wt a y 2 y i q n . Ý igZ q
b Ž wt b y 1 y m . zynqiqm u 2
;
mGyn
Ž 4.13. n
¦
s u9,
;
Ý aŽ wt a y 1 y i . b Ž wt b y 1 q i . u is0
¦
q u9,
n
Ž 4.14.
;
Ý aŽ wt a y 1 q i . b Ž wt b y 1 y i . u is1
.
Ž 4.15.
84
DONG, LI, AND MASON
Note that the A nŽ V .-module structure on U is equivalent to o Ž a . o Ž b . u s a Ž wt a y 1 . b Ž wt b y 1 . u wt aqn
n
s
Ý
Ž y1.
ms0
m
ž
m q n o Res Y Ž a, z . b Ž 1 q z . z n z mqnq1
ž
/
/
u.
By Ž4.5. with s s 2, a1 s a, a2 s b, p1 s k s yp 2 Ž k ) 0. we see that ² u9, ok Ž a . oyk Ž b . u: s ² u9, a Ž wt a y 1 y k . b Ž wt b y 1 q k . u: wt aqn
nyk
¦
s u9,
Ý
Ž y1.
m
ms0
ž
m q n q k o Res Y Ž a, z . b Ž 1 q z . z m z mqnq1qk
/;
ž
/
u .
Ž 4.16. Thus n
¦
u9,
;
Ý aŽ wt a y 1 y k . b Ž wt b y 1 q k . u ks0 n
¦
s u9,
wt aqn
nyk
Ý Ý Ž y1.
m
ks0 ms0
ž
m q n q k o Res Y Ž a, z . b Ž 1 q z . z m z mqnq1qk
ž
/
/;
u .
Use Lie algebra bracket Ž3.2. to get a Ž wt a y 1 q k . b Ž wt b y 1 y k . s b Ž wt b y 1 y k . a Ž wt a y 1 q k . qÝ iG0
ž
wt a y 1 q k Ž ai b . Ž wt a q wt b y 2 y i . . i
/
By Ž4.16., ² u9, b Ž wt b y 1 y k . a Ž wt a y 1 q k . u: s
¦
wt bqn
nyk
Ý Ž y1.
ms0
m
ž
m q n q k o Res Y Ž b, z . a Ž 1 q z . z m z mqnq1qk
/
ž
/;
u .
A proof similar to that of Lemma 2.1Žii. shows that wt bqn
nyk
Ý Ž y1. ms0
Ý ms0
ž
m q n q k Res Y Ž b, z . a Ž 1 q z . z m z mqnq1qk
/
wt aqmqky1
nyk
y
m
ž
m q n q k Ž y1 . nq k Res Y Ž a, z . b Ž 1 q z . z m z 1qmqnqk
/
g On Ž V . .
85
VERTEX OPERATOR ALGEBRAS
We now have
¦
u9,
n
;
Ý aŽ wt a y 1 q k . b Ž wt b y 1 y k . u ks1 n
s
nyk
ž m qmn q k / Ž y1.
Ý Ý ks1 ms0
¦ ž
= u9, o Res z Y Ž a, z . b n
q
Ý Ý ks1 iG0
n
s
nyk
Ž1 q z.
ks1 ms0
wt aqmqky1
z 1qmqnqk
/; u
wt a y 1 q k ² u9, Ž a i b . Ž wt a q wt b y 2 y i . u: i
ž
/
ž m qmn q k / Ž y1.
Ý Ý
nq k
¦ ž
= u9, o Res z Y Ž a, z . b
nq k
Ž1 q z.
wt aqmqky1
z 1qmqnqk
/; u
n
q
Ý ² u9, o Ž Res z Y Ž a, z . b Ž 1 q z . wt ay1qk . u: . ks1
So it is enough to show the identity n
wt aqn
nyk
Ž y1.
Ý Ý
m
ks0 ms0 n
q
ž
m q n q k Ž y1 . nq k Ž 1 q z . m z 1q mqnqk
Ý Ý
/
wt aqmqky1
nyk
ks1 ms0 n
q
ž
m q n q k Ž1 q z. m z mq nq1qk
/
Ý Ž 1 q z . wt ay1qk
ks1
Ž1 q z.
s
wt aqn
,
z
or equivalently, n
n
nyk
Ý Ý Ž y1.
m
ks0 ms0 n
q
ž
m q n q k Ž1 q z. m z mq nqk
ž
m q n q k Ž y1 . nq k Ž 1 q z . m z mq nqk
mq ky1
nyk
Ý Ý ks1 ms0
/
/
s 1. This identity is proved in Proposition 5.1 in the Appendix.
86
DONG, LI, AND MASON
Proposition 4.3 is a consequence of the next lemma. LEMMA 4.5.
For all m g Z we ha¨ e
Res z 0 z 0m Ž z 0 q z 2 .
wt aqmqj
s Res z 0 z 0m Ž z 2 q z 0 .
² u9, Y Ž a, z 0 q z 2 . Y Ž b, z 2 . u: wt aqmqj
² u9, Y Ž Y Ž a, z 0 . b, z 2 . u: .
Proof. This is true for m G y1 by Lemma 4.4. Let us write m s yk q i with i g Zq and proceed by induction k. Induction yields Res z 0 zyk 0 Ž z0 q z2 .
wt aqmqj
² u9, Y Ž L Ž y1 . a, z 0 q z 2 . Y Ž b, z 2 . u:
s Res z 0 zyk 0 Ž z2 q z0 .
wt aqmqj
² u9, Y Ž Y Ž L Ž y1 . a, z 0 . b, z 2 . u: .
Using the residue property Res z f 9Ž z . g Ž z . q Res z f Ž z . g 9Ž z . s 0 and the LŽy1.-derivation property Y Ž LŽy1. a, z . s dzd Y Ž a, z . we have Res z 0 zyk 0 Ž z0 q z2 . s yRes z 0
ž
wt aq1qmqj
z0
² u9, Y Ž L Ž y1 . a, z 0 q z 2 . Y Ž b, z 2 . u:
zyk 0 Ž z0 q z2 .
s Res z 0 kzyky1 Ž z0 q z2 . 0
wt aq1qmqj
wt aq1qmqj
/
² u9, Y Ž a, z 0 q z 2 . Y Ž b, z 2 . u:
² u9, Y Ž a, z 0 q z 2 . Y Ž b, z 2 . u:
y Res z 0Ž wt a q 1 q m q j . zyk 0 Ž z0 q z2 .
wt aqmqj
= ² u9, Y Ž a, z 0 q z 2 . Y Ž b, z 2 . u: s Res z 0 kzyky1 z2 Ž z0 q z2 . 0 q Res z 0 kzyk 0 Ž z0 q z2 .
wt aqmqj
wt aqmqj
² u9, Y Ž a, z 0 q z 2 . Y Ž b, z 2 . u:
² u9, Y Ž a, z 0 q z 2 . Y Ž b, z 2 . u:
y Res z 0Ž wt a q 1 q m q j . zyk 0 Ž z2 q z0 .
wt aqmqj
= ² u9, Y Ž Y Ž a, z 0 . b, z 2 . u: s Res z 0 kzyky1 z2 Ž z0 q z2 . 0 q Res z 0 kzyk 0 Ž z2 q z0 .
wt aqmqj
wt aqmqj
² u9, Y Ž a, z 0 q z 2 . Y Ž b, z 2 . u:
² u9, Y Ž Y Ž a, z 0 . b, z 2 . u:
y Res z 0Ž wt a q 1 q m q j . zyk 0 Ž z2 q z0 . = ² u9, Y Ž Y Ž a, z 0 . b, z 2 . u: ,
wt aqmqj
87
VERTEX OPERATOR ALGEBRAS
and Res z 0 zyk 0 Ž z2 q z0 . s yRes z 0
ž
wt aq1qmqj
z0
² u9, Y Ž Y Ž L Ž y1 . a, z 0 . b, z 2 . u:
zyk 0 Ž z2 q z0 .
s Res z 0 kzyky1 Ž z2 q z0 . 0
wt aq1qmqj
wt aq1qmqj
/
² u9, Y Ž Y Ž a, z 0 . b, z 2 . u:
² u9, Y Ž Y Ž a, z 0 . b, z 2 . : u:
y Res z 0Ž wt a q 1 q m q j . zyk 0 Ž z2 q z0 .
wt aqmqj
= ² u9, Y Ž Y Ž a, z 0 . b, z 2 . u: s Res z 0 kz 2 zyky1 Ž z2 q z0 . 0 q Res z 0 kzyk 0 Ž z2 q z0 .
wt aqmqj
wt aqmqj
² u9, Y Ž Y Ž a, z 0 . b, z 2 . u:
² u9, Y Ž Y Ž a, z 0 . b, z 2 . u:
y Res z 0Ž wt a q 1 q m q j . zyk 0 Ž z2 q z0 .
wt aqmqj
= ² u9, Y Ž Y Ž a, z 0 . b, z 2 . u: . This yields the identity Res z 0 zyky1 Ž z0 q z2 . 0
wt aqmqj
s Res z 0 zyky1 Ž z2 q z0 . 0
² u9, Y Ž a, z 0 q z 2 . Y Ž b, z 2 . u: wt aqmqj
² u9, Y Ž Y Ž a, z 0 . b, z 2 . u: ,
and the lemma is proved.
ˆ Let us now introduce an arbitrary Z-graded V-module M s [m g Z M Ž m.. As before we extend M Ž n.* to M by letting it annihilate M Ž m. for m / n. The proof of Proposition of 6.1 of wDLMx with ² u9, ? : suitably inserted gives: PROPOSITION 4.6. M Ž n.9 such that Ži. Žii.
Let U be a subspace of M Ž n. and U9 a subspace of
M s UŽ Vˆ .U. For a g V and u g U there is k g Z such that ² u9, Ž z 0 q z 2 . kq n Y Ž a, z 0 q z 2 . Y Ž b, z 2 . u: s ² u9, Ž z 2 q z 0 .
kq n
Y Ž Y Ž a, z 0 . b, z 2 . u:
for any b g V, u9 g U9. Then in fact Ž4.17. holds for any u g M.
Ž 4.17.
88
DONG, LI, AND MASON
PROPOSITION 4.7. Let M be as in Proposition 4.6. Then for any x g UŽ Vˆ ., a g V, u g M, there is an integer k such that ² u9, Ž z 0 q z 2 . kq n x ? Y Ž a, z 0 q z 2 . Y Ž b, z 2 . u: s ² u9, Ž z 2 q z 0 .
kq n
x ? Y Ž Y Ž a, z 0 . b, z 2 . u:
Ž 4.18.
for any b g V, u9 g U9. Proof. Let L be the subspace of UŽ Vˆ . consisting of those x for which Ž4.18. holds. Let x g L, let c be any homogeneous element of V, and let m g Z. Then from Ž4.8. we have ² u9, xc Ž m . Y Ž a, z 0 q z 2 . Y Ž b, z 2 . u: Ž z 0 q z 2 . kq n s
`
Ý is0
q
m i
z0 q z2 .
ž /Ž Ýž / `
is0
kq nqmyi
² u9, xY Ž c Ž i . a, z 0 q z 2 . Y Ž b, z 2 . u:
kq n m my i ² u9, xY Ž a, z 0 q z 2 . Y Ž c Ž i . b, z 2 . u: z2 Ž z0 q z2 . i
qŽ z0 q z2 .
kq n
² u9, xY Ž a, z 0 q z 2 . Y Ž b, z 2 . c Ž m . u: .
Ž 4.19.
The same method that was used in the proof of Proposition 4.6 shows that xcŽ m. g L. Since UŽ Vˆ . is generated by all such cŽ n.’s, and since Ž4.18. holds for x s 1 by Proposition 4.6, we conclude that L s UŽ Vˆ ., as desired. We can now finish the proof of Theorems 4.1 and Theorem 4.2. We can take M s MnŽU . in Proposition 4.7, as we may since MnŽU . certainly satisfies the conditions placed on M prior to Proposition 4.6 and in Proposition 4.6. Then from the definition of W Ž4.3. and Propositions 4.3, 4.6, and 4.7 we conclude that UŽ Vˆ .W ; J. It is clear that LŽU . is a quotient of MnŽU . and hence an admissible V-module. Note that J l U s 0. So LŽU .Ž n. contains U as an A nŽ V .-submodule. This shows that MnŽU .Ž n. ( U as A nŽ V .-modules. If MnŽU .Ž0. s 0 then U will be an A ny 1Ž V .-module, contradicting the assumption on U. This finishes the proof of Theorem 4.1. Theorem 4.2 is now obvious. At this point we have a pair of functors V n , L n defined on appropriate module categories. It is clear that V nrV ny1 ( L n is equivalent to the identity. LEMMA 4.8. Suppose that U is a simple A nŽ V .-module. Then L nŽU . is a simple admissible V-module. Proof. If 0 / W ; L nŽU . is an admissible submodule then, by the definition of L nŽU ., we have W Ž n. s W l L nŽU .Ž n. / 0. As W Ž n. is an
VERTEX OPERATOR ALGEBRAS
89
A nŽ V .-submodule of U s L nŽU .Ž n. by Theorem 3.2 then U s W Ž n., whence W > UŽ Vˆ .W Ž n. s UŽ Vˆ .U s L nŽU .. THEOREM 4.9. L n and V nrOny1 are equi¨ alences when restricted to the full subcategories of completely reducible A nŽ V .-modules whose irreducible components cannot factor through A ny 1Ž V . and completely reducible admissible V-modules, respecti¨ ely. In particular, L n and V nrV ny1 induce mutually in¨ erse bijections on the isomorphism classes of simple objects in the category of A nŽ V .-modules which cannot factor through A ny1Ž V . and admissible V-modules, respecti¨ ely. Proof. We have V nrOny1Ž LŽU .. ( U for any A nŽ V .-module by Theorem 4.2. If M is a completely reducible admissible V-module we must show L nŽ V nrV ny1Ž M .. ( M. For this we may take M simple, whence V nrV ny1Ž M . is simple by Proposition 3.4Žii. and then L nŽ V nrV ny1Ž M .. is simple by Lemma 4.8. Since both M and L nŽ V nrV ny1Ž M .. are simple quotients of the universal object MnŽ V nrV ny1Ž M .. then they are isomorphic by Theorems 4.1 and 4.2. The following theorem is a generalization of Theorem 8.1 of wDLMx. THEOREM 4.10. Suppose that V is a rational ¨ ertex operator algebra. Then the following hold: Ža. A nŽ V . is a finite-dimensional, semisimple associati¨ e algebra. Žb. The functors L n , V nrOny1 are mutually in¨ erse categorical equi¨ alences between the category of A nŽ V .-modules whose irreducible components cannot factor through A ny 1Ž V . and the category of admissible V-modules. Žc. The functors L n , V nrV ny1 induce mutually in¨ erse categorical equi¨ alences between the category of finite-dimensional A nŽ V .-modules whose irreducible components cannot factor through A ny 1Ž V . and the category of ordinary V-modules. Proof. Part Žb. follows from Theorem 4.9 and Ža.. Since V is rational any irreducible admissible V-module is an ordinary module by Theorem 8.1 of wDLMx. Now Žc. follows from Žb.. It remains to prove Ži.. Let W be an A nŽ V .-module. Set U s W [ V Ž n.. Then U is an A nŽ V .module which cannot factor through A ny 1Ž V .. Now L nŽU . is admissible and hence a direct sum of irreducible ordinary V-modules. Thus V nŽ L nŽU ..rV ny1Ž L nŽU .. , U is a direct sum of finite-dimensional irreducible A nŽ V .-modules and so is W. It is believed that if AŽ V . s A 0 Ž V . is semisimple then V is rational. We cannot solve this problem completely in this paper. But we have some partial results which are applications of A nŽ V .-theory.
90 THEOREM 4.11. then V is rational.
DONG, LI, AND MASON
If all A nŽ V . are finite-dimensional semisimple algebras
Proof. Since AŽ V . is semisimple V has only finitely many irreducible admissible modules which are necessarily ordinary V-modules. For any l g C let Ml be the set of irreducible admissible modules whose weights are congruent to l module Z. Then for each W g Ml we have W s [ng Z q Wlqn W qn s [ng Z q W Ž n. where nW g Z and Wlqn W qn s W Ž n.. Since LŽy1.: W Ž n. ª W Ž n q 1. is injective if n is large Žsee wL1x. there exists an ml g N such that the weight space Wlqm / 0 for any W g Wl and m G ml. Consider any admissible module M whose weights are in l q Z and whose homogeneous subspace Mlqm with some m G ml is 0. Let U be an irreducible AŽ V .-submodule of M Ž0.. Then L0 ŽU . s LŽU . is an irreducible V-module such that LŽU .Ž0. s U and LŽU .lqm s 0. Thus LŽU . s 0 and U s 0. This implies that M s 0. Now take an admissible module M s [k g Z q M Ž k .. Then M Ž0. is a direct sum of simple AŽ V .-modules as AŽ V . is semisimple. Let U be an AŽ V .-submodule of M Ž0. isomorphic to W Ž0. s Wlqn W for some W g Ml . We assert that the submodule N of M generated by U is irreducible and necessarily isomorphic to W. First note that N has an irreducible quotient isomorphic to W. Take n g N such that n q nW G ml . Observe that MnŽW Ž n..rJ s L nŽW Ž n.. is isomorphic to W where J is a maximal submodule of MnŽW Ž n.. such that J l W Ž n. s 0. Since Jlqn W qn s 0 we see that J s 0 and MnŽW Ž n.. s L nŽW Ž n.. , W. Write N Ž n. as a direct sum of W Ž n. and another A nŽ V .-submodule N Ž n.9 of N Ž n. as A nŽ V . is semisimple. Clearly the submodule of N Ž n. generated by W Ž n. is isomorphic to W. This shows that N must be isomorphic to W, as claimed. It is obvious now that the submodule UŽ Vˆ . M Ž0. generated by M Ž0. is completely reducible. Using the semisimplicity of A1Ž V . we can decompose M Ž1. into a direct sum of A1Ž V .-modules ŽUŽ Vˆ . M Ž0..Ž1. [ M Ž1.9. The same argument shows that UŽ Vˆ . M Ž1.9 is a completely reducible submodule of M. Continuing in this way proves that M is completely reducible. Remark 4.12. From the proof of Theorem 4.11, we see, in fact, that we can weaken the assumption in Theorem 4.11. Namely we only need to assume that A nŽ V . is semisimple if n is large. 5. APPENDIX In this appendix we prove several combinatorial identities which are used in the previous sections.
91
VERTEX OPERATOR ALGEBRAS
For n G 0 define n
AnŽ z . s
n
nyk
Ý Ý Ž y1.
m
ž
ks0 ms0 n
q
Ý Ý
/
mq ky1
nyk
ks1 ms0
m q n q k Ž1 q z. m z mqnqk
ž
m q n q k Ž y1 . nq k Ž 1 q z . m z mq nqk
/
Using the well-known identity i
Ý Ž y1. k ks0
i ny1 n s Ž y1 . k i
ž /
ž
/
we can rewrite A nŽ z . as n
AnŽ z . s
k
Ý Ý Ž y1.
m
ks0 ms0 n
q
Ý Ý
n
s
Ý
Ž y1.
k
ks0
y Ž y1 .
ž
n q k y 1 Ž1 q z. k z nq k
/
Ý ks1
PROPOSITION 5.1.
/
ž
n
/
n
n
n
n q k Ž y1 . nq kqm Ž 1 q z . m z nq k
ky1
ks1 ms0
ž
n q k Ž1 q z. m z nqk
ž
n q k y 1 Ž1 q z. ky1 z nq k
ky 1
/
.
A nŽ z . s 1 for all n G 0.
Proof. Set n
Bn Ž z . s
Ý Ž y1. ks0 n
Cn Ž z . s
Ý ks1
ž
k
ž
n q k y 1 Ž1 q z. k z nqk
/
n q k y 1 Ž1 q z. ky1 z nqk
/
ky 1
.
n
ky 1
.
92
DONG, LI, AND MASON
Then ny1
Bn Ž z . s
k
Ž y1.
Ý ks0
q Ž y1 . s
1qz z
n
/ ž
2 n y 1 Ž1 q z. n z2n
ž
Ý Ž y1.
kq 1
ks0
1qz z
//
z nqk
/
Bny 1 Ž z . q n
n
Ž1 q z.
n
ny2
q Ž y1 . s
žž
nqky2 nqky2 q k ky1
ž
2 n y 1 Ž1 q z. n z2n
ž
n q k y 1 Ž1 q z. k z nqkq1
n
/
n
/
Bny 1 Ž z . y
1 z
Bn Ž z . q Ž y1 .
ny 1
ž
2 n y 2 Ž1 q z. ny1 z2n
/
nq 1
q Ž y1 .
n
ž
2 n y 1 Ž1 q z. n z 2 nq1
/
.
Solving BnŽ z . gives ny 1
Bn Ž z . s Bny1 Ž z . q Ž y1 . q Ž y1 .
n
Ž1 q z. z
n
Ž1 q z. z2n
ny 1
2 n y 1. ny1
žŽ
2 ny1
/
ž 2 n ny 1 / .
Similarly, ny1
Cn Ž z . s
Ý ks1
žž
nqky2 nqky2 q ky1 ky2
/ ž
2 n y 1 Ž1 q z. q ny1 z2n
ž
s
1 z
ny2
s
1 z
Ý ks0
ž
ž
n q k y 1 Ž1 q z. ky1 z nqkq1
k
/
ny 1
/
1qz z
2 n y 2 Ž1 q z. Cn Ž z . q ny1 z2n
ž
n
ž
z nqk
ny 1
2 n y 1 Ž1 q z. ny1 z2n
Cny 1 Ž z . q
y
ky 1
/
Cny 1 Ž z . q
q
//
Ž1 q z.
2 n y 1 Ž1 q z. . ny1 z 2 nq1
/
/
ny 1
n
93
VERTEX OPERATOR ALGEBRAS
Thus ny 1
Ž y1.
nq1
n
Cn Ž z . s Ž y1 . Cny1 Ž z . q Ž y1 .
n
ž
2 n y 2 Ž1 q z. ny1 z 2 ny1
/
n
q Ž y1 .
ny 1
ž
2 n y 1 Ž1 q z. . ny1 z2n
/
Thus A n Ž z . s Bn Ž z . q Ž y1 .
ny 1
Cn Ž z . s A ny1 .
Note that A 0 Ž z . s 1 and the proposition follows. For n G 0 we define m
n
Fn Ž z . s
Ý ms0
PROPOSITION 5.2.
ž
n
nq1
y Ž y1 . Ž 1 q z . m q n Ž y1 . Ž 1 q z . n z nqmq1
/
FnŽ z . s 1 for all n.
Proof. Set nq 1
ž
m q n Ž y1 . m Ž 1 q z . n z nqmq1
ž
m q n Ž1 q z. . n z nqmq1
n
Dn Ž z . s
Ý ms0
m
n
En Ž z . s
Ý ms0
/ /
Then nq 1
Dn Ž z . s Bnq1 Ž z . q Ž y1 . s Bn Ž z . q Ž y1 . q Ž y1 .
nq 1
q Ž y1 .
n
n
n
ž
2 n q 1 Ž1 q z. n z 2 nq2
/
Ž1 q z.
n
ž 2nn /
z 2 nq1
Ž1 q z.
nq 1
z 2 nq2
ž
2n q 1 nq1 nq 1
ž
2 n q 1 Ž1 q z. n z 2 nq2
s Bn Ž z . q Ž y1 .
/
n
2 n Ž1 q z. n z 2 nq1
n
ž /
and n
Ž1 q z. En Ž z . s Cn Ž z . q 2 n . n z 2 nq1
ž /
/
m
.
94
DONG, LI, AND MASON
Thus Fn Ž z . s Dn Ž z . q Ž y1 .
nq 1
En Ž z . s A n Ž z . s 1,
as required. For n G 0 define n
a n Ž w, z . s
m Ý Ž y1.
ms0
ž m qn n /
nym
=
ž
Ý Ý is0 jG0
ž
ym y n y 1 i
w iqj 1 m i y1 . iqm y m . Ž j z z
/
/ž /
. s Žy1. k Ž p q kk y 1 .. We can rewrite Note that if p ) 0, k ) 0 then Ž yp k a nŽ w, z . as n
a n Ž w, z . s
Ý
Ž y1 .
m
ms0
ž
mqn n
nym
/ž Ý
Ý
is0 jG0
ž
mqnqi i
iqj 1 m w y m . iqm j z z
/
/ž /
PROPOSITION 5.3. The a nŽ w, z . s 0 for all n G 0. Proof. Regarding a nŽ w, z . as a polynomial in zy1 , the coefficient of z in a nŽ w, z . Ž0 F p F n. is equal to Žsetting m q i s p . yp
p
m Ý Ž y1.
ms0
ž m qn n / Ý ž pn yq mp / ž mj / w
py mqj
y Ž y1 .
jG0
p
swp
m Ý Ž y1.
ms0
ž m qn n / ž pn yq mp / Ž 1 q 1rw .
m
p
ž
pqn n
y Ž y1 .
p
ž
/ pqn . n
So the coefficient of zyp w 0 in a nŽ w, z . equals 0. If 0 F q F p y 1, the coefficient of zyp w pyq in a nŽ w, z . is equal to p
c n Ž p, q . s
m Ý Ž y1.
ms0
ž m qn n / ž nnqqmp / ž mq /
/
95
VERTEX OPERATOR ALGEBRAS
which is defined for any n, p, q G 0. So we must prove that a nŽ p, q . s 0 1. Ž . for 1 F q q 1 F p F n. Recall Ž kl . s Ž l yk 1 . q Ž kl y y 1 . Then c n p, q is equal to p
m Ý Ž y1.
ms0
ž m qn n / ž ž n qn qp ym 1 / q ž nnqqmp yy11 / / ž mq /
s Ž y1 .
p
ž
pqn n
p q q
/ž /
py1
m Ý Ž y1.
ms0
ž m qn n / ž nnqqmp yy11 / ž mq /
qc n Ž p y 1, q . py1
s
m Ý Ž y1.
ms0
q Ž y1 .
p
ž
žž
mqny1 q mqny1 n ny1
pqn n
/ ž
p q c n Ž p y 1, q . q
/ž /
s c ny 1 Ž p, q . q c n Ž p y 1, q . y Ž y1 . q Ž y1 .
p
ž
pqn n
py1
y
Ý Ž y1. my 1 ms0
=
žž
pqny1 ny1
my1 my1 q q qy1
/ ž
my1s0
// p
ž
pqny1 n
/ ž
//
s c ny 1 Ž p, q . y c n Ž p y 1, q y 1 . q Ž y1 . py 1
ž
p q
/ž /
ž m qnn y 1 / ž nnqqmp yy11 /
my1 my1 q q qy1
q Ž y1 .
p q
/ž /
ž m qnn y 1 / ž nnqqmp yy11 /
Ý Ž y1. my 1
žž
ž
p q
py2
=
p
/ž /
s c ny 1 Ž p, q . q c n Ž p y 1, q . q Ž y1 . y
/ / ž nnqqmp yy11 / ž mq /
py1qn n
/žž
p
ž
pqny1 n
py1 py1 q q qy1
s c ny 1 Ž p, q . y c n Ž p y 1, q y 1 . .
/ ž
//
p q
/ž /
96
DONG, LI, AND MASON
That is, c n Ž p, q . s c ny1 Ž p, q . y c n Ž p y 1, q y 1 . . so by induction it is enough to show that c 0 Ž p, q . s 0 and c nŽ p, 0. s 0 if p ) q. But this is clear from the definition.
REFERENCES wBx
R. E. Borcherds, Vertex algebras, Kac]Moody algebras, and the Monster, Proc. Nat. Acad. Sci. U.S. A. 83 Ž1986., 3068]3071. wDLMx C. Dong, H. Li, and G. Mason, Twisted representations of vertex operator algebras, Math. Ann. 310 Ž1998., 571]600. wDLMMx C. Dong, H. Li, G. Mason, and P. Montague, The radical of a vertex operator algebra, in ‘‘Proc. of the Conference on the Monster and Lie algebras at The Ohio State University, May 1996’’ ŽJ. Ferrar and K. Harada, Eds.., de Gruyter, BerlinrNew York. wDLx C. Dong and Z. Lin, Induced modules for vertex operator algebras, Comm. Math. Phys. 179 Ž1996., 157]184. wDMx C. Dong and G. Mason, On quantum Galois theory, Duke Math. J. 86 Ž1997., 305]321. wFHLx I. Frenkel, Y. Huang, and J. Lepowsky, On axiomatic approaches to vertex operator algebras and modules, Mem. Amer. Math. Soc. 104 Ž1993.. wFLMx I. Frenkel, J. Lepowsky, and A. Meurman, ‘‘Vertex Operator Algebras and the Monster,’’ Pure and Appl. Math., Vol. 134, Academic Press, Boston, 1988. wL1x H. Li, Symmetric invariant bilinear forms on vertex operator algebras, J. Pure Appl. Algebra 96 Ž1994., 279]297. wL2x H. Li, ‘‘Representation Theory and Tensor Product Theory for Vertex Operator Algebras,’’ Ph.D. thesis, Rutgers University, 1994. wZx Y. Zhu, Modular invariance of characters of vertex operator algebras, J. Amer. Math. Soc. 9 Ž1996., 237]302.