## An analogue of the Hom functor and a generalized nuclear democracy theorem

An analogue of the Hom functor and a generalized nuclear democracy theorem

Haisheng Li Department of Mathematics, Rutgers University-Camden, Camden, NJ 08102 Abstract. We give an analogue of the Hom functor and prove a generalized form of

arXiv:q-alg/9706012v1 12 Jun 1997

the nuclear democracy theorem of Tsuchiya and Kanie by using a notion of tensor product for two modules for a vertex operator algebra.

1

Introduction

The notion of vertex operator algebra ([B], [FHL], [FLM]) is the algebraic counterpart of the notion of what is now usually called “chiral algebra” in conformal ?eld theory, and vertex operator algebra theory generalizes the theories of a?ne Lie algebras, the Virasoro algebra and representations (cf. [B], [DL], [FLM], [FZ]). It has been well known (cf. [FZ], [L1]) that the irreducible highest weight modules (usually called the vacuum ? of level ? and L(c, 0) for the Virasoro representations) L(?, 0) for an a?ne Lie algebra g algebra with central charge c have natural vertex operator algebra structures. If ? is a positive integer, it was proved ([DL], [FZ], [L1]) that the category of L(?, 0)-modules is a semi-simple category whose irreducible objects are irreducible highest weight integrable ? -modules of level ? (cf. [K]). If c = 1 ? g

6(p?q )2 , pq

where p, q ∈ {2, 3, · · ·} are relatively

prime, it was proved ([DMZ], [W]) that the category of L(c, 0)-modules is also a semisimple category whose irreducible objects are exactly those irreducible Virasoro algebra modules L(c, h) listed in [BPZ]. These give the rationality (de?ned in Section 2) of L(?, 0) and L(c, 0). To state our results, let us start with de?nitions of intertwining operator. In the minimal models, an intertwining operator from L(c, h2 ) to L(c, h3 ) was de?ned in [BPZ] 1

to be a primary ?eld operator Φ(x) of weight h, i.e., Φ(x) ∈ HomC (L(c, h2 ), L(c, h3 )){x} satisfying the following relation: [L(m), Φ(x)] = xm (m + 1)h + x d Φ(x) dx (1.1)

j3 was jj2 de?ned (cf. [TK]) as a linear map Φ(u, x) ∈ Hom(L(?, j2 ), L(?, j3 )){x} such that (1.1) for m ∈ Z. For WZW models with g = sl2 , an intertwining operator of type holds with h =

j (j +2) 4(?+2)

and (1.2)

[a(m), Φ(u, x)] = xm Φ(au, x) for m ∈ Z, a ∈ g, u ∈ L(j ),

where L(j ) is the irreducible sl2 -module with highest weight j . By employing singular vectors, Tsuchiya and Kanie proved in [TK] that such an intertwining operator Φ(·, x) on L(j ) can be uniquely and naturally extended to an intertwining operator on L(?, j ). This is the so-called nuclear democracy theorem of Tsuchiya and Kanie. On the other hand, in the context of vertex operator algebra, an intertwining operator W3 of type , where Wi (i = 1, 2, 3) are modules for a vertex operator algebra V , is W1 W2 de?ned in [FHL] to be a linear map I (·, x) from W1 to (Hom(W2 , W3 )) {x} satisfying the L(?1)-bracket formula (1.1) with m = ?1 and the Jacobi identity (2.1) (together with the truncation condition (I1) in Section 2). An intertwining operator in the sense of [FHL] restricted to W1 (0) gives an intertwining operator on W1 (0) in the sense of [TK] and [BPZ] for the WZW and minimal models. For WZW models, Tsuchiya and Kanie’s nuclear democracy theorem implies that the two de?nitions de?ne the same fusion rules. The question is: do we have a generalized form of the nuclear democracy theorem for an arbitrary vertex operator algebra? If V is not rational, the answer is negative. (See the appendix for a counterexample.) As the main result of this paper we prove a generalized form of the nuclear democracy theorem for a rational vertex operator algebra so that for all rational models, the fusion rules de?ned 2

in the context of vertex operator algebra coincide with those de?ned in the context of conformal ?eld theory. ? available so that one can make For WZW models, one has an a?ne Lie algebra g use of the notion of Verma module and singular vectors. To any vertex operator algebra V , we associate a Z-graded Lie algebra g (V ) = ⊕n∈Z g (V )n with generators tn ? a (linearly in a) for a ∈ V, n ∈ Z and with Borcherds’ commutator formula (2.4) and the L(?1)-bracket formula as its de?ning relations (see also [B],[FFR]). Since L(0) is a central element in g (V )0 , using the triangular decomposition with respect to the Z-grading we have the notions of generalized Verma g (V )-module [Le] (or Weyl module) and lowest weight module. Then any V -module M is a natural g (V )-module such that any weight space M(h) is a natural g (V )0 -module where twta?1 ? a is represented by awta?1 for a ∈ V . But a lowest weight, or even an irreducible lowest weight g (V )-module is not necessarily a weak V -module. To formulate a nuclear democracy theorem for arbitrary rational vertex operator algebra, we notice that (1.2) is a special case of the general commutator formula (2.4). Since (1.2) does not hold if a is not a weight-one element, we have to use a certain cross product [FLM]. Here is our generalized form of the nuclear democracy theorem or brie?y GNDT: Let V be a rational vertex operator algebra and Wi (i = 1, 2, 3) be three irreducible V modules with lowest weights hi , respectively. Let Wi (0) be the lowest weight subspace of Wi (with weight hi ). Let Φ(·, x) be a linear map from W1 (0) to HomC (W2 , W3 ){x} satisfying the L(?1)-bracket formula and (x1 ? x2 )n?1 Y (a, x1 )Φ(u, x2 ) ? (?x2 + x1 )n?1 Φ(u, x2 )Y (a, x1 )

1 = x? 2 δ

x1 Φ(an?1 u, x2 ) x2

(1.3)

for any a ∈ V(n) , u ∈ W1 (0). Then there exists a unique intertwining operator I (·, x) from W1 ? W2 to W3 in the sense of [FHL], which extends Φ(·, x). To prove this GNDT, we notice that if it is true, then I (·, x) will be an injective map 3

on W1 so that W1 (0) can be identi?ed as the space Φ(W1 (0), x) consisting of Φ(u, x) for u ∈ U because W1 is an irreducible V -module. For any u ∈ W1 , a ∈ V , I (u, x) satis?es the L(?1)-bracket formula, but (1.3) is not true for an arbitrary u ∈ W1 . However, the local property holds, i.e., for any a ∈ V, u ∈ W1 , there is a positive integer k such that (x1 ? x2 )k Y (a, x1 )Φ(u, x2 ) = (x1 ? x2 )k Φ(u, x2 )k Y (a, x1 ) (cf. [DL, formula (9.37)]). A ?eld operator Φ(x) from W2 to W3 satisfying the L(?1)bracket formula and the local property is called a generalized intertwining operators. Collecting all generalized intertwining operators Φ(x) from W2 to W3 we get a vector space G(W2 , W3 ). Then we prove (Theorem 4.6) that G(W2 , W3 ) becomes a V -module under a natural action that comes from the Jacobi identity. Then GNDT follows. We also prove that G(W2 , W3 ) satis?es the universal property: For any V -module W and any intertwining operator I (·, x) from W ? W2 to W3 , there exists a unique V -homomorphism ψ from W to G(W2 , W3 ) such that I (u, x) = ψ (u)(x) for u ∈ W . It follows from the universal property that there is a natural linear isomorphism from HomV (W, G(W2 , W3 )) W3 onto I , the space of intertwining operators of the indicated type. W W2 For WZW models, there is another notion of intertwining operator involving homomorphisms from the tensor product module of a loop module with a highest weight module to another highest weight module for an a?ne Lie algebra ? g. By using the generalized form of the nuclear democracy theorem we prove (Proposition 4.15) that this notion is essentially equivalent to the notion in [FHL]. The notion of G(W2 , W3 ) is clearly analogous to the notion of “Hom”-functor. In Lie algebra theory, if Ui (i = 1, 2, 3) are modules for a Lie algebra g, the space HomC (U1 , U2 ) is a natural g-module and we have the following natural inclusion relations: (U1 )? ? U2 ?→ HomC (U1 , U2 ) ?→ (U1 ? (U2 )? )? . (1.4)

If both U1 and U2 are ?nite-dimensional, the arrows are isomorphisms so that the space 4

of linear homomorphisms gives a construction of tensor product modules. In vertex operator algebra theory, a tensor product theory has been recently developed [HL0-4]. (In the a?ne Lie algebra level, a theory of tensor product was developed in [KL0-2] for modules of certain levels for an a?ne Lie algebra and part of this theory was extended to positive integral levels in [F].) In [HL0-4], in addition to the notion of intertwining operator, a notion called intertwining map was also used. An intertwining map was proved to be essentially equivalent to an intertwining operator and could be viewed as an operator-valued functional instead of a formal series of operators. As one of our results in this paper we give a de?nition and a construction of tensor product in terms of formal variable language. Motivated by the classical tensor product theory, we formulate a de?nition of tensor product of an ordered pair of two V -modules in terms of intertwining operators and a certain universal property. As an analogue of the construction of the classical tensor product we give a construction of tensor product for a rational vertex operator algebra V . Roughly speaking, our tensor product module T (W1 , W2 ) is constructed as the quotient space of the tensor product vector space C[t, t?1 ] ? W1 ? W2 (symbolically the linear span of all coe?cients of Y (u1 , x)u2 for ui ∈ Wi ) modulo all the axioms for an intertwining operator of a certain type. It is very natural that the tensor product vector space C[t, t?1 ] ? W1 ? W2 modulo all the axioms for an intertwining operator of a certain type is a weak V -module. By using universal properties, it can be proved that the tensor product module from this construction is isomorphic to those (depending on z ∈ C× ) constructed in [HL0-4] in the category of V -modules. Analogous to the classical result, if V satis?es certain “?niteness” and “semisimplicity” conditions, we prove that there exists a unique maximal submodule ?(W1 , W2 ) inside the weak module G(W1 , W2 ) (Proposition 4.9) such that ?(W1 , W2 )′ is a tensor product

′ module for the ordered pair (W1 , W2 ) (Theorem 4.10).

5

This paper is organized as follows: Section 2 is preliminary. In Section 3 we formulate a de?nition of tensor product and give a construction of a tensor product. In Section 4, we prove a generalized form of the nuclear democracy theorem by using an analogue of “Hom”- functor. Acknowledgment. This paper is based on some Chapters of my Ph.D. thesis written under the direction of Professors James Lepowsky and Robert Wilson at Rutgers University, 1994. I would like to thank Professors Lepowsky and Wilson for their insightful advice. I am greatly indebted to Professor Lepowsky for helping me clarify many subtle points. I am grateful to Professor Chongying Dong for reading this paper and for giving many useful suggestions.

6

2

Preliminaries

In this section we ?rst review some necessary de?nitions from [B], [FHL] and [FLM]. Then we present some elementary results about certain Lie algebras and modules related to a vertex (operator) algebra. We use standard notations and de?nitions of [FHL], [FLM] and [FZ]. De?nition 2.1 A vertex operator algebra is a quadruple (V, Y, 1, ω ) where V = ⊕n∈Z V(n) is a Z-graded vector space, Y (·, x) is a linear map from V to (EndV )[[x, x?1 ]], 1 and ω are ?xed elements of V such that the following conditions hold: (V0) dim V(n) < ∞ for any n ∈ Z and V(n) = 0 for n su?ciently small; (V1) Y (1, x) = 1; (V2) Y (a, x)1 ∈ (EndV )[[x]] and lim Y (a, x)1 = a for any a ∈ V ;

x→ 0

(V3)

For any a, b ∈ V , Y (a, x)b ∈ V ((x)) and for any a, b, c ∈ V , the following

Jacobi identity holds: ?x2 + x1 x1 ? x2 1 Y (a, x1 )Y (b, x2 )c ? x? Y (b, x2 )Y (a, x1 )c 0 δ x0 x0 x1 ? x0 1 = x? Y (Y (a, x0 )b, x2 )c. (2.1) 2 δ x2

1 x? 0 δ

For a ∈ V , Y (a, x) =

n∈Z

an x?n?1 is called the vertex operator associated to a; L(n)x?n?1 . Then we have (m3 ? m) δm+n,0 rankV 12 (2.2)

(V4) Set Y (ω, x) =

n∈Z

[L(m), L(n)] = (m ? n)L(m + n) +

for m, n ∈ Z, where rankV is a ?xed complex number, called the rank of V ; Y (L(?1)a, x) = d Y (a, x) dx for any a ∈ V ; (2.3)

and L(0)u = nu := (wtu)u for u ∈ V(n) , n ∈ Z. This completes the de?nition of vertex operator algebra. 7

Remark 2.2 If a triple (V, Y, 1) satis?es the axioms (V1)-(V3) (without assuming the

Z-grading

and the existence of Virasoro algebra), (V, Y, 1) is called a vertex algebra. It

can be proved (cf. [L1]) that this de?nition is equivalent to Borcherds’ de?nition in [B]. As a consequence of the Jacobi identity we have the following commutator formula [B]:

1 [Y (a, x1 ), Y (b, x2 )] = Resx0 x? 2 δ

x1 ? x0 Y (Y (a, x0 )b, x2 ). x2

(2.4)

De?nition 2.3 A module for a vertex operator algebra V is a pair (M, YM ) where M = ⊕h∈C M(h) is a C-graded vector space and Y M(·, x) is a linear map from V to (EndM )[[x, x?1 ]] satisfying the following conditions: (M0) For any h ∈ C, L(0)u = hu for u ∈ M(h) , dim M(h) < ∞ and M(n+h) = 0 for n ∈ Z su?ciently small; (M1) YM (1, x) = 1; d YM (a, x) for any a ∈ V ; dx (M3) YM (a, x)u ∈ M ((x)) for any a ∈ V, u ∈ M and for any a, b ∈ V, u ∈ M , the (M2) YM (L(?1)a, x) = following Jacobi identity holds: ?x2 + x1 x1 ? x2 1 YM (a, x1 )YM (b, x2 )u ? x? YM (b, x2 )YM (a, x1 )u 0 δ x0 x0 x1 ? x0 1 YM (Y (a, x0 )b, x2 )u. (2.5) = x? 2 δ x2

1 x? 0 δ

By a weak V -module we mean a pair (M, YM ) satisfying the axioms (M1)-(M3). A weak V -module M is said to be N-gradable if there exists an N-gradation M = ⊕n∈N M (n) such that an M (k ) ? M (m + n ? 1 + k ) for m, n, k ∈ Z, a ∈ V(m) , (2.6)

where N is the set of nonnegative integers and M (n) = 0 for n < 0 by de?nition. The notions of submodule, irreducible module, quotient module and module homomorphism 8

can be de?ned in the obvious way. A vertex operator algebra V is said to be rational if any

N-gradable

weak V -module is a direct sum of irreducible N-gradable weak V -modules. If

V is rational, it was proved [DLM1] that there are only ?nitely many irreducible modules up to equivalence and that any irreducible N-gradable weak V -module is a module so that L(0) acts semisimply on any N-gradable weak V -module. Then this de?nition of rationality is equivalent to Zhu’s de?nition [Z] of rationality. There are also other variant de?nitions of rationality. For example, the de?nition of rationality in [HL0-4] is di?erent from the current de?nition. Let M = ⊕h∈C M(h) be a V -module. Set M ′ = ⊕h∈C M(?h) and de?ne Y (a, x)u′ , v = u′ , Y (exL(1) (?x2 )L(0) a, x?1 )v (2.7)

for u′ ∈ M ′ , v ∈ M . Then it was proved in [FHL] that M ′ is a V -module, called the contragredient module, and that (M ′ )′ = M . If f is a V -homomorphism from a V -module W to M , then we have a V -homomorphism f ′ from M ′ to W ′ such that f ′ (u ′ ), v = u ′ , f (v ) Furthermore, we have (f ′ )′ = f [HL0-4]. De?nition 2.4 Let W1 , W2 and W3 be three weak V -modules. An intertwining operator W3 of type is a linear map W1 W2 I (·, x) : W1 → (Hom(W2 , W3 )){x}, u → I (u, x) =

α∈C

for u′ ∈ M ′ , v ∈ W.

(2.8)

uα x?α?1

(2.9)

satisfying the following conditions: (I1) For any ?xed u ∈ W1 , v ∈ W2 , α ∈ C, uα+n v = 0 for n ∈ Z su?ciently large; d I (u, x)v for u ∈ W1 , v ∈ W2 ; (I2) I (L(?1)u, x)v = dx (I3) For a ∈ V, u ∈ W1 , v ∈ W2 , the modi?ed Jacobi identity (2.1) where Y (b, x2 ) and Y (Y (a, x0 )b, x2 ) are replaced by I (u, x2 ) and I (Y (a, x0 )u, x2 ), respectively, holds. 9

W3 the vector space of all intertwining operators of the indiW1 W2 cated type and we call the dimension of this vector space the fusion rule of the correWe denote by I sponding type. The following proposition was proved in [FHL] and [FZ]: Proposition 2.5 Let Wi = ⊕∞ |Wi (n) n=0 Wi (n) (i = 1, 2, 3) be weak V -modules such that L(0) W3 = (hi + n)id (i = 1, 2, 3) and let I (·, x) be an intertwining operator of type . W1 W2 Then I o (u, x) := xh1 +h2 ?h3 I (u, x) ∈ (Hom(W2 , W3 ))[[x, x?1 ]]. Set I o (u, x) =

?n?1 . n∈Z Iu (n)x

(2.10)

Then for any k ∈ N, u ∈ W1 (k ), m, n ∈ N, (2.11)

Iu (n)W2 (m) ? W3 (m + k ? n ? 1). In particular, Iu (k + m + i)W2 (m) = 0 for all i ≥ 0.

(2.12)

Let Wi (i = 1, 2, 3) be V -modules and let I (·, x) be an intertwining operator of type W3 . The transpose operator I t (·, x) is de?ned by: W1 W2 I t (·, x) : W2 ? W1 → W3 {x} I t (u2 , x)u1 = exL(?1) I (u1 , eπi x)u2 for u1 ∈ W1 , u2 ∈ W2 . The adjoint operator I ′ (·, x) is de?ned by:

′ ′ I ′ (·, x) : W1 ? W3 → W2 {x}

(2.13)

I ′ (u1, x)u′3 , u2 = u′3 , I (exL(1) (eπi x?2 )L(0) u1 , x?1 )u2

(2.14)

′ for u1 ∈ W1 , u2 ∈ W2 , u′3 ∈ W3 . The following proposition was proved in [HL0-4] (see also

[FHL], [L2]). 10

Proposition 2.6 The transpose operator I t (·, x) and the adjoint operator I ′ (·, x) are intertwining operators of corresponding types. Notice that the transpose operator I t (·, x) can be de?ned more generally for weak V -modules Wi for i = 1, 2, 3 and it follows from the same proof that it is an intertwining operator. The following Borcherds’ examples of vertex algebras [B] show that the notion of vertex algebra is really a generalization of the notion of commutative associative algebra. Example 2.7 Let A be a commutative associative algebra with identity together with a derivation d. De?ne Y (a, x)b = exd a b for any a, b ∈ A. (2.15)

Then (A, Y, 1) is a vertex algebra. Furthermore, let M be a module for A viewed as an associative algebra. De?ne YM (a, x)u = exd a u for a ∈ V, u ∈ M . Then (M, YM ) is a d module for the vertex algebra (A, Y, 1). In particular, let A = C((t)) and d = . Then dt (C((t)), Y, 1) is a vertex algebra. By de?nition, we have Y (f (t), x) = ex dt f (t) = f (t + x)

d

for f (t) ∈ C((t)).

(2.16)

It is clear that the Laurent polynomial ring C[t, t?1 ] is a vertex subalgebra. For convenience in the following we shall associate Lie algebras g0 (V ) and g (V ) to a vertex algebra V . The following lemma could be found in [B]: Lemma 2.8 Let (V, Y, 1) be a vertex algebra and let d be the endomorphism of V de?ned by d(a) = a?2 1 for a ∈ V . Then the quotient space g0 (V ) := V /dV is a Lie algebra with the bilinear product: [? a, ? b] = a0 b for a, b ∈ V . Furthermore, any V -module M is a g0 (V )-module with the action given by: au = a0 u for a ∈ V, u ∈ M . 11

? := C[t, t?1 ] ? V Let V be any vertex algebra. Then from [FHL] (see also [B]), V has a vertex algebra structure with Y (f (t) ? u, x) = Y (f (t), x) ? Y (u, x) for any f (t) ∈

C[t, t?1 ], u

∈ V , and 1 = 1 ? 1V . (The a?nization of a vertex operator algebra has also

d dt

? := been used in [HL0-4].) Set d

?(u) = u?2 1 for u ∈ V ? . Then from ? 1 + 1 ? dV . Then d

?V ?) = V ? /d ? is a Lie algebra. For any m, n ∈ Z, a ∈ V , by de?nition we Lemma 2.8 g0 (V have (tm ? a)n = Resx xn Y (tm ? a, x) = Resx xn (t + x)n ? Y (a, x)

∞

=

i=0

m m+n?i t ? ai . i

(2.17)

Thus

∞

[(tm ? a), (tn ? b)] = (tm ? a)0 (tn ? b) =

i=0

m m+n?i t ? ai b i

(2.18)

? to g0 (V ? ). for any a, b ∈ V, m, n ∈ Z, where “bar” denotes the natural quotient map from V Therefore, we have (see also [B]) ? ) is Proposition 2.9 Let V be any vertex algebra. Then the quotient space g (V ) := g0 (V a Lie algebra with the bilinear operation:

∞

[tm

?

a, tn

? b] =

i=0

m m+n?i t ? ai b. i

(2.19)

(This Lie algebra g (V ) has been also studied in [FFR].) We also use a(m) for tm ? a through the paper. It is clear that 1(?1) is a central element of g (V ). If 1(?1) acts as a scalar k on a g (V )-module M , we call M a g (V )-module of level k . (This corresponds to level for a?ne Lie algebras.) A g (V )-module M is said to be restricted if for any a ∈ V, u ∈ M , a(n)u = 0 for n su?ciently large. Then any weak V -module M is a restricted g (V )-module of level one, where a(n) is represented by an . (However, a restricted g (V )-module is not necessarily a weak V -module.) Then we obtain a functor 12

F from the category of weak V -modules to the category of restricted g (V )-modules. For any restricted g (V )-module M , we de?ne J (M ) to be the intersection of all ker f , where f runs through all g (V )-homomorphisms from M to weak V -modules. Then M is a weak V -module if and only if J (M ) = 0. Furthermore, M/J (M ) is a weak V -module and M/J (M ) is a universal from M to the functor F [J]. To summarize, for any vertex algebra V we have two Lie algebras g0 (V ) and g (V ) which are related by the following inclusion relations: ? ) ? g (V ?) ? ···. g 0 (V ) ? g (V ) ? g 0 (V Let V be a vertex operator algebra. For any a ∈ V(m) , m, n ∈ Z, we de?ne deg a(n) = deg (tn ? a) = wta ? n ? 1 = m ? n ? 1. (2.21) (2.20)

Then g (V ) becomes a Z-graded Lie algebra. Denote by g (V )0 the degree-zero subalgebra. Then we obtain a triangular decomposition g (V ) = g (V )+ ⊕ g (V )0 ⊕ g (V )? . Lemma 2.10 Let V be a vertex algebra, let M be a V -module and let z be any nonzero complex number. For any a ∈ V, u ∈ M, m, n ∈ Z, de?ne

∞

a(m)(t ? u) =

i=0

n

m m?i m+n?i z (t ? ai u). i

(2.22)

? := C[t, t?1 ] ? M . Then this de?nes a g (V )-module (of level zero) structure on M Proof. Let ψ be the automorphism of the associative algebra C[t, t?1 ] such that ψ (f (t)) = f (zt) for f (t) ∈ C[t, t?1 ]. Set C[t, t?1 ]ψ = C[t, t?1 ]. Then C[t, t?1 ]ψ is a C[t, t?1 ]module with the following action: f (t)u = ψ (f (t))u = f (zt)u for f (t) ∈ C[t, t?1 ], u ∈ C[t, t?1 ]ψ .

By Example 2.7 C[t, t?1 ]ψ is a module for the vertex algebra C[t, t?1 ] such that Y (f (t), x)u = ψ ex dt f (t) u = f (zt + x)u 13

d

(2.23)

? -module, so that it is a g (V ) for f (t) ∈ C[t, t?1 ], u ∈ C[t, t?1 ]ψ . Then C[t, t?1 ]ψ ? M is a V ? ))-module (of level zero). Then the lemma follows (2.17) immediately. (= g0 (V 2

Let V be a vertex algebra and let M be a V -module. For any nonzero complex number z , let Cz be the evaluation module for the associative algebra C[t, t?1 ] with t acting as a scalar z . Then from Example 2.7 Cz is a module for vertex algebra C[t, t?1 ], so that

Cz

? -module. Therefore Cz ? M is a g (V ) = g0 (V ? )-module (by Lemma 2.1). ? M is a V

From (2.17) we have

∞

a(m) · (1 ? u) =

i=0

m m?i z (1 ? ai u) for a ∈ V, u ∈ M. i

(2.24)

Denote this g (V )-module by Mz . Then we obtain Proposition 2.11 Let V be a vertex algebra, let M be a V -module and let z be any nonzero complex number. De?ne ρ : g (V ) → EndC M as follows:

∞

ρ(a(m))u =

i=0

m m?i z ai u for a ∈ V, u ∈ M. i

(2.25)

Then ρ is a representation of g (V ) (of level zero) on M . m m?i z ai is an in?nite sum (although it is a ?nite sum after applied i i=0 to each vector u of M ), we may consider a certain completion of g (V ). By considering Noticing that the tensor product vertex algebra C((t)) ? V we obtain a Lie algebra g0 (C((t)) ? V ) (from Lemma 2.1). It is clear that this Lie algebra is the completion of g (V ) with respect to a certain topology for g (V ). We denote this Lie algebra by g ?(V ). For any f (t) =

m m≥k cm t ∞ ∞

∈ C((t)), since the following sum:

∞ i=0

cm

m≥k i=0

m m?i i z t = i

? ?

m≥k

m cm z m?i ? ti i

?

(2.26)

may not be a well-de?ned element of C((t)), we cannot extend an evaluation g (V )-module Mz to a g ?(V )-module. 14

De?ne a linear map ?z as follows: ?z :

C[t, t?1 ] ? V

→ (C((t)) ? V ) ? (C((t)) ? V ); (2.27)

f (t) ? a → 1 ? (f (t) ? a) + (f (z + t) ? a) ? 1.

Proposition 2.12 ?z induces an associative algebra homomorphism from U (g (V )) to U (? g (V )) ? U (? g (V )).

?1 Proof. De?ne a linear map ?1 z from C[t, t ] ? V to C((t)) ? V as follows: ?1 ?1 z (f (t) ? a) = f (z + t) ? a for f (t) ∈ C[t, t ], a ∈ V.

(2.28)

1 Then ?z = ?1 z ? 1 + 1 ? id. Therefore it su?ces to prove that ?z induces a Lie algebra

homomorphism from g (V ) to g ?(V ). Let ψz be the algebra homomorphism from C[t, t?1 ] to C((t)) de?ned by: ψz (f (t)) = f (z + t) for f (t) ∈ C((t)). From Examples 2.4 ψz is a vertex algebra homomorphism from C[t, t?1 ] to C((t)), so that ψz ? id is a vertex algebra homomorphism from C[t, t?1 ] ? V to C((t)) ? V . By de?nition ?1 z = ψz ? id. Therefore ?1 ?(V ). z induces a Lie algebra homomorphism from g (V ) to g 2

Remark 2.13 The Hopf-like algebra (U (g (V )), U (? g (V )), ?z ) is implicitly used in many references such as [HL0-4], [KL0-2] and [MS].

3

A de?nition of tensor product and a construction

In this section we shall ?rst formulate a de?nition of a tensor product in terms of a certain universal property as an analogue of the notion of the classical tensor product. Then we give a construction of a tensor product for an ordered pair of modules for a rational vertex operator algebra. Throughout this section, V will be a ?xed vertex operator algebra. Let C be the category of weak V -modules where a morphism f from W to M is a linear map such that 15

f (Y (a, x)u) = Y (a, x)f (u) for any a ∈ V, u ∈ W. Let C0 be the subcategory of C where objects of C0 are weak V -modules satisfying all the axioms of a module except that in (M0), in?nite-dimensional homogeneous subspaces are allowed. De?nition 3.1 Let D be either the category C or C0 and let W1 and W2 be objects of D . A tensor product for the ordered pair (W1 , W2 ) is a pair (M, F (·, x)) consisting of M an object M of D and an intertwining operator F (·, x) of type satisfying the W1 W2 following universal property: For any object W of D and any intertwining operator I (·, x) W of type , there exists a unique V -homomorphism ψ from M to W such that W1 W2 I (·, x) = ψ ? F (·, x). (Here ψ extends canonically to a linear map from M {x} to W {x}.) Remark 3.2 Just as in the classical algebra theory, it follows from the universal property that if there exists a tensor product (M, F (·, x)) in the category C or C0 for the ordered pair (W1 , W2 ), then it is unique up to V -module isomorphism, i.e., if (W, G(·, x)) is another tensor product, then there is a V -module isomorphism ψ from M to W such that G = ψ ?F . Conversely, let (M, F (·, x)) be a tensor product for the ordered pair (W1 , W2 ) and let σ be an automorphism of the V -module M . Then (M, σ ? F (·, x)) is a tensor product for (W1 , W2 ). Lemma 3.3 Let (W, F (·, x)) is a tensor product in the category C or C0 for the ordered pair (W1 , W2 ). Then F (·, x) is surjective in the sense that all the coe?cients of F (u1, x)u2 for ui ∈ Wi linearly span W . Proof. Let W be the linear span of all the coe?cients of F (u1, x)u2 for ui ∈ Wi . Then it follows from the commutator formula (2.4) that W is a submodule of W and F (·, x) W is an intertwining operator of type . It follows from the universal property of W1 W2 (W, F (·, x)) that there is a unique V -module homomorphism ψ from W to W such that F (u1 , x)u2 = ψ (F (u1, x)u2 ) 16 for u1 ∈ W1 , u2 ∈ W2 . (3.1)

? is a submodule of W , ψ may be viewed as a V -homomorphism from W to W . Since W ? . Then It follows from the universal property of (W, F (·, x)) that ψ = 1. Thus W = W the proof is complete. 2

Corollary 3.4 If (M, F (·, x)) is a tensor product in the category C or C0 for the ordered pair (W1 , W2 ), then for any weak V -module W3 in the same category, HomV (M, W3 ) is W3 naturally isomorphic to the space of intertwining operators of type . W1 W2 Proof. Let φ be any V -homomorphism from M to W3 . Then φF (·, x) is an interW3 twining operator of type . Thus we obtain a linear map π from HomV (M, W3 ) W1 W2 W3 to I de?ned by π (φ) = φF (·, x). Since F (·, x) is surjective (Lemma 3.3), π W1 W2 is injective. On the other hand, the universal property of (W, F (·, x)) implies that π is surjective. 2

Remark 3.5 If (M, F (·, x)) is a tensor product in the category C or C0 for the ordered pair (W1 , W2 ), then one can show that (M, F t (·, x)) is a tensor product in the same category for the ordered pair (W2 , W1 ). This gives a sort of commutativity of tensor product. It is important to notice that it should not be confused with the symmetric property of a tensor category. As a matter of fact, the tensor category of V -modules is not a symmetric tensor category [HL0-4]. If (M, YM (·, x)) is a V -module, one can show that (M, YM (·, x)) is a tensor product for (V, M ). This shows that the adjoint module V satis?es a certain unital property. Next toward a construction of a tensor product we shall construct an N-gradable weak V -module T (W1 , W2 ) for an ordered pair (W1 , W2 ) of N-gradable weak V -modules. First form the vector space F0 (W1 , W2 ) = C[t, t?1 ] ? W1 ? W2 (3.2)

17

and set Yt (u, x) =

n∈Z

(tn ? u)x?n?1 for u ∈ W1 .

(3.3)

Then C[t, t?1 ] ? W1 is linearly spanned by the coe?cients of all Yt (u, x) for u ∈ W1 . Fix a gradation Wi = ⊕n∈N Wi (n) for each Wi (i = 1, 2). Later we will show that if V is rational, there is a canonical gradation for any N-gradable weak V -module. We de?ne a Z-grading for F0 (W1 , W2 ) as follows: For k ∈ Z; m, n ∈ N, u ∈ W1 (m), v ∈ W2 (n), we de?ne deg (tk ? u ? v ) = m + n ? k ? 1. (3.4)

? = C[t, t?1 ] ? V on F0 (W1 , W2 ) as follows: For a ∈ V, u ∈ W1 , v ∈ De?ne an action of V W2 , we de?ne Yt (a, x1 )(Yt (u, x2 ) ? v )

1 = Yt (u, x2 ) ? Y (a, x1 )v + Resx0 x? 2 δ

x1 ? x0 Yt (Y (a, x0 )u, x2 ) ? v. x2

(3.5)

? , F0 (W1 , W2 ) becomes a Z-graded Proposition 3.6 Under the above de?ned action of V g (V )-module of level one, i.e., Yt (1, x) = 1, Yt (L(?1)a, x) = d Yt (a, x) for a ∈ V ; dx (3.6) (3.7)

deg a(n) = wt a ? n ? 1 for each homogeneous a ∈ V, n ∈ Z;

1 [Yt (a, x1 ), Yt (b, x2 )] = Resx0 x? 2 δ

x1 ? x0 Yt (Y (a, x0 )b, x2 ) for a, b ∈ V. (3.8) x2

Proof. Writing (3.5) into components we get (tm ? a)(tn ? u ? v ) = tn ? u ? am u

n ?1 ?1 +Resx0 Resx1 Resx2 xm 1 x2 x1 x1 δ

x2 + x0 Yt (Y (a, x0 )u, x2 ) ? v x1

18

∞

= tn ? u ? am u + Resx2

i=0 ∞

m i

m+n?i x2 Yt (ai u, x2 ) ? v

= tn ? u ? am u +

i=0

m i

(tm+n?i ? ai u ? v ).

(3.9)

It follows from Lemma 2.10 that (3.5) de?nes a g (V )-module structure on C[t, t?1 ] ? W1 ? W2 , which is a tensor product module of level-zero g (V )-module C[t, t?1 ] ? W1 with the level-one g (V )-module W2 . 2

De?ne J0 to be the g (V )-submodule of F0 (W1 , W2 ) generated by the following subspaces: tm+n+i ? W1 (m) ? W2 (n) for m, n, i ∈ N. Set F1 (W1 , W2 ) = F0 (W1 , W2 )/J0 . (3.11) (3.10)

Remark 3.7 The space F1 (W1 , W2 ) is an N-gradable g (V )-module of level one, so that the axioms (M1), (M2) and the commutator formula (2.4) automatically hold. Furthermore, for any a ∈ V, w ∈ F1 (W1 , W2 ), Yt (a, x)w involves only ?nitely many negative powers of x.

Remark 3.8 Notice that the action (3.5) of g (V ) on F0 (W1 , W2 ) only re?ects the commutator formula (2.4), which is weaker than the Jacobi identity, unlike the situation in the classical Lie algebra theory. In the next step, we consider the whole Jacobi identity relation for an intertwining operator. This step in our approach might be related to the “compatibility condition” in Huang and Lepowsky’s approach [HL0-4]. Let π1 be the quotient map from F0 (W1 , W2 ) onto F1 (W1 , W2 ) and let J1 be the

n k subspace of F1 (W1 , W2 ), linearly spanned by all coe?cients of monomials xm 0 x1 x2 in the

19

following expressions: x1 ? x2 Yt (a, x1 )π1 (Yt (u, x2 ) ? v ) x0 x2 ? x1 1 π1 (Yt (u, x2 ) ? Y (a, x1 )v ) ?x? 0 δ ?x0 x1 ? x0 1 ? x? π1 (Yt (Y (a, x0 )u, x2 )) ? v 2 δ x2

1 x? 0 δ

(3.12)

for a ∈ V, u ∈ W1 , v ∈ W2 . Proposition 3.9 The subspace J1 is a graded g (V )-submodule of F1 (W1 , W2 ). Proof. For a, b ∈ V, u ∈ W1 , v ∈ W2 , we have x1 ? x2 Yt (b, x3 )Yt (a, x1 )π1 (Yt (u, x2 ) ? v ) x0 x1 ? x2 1 = x? Yt (a, x1 )Yt (b, x3 )π1 (Yt (u, x2 ) ? v ) 0 δ x0 x1 ? x2 ?1 x3 ? x4 1 +Resx4 x? x1 δ Yt (Y (b, x4 )a, x1 )π1 (Yt (u, x2 ) ? v ) 0 δ x0 x1 x1 ? x2 1 Yt (a, x1 )π1 (Yt (u, x2 ) ? Y (b, x3 )v ) = x? 0 δ x0 x1 ? x2 ?1 x3 ? x4 1 +Resx4 x? x2 δ Yt (a, x1 )π1 (Yt (Y (b, x4 )u, x2 ) ? v ) 0 δ x0 x2 x1 ? x2 ?1 x3 ? x4 1 x1 δ Yt (Y (b, x4 )a, x1 )π1 (Yt (u, x2 ) ? v ); +Resx4 x? 0 δ x0 x1

1 x? 0 δ

(3.13)

Yt (b, x3 )π1 (Yt (u, x2 ) ? Y (a, x1 )v ) = π1 (Yt (u, x2 ) ? Y (b, x3 )Y (a, x1 )v )

1 +Resx4 x? 2 δ

x3 ? x4 π1 (Yt (Y (b, x4 )u, x2 ) ? Y (a, x1 )v ) x2

= π1 (Yt (u, x2 ) ? Y (a, x1 )Y (b, x3 )v ) x3 ? x4 π1 (Yt (u, x2 ) ? Y (Y (b, x4 )a, x1 )v ) x1 x3 ? x4 1 π1 (Yt (Y (b, x4 )u, x2 ) ? Y (a, x1 )v ); +Resx4 x? 2 δ x2

1 +Resx4 x? 1 δ

(3.14)

20

and x1 ? x0 Yt (b, x3 )π1 (Yt (Y (a, x0 )u, x2 ) ? v ) x2 x1 ? x0 1 π1 (Yt (Y (a, x0 )u, x2 ) ? Y (b, x4 )v ) = x? 2 δ x2 x1 ? x0 ?1 x3 ? x4 1 +Resx4 x? x2 δ π1 (Yt (Y (b, x4 )Y (a, x0 )u, x2 ) ? v ) 2 δ x2 x2 x1 ? x0 1 = x? Yt (b, x3 )π1 (Yt (Y (a, x0 )u, x2 ) ? v ) 2 δ x2 x1 ? x0 ?1 x3 ? x4 1 x2 δ π1 (Yt (Y (a, x0 )Y (b, x4 )u, x2 ) ? v ) +Resx4 x? 2 δ x2 x2 x1 ? x0 ?1 x3 ? x4 ?1 x4 ? x5 1 +Resx4 Resx5 x? x2 δ x0 δ · 2 δ x2 x2 x0

1 x? 2 δ

·π1 (Yt (Y (Y (b, x5 )a, x0 )u, x2 ) ? v )

1 = x? 2 δ

x1 ? x0 Yt (b, x3 )π1 (Yt (Y (a, x0 )u, x2 ) ? v ) x2 x1 ? x0 ?1 x3 ? x4 ?1 x3 ? x4 1 +Resx4 x? x2 δ x2 δ 2 δ x2 x2 x2 x1 ? x0 ?1 x3 ? x0 ? x5 x2 δ · x2 x2

·π1 (Yt (Y (a, x0 )Y (b, x4 )u, x2 ) ? v )

1 +Resx5 x? 2 δ

·π1 (Yt (Y (Y (b, x5 )a, x0 )u, x2 ) ? v )

1 = x? 2 δ

x1 ? x0 Yt (b, x3 )π1 (Yt (Y (a, x0 )u, x2 ) ? v ) x2 x1 ? x0 ?1 x3 ? x4 ?1 x3 ? x4 1 +Resx4 x? x2 δ x2 δ 2 δ x2 x2 x2 x1 ? x0 ?1 x3 ? x4 x1 δ π1 (Yt (Y (Y (b, x5 )a, x0 )u, x2 ) ? v ). x2 x1 (3.15)

·π1 (Yt (Y (a, x0 )Y (b, x4 )u, x2 ) ? v )

1 +Resx4 x? 2 δ

Then it is clear that J1 is stable under the action of Yt (b, x) for any b ∈ V .

2

Theorem 3.10 The quotient space F2 (W1 , W2 ) := F1 (W1 , W2 )/J1 is an N-gradable weak V -module. Proof. We only need to prove the Jacobi identity. Let π2 be the natural quotient 21

map from F0 (W1 , W2 ) onto F2 (W1 , W2 ). For a, b ∈ V, u ∈ W1 , v ∈ W2 , we have x1 ? x2 Yt (a, x1 )Yt (b, x2 )π2 (Yt (u, x3 ) ? v ) x0 x1 ? x2 1 Yt (a, x1 )π2 (Yt (u, x3 ) ? Y (b, x2 )v ) = x? 0 δ x0 x1 ? x2 ?1 x2 ? x4 1 +Resx4 x? x3 δ Yt (a, x1 )π2 (Yt (Y (b, x4 )u, x3 ) ? v ) 0 δ x0 x3 x1 ? x2 1 π2 (Yt (u, x3 ) ? Y (a, x1 )Y (b, x2 )v ) (3.16) = x? 0 δ x0 x1 ? x2 ?1 x1 ? x4 1 +Resx4 x? x3 δ π2 (Yt (Y (a, x4 )u, x3 ) ? Y (b, x2 )v ) 0 δ x0 x3

1 x? 0 δ

(3.17)

1 +Resx4 x? 0 δ

x1 ? x2 ?1 x2 ? x4 x3 δ Yt (a, x1 )π2 (Yt (Y (b, x4 )u, x3 ) ? v ); x0 x3 (3.18)

?x2 + x1 Yt (b, x2 )Yt (a, x1 )π2 (Yt (u, x3 ) ? v ) x0 ?x2 + x1 1 π2 (Yt (u, x3 ) ? Y (b, x2 )Y (a, x1 )v ) (3.19) = x? 0 δ x0 ?x2 + x1 ?1 x2 ? x4 1 +Resx4 x? x3 δ π2 (Yt (Y (b, x4 )u, x3 ) ? Y (a, x1 )v ) 0 δ x0 x3

1 x? 0 δ

(3.20)

1 +Resx4 x? 0 δ

?x2 + x1 ?1 x1 ? x4 x3 δ Yt (b, x2 )π2 (Yt (Y (a, x4 )u, x3 ) ? v ); x0 x3 (3.21)

and x1 ? x0 Yt (Y (a, x0 )b, x2 )π2 (Yt (u, x3 ) ? v ) x2 x1 ? x0 1 = x? π2 (Yt (u, x3 ) ? Y (Y (a, x0 )b, x2 )v ) (3.22) 2 δ x2 x1 ? x0 ?1 x2 ? x4 1 x3 δ π2 (Yt (Y (Y (a, x0 )b, x4 )u, x3 ) ? v ). +Resx4 x? 2 δ x2 x3

1 x? 2 δ

(3.23) It follows from the Jacobi identity of W2 that (3.16) ? (3.19) = (3.22). Since

1 x? 0 δ

x1 ? x2 ?1 x2 ? x4 x3 δ x0 x3 22

1 = x? 0 δ

x1 ? x3 ? x4 ?1 x2 ? x4 x3 δ x0 x3 x + x x2 ? x4 0 4 1 = (x1 ? x3 )?1 δ x? 3 δ x1 ? x3 x3 x ? x4 x ? x 2 1 3 1 x? ; = (x0 + x4 )?1 δ 3 δ x0 + x4 x3

1 x? 0 δ

(3.24)

?x2 + x1 ?1 x2 ? x4 x3 δ x0 x3 x2 ? x4 ? x ? x + x 3 4 1 1 1 x? = x? 3 δ 0 δ x0 x3 ?x3 + x1 ?1 x2 ? x4 = (x0 + x4 )?1 δ x3 δ , x0 + x4 x3 by the J1 -de?ning relation (3.12), we have (3.18) ? (3.20)

1 = Resx4 x? 3 δ

(3.25)

x2 ? x4 ?1 x1 ? x0 ? x4 x3 δ x3 x3 x0 + x4 ?1 x2 ? x4 ?1 x1 ? x5 x3 δ x3 δ x5 x3 x3 x5 ? x4 ?1 x2 ? x4 ?1 x1 ? x5 x3 δ x3 δ x0 x3 x3 (3.26)

·π2 (Yt (Y (a, x0 + x4 )Y (b, x4 )u, x3 ) ? v )

1 = Resx4 Resx5 x? 5 δ

·π2 (Yt (Y (a, x5 )Y (b, x4 )u, x3 ) ? v )

1 = Resx4 Resx5 x? 0 δ

·π2 (Yt (Y (a, x5 )Y (b, x4 )u, x3 ) ? v ). Similarly, we obtain (3.17) ? (3.21)

1 = ?Resx4 x? 3 δ

x1 ? x4 ?1 x1 + x0 ? x4 x3 δ x3 x3 ?x0 + x4 ?1 x1 ? x4 ?1 x2 ? x5 x3 δ x3 δ x5 x3 x3 ?x5 + x4 ?1 x1 ? x4 ?1 x2 ? x5 x3 δ x3 δ x0 x3 x3 23

·π2 (Yt (Y (a, ?x0 + x4 )Y (b, x4 )u, x3 ) ? v )

1 = ?Resx4 Resx5 x? 5 δ

·π2 (Yt (Y (b, x4 )Y (a, x5 )u, x3 ) ? v )

1 = ?Resx4 Resx5 x? 0 δ

·π2 (Yt (Y (b, x4 )Y (a, x5 )u, x3 ) ? v )

1 = ?Resx5 Resx4 x? 0 δ

?x4 + x5 ?1 x1 ? x5 ?1 x2 ? x4 x3 δ x3 δ x0 x3 x3 (3.27)

·π2 (Yt (Y (b, x5 )Y (a, x4 )u, x3 ) ? v ). Therefore, we have (3.17) + (3.18) ? (3.20) ? (3.21)

1 = Resx4 Resx5 x? 4 δ

x5 ? x0 ?1 x2 ? x4 ?1 x1 ? x5 x3 δ x3 δ x4 x3 x3

·π2 (Yt (Y (Y (a, x0 )b, x4 )u, x3 ) ? v )

1 = Resx4 x? 3 δ

x2 ? x4 ?1 x1 ? x0 x2 δ π2 (Yt (Y (Y (a, x0 )b, x4 )u, x3 ) ? v ) x3 x2 (3.28)

= (3.23). Here we have used the following fact: x5 ? x0 x4 x ? x4 2 1 = Resx5 x? 3 δ x3 x ? x 2 4 1 1 x? = x? 1 δ 3 δ x3 x2 ? x4 ?1 1 = x? x2 δ 3 δ x3

1 Resx5 x? 4 δ

x2 ? x4 ?1 x1 ? x5 x3 δ x3 x3 x ? x ? x x4 + x0 1 4 0 1 1 x? x? 3 δ 5 δ x3 x5 x2 ? x4 + x4 + x0 x1 x1 ? x0 . x2

1 x? 3 δ

(3.29)

Then the Jacobi identity is proved.

2

Since F2 (W1 , W2 ) is a weak V -module, we will freely use Y (a, x) for Yt (a, x). Recall from Section 2 that any weak V -module M is a restricted g (V )-module and that for ? := M/J (M ) is a weak V -module, where J (M ) is the any restricted g (V )-module M , M intersection of all ker f with f running through all g (V )-homomorphisms from M to weak V -modules. Then Theorem 3.10 says that J1 = J (F1 (W1 , W2 )). Remark 3.11 If Wi for i = 1, 2, 3 are just restricted g (V )-modules, we can also de?ne intertwining operator by using the same axioms as in De?nition 2.4. Then following the 24

proof given in [FHL] one can easily see that the transpose operator I t (·, x) is well de?ned and it is an intertwining operator.

Proposition 3.12 Let W1 and W3 be weak V -modules, let M be a restricted g (V )-module W3 and let I (·, x) be an intertwining operator of type . Then I (·, x)J (M ) = 0, so that W1 M W3 we obtain an intertwining operator of type ? . W1 M Proof. In the proof of Theorem 3.10, replace W2 , F2 (W1 , W2 ) and Yt (·, x) by M , W3 and I (·, x), respectively. Then the J1 -de?ning relation (3.12) or Jacobi identity for I (·, x) and the Jacobi identity for W3 : (3.17) + (3.18) ? (3.20) ? (3.21) = (3.23) are given. Following the proof of Theorem 3.10, we obtain (3.16) ? (3.19) = (3.22). That is, I (·, x)J (M ) = 0. Then the proof is complete. Symmetrically, we have Proposition 3.13 Let W2 and W3 be weak V -modules, let M be a restricted g (V )-module W3 and let I (·, x) be an intertwining operator of type . Then I (J (M ), x) = 0, so MW2 W3 that we obtain an intertwining operator of type ? W2 . M Proof. The proof of this proposition does not directly follow from the proof of Theorem 3.10, but it follows from Proposition 3.12 and the notion of transpose intertwining W3 operator. Since I t (·, x) is an intertwining operator of type , by Proposition 3.12 W2 M we get I t (·, x)J (M ) = 0. Thus I (J (M ), x) = 0. 2 To construct a tensor product out of the weak V -module F2 (W1 , W2 ), we shall study the universal property. For simplicity, from now on we assume that W1 and W2 are weak V -modules in the category C0 such that Wi = ⊕n∈N (Wi )(n+hi ) for i = 1, 2. 25 2

Let W be a weak V -module in the category C0 such that W = ⊕∞ n=0 W(n+h) for some h. W Let I (·, x) be an intertwining operator of type . Let I o (·, x) = xh1 +h2 ?h I (·, x) W1 W2 be the normalization. Then we de?ne ψI : F0 (W1 , W2 ) → W, tn ? u ? v → Iu (n)v for u ∈ W1 , v ∈ W2 , n ∈ Z. In terms of generating elements, ψI can be written as: ψI (Yt (u, x) ? v ) = I o (u, x)v for u ∈ W1 , v ∈ W2 . (3.31) (3.30)

Lemma 3.14 The linear map ψI is a g (V )-homomorphism. In other words, ψI (Yt (a, x)w ) = Y (a, x)ψI (w ) for a ∈ V, w ∈ F0 (W1 , W2 ). Proof. For a ∈ V, u ∈ W1 , v ∈ W2 , we have ψI (Yt (a, x1 )(Yt (u, x2 ) ? v ))

1 = ψI (Yt (u, x2 ) ? Y (a, x1 )v ) + Resx0 x? 2 δ 1 = I o (u, x2 )Y (a, x1 )v + Resx0 x? 2 δ

(3.32)

x1 ? x0 o I (Y (a, x0 )u, x2 )v x2

x1 ? x0 ψI (Yt (Y (a, x0 )u, x2 ) ? v ) x2

= Y (a, x1 )I o (u, x2 )v = Y (a, x1 )ψI (Yt (u, x2 ) ? v ). 2

?I from F2 (W1 , W2 ) to Corollary 3.15 The linear map ψI induces a V -homomorphism ψ ?I preserves the N-gradation and ψ ?I = π2 ψI , where π2 is the quotient map W such that ψ from F0 (W1 , W2 ) to F2 (W1 , W2 ). Proof. From Proposition 2.5 and the Jacobi identity for a V -module and for an ?I intertwining operator we get: J0 + J1 ? ker ψI . Then we have an induced linear map ψ ?I is a V -homomorphism. from F2 (W1 , W2 ) to W . From (3.32) ψ 26 2

Let W = ⊕n∈N W(n+h) be given as before. Let Hom0 V (F2 (W1 , W2 ), W ) be the space of all V -homomorphisms from F2 (W1 , W2 ) to W which preserve the given N-gradation. Then we de?ne the following linear map: ?: ψ I W W1 W2 → Hom0 V (F2 (W1 , W2 ), W ) (3.33)

?I , I (·, x) → ψ W W1 W2

? : I Proposition 3.16 The map ψ linear isomorphism.

? → Hom0 V (F2 (W1 , W2 ), W ); I → ψI is a

? is injective. Let f be a V -homomorphism from F2 (W1 , W2 ) to Proof. It is clear that ψ W that preserves the Z-grading. De?ne a linear map I (·, x) from W1 to Hom(W2 , W ){x} as follows: I (u1 , x)u2 = xh?h1 ?h2 f π2 (Yt (u1 , x) ? u2 ) (3.34)

for any ui ∈ Wi . It follows from the de?ning relations J0 and J1 that I (·, x) satis?es the axioms (I1) and (I3). If we prove (I2), then I (·, x) is an intertwining operator satisfying ?I = f . For k ∈ Z, m, n ∈ N; u ∈ W1 (m), v ∈ W2 , we have ψ deg (tk ? u ? v ) = m + n ? k ? 1. Therefore L(0)f π (tk ? u ? v ) = (h + m + n ? k ? 1)f π (tk ? u ? v ). By formula (3.5), we obtain L(0)(tk ? u ? v ) = tk ? u ? L(0)v + tk+1 ? u ? v + tk ? L(0)u ? v = tk+1 ? L(?1)u ? v + (h1 + h2 + m + n)tk ? u ? v. 27 (3.36) (3.35)

Therefore f π tk+1 ? L(?1)u ? v + (h1 + h2 ? h + k + 1)tk ? u ? v = 0. (3.37) 2

This is exactly the axiom (I2) in terms of components. Then the proof is complete.

For any nonzero N-gradable weak V -module with a ?xed a gradation M = ⊕n∈N M (n) such that M (0) = 0, we de?ne the radical of M to be the maximal graded submodule R(M ) such that R(M ) ∩ M (0) = 0. De?nition 3.17 De?ne T (W1 , W2 ) to be the quotient module of F2 (W1 , W2 ) modulo the radical of F2 (W1 , W2 ) with respect to the given gradation. As a corollary of Proposition 3.16 we get ?:I Corollary 3.18 The linear isomorphism ψ W → Hom0 V (F2 (W1 , W2 ), W ); I → W1 W2 W ?I gives rise to a linear isomorphism from I ψ to Hom0 V (T (W1 , W2 ), W ), the space W1 W2 of all V -homomorphisms which preserve the given gradation. From now on we shall assume that V is rational. Then up to equivalence, V has only ?nitely many irreducible modules. Let λ1 , · · · , λk be all the distinct lowest weights of irreducible V -modules. For any N-gradable weak V -module W , let W (i) be the sum of all irreducible submodules of W with lowest weight λi . Then we obtain a canonical

(i) (i) decomposition W = ⊕k is a direct sum of irreducible V -modules with i=1 W . Since W

lowest weight λi , any submodule of W (i) is a direct sum of irreducible modules with lowest weight λi . Thus W (i) = ⊕n∈N W(n+λi ) := ⊕n∈N W (i) (n). Then W = ⊕n∈N ⊕k i=1 W(n+λi ) . Then W has a canonical N-gradation with W (n) = ⊕k i=1 W(n+λi ) for n ∈ N. It is clear that the radical of W is zero with respect to this gradation. In particular, T (W1 , W2 ) = T (W1 , W2 )(1) ⊕ · · · ⊕ T (W1 , W2 )(k) .

(i) (i) (i)

28

Since F2 (W1 , W2 ) is completely reducible, T (W1 , W2 ) is isomorphic to the submodule generated by the degree-zero subspace F2 (W1 , W2 )(0). Let Pi be the projection map of T (W1 , W2 ) onto T (W1 , W2 )(i) and let π be the natural quotient map from F0 (W1 , W2 ) onto T (W1 , W2 ). Then we de?ne F (·, x) : W1 → (HomC (W2 , T (W1 , W2 ))) {x}; u1 → F (u1, x) where F (u1, x)u2 =

k i=1

for u1 ∈ W1

(3.38)

xλi ?h1 ?h2 Pi π (Yt (u1 , x) ? u2 ) for u1 ∈ W1 , u2 ∈ W2 .

Proposition 3.19 Suppose that V is rational. Then the de?ned map F (·, x) is an interT (W1 , W2 ) twining operator of type . W1 W2 Proof. Let Fi (u1 , x)u2 = xλi ?h1 ?h2 Pi π (Yt (u1 , x) ? u2 ). Then it follows from ProposiT (W1 , W2 )(i) . Then it tion 3.16 that each Fi (·, x) is an intertwining operator of type W1 W2 follows immediately. 2 Theorem 3.20 If V is rational and Wi (i=1,2,3) are irreducible weak V -modules in the category C0 , then the pair (T (W1 , W2 ), F (·, x)) is a tensor product in the category C0 for the ordered pair (W1 , W2 ). Proof. Let W be a V -module and let I (·, x) be any intertwining operator of type W . Let Di be the projection of W onto W (i) for i = 1, · · · , k . Then Di I (·, x) W1 W2 W (i) . By Corollary 3.18, we obtain a V is an intertwining operator of type W1 W2 homomorphism gi from T (W1 , W2 ) to W (i) satisfying the condition: gi π (Yt (u1 , x) ? u2 ) = Di I o (u1 , x)u2 for u1 ∈ W1 , u2 ∈ W2 . (3.39)

Since gi Pj = 0 for j = i, we obtain gi ? F (u1 , x)u2 = Di I (u1 , x)u2 for u1 ∈ W1 , u2 ∈ W2 . Set g = g1 ⊕ · · · ⊕ gk . Then g ? F (u1, x)u2 = I (u1, x)u2 29 for u1 ∈ W1 , u2 ∈ W2 .

From the construction of T (W1 , W2 ), F (·, x) is surjective in the sense of Lemma 3.3, i,e., all the coe?cients of F (u1 , x)u2 for ui ∈ Wi linearly span T (W1 , W2 ). Thus such a g is unique. Then the pair (T (W1 , W2 ), F (·, x)) is a tensor product for the ordered pair (W1 , W2 ). 2

4

An analogue of the “Hom”-functor and a generalized nuclear democracy theorem

In this section we shall introduce the notion of what we call “generalized intertwining operator” from a V -module W1 to another V -module W2 . The notion of generalized intertwining operator can be considered as a generalization of the physicists’ notion of “primary ?eld” (cf. [BPZ], [MS] and [TK]) to the notion of general (non-primary) ?eld. On the other hand, it exactly re?ects the main features of I (u, x) for u ∈ M , where M W2 is a V -module and I (·, x) is an intertwining operator of type . We prove that MW1 G(W1 , W2 ), the space of all generalized intertwining operators, is a weak V -module (Theorem 4.6), which satis?es a certain universal property in terms of the space of intertwining operators of a certain type (Theorem 4.7). If the vertex operator algebra V satis?es certain ?niteness and semisimplicity conditions, we prove that there exists a unique maximal submodule ?(W1 , W2 ) of G(W1 , W2 ) so that the contragredient module of ?(W1 , W2 ) is

′ a tensor product module for the ordered pair (W1 , W2 ) (Theorem 4.10). Using Theorem

4.7 we derive a generalized form of the nuclear democracy theorem of Tsuchiya and Kanie [TK] (Theorem 4.12). All these results show that the notion of G(W1 , W2 ) is an analogue of the classical “Hom”-functor. Throughout this section, V will be a ?xed vertex operator algebra. De?nition 4.1 Let W1 and W2 be V -modules. A generalized intertwining operator from W1 to W2 is an element Φ(x) =

α∈C

Φα x?α?1 ∈ (Hom(W1 , W2 )) {x} satisfying the follow-

ing conditions (G1)-(G3): 30

(G1) For any α ∈ C, u1 ∈ W1 , Φα+n u1 = 0 for n ∈ Z su?ciently large; d (G2) [L(?1), Φ(x)] = Φ′ (x) = Φ(x) ; dx (G3) For any a ∈ V , there exists a positive integer k such that (x1 ? x2 )k YW2 (a, x1 )Φ(x2 ) = (x1 ? x2 )k Φ(x2 )YW1 (a, x1 ). (4.1)

Denote by G(W1 , W2 ) the space of all generalized intertwining operators from W1 to W2 . A generalized intertwining operator Φ(x) is said to be homogeneous of weight h if it satis?es the following condition: [L(0), Φ(x)] = h + x d Φ(x). dx (4.2)

A generalized intertwining operator Φ(x) of weight h is said to be primary if the following condition holds: [L(m), Φ(x)] = xm (m + 1)h + x d Φ(x) dx for m ∈ Z. (4.3)

Denote by G(W1 , W2 )(h) the space of all weight-h homogeneous generalized intertwining operators from W1 to W2 and set G0 (W1 , W2 ) = ⊕h∈C G(W1 , W2 )(h) . (4.4)

Let W (W1 , W2 ) be the space consisting of each element Φ(x) ∈ (HomC (W1 , W2 )) which satis?es the condition (G1) and let E (W1 , W2 ) be the space consisting of each element Φ(x) ∈ (HomC (W1 , W2 )) which satis?es the conditions (G1) and (G3). For any a ∈ V , we ? on W (W1 , W2 ) as follows: de?ne the left and the right actions of V

1 Yt (a, x0 ) ? Φ(x2 ) : = Resx1 x? 0 δ

x1 ? x2 YW2 (a, x1 )Φ(x2 ) x0 ?x2 + x1 Φ(x2 )YW1 (a, x1 ) x0

(4.5) (4.6) (4.7) (4.8)

= YW2 (a, x0 + x2 )Φ(x2 ).

1 Φ(x2 ) ? Yt (a, x0 ) : = Resx1 x? 0 δ

= Φ(x2 )(YW1 (a, x0 + x2 ) ? YW1 (a, x2 + x0 )). 31

Proposition 4.2 a) W (W1 , W2 ) is a left g (V )-module of level one under the de?ned left action. b) W (W1 , W2 ) is a right g (V )-module of level zero under the de?ned right action. Proof. a) First we check that W (W1 , W2 ) is closed under the left action. For any a ∈ V, m ∈ Z, Φ(x) ∈ W (W1 , W2 ), u ∈ W1 , by de?nition we have: ((tm ? a) ? Φ(x)) u = Resx0 xm 0 YW2 (a, x0 + x)Φ(x2 )u

∞

=

i=0

?m + i ? 1 i x am?i Φ(x)u. i

(4.9)

Then it is clear that (tm ? a) ? Φ(x2 ) satis?es (G1). Next, we check the de?ning relations for g (V ). By de?nition we have Yt (1, x0 ) ? Φ(x2 ) = YW2 (1, x0 + x2 )Φ(x2 ) = Φ(x2 ) and Yt (L(?1)a, x0 ) ? Φ(x2 ) = YW2 (L(?1)a, x0 + x2 )Φ(x2 ) ? YW (a, x0 + x2 )Φ(x2 ) ?x0 2 ? = Yt (a, x0 ) ? Φ(x2 ). ?x0 = Furthermore, for any a, b ∈ V , we have Yt (a, x1 ) ? Yt (b, x2 ) ? Φ(x3 ) = Yt (a, x1 ) ? (YW2 (b, x2 + x3 )Φ(x3 )) = YW2 (a, x1 + x3 )YW2 (b, x2 + x3 )Φ(x3 ). Similarly, we have Yt (b, x2 ) ? Yt (a, x1 ) ? Φ(x3 ) = YW2 (b, x2 + x3 )YW2 (a, x1 + x3 )Φ(x3 ). (4.13) (4.12) (4.10)

(4.11)

32

Therefore Yt (a, x1 ) ? Yt (b, x2 ) ? Φ(x3 ) ? Yt (b, x2 ) ? Yt (a, x1 ) ? Φ(x3 ) = Resx0 (x2 + x3 )?1 δ

1 = Resx0 x? 1 δ

x2 + x0 YW2 (Y (a, x0 )b, x2 + x3 )Φ(x3 ) x1 x1 ? x0 1 Yt (Y (a, x0 )b, x2 ) ? Φ(x3 ). = Resx0 x? 2 δ x2

x1 + x3 ? x0 YW2 (Y (a, x0 )b, x2 + x3 )Φ(x3 ) x2 + x3

(4.14)

Then a) is proved. The proof of b) is similar to the proof of a), but for completeness, we also write the details. For any a ∈ V, Φ(x) ∈ W (W1 , W2 ), by de?nition we have Φ(x2 ) ? Yt (L(?1)a, x0 ) = Φ(x2 )(YW1 (L(?1)a, x0 + x2 ) ? YW1 (L(?1)a, x2 + x0 )) = ? (Φ(x2 )(YW1 (a, x0 + x2 ) ? YW1 (a, x2 + x0 )) ?x0 ? = Φ(x2 ) ? Yt (a, x0 ). ?x0

(4.15)

For any a, b ∈ V , we have Φ(x3 ) ? Yt (a, x1 ) ? Yt (b, x2 ) ?x3 + x4 (Φ(x3 )YW1 (a, x4 )) ? Yt (b, x2 ) x1 ?x3 + x4 ?1 ?x3 + x5 1 = Resx4 Resx5 x? x2 δ Φ(x3 )YW1 (a, x4 )YW1 (b, x5 ). 1 δ x1 x2

1 = Resx4 x? 1 δ

(4.16) Similarly, we have Φ(x3 ) ? Yt (b, x2 ) ? Yt (a, x1 )

1 = Resx4 Resx5 x? 1 δ

?x3 + x4 ?1 ?x3 + x5 x2 δ Φ(x3 )YW1 (b, x5 )YW1 (a, x4 ). x1 x2 (4.17) 33

Thus Φ(x3 ) ? Yt (a, x1 ) ? Yt (b, x2 ) ? Φ(x3 ) ? Yt (b, x2 ) ? Yt (a, x1 )

1 = Resx4 Resx5 Resx0 x? 1 δ

?x3 + x4 ?1 ?x3 + x5 ?1 x4 ? x0 x2 δ x5 δ x1 x2 x5

·Φ(x3 )YW1 (Y (a, x0 )b, x5 )

1 = Resx5 Resx0 x? 1 δ

?x3 + x5 + x0 ?1 ?x3 + x5 x2 δ Φ(x3 )YW1 (Y (a, x0 )b, x5 ) x1 x2 x2 + x0 ?1 ?x3 + x5 1 = Resx5 Resx0 x? x2 δ Φ(x3 )YW1 (Y (a, x0 )b, x5 ) 1 δ x1 x2 x1 ? x0 1 = Φ(x3 ) ? Resx0 x? Yt (Y (a, x0 )b, x2 ). (4.18) 2 δ x2 2

Then the proof is complete.

For any a ∈ V, Φ(x) ∈ W (W1 , W2 ), we de?ne Yt (a, x0 ) ? Φ(x2 ) := = Yt (a, x0 ) ? Φ(x2 ) ? Φ(x2 ) ? Yt (a, x0 )

1 = Resx1 x? 0 δ

x2 ? x1 x1 ? x2 1 YW2 (a, x1 )Φ(x2 ) ? x? Φ(x2 )YW1 (a, x1 ) ,(4.19) 0 δ x0 ?x0

or equivalently a(m) ? Φ(x2 ) = Resx1 ((x1 ? x2 )m YW2 (a, x1 )Φ(x2 ) ? (?x2 + x1 )m Φ(x2 )YW1 (a, x1 )) (4.20) for any m ∈ Z. From the classical Lie algebra theory, we have Corollary 4.3 Under the de?ned action ′′ ?′′ , W (W1 , W2 ) becomes a g (V )-module (of level one).

Lemma 4.4 Let Φ(x) ∈ W (W1 , W2 ) satisfying (4.2) for some complex number h and let a be any homogeneous element of V . Then [L(0), Yt (a, x0 ) ? Φ(x2 )] = wta + h + x0 34 ? ? + x2 Yt (a, x0 ) ? Φ(x2 ). ?x0 ?x2 (4.21)

Proof. By de?nition we have [L(0), Yt (a, x0 ) ? Φ(x2 )]

1 = Resx1 x? 0 δ

x1 ? x2 [L(0), Y (a, x1 )Φ(x2 )] x0 ?x2 + x1 1 [L(0), Φ(x2 )Y (a, x1 )] ?Resx1 x? 0 δ x0 x1 ? x2 ? ? 1 = Resx1 x? wta + x1 + h + x2 Y (a, x1 )Φ(x2 ) 0 δ x0 ?x1 ?x2 x2 ? x1 ? ? 1 ?Resx1 x? wta + x1 Φ(x2 )Y (a, x1 ) + h + x2 0 δ ?x0 ?x1 ?x2 = (wta + h)Yt (a, x0 ) ? Φ(x2 ) ?Resx1 ? 1 x1 x? 0 δ ?x1 x1 ? x2 1 +Resx1 x? 0 δ x0 x2 ? x1 1 ?Resx1 x? 0 δ ?x0 ? 1 x1 x? +Resx1 0 δ ?x1 x1 ? x2 Y (a, x1 )Φ(x2 ) x0 ? x2 Y (a, x1 )Φ(x2 ) ?x2 ? x2 Φ(x2 )Y (a, x1 ) ?x2 x2 ? x1 Φ(x2 )Y (a, x1 ). ?x0

(4.22)

Since ? ?1 x1 ? x2 x δ ?x0 0 x0 we have ? x1 ? x2 1 x1 x? 0 δ ?x1 x0 x1 ? x2 ? 1 (x0 + x2 )x? = 0 δ ?x1 x0 ? ?1 x1 ? x2 ? ?1 x1 ? x2 x0 δ + x2 x δ = x0 ?x1 x0 ?x1 0 x0 ? ?1 x1 ? x2 ? ?1 x1 ? x2 = ?x0 x δ ? x2 x δ . ?x0 0 x0 ?x2 0 x0 Similarly, we have x2 ? x1 ? 1 x1 x? 0 δ ?x1 ?x0 = ? x0 ? ? x2 ? x1 1 x1 x? . + x2 0 δ ?x0 ?x2 ?x0 35 (4.25) = ? ?1 x1 ? x2 x δ ?x2 0 x0 =? ? ?1 x1 ? x2 , x δ ?x1 0 x0 (4.23)

(4.24)

Therefore, we obtain [L(0), Yt (a, x0 ) ? Φ(x2 )] = wta + h + x0 ? ? Yt (a, x0 ) ? Φ(x2 ). + x2 ?x0 ?x2 2 (4.26)

Proposition 4.5 The subspaces E (W1 , W2) and G(W1 , W2) are restricted g (V )-submodules of W (W1 , W2 ) and G0 (W1 , W2 ) is a C-graded g (V )-module. Proof. For any a ∈ V, m ∈ Z, Φ(x) ∈ E (W1 , W2 ), it follows from the proof of Proposition 3.2.7 in [L1] (for an analogous result) that a(m) ? Φ(x) ∈ E (W1 , W2 ). Thus E (W1 , W2 ) is a submodule of W (W1 , W2 ). For Φ(x) ∈ G(W1 , W2 ), since [L(?1), Yt (a, x0 ) ? Φ(x2 )] = [L(?1), YW2 (a, x0 + x2 )Φ(x2 )] = [L(?1), YW2 (a, x0 + x2 )]Φ(x2 ) + YW2 (a, x0 + x2 )[L(?1), Φ(x2 )] ? (YW2 (a, x0 + x2 )Φ(x2 )) ?x2 ? = Yt (a, x0 ) ? Φ(x2 ), ?x2 = a(m) ? Φ(x2 ) satis?es (G2). Thus a(m) ? Φ(x2 ) ∈ G(W1 , W2 ). Similarly, since ? (Φ(x2 ) ? Yt (a, x0 )) ?x2 ? = (Φ(x2 )(YW1 (a, x0 + x2 ) ? YW1 (a, x2 + x0 ))) ?x2 = Φ′ (x2 )(YW1 (a, x0 + x2 ) ? YW1 (a, x2 + x0 )) +Φ(x2 )(YW1 (L(?1)a, x0 + x2 ) ? YW1 (L(?1)a, x2 + x0 )) = [L(?1), Φ(x2 ) ? Yt (a, x0 )], (4.28)

(4.27)

we obtain Φ(x2 ) ? a(m) ∈ G(W1 , W2 ). Therefore a(m) ? Φ(x) ∈ G(W1 , W2 ). Thus G(W1 , W2 ) is a submodule. That G0 (W1 , W2 ) is a C-graded g (V )-module follows from 36

Lemma 4.4. It follows from (4.20) and (G3) that E (W1 , W2 ) is a restricted g (V )-module and so are G(W1 , W2 ) and G0 (W1 , W2 ). Then the proof is complete. 2

De?ne a linear map F (·, x) from E (W1 , W2 ) to Hom(W1 , W2 ){x} as follows: F (Φ, x)u1 = Φ(x)u1 For a ∈ V, Φ ∈ E (W1 , W2 ), we have F (Y (a, x0 )Φ, x2 ) = (Y (a, x0 )Φ)(x)|x=x2

1 = Resx1 x? 0 δ

for Φ ∈ E (W1 , W2 ), u1 ∈ W1 .

(4.29)

= Resx1

x1 ? x ?x + x1 1 Y (a, x1 )Φ(x) ? x? Φ(x)Y (a, x1 ) |x=x2 0 δ x0 x0 x2 ? x1 x1 ? x2 1 1 Y (a, x1 )F (Φ, x2 ) ? x? F (Φ, x2 )Y (a, x1 ) . x? 0 δ 0 δ x0 ?x0 (4.30)

It is well-known that this iterate formula implies the associativity [FHL]. Furthermore, (G3) gives the commutativity for F (·, x). Therefore, F (·, x) satis?es the Jacobi identity ([DL], [FHL], [L1]). Thus F (·, x) is a weak intertwining operator. It is clear that F (·, x) is injective in the sense that F (Φ, x) = 0 implies Φ = 0 for Φ ∈ E (W1 , W2 ). Furthermore, if Φ(x) ∈ G(W1 , W2 ), by de?nition we have F (L(?1)Φ, x)u1 = (L(?1)Φ)(x)u1 = (L(?1) ? Φ(x))u1 = d d Φ(x)u1 = F (Φ, x)u1 . dx dx (4.31) Therefore, F (·, x) is an intertwining operator of type to G(W1 , W2 ). Theorem 4.6 The g (V )-module E (W1 , W2 ) and G(W1 , W2 ) are weak V -modules. Proof 1 . By Proposition 3.12, we get F (Φ, x) = 0

1

W2 G(W1 , W2 )W1

after restricted

for any Φ ∈ J (E (W1 , W2 )).

This was proved directly in [L1].

37

Since F (·, x) injective, J (E (W1 , W2 )) = 0. That is, E (W1 , W2 ) is a weak V -module.

2

Let M be another V -module and let f ∈ HomV (M, G(W1 , W2 )). Then F (f ·, x) is an W2 intertwining operator of type . Since F (·, x) is injective, we obtain an injective MW1 linear map θ: HomV (M, G(W1 , W2 )) → I f → F (f ·, x). On the other hand, for any intertwining operator I (·, x) of type W2 MW1 (4.32)

W2 , it is clear MW1 that I (u, x) ∈ G(W1 , W2 ) for any u ∈ M . Then we obtain a linear map fI from M to G(W1 , W2 ) de?ned by fI (u) = I (u, x). For any a ∈ V, u ∈ M , from the de?nition of Y (a, x0 ) ? I (u, x) we get fI (Y (a, x0 )u) = I (Y (a, x0 )u, x)

1 = Resx1 x? 0 δ

x ? x1 x1 ? x 1 Y (a, x1 )I (u, x) ? x? I (u, x)Y (a, x1 ) 0 δ x0 ?x0

= Y (a, x0 ) ? I (u, x) = Y (a, x0 )fI (u). (4.33)

Thus fI is a V -homomorphism such that F (fI ·, x) = I (·, x). Since F (·, x) is injective, such an fI is unique. Therefore we obtain Theorem 4.7 Let W1 and W2 be V -modules. Then (a) For any V -module M and any W2 intertwining operator I (·, x) of type , there exists a unique V -homomorphism f MW1 from M to G(W1 , W2 ) such that I (u, x) = F (f (u), x) for u ∈ M . W2 (b) The linear space HomV (M, G(W1 , W2 )) is naturally isomorphic to I for MW1 any V -module M . The universal property in Theorem 4.7 looks very much like the universal property for a tensor product in De?nition 3.1 and also in [HL1]. Next, we study the relation between 38

′ G(W1 , W2 ) and the contragredient module of tensor product of W1 and W2 .

Remark 4.8 Let M be any V -module. Then it was proved in [L1] that G(V, M ) ? M . If M = V , then V = G(V, V ). That is, any generalized intertwining operator is a vertex operator. In this special case, this has been proved in [G]. For any two V -modules W1 and W2 , let ?(W1 , W2 ) be the sum of all V -modules inside the weak V -module G(W1 , W2 ). Proposition 4.9

2

Let V be a vertex operator algebra satisfying the following conditions:

(1) There are ?nitely many inequivalent irreducible V -modules. (2) Any V -module is completely reducible. (3) Any fusion rule for three modules is ?nite. Then for any V modules W1 and W2 , ?(W1 , W2 ) is the unique maximal V -module inside the weak module G(W1 , W2 ). Proof. It follows from the condition (2) that ?(W1 , W2 ) is a direct sum of irreducible V -modules. It follows from Theorem 4.7 and the condition (3) that the multiplicity of each irreducible V -module in ?(W1 , W2 ) is ?nite. Therefore ?(W1 , W2 ) is a direct sum of ?nitely many irreducible V -modules. That is, ?(W1 , W2 ) is a V -module. By the de?nition of ?(W1 , W2 ), it is clear that ?(W1 , W2 ) is the unique maximal V -module inside the weak V -module G(W1 , W2 ). 2

Let V be a vertex operator algebra satisfying the conditions (1)-(3) of Proposition 4.9 ? (·, x) be the restriction of F (·, x) on and let W1 and W2 be any two V -modules. Let F W2 ? (·, x) is an intertwining operator of type ?(W1 , W2 ) so that F such that ?(W1 , W2 )W1 ? (Φ, x) = Φ(x) F for any Φ ∈ ?(W1 , W2 ). (4.34)

? t (·, x) of F ? (·, x) is an intertwining Then by Proposition 2.6, the transpose operator F W2 operator of type . Furthermore, it follows from Proposition 2.6 that W1 ?(W1 , W2 )

2

A similar result has also been obtained in [HL0-4].

39

? t )′ (·, x) is an intertwining operator of type (F

(?(W1 , W2 ))′ . ′ W1 W2

Theorem 4.10 If V satis?es the conditions (1)-(3) of Proposition 4.9, then the pair ? t )′ (·, x)) is a tensor product for the ordered pair (W1 , W ′ ) in the category ((?(W1 , W2 )′ , (F 2 of V -modules. Proof. Let W be any V -module and let I (·, x) be any intertwining operator of type W ′ t ′ . It follows from Proposition 2.6 that (I ) (·, x) is an intertwining operator of W1 W2 W2 type . From Theorem 4.6, there exists a (unique) V -homomorphism ψ from W ′ W ′ W1 ? (ψ (w ′ ), x) for any w ′ ∈ M ′ . It follows from the to G(W1 , W2 ) such that (I ′ )t (w ′, x) = F de?nition of ?(W1 , W2 ) that ψ is a V -homomorphism from W ′ to ?(W1 , W2 ). Therefore, we obtain a V -homomorphism ψ ′ from (?(W1 , W2 ))′ to W . For any w ′ ∈ W ′, u1 ∈

′ W1 , u′2 ∈ W2 , by using FLM’s conjugation formulas [FHL] we obtain

? t )′ (u1 , x)u′ w ′ , ψ ′ (F 2 = = = = ? t (exL(1) (eπi x?2 )L(0) u1 , x?1 )ψw ′ , u′2 F I ′ (exL(1) (eπi x?2 )L(0) u1 , x?1 )w ′, u′2 w ′ , I (ex

?1 L(1)

(eπi x2 )L(0) exL(1) (eπi x?2 )L(0) u1 , x)u′2 (4.35)

w ′ , I (e2πiL(0) u1 , x)u′2 .

For any V -module M , we de?ne a linear endomorphism tM of M by: tM (u) = e2πiL(0) u for u ∈ M . Then one can easily prove that tM is a V -automorphism of M so that tM is a ? t )′ (·, x) = I (·, x). The uniqueness scalar if M is irreducible. Let tW1 = α. Then α?1 ψ ′ (F of α?1 ψ ′ follows from the uniqueness of ψ . Then the proof is complete. 2

Remark 4.11 It was was proved in [DLM2] that the category C of all weak V -modules is a semisimple category for vertex operator algebras L(?, 0), associated to an integrable highest weight module of level ? for an a?ne Lie algebra, L(cp,q , 0), associated to the 40

irreducible highest weight module for the Virasoro algebra with central charge cp,q = 1 ?

6(p?q )2 , pq

the moonshine module vertex operator algebra V ? and VL , associated to any even

positive-de?nite lattice L. Thus, for these vertex operator algebras, we have G(W1 , W2 ) = ?(W1 , W2 ). Let U be an irreducible g (V )0 -module on which L(0) acts as a scalar h. De?ne g (V )? U = 0. Then U becomes a (g (V )? + g (V )0 )-module. Form the induced g (V )module Ind(U ) = U (g (V )) ?U (g(V )? +g(V )0 ) U . Set V (U ) = Ind(U )/J (Ind(U )). Then V (U ) is a lowest weight weak V -module. If V is rational, it follows from the complete reducibility of V (U ) that V (U ) is irreducible. The following is our generalized nuclear democracy theorem of Tsuchiya and Kanie [TK]. Theorem 4.12 Let W1 and W2 be V -modules. Let U be a g (V )0 -module on which L(0) acts as a scalar h and let I0 (·, x) be a linear injective map from U to (HomC (W1 , W2 )) {x} such that for any u ∈ U , I0 (u, x) satis?es the truncation condition (G1), the L(?1)-bracket formula (G2) and the following condition: (x1 ? x2 )wta?1 YW2 (a, x1 )I0 (u, x2 ) ? (?x2 + x1 )wta?1 I0 (u, x2 )YW1 (a, x1 )

1 = x? 1 δ

x2 I0 (awta?1 u, x2 ) x1

(4.36)

for any a ∈ V, u ∈ U . Then there exists a lowest weight weak V -module W with U as its lowest weight subspace generating W and there is a unique intertwining operator I (·, x) W2 of type extending I0 (·, x). In particular, if V is rational and U is irreducible, W W1 W is irreducible. Proof. Since (x1 ? x2 )δ x2 x1 = 0, we have (4.37)

(x1 ? x2 )wta+i YW2 (a, x1 )I0 (u, x2 ) = (?x2 + x1 )wta+i I0 (u, x2 )YW1 (a, x1 )

41

for a ∈ V, u ∈ U, i ∈ N. Then by de?nition I0 (u, x) ∈ G(W1 , W2 ) for any u ∈ U and am ? I0 (u, x) = 0 for m ≥ wta, awta?1 ? I0 (u, x) = I0 (awta?1 u, x). (4.38) (4.39)

? := {I (u, x)|u ∈ U } ? G(W1 , W2 ). Then U ? is a g (V )0 -submodule of G(W1 , W2 ) Set U ? as a g (V )0 -module is isomorphic to U . Let W = U (g (V ))U ? be the V or g (V )and U ? is a lower-truncated Z-graded weak V submodule of G(W1 , W2 ). Then W = U (g+ )U W2 module generated by U . Then we have a natural intertwining operator of type . W W1 The uniqueness is clear. Then the proof is complete. 2. Let g be a ?nite-dimensional simple Lie algebra, let h be a Cartan subalgebra, let ? be the root system of g and let ·, · be the normalized Killing form on g [K]. For any linear functional λ ∈ h? , we denote by L(λ) the irreducible highest weight g-module with highest weight λ. ? = C[t, t?1 ] ? g ⊕ Cc be the corresponding a?ne Lie algebra and let g ?=g ? ⊕ Cd be Let g the extended a?ne algebra. For any ? ∈ C, λ ∈ h? , let L(?, λ) be the irreducible highest ? be the loop g ? -module of level ?. For any g-module U , let U ? -module C[t, t?1 ] ? U weight g of level 0. It is well known (cf. [FZ], [L1]) that each L(?, 0) has a vertex operator algebra structure except when ? is the negative dual Coxeter number. Then we have the following nuclear democracy theorem of Tsuchiya and Kanie. (To be precise, this was proved only for g = sl2 in [TK].) Proposition 4.13 Let ? be a positive integer and let W2 , W3 be L(?, 0)-modules. Let λ be a linear functional on h, let L(λ) be the irreducible highest weight g-module with highest weight λ and let Φ(·, x) be a nonzero linear map from L(λ) to Hom(W2 , W3 ){x} such that [a(m), Φ(u, x)] = xm Φ(a(0)u, x); [L(?1), Φ(u, x)] = 42 d Φ(u, x) dx (4.40) (4.41)

for any a ∈ g ? L(?, 0), u ∈ L(λ), m ∈ Z, Then L(?, λ) is an irreducible L(?, 0)-module and there is a unique intertwining operator I (·, x) on L(?, λ) in the sense of [FHL] extending Φ(·, x). Proof. Writing (4.40) in terms of generating functions, we obtain

1 [Y (a, x1 ), Φ(u, x2 )] = x? 2 δ

x1 Φ(a0 u, x2 ). x2

(4.42)

Since (x1 ? x2 )δ

x1 x2

= 0, we get (x1 ? x2 )[Y (a, x1 ), Φ(u, x2 )] = 0 (4.43)

for any a ∈ g, u ∈ L(λ). Since g generates L(?, 0) as a vertex operator algebra, similar to the proof of Proposition 4.5 it follows from the proof of Proposition 3.2.7 in [L1] that I (u, x) satis?es (G3) for any a ∈ L(?, 0). Furthermore, (4.40) implies (G2). Therefore Φ(u, x) ∈ G(W2 , W3 ) for u ∈ L(λ). From (4.20) and (4.43) we obtain a(0) ? Φ(u, x) = [a(0), Φ(u, x)] = Φ(a(0)u, x); a(m) ? Φ(u, x) = 0 for any a ∈ g, m > 0, u ∈ L(λ). (4.44) (4.45)

Then Φ is a g-homomorphism. Consequently, L(λ) is embedded into G(W2 , W3 ) by Φ. Let W be the V -submodule generated by L(λ). Then W is a certain quotient module of M (?, λ). From the rationality of L(?, 0), we get W = L(?, λ). By Theorem 4.7, we obtain W3 ) an intertwining vertex operator I (·, x) of type . The uniqueness is clear. L(?, λ)W2 ) Then the proof is complete. 2 Remark 4.14 Under the conditions of Proposition 4.13, we obtain an intertwining opW3 erator I (·, x) of type . It follows from commutator formula (2.4) that L(?, λ)W2 [L(m), I (u, x)] = xm (m + 1)h + x 43 d I (u, x) dx

for u ∈ L(λ), where h is the lowest weight of L(?, λ). Thus [L(m), Φ(u, x)] = xm (m + 1)h + x d Φ(u, x) dx for u ∈ L(λ), m ∈ Z. (4.46)

? -module was used to de?ne intertwining operIn many references the notion of loop g ators. Next we shall discuss this issue. Suppose L(?, λi ) (i = 1, 2, 3) are L(?, 0)-modules. Let I (·, x) be an intertwining operL(?, λ3 ) ator of type . As before, we set L(?, λ1 )L(?, λ2 ) I o (u1 , x) = xh1 +h2 ?h3 I (u1 , x) =

n∈Z

Iu1 (n)x?n?1

for any u1 ∈ L(?, λ1 ).

(4.47)

Then (the second identity follows from Proposition 2.5) [a(m), Iu (n)] = Iau (m + n); [L(0), Iu (n)] = (h3 ? h2 ? n ? 1)Iu (n) (4.48) (4.49)

for a ∈ g, u ∈ L(λ1 ), m, n ∈ Z. Then I o (·, x) naturally gives rise to a linear map RI from

C[t, t?1 ] ? L(λ1 ) ? L(?, λ2 )

to L(?, λ3 ) such that

RI (tn ? u1 ? u2 ) = Iu1 (n)u2 for u1 ∈ L(λ1 ), u2 ∈ L(?, λ2 ), n ∈ Z. ? (λ1 ) ? L(?, λ2 ) ? -homomorphism from L (4.48) is equivalent to say that the map RI is a g to L(?, λ3 ). From (4.49) we get L(0)(tn ? u1 ? u2 ) = tn ? u1 ? L(0)u2 + (h3 ? h2 ? n ? 1)(tn ? u1 ? u2 ) for u1 ∈ L(λ1 ), u2 ∈ L(?, λ2 ), n ∈ Z. Then (h3 ? L(0))(tn ? u1 ? u2 ) = tn ? u1 ? (h2 ? L(0))u2 + (n + 1)(tn ? u1 ? u2 ). (4.51)

d ? (λ1 ) as a g ? -module ? -module with d = (1 + t dt ) ? 1 and view L(?, λ) as a g View L

(4.50)

with d = h ? L(0) where h is the lowest weight. Then it follows from (4.51) that RI is a 44

? -homomorphism. Then we obtain a linear map: g R: I L(?, λ3 ) L(?, λ1 )L(?, λ2 ) ? → Homg ? (L(λ1 ) ? L(?, λ2 ), L(?, λ3 )); (4.52)

I (·, x) → RI . In some references, an intertwining operator of type

L(?, λ3 ) is de?ned to L(?, λ1 )L(?, λ2 ) ? (λ1 ) ? L(?, λ2 ) to L(?, λ3 ). The following proposition ? -module homomorphism from L be a g asserts that this de?nition is equivalent to FHL’s de?nition. L(?, λ3 ) is naturally isoL(?, λ1 )L(?, λ2 ) ? (λ) ? L(?, λ2 ) to L(?, λ3 ). ? -homomorphisms from L morphic to the space of g Proposition 4.15 The intertwining operator space I Proof. From the above discussion we see that for any intertwining operator I (·, x) L(?, λ3 ) ? -homomorphism RI . Conversely, let f be a of type , we obtain a g L(?, λ1 )L(?, λ2 ) ? (λ) ? L(?, λ2 ) to L(?, λ3 ). Then we de?ne a linear map Φ(·, x) ? -homomorphism from L g from L(λ1 ) to Hom(L(?, λ2 ), L(?, λ2 )){x} such that Φ(u1 , x)u2 = xh3 ?h1 ?h2

n∈Z

f (tn ? u1 ? u2 )x?n?1

(4.53)

for u1 ∈ L(λ1 ), u2 ∈ L(?, λ2 ). Then Φ(·, x) satis?es (4.40) and [L(0), Φ(u1 , x)] = h1 + x d Φ(u1 , x) dx for u1 ∈ L(λ1 ). (4.54)

Then Φ(u1 , x) ∈ E (L(?, λ2 ), L(?, λ3 )) for any u1 ∈ L(λ1 ). Similar to the proof of Proposition 4.13, L(?, λ1 ) is a submodule of E (L(?, λ2 ), L(?, λ3 )) generated by L(λ1 ) and there is a weak intertwining operator I (·, x) from L(?, λ) to Hom(L(?, λ2 ), L(?, λ3)){x}. It is well known (cf. [HL0-4], [FLM]) that under the commutator formula (2.4), the L(?1)-bracket formula (I2) is equivalent to the L(0)-bracket formula. Thus Φ(u1 , x) ∈ G(L(?, λ2 ), L(?, λ3 )) for u1 ∈ L(λ1 ). Since L(λ1 ) generates L(?, λ1 ) by U (? g), it follows from Proposition 4.5 that L(?, λ1 ) ? G(L(?, λ2 ), L(?, λ3 )). Thus I (·, x) is an intertwining operator. Then the proof is complete. 2 45

Let L(c, h) be the irreducible module of the Virasoro algebra Vir with central charge c and lowest weight h. It is well known (cf. [FZ], [H1], [L1]) that L(c, 0) is a vertex operator algebra. Suppose that L(c, h1 ) and L(c, h2 ) are two modules for the vertex operator algebra L(c, 0). Let Φ(x) ∈ (HomC (L(c, h1 ), L(c, h2 ))) {x} such that [L(m), Φ(x)] = xm (m + 1)h + x for some complex number h. That is,

1 [Y (ω, x1 ), Φ(x2 )] = x? 1 δ

d Φ(x) dx

(4.55)

x2 x1

d 2 ′ x2 Φ(x2 ) + hx? Φ(x2 ). 1 δ dx2 x1

(4.56)

Similarly to the proof of Proposition 4.9 we get Φ(x) ∈ G(L(c, h1 ), L(c, h2 )) and Φ(x) generates a L(c, 0)-module M which is a lowest weight Virasoro algebra module with lowest weight h in G(L(c, h1 ), L(c, h2 )). If c = 1 ?

6(p?q )2 , pq

where p, q ∈ {2, 3 · · ·} are

relatively prime, L(c, 0) is rational ([DMZ], [W]). Therefore M = L(c, h). Then we obtain L(c, h2 ) an intertwining vertex operator of type . Thus we have L(c, h)L(c, h2 ) Proposition 4.16 If c = 1 ?

6(p?q )2 , pq

where p, q ∈ {2, 3 · · ·} are relatively prime, let

L(c, h1 ), L(c, h2 ) be L(c, 0)-modules and let Φ(x) satisfy (4.55). Then there exists a unique L(c, h2 ) intertwining vertex operator of type extending Φ(x). L(c, h)L(c, h2 )

5

Appendix

The main purpose of this appendix is to give an example to show that the generalized form of the nuclear democracy theorem may not be true if V is not rational. We use the same notions as in Section 4. Let ? be a positive integer and let C? be the (C[t] ? g + Cc)-module such that c acts as ? and C[t] ? g acts as zero. Set M (?, C) = U (g)U (C[t]⊕g+Cc)C? . ? -modules of level ? is a Then M (?, C) is a vertex operator algebra and any restricted g M (?, C)-module (cf. [FZ], [L1]). Consequently, M (?, C) is irrational. Since L(?, 0) is 46

rational, we may choose an λ such that L(?, λ) is not a L(?, 0)-module. Let Φ(x) be the identity map from L(?, λ) to L(?, λ). Then Φ satis?es all the conditions in Theorem 4.12. If we could extend Φ to an intertwining operator on L(?, 0), then we would have an L(?, λ) intertwining operator of type so that L(?, λ) would be a L(?, 0)-module. L(?, 0)L(?, λ) This would contradict the assumption that L(?, λ) is not a L(?, 0)-module.

References

[BPZ] A. Belavin, A. M. Polyakov, A. B. Zamolodchikov, In?nite conformal symmetries in two-dimensional quantum ?eld theory, Nucl. Phys. B 241 (1984), 333-380. [B] R. E. Borcherds, Vertex algebras, Kac-Moody algebras, and the Monster, Proc. Natl. Acad. Sci. USA 83 (1986), 3068-3071. [D1] C. Dong, Vertex algebras associated with even lattices, J. Alg. 161 (1993), 245265. [D2] C. Dong, Representations of the moonshine module vertex operator algebra, Contemporary Math. 175 (1994), 27-35. [DL] C. Dong and J. Lepowsky, Generalized Vertex Algebras and Relative Vertex Operators, Progress in Math., Vol. 112, Birkh¨ auser, Boston, 1993. [DLM1] C. Dong, H. Li and G. Mason, Twisted representations of vertex operator algebras, preprint, q-alg/9509005. [DLM2] C. Dong, H. Li and G. Mason, Regularity of rational vertex operator algebras, Adv. in Math. 125 (1997). [DLM3] C. Dong, H. Li and G. Mason, Hom functor and the associativity of tensor products of modules for vertex operator algebras, J. Alg. 188 (1997), 443-475.

47

[DMZ] C. Dong, G. Mason and Y.-C. Zhu, Discrete series of the Virasoro algebra and the moonshine module, Vol. 56, Part 2, Algebraic Groups and Their Generalizations: Quantum and In?nite-Dimensional Methods, William J. Haboush, Brian J. Parshall, editors, American Math. Soc., Providence, RI 1994, pp. 295-316. [FFR] Alex J. Feingold, Igor B. Frenkel and John F. X. Ries, Spinor Construction of Vertex Operator Algebras, Triality, and E8 , Contemporary Math. 121 (1991). [F] [FHL] M. Finkelberg, Fusion categories, Ph.D. thesis, Harvard University, 1993. I. Frenkel, Y.-Z. Huang and J. Lepowsky, On axiomatic approaches to vertex operator algebras and modules, preprint, 1989; Memoirs Amer. Math. Soc. 104, 1993. [FLM] I. Frenkel, J. Lepowsky and A. Meurman, Vertex Operator Algebras and the Monster, Pure and Appl. Math., Vol. 134, Academic Press, Boston, 1988. [FZ] I. Frenkel and Y.-C. Zhu, Vertex operator algebras associated to representations of a?ne and Virasoro algebras, Duke Math. J. 66 (1992), 123-168. [G] P. Goddard, Meromorphic conformal ?eld theory, in: Kac V.G. (ed.) In?nitedimensional Lie Algebras and Lie Groups. Proceedings of CIRM-Luminy Conference 1988, p. 556. Singapore, New Jersey, Hong Kong: World Scienti?c 1989. [H1] Y.-Z. Huang, Vertex operator algebras and conformal ?eld theory, Intl. J. Mod. Phys. A7 (1992), 2109-2151. [H2] Y.-Z. Huang, A theory of tensor products for module categories for a vertex operator algebra, IV, J. Pure Appl. Alg. 100 (1995), 173-216. [HL0] Y.-Z. Huang and J. Lepowsky, Toward a theory of tensor product for representations for a vertex operator algebra, in Proc. 20th International Conference on 48

(1)

Di?erential Geometric Methods in Theoretical Physics, New York, 1991, ed. S. Catto and A. Rocha, World Scienti?c, Singapore, 1992, Vol. 1, 344-354. [HL1] Y.-Z. Huang and J. Lepowsky, A theory of tensor product for module category of a vertex operator algebra, I, Selecta Mathematica, Vol. 1 (1995), 699-756. [HL2] Y.-Z. Huang and J. Lepowsky, A theory of tensor product for module category of a vertex operator algebra, II, Selecta Mathematica, Vol. 1 (1995), 757-786. [HL3] Y.-Z. Huang and J. Lepowsky, A theory of tensor product for module category of a vertex operator algebra, III, J. Pure Appl. Alg. 100 (1995), 141-171. [HL4] Y.-Z. Huang and J. Lepowsky, Tensor products of modules for a vertex operator algebra and vertex tensor categories, in: Lie Theory and Geometry, in honor of Bertram Kostant, ed. R. Brylinski, J.-L. Brylinski, V. Guillemin, V. Kac, Birkh¨ auser, 1994, 349-383. [J] N. Jacobson, Basic Algebra, Vol. II, W. H. Freeman and Company, San Francisco, 1980. [KL0] D. Kazhdan and G. Lusztig, A?ne Lie algebras and quantum groups, Duke Math. J. 2 (1991), 21-29. [KL1] D. Kazhdan and G. Lusztig, Tensor structures arising from a?ne Lie algebras, I, J. Amer. Math. Soci. Vol. 64 (1993), 905-9047 [KL2] D. Kazhdan and G. Lusztig, Tensor structures arising from a?ne Lie algebras, II, J. Amer. Math. Soci. Vol. 64 (1993), 9047-1011. [Le] J. Lepowsky, Generalized Verma modules, loop space cohomology and Macdonald-type identities, Ann. Sci. Ecole Norm. Sup. 12 (1979), 169-234.

49

[L1]

H.-S. Li, Local systems of vertex operators, vertex superalgebras and modules, J. Pure Appl. Alg. 109 (1996), 143-195.

[L2]

H.-S. Li, Representation theory and tensor product theory for vertex operator algebras, Ph.D. thesis, Rutgers University, 1994.

[MS]

G. Moore and N. Seiberg, Classical and quantum conformal ?eld theory, Comm. Math. Phys. 123 (1989), 177-254.

[TK]

A. Tsuchiya and Y. Kanie, Vertex operators in conformal ?eld theory on P1 and monodromy representations of braid group, in: Conformal Field Theory and Solvable Lattice Models, Advanced Studies in Pure Math., Vol. 16, Kinokuniya Company Ltd., Tokyo, 1988, 297-372.

[W]

W.-Q. Wang, Rationality of Virasoro vertex operator algebras, Duke Math. J. IMRN, Vol. 71, No. 1 (1993), 197-211.

[Z]

Y.-C. Zhu, Vertex operator algebras, elliptic functions and modular forms, Ph.D. thesis, Yale University, 1990.

50

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