Bimodules associated to vertex operator algebras Chongying Dong1 Department of Mathematics 6 University of California 0 Santa Cruz, CA 95064 0 2 Cuipo Jiang2 n Department of Mathematics a Shanghai Jiaotong University J 7 Shanghai 200030 China 2 ] A Abstract Q . Let V be a vertex operator algebra and m,n ≥ 0. We construct an A (V)- h n t A (V)-bimodule A (V) which determines the action of V from the level m sub- a m n,m m space to level n subspace of an admissible V-module. We show how to use A (V) n,m [ to construct naturally admissible V-modules from Am(V)-modules. We also deter- mine the structure of A (V) when V is rational. 2000MSC:17B69 2 n,m v 6 2 1 Introduction 6 1 0 The representation theory for a vertex operator algebra [B], [FLM] has been studied 6 largely in terms of representation theory for various associative algebras associated to the 0 / vertex operator algebra (see [Z], [KW], [DLM2]-[DLM4], [MT], [DZ], [X]). A sequence of h t associative algebras An(V) for n ≥ 0 was introduced in [DLM3] to deal with the first n+1 a m homogeneous subspaces of an admissible module. These algebras extend and generalize the associative algebra A(V) constructed in [Z]. The main idea of A (V) theory is how : n v to use the first few homogeneous subspaces of a module to determine the whole module. i X From this point of view, the A (V) theory is an analogue of the highest weight module n r a theory for semisimple Lie algebras in the field of vertex operator algebra. Let M = ∞ M(n) be an admissible V-module with M(0) 6= 0. (cf. [DLM2]).Then n=0 each M(k) is an A (V)-module for k ≤ n [DLM3]. On the other hand, given an A (V)- n n L module U which cannot factor through A (V) one can construct a Verma type admis- n−1 sible V-module M¯(U) such that M¯(U)(n) = U. Also V is rational if and only if A (V) n is semisimple for all n. So the collection of associative algebras A (V) determine the n representation theory of V in some sense. However, A (V) preserves each homogeneous n subspace M(m) for m ≤ n and cannot map M(s) to M(t) if s 6= t. The goals of the present paper are to alleviate this situation. 1 Supported by NSF grants, China NSF grant 10328102 and a Faculty research grant from the Uni- versity of California at Santa Cruz. 2 Supported in part by China NSF grant 10571119. 1 Given two nonnegative integers m,n we will construct an A (V)-A (V)-bimoulde n m A (V) with the property that for any A (V)-module U which cannot factor through n,m m A (V) one can associate a Verma type admissible V-module M(U) = ∞ M(U)(n) m−1 k=0 such that M(U)(n) = A (V) ⊗ U. The action of V on M(U)(n) is determined n,m Am(V) L by a canonical bimodule homomorphism from A (V)⊗ A (V) to A (V). Also, p,n An(V) n,m p,m for a given admissible V-module W = W(k) with W(0) 6= 0, there is an A (V)- k≥0 n Am(V)-bimodule homomorphism from An,m(V) to HomC(W(m),W(n)). So the collection of A (V) for all m,n ∈ Z determiLne the action of vertex operator algebra V on n,m + its admissible module W completely. This, in fact, is our original motivation to define A (V). n,m If V is a rational vertex operator algebra, then V has only finitely many irreducible admissible V-modules up to isomorphism and each irreducible V-module is ordinary (see [DLM2]). In this case we let W1,...,Ws be the inequivalent irreducible admissible V- modules such that Wi(0) 6= 0. Then A (V) is the direct sum of full matrix algebras n An(V) = si=1 nk=0EndC(Wi(k)). We show in this paper that if V is rational then L L min{m,n} s An,m(V) ∼= HomC(Wi(m−l),Wi(n−l)) . ! l=0 i=1 M M The structure of A (V) for general V will be studied in a sequel to this paper. n,m We have already mentioned the A (V) theory. In fact, the Verma type admissible n V-module M(U) has been constructed and denoted by M¯(U) in [DLM3] using the idea of induced module in Lie theory. But our work in this paper leads to a strengthening of this old construction. While the old construction in [DLM3] was given as an abstract quotient of certain induced module for certain Lie algebras, the new construction is explicit and each homogeneous subspace M(U)(n) is obvious. In the case that U = A (V) we see m immediately that A (V) is an admissible V-module. We expect that the study of n≥0 n,m bimodules A (V) will lead to a proof of some well known conjecture in representation n,m L theory. The result in this paper is also related to some results obtained in [MNT] where the universal enveloping algebraof avertex operatoralgebra isused instead ofvertex operator algebra itself in this paper. The paper is organized as follows: In Section 2 we introduce the A (V)-A (V)- n m bimodule A (V) with lot of technical calculations. In Section 3 we discuss the various n,m properties of A (V) such as the A (V)-A (V)-bimodule epimorphism from A (V) n,m n m n,m to A (V) induced from the identity map on V, isomorphism between A (V) n−1,m−1 n,m and A (V) and bimodule homomorphism from A (V) ⊗ A (V) to A (V). m,n n,p Ap(V) p,m n,m In Section 4 we first give an A (V)-A (V)-bimodule homomorphism from A (V) to n m n,m HomC(W(m),W(n)) for any admissible V-module W = ∞ W(k). We also show how k=0 to construct anadmissible V-modulefromanA (V)-modulewhich cannot factor through m L A (V) by using A (V). In addition we show that A (V) and A (V) are the same m−1 n,m n n,n although A (V) as a quotient space of A (V) seems much smaller from the definition. n,n n The explicit structure of A (V) is determined if V is rational. n,m 2 We assume that the reader is familiar with the basic knowledge on the representation theory such as of weak modules, admissible modules and (ordinary) modules as presented in [DLM1]-[DLM2] (also see [FLM], [LL]). 2 A (V )-A (V )-bimodule A (V ) n m n,m Let V = (V,Y,1,ω) be a vertex operator algebra. An associative algebra A (V) for any n nonnegative integer n has been constructed in [DLM3] to study the representation theory for vertex operator algebras (see below). For m,n ∈ Z , we will construct an A (V)- + n A (V)-bimodule A (V) in this section. It is hard to see in this section why A (V) is m n,m n,m so defined and the motivation for defining A (V) comes from the representation theory n,m of V (see Section 4 below). For homogeneous u ∈ V, v ∈ V and m,n,p ∈ Z , define the product ∗n on V as + m,p follows p m+n−p+i (1+z)wtu+m u∗n v = (−1)i Res Y(u,z)v. m,p i z zm+n−p+i+1 i=0 (cid:18) (cid:19) X If n = p, we denote ∗n by ∗¯n, and if m = p, we denote ∗n by ∗n, i.e., m,p m m,p m m n+i (1+z)wtu+m u∗n v = (−1)i Res Y(u,z)v, m i z zn+i+1 i=0 (cid:18) (cid:19) X n m+i (1+z)wtu+m u∗¯nv = (−1)i Res Y(u,z)v. m i z zm+i+1 i=0 (cid:18) (cid:19) X The products u∗n v andu∗¯nv will induce the right A (V)-moduleandleft A (V)-module m m m n structure on A (V) which will be defined later. n,m If m = n, then u∗n v and u∗¯nv are equal, and have been defined in [DLM3]. As in m m [DLM3] we will denote the product by u∗ v in this case. n Let O′ (V) be the linear span of all u◦n v and L(−1)u+(L(0)+m−n)u, where for n,m m homogeneous u ∈ V and v ∈ V, (1+z)wtu+m u◦n v = Res Y(u,z)v. m z zn+m+2 Again if m = n, u◦n v has been defined in [DLM3] where it was denoted by u◦ v. Let m n O (V) = O′ (V) (see [DLM3]). The following theorem is obtained in [DLM3]. n n,n Theorem 2.1. The A (V) is an associative algebra with product ∗ with identity 1 + n n O (V). n We will present more results on A (V) and its connection with the representation n theory of V from [DLM3] later on when necessary. In order to define A (V) we need n,m several lemmas. In the case that m = n most of these lemmas have been proved in [DLM3]. But when m 6= n even if the old proofs given in [DLM3] work but they are much more complicated and need a lot of modifications. Sometimes we need to find totally new proofs. 3 Lemma 2.2. For any u,v ∈ V, u◦n v lies in O (V)∗¯nV. m n m Proof: Let u,v ∈ V. Then n m+i (1+z)wtu+1+m d (L(−1)u)∗¯nv = (−1)i Res Y(u,z)v m i z zm+i+1 dz i=0 (cid:18) (cid:19) X n m+i (1+z)wtu+m = −(wtu+m+1) (−1)i Res Y(u,z)v i z zm+i+1 i=0 (cid:18) (cid:19) X n m+i (1+z)wtu+m+1 + (m+i+1)(−1)i Res Y(u,z)v i z zm+i+2 i=0 (cid:18) (cid:19) X n m+i (1+z)wtu+m = −(wtu+m+1) (−1)i Res Y(u,z)v z i zm+i+1 i=0 (cid:18) (cid:19) X n m+i (1+z)wtu+m + (m+i+1)(−1)i Res Y(u,z)v i z zm+i+1 i=0 (cid:18) (cid:19) X n m+i (1+z)wtu+m + (m+i+1)(−1)i Res Y(u,z)v z i zm+i+2 i=0 (cid:18) (cid:19) X Thus (L(−1)u+L(0)u)∗¯nv m n m+i (1+z)wtu+m = i(−1)i Res Y(u,z)v z i zm+i+1 i=0 (cid:18) (cid:19) X n m+i (1+z)wtu+m + (m+i+1)(−1)i Res Y(u,z)v i z zm+i+2 i=0 (cid:18) (cid:19) X n+m (1+z)wtu+m = (n+m+1)(−1)n Res Y(u,z)v m z zn+m+2 (cid:18) (cid:19) n+m = (n+m+1)(−1)n u◦n v. m m (cid:18) (cid:19) Note that L(−1)u+L(0)u ∈ O (V). The proof is complete. n The next lemma is motivated by the commutator relation of vertex operators and will relate the two products u∗¯nv and u∗n v. m m Lemma 2.3. If u,v ∈ V, p ,p ,m ∈ Z with p +p −m ≥ 0, then 1 2 + 1 2 u∗p1+p2−mv −v ∗p1+p2−m u−Res (1+z)wtu−1+m−p2Y(u,z)v ∈ O′ (V). m,p1 m,p2 z p1+p2−m,m Proof: From the definition of O′ (V), one can easily verify that p1+p2−m,m −z Y(v,z)u ≡ (1+z)−wtu−wtv−2m+p1+p2Y(u, )v 1+z 4 modulo O′ (V) (cf. [Z] and [DLM2]). Hence p1+p2−m,m p2 p +i (1+z)wtv+m v ∗p1+p2−m u = (−1)i 1 Res Y(v,z)u m,p2 i z zp1+i+1 i=0 (cid:18) (cid:19) X p2 p +i (1+z)wtv+m −z ≡ (−1)i 1 Res (1+z)−wtu−wtv−2m+p1+p2Y(u, )v i z zp1+i+1 1+z i=0 (cid:18) (cid:19) X (mod O′ (V)) p1+p2−m,m p2 p +i (1+z)wtu+i−1+m−p2 = (−1)p1 1 Res Y(u,z)v. z i zp1+i+1 i=0 (cid:18) (cid:19) X Recall the definition of u∗p1+p2−mv. Then m,p1 u∗p1+p2−mv −v∗p1+p2−m u ≡ Res A (z)(1+z)wtu−1+m−p2Y(u,z)v, m,p1 m,p2 z p1,p2 where p1 p +i (1+z)p2+1 p2 p +i (1+z)i A (z) = (−1)i 2 − (−1)p1 1 . p1,p2 i zp2+i+1 i zp1+i+1 i=0 (cid:18) (cid:19) i=0 (cid:18) (cid:19) X X The lemma now follows from Proposition 5.1 in the Appendix. The proof of the following lemma is fairly standard (cf. [DLM3] and [Z]). Lemma 2.4. For homogeneous u,v ∈ V, and integers k ≥ s ≥ 0, (1+z)wtu+m+s Res Y(u,z)v ∈ O′ (V). z zn+m+2+k n,m Lemma 2.5. We have V¯∗nO′ (V) ⊆ O′ (V), O′ (V)∗n V ⊆ O′ (V). m n,m n,m n,m m n,m Proof: For homogeneous u,v ∈ V and w ∈ V, u∗¯n(v◦n w) m m n m+i (1+z )wtv+m (1+z )wtv+m ≡ (−1)i Res 1 Y(u,z )Res 2 Y(v,z )w i z1 zm+i+1 1 z2 zn+m+2 2 i=0 (cid:18) (cid:19) 1 2 X n m+i (1+z )wtv+m (1+z )wtv+m − (−1)i Res 2 Y(v,z )Res 1 Y(u,z )w i z2 zn+m+2 2 z1 zm+i+1 1 i=0 (cid:18) (cid:19) 2 1 X n ∞ m+i wtu+m −m−i−1 = (−1)i i j k i=0 (cid:18) (cid:19) j≥0 (cid:18) (cid:19)k=0(cid:18) (cid:19) X X X (1+z )wtu+wtv+2m−j(z −z )j+k 2 1 2 ·Res Res Y(Y(u,z −z )v,z )w z2 z1−z2 z2m+n+i+3+k 1 2 2 2 n ∞ m+i wtu+m −m−i−1 = (−1)i i j k i=0 (cid:18) (cid:19) j≥0 (cid:18) (cid:19)k=0(cid:18) (cid:19) X X X (1+z )wtu+wtv+2m−j 2 ·Res Y(u v,z )w. z2 z2m+n+i+3+k j+k 2 2 5 Note that the weight of u v is wtu + wtv − j − k − 1. By Lemma 2.4 we see that j+k u∗¯n(v ◦n w) lies in O′ (V). m m n,m By Lemma 2.3, we have u∗¯n(v ◦n w)−(v ◦n w)∗nu m m m m ≡ Res (1+z)wtu−1Y(u,z)(v ◦n w) z m (1+z )wtv+m = Res (1+z )wtu−1Res 2 Y(u,z )Y(v,z )w z1 1 z2 zn+m+2 1 2 2 wtu−1 (1+z )wtu+wtv+m−1−j(z −z )j 2 1 2 ≡ Res Res j z2 z1−z2 zn+m+2 j≥0 (cid:18) (cid:19) 2 X ·Y(Y(u,z −z )v,z )w 1 2 2 wtu−1 (1+z )wtu+wtv+m−1−j = Res 2 Y(u v,z )w ∈ O′ (V) j z2 zn+m+2 j 2 n,m j≥0 (cid:18) (cid:19) 2 X which is a vector in O′ (V) by the definition of O′ (V). As a result, (v ◦n w)∗nu ∈ n,m n,m m m O′ (V). n,m Next we deal with L(−1)u+(L(0)+m−n)u ∈ O′ (V). As before we assume that n,m u is homogeneous. Then (L(−1)u+(L(0)+m−n)u)∗nv m m n+i (1+z)wtu+m+1 = (−1)i Res Y(L(−1)u,z)v i z zn+i+1 i=0 (cid:18) (cid:19) X m n+i (1+z)wtu+m + (−1)i (wtu+m−n)Res Y(u,z)v z i zn+i+1 i=0 (cid:18) (cid:19) X m n+i (1+z)wtu+m = (−1)i+1 (wtu+m+1)Res Y(u,z)v i z zn+i+1 i=0 (cid:18) (cid:19) X m n+i (1+z)wtu+m+1 + (−1)i (n+i+1)Res Y(u,z)v z i zn+i+2 i=0 (cid:18) (cid:19) X m n+i (1+z)wtu+m + (−1)i (wtu+m−n)Res Y(u,z)v i z zn+i+1 i=0 (cid:18) (cid:19) X m n+i (1+z)wtu+m = (−1)i+1 (n+1)Res Y(u,z)v z i zn+i+1 i=0 (cid:18) (cid:19) X m n+i (1+z)wtu+m + (−1)i (n+i+1)Res Y(u,z)v i z zn+i+1 i=0 (cid:18) (cid:19) X m n+i (1+z)wtu+m + (−1)i (n+i+1)Res Y(u,z)v. i z zn+i+2 i=0 (cid:18) (cid:19) X 6 It is easy to show that the last expression is equal to (−1)m(n+m+1) m+n u◦ v. By m n Lemma 2.2, (L(−1)u+(L(0)+m−n)u)∗nv belongs to O′ (V). m n,m (cid:0) (cid:1) Finally for v¯∗n(L(−1)u+(L(0)+m−n)u) we use Lemma 2.3 to yield m v¯∗n(L(−1)u+(L(0)+m−n)u)−(L(−1)u+(L(0)+m−n)u)∗nv m m = Res (1+z)wtv−1Y(v,z)(L(−1)u+(L(0)+m−n)u) z wtv −1 wtv−1 = v L(−1)u+(wtu+m−n) v u i i i i i≥0 (cid:18) (cid:19) i≥0 (cid:18) (cid:19) X X wtv −1 wtv −1 = L(−1) v u+ iv u i i−1 i i i≥0 (cid:18) (cid:19) i≥0 (cid:18) (cid:19) X X wtv−1 +(wtu+m−n) v u i i i≥0 (cid:18) (cid:19) X wtv −1 wtv −1 = L(−1) v u+ (i+1)v u i i i i+1 i≥0 (cid:18) (cid:19) i≥0 (cid:18) (cid:19) X X wtv−1 +(wtu+m−n) v u i i i≥0 (cid:18) (cid:19) X wtv −1 = (L(−1)+wtv −i−1+wtu+m−n)v u i i i≥0 (cid:18) (cid:19) X wtv −1 = (L(−1)v u+L(0)v u+(m−n)v u) i i i i i≥0 (cid:18) (cid:19) X which is in O′ (V). So v¯∗n(L(−1)u+(L(0)+m−n)u) ∈ O′ (V), as desired. n,m m n,m We should remind the reader that our goal is to construct an A (V)-A (V)-bimodule n m A (V) with the left action∗¯n of A (V) and the right action∗n of A (V). The following n,m m n m m lemma claims that the left action ∗¯n and the right action ∗n commute. On the other m m hand, we do not need to prove this lemma as a bigger subspace O (V) of V containing n,m O′ (V) will be modulo out. In fact, (a∗¯nb)∗n c−a∗¯n(b∗n c) is an element of O (V) n,m m m m m n,m (see Lemma 2.6 below). But eventually we expect to prove that O (V) and O′ (V) m,n n,m are the same although we cannot achieve this in the paper. Lemma 2.6. We have (a∗¯nb)∗nc−a∗¯n(b∗nc) lies in O′ (V) forhomogeneousa,b,c ∈ V. m m m m n,m Proof: The proof of this lemma is similar to that of Theorem 2.4 of [DLM3]. In fact, if m = n, the lemma is exactly the associativity of product ∗ in A (V). n n 7 A straightforward calculation using Lemma 2.4 gives: (a∗¯nb)∗n c m m m n n+k m+i wta+m = (−1)k (−1)i k i j k=0 (cid:18) (cid:19) i=0 (cid:18) (cid:19) j≥0 (cid:18) (cid:19) X X X (1+z)wta+wtb+2m−j+i ·Res Y(a b,z)c z zn+k+1 j−m−i−1 m n n+k m+i wta+m = (−1)k (−1)i k i j k=0 (cid:18) (cid:19) i=0 (cid:18) (cid:19) j≥0 (cid:18) (cid:19) X X X (1+z )wta+wtb+2m−j+i(z −z )j−m−i−1 2 1 2 ·Res Res Y(Y(a,z −z )b,z )c z2 z1−z2 zn+k+1 1 2 2 2 m n n+k m+i = (−1)k (−1)i Res Res k i z2 z1−z2 k=0 (cid:18) (cid:19) i=0 (cid:18) (cid:19) X X (1+z )wta+m(1+z )wtb+m+i(z −z )−m−i−1 1 2 1 2 · Y(Y(a,z −z )b,z )c zn+k+1 1 2 2 2 m n n+k m+i −m−i−1 = (−1)k (−1)i k i j k=0 (cid:18) (cid:19) i=0 (cid:18) (cid:19) j≥0 (cid:18) (cid:19) X X X (1+z )wta+m(1+z )wtb+m+i(−z )j 1 2 2 ·Res Res Y(a,z )Y(b,z )c z1 z2 zm+i+1+jzn+k+1 1 2 1 2 m n n+k m+i −m−i−1 − (−1)k (−1)i k i j k=0 (cid:18) (cid:19) i=0 (cid:18) (cid:19) j≥0 (cid:18) (cid:19) X X X (1+z )wta+m(1+z )wtb+m+izj ·Res Res 1 2 1Y(b,z )Y(a,z )c z2 z1 (−z )m+i+1+jzn+k+1 2 1 2 2 m n n+k m+i ≡ a∗¯n(b∗n c)+ (−1)k (−1)i m m k i k=0 (cid:18) (cid:19) i=0 (cid:18) (cid:19) X X n−i −m−i−1 i i zj+l 1 ·Res Res (−1)j 2 − z1 z2"j=0 (cid:18) j (cid:19) l=0 (cid:18)l(cid:19)z1j+i z1i# X X (1+z )wta+m(1+z )wtb+m 1 2 · Y(a,z )Y(b,z )c. zm+1zn+k+1 1 2 1 2 The lemma then follows from Proposition 5.2 in the Appendix. In order to construct A (V) we need to introduce more subspaces of V. Let O′′ (V) n,m n,m be the linear span of u∗n ((a∗p3 b)∗p3 c−a∗p3 (b∗p2 c)),for a,b,c,u ∈ V,p ,p ,p ∈ m,p3 p1,p2 m,p1 m,p2 m,p1 1 2 3 Z , and O′′′ (V) = (V ∗n O (V))∗n V. Set + n,m p∈Z+ p p m,p PO (V) = O′ (V)+O′′ (V)+O′′′ (V). n,m n,m n,m n,m 8 Lemma 2.7. For p ,p ,m,n ∈ Z , we have (V ∗n O′ (V))∗n V ⊆ O (V). 1 2 + p1,p2 p2,p1 m,p1 n,m Proof: We first prove that (L(−1)u + (L(0) +p −p )u)∗p2 v ∈ O (V) ∗p2 V, for 1 2 m,p1 p2 m,p2 homogeneous u ∈ V and v ∈ V. In fact, (L(−1)u+(L(0)+p −p )u)∗p2 v 1 2 m,p1 p1 m+p −p +i (1+z)wtu+m+1 = (−1)i 2 1 Res Y(L(−1)u,z)v i z zm+p2−p1+i+1 i=0 (cid:18) (cid:19) X p1 m+p −p +i (1+z)wtu+m + (−1)i 2 1 (wtu+p −p )Res Y(u,z)v i 1 2 zzm+p2−p1+i+1 i=0 (cid:18) (cid:19) X p1 m+p −p +i (1+z)wtu+m = (−1)i+1 2 1 (wtu+m+1)Res Y(u,z)v i zzm+p2−p1+i+1 i=0 (cid:18) (cid:19) X p1 m+p −p +i (1+z)wtu+m+1 + (−1)i 2 1 (m+p −p +i+1)Res Y(u,z)v i 2 1 z zm+p2−p1+i+2 i=0 (cid:18) (cid:19) X p1 m+p −p +i (1+z)wtu+m + (−1)i 2 1 (wtu+p −p )Res Y(u,z)v i 1 2 zzm+p2−p1+i+1 i=0 (cid:18) (cid:19) X m+p (1+z)wtu+m = (−1)p1 2 (m+p +1)Res Y(u,z)v. p 2 z zm+p2+2 (cid:18) 1 (cid:19) So by Lemma 2.2, (L(−1)u+(L(0)+p −p )u)∗p2 v ∈ O (V)∗p2 V. Hence by the 1 2 m,p1 p2 m,p2 definitions of O′ (V) and O (V), and Lemma 2.2, we have n,m n,m (V ∗n O′ (V))∗n V ⊆ V ∗n (O′ (V)∗p2 V)+O (V) p1,p2 p2,p1 m,p1 m,p2 p2,p1 m,p1 n,m ⊆ V ∗n ((O (V)∗p2 V)∗p2 V +O (V)∗p2 V)+O (V) m,p2 p2 p1,p2 m,p1 p2 m,p2 n,m ⊆ V ∗n (O (V)∗p2 (V ∗p2 V))+(V ∗n O (V))∗n V +O (V) m,p2 p2 m,p2 m,p1 p2 p2 m,p2 n,m ⊆ (V ∗n O (V))∗n (V ∗p2 V)+O (V) p2 p2 m,p2 m,p1 n,m ⊆ O (V), n,m as required. Lemma 2.8. For any m,n,p ∈ Z , we have V∗n O (V) ⊆ O (V), O (V)∗n V ⊆ + m,p p,m n,m n,p m,p O (V). In particular, V∗¯nO (V) ⊆ O (V), O (V)∗n V ⊆ O (V). n,m m n,m n,m n,m m n,m Proof: By the definition of O (V) and Lemma 2.7, it suffices to prove that n,m V∗n ((V ∗p O (V))∗p V +O′′ (V)) ⊆ O (V) (2.1) m,p p1 p1 m,p1 p,m n,m and ((V ∗n O (V))∗n V +O′′ (V))∗n V ⊆ O (V) (2.2) p1 p1 p,p1 n,p m,p n,m for p ,p ∈ Z . 1 + 9 We first prove (2.1). It is clear that V∗n ((V ∗p O (V))∗p V) ⊆ V∗n (V ∗p (O (V)∗p1 V))+O (V) m,p p1 p1 m,p1 m,p m,p1 p1 m,p1 n,m ⊆ (V∗n V)∗n (O (V)∗p1 V)+O (V) p1,p m,p1 p1 m,p1 n,m ⊆ ((V∗n V)∗n O (V))∗n V +O (V) ⊆ O (V). p1,p p1 p1 m,p1 n,m n,m It remains to prove that V∗n O′′ (V) ⊆ O (V). With u = 1 in the definition of m,p p,m n,m O′′ (V) we have (a∗n b)∗n c−a∗n (b∗p2 c) ∈ O (V). Thus n,m p1,p2 m,p1 m,p2 m,p1 n,m v∗n (u∗p ((a∗p3 b)∗p3 c−a∗p3 (b∗p2 c))) m,p m,p3 p1,p2 m,p1 m,p2 m,p1 ≡ (v∗n u)∗n ((a∗p3 b)∗p3 c−a∗p3 (b∗p2 c)) p3,p m,p3 p1,p2 m,p1 m,p2 m,p1 ≡ 0 (mod O (V)). n,m So (2.1) is true. For(2.2),itiseasytoseethat((V∗n O (V))∗n V)∗n V ⊆ (V∗n O (V))∗n (V∗p1 p1 p1 p,p1 m,p p1 p1 m,p1 m,p V)+O (V) ⊆ O (V). Let a,b,c,u,v ∈ V and p ,p ,p ,p ∈ Z , then n,m n,m 1 2 3 + (u∗n ((a∗p3 b)∗p3 c−a∗p3 (b∗p2 c)))∗n v p,p3 p1,p2 p,p1 p,p2 p,p1 m,p ≡ u∗n (((a∗p3 b)∗p3 c−a∗p3 (b∗p2 c))∗p3 v) m,p3 p1,p2 p,p1 p,p2 p,p1 m,p ≡ u∗n ((a∗p3 b)∗p3 (c∗p1 v))−u∗n (a∗p3 ((b∗p2 c)∗p2 v)) m,p3 p1,p2 m,p1 m,p m,p3 m,p2 p,p1 m,p ≡ u∗n (a∗p3 (b∗p2 (c∗p1 v)))−(u∗n a)∗n ((b∗p2 c)∗p2 v) m,p3 m,p2 m,p1 m,p p2,p3 m,p2 p,p1 m,p ≡ (u∗n a)∗n (b∗p2 (c∗p1 v))−(u∗n a)∗n ((b∗p2 c)∗p2 v) p2,p3 m,p2 m,p1 m,p p2,p3 m,p2 p,p1 m,p ≡ 0 (mod O (V)). n,m The lemma is proved. We now define A (V) = V/O (V). n,m n,m The reason for this definition will become clear from the representation theory of V discussed later. The following is the first main theorem in this paper. Theorem 2.9. Let V be a vertex operator algebra and m,n nonnegative integers. Then A (V) is an A (V)-A (V)-bimodule such that the left and right actions of A (V) and n,m n m n A (V) are given by ¯∗n and ∗n. m m m Proof: First, both actions are well defined from the definition of O (V) and Lemma n,m 2.8. The left A (V)-module and right A (V)-module structures then follow from the n m fact that O′′ (V) is a subspace of O (V). The commutativity of two actions proved in n,m n,m Lemma 2.6 asserts that A (V) is an A (V)-A (V)-bimodule. n,m n m Remark 2.10. We will prove in Section 4 that if m = n, the A (V) defined here is the n,n same as A (V) discussed before. In particular, O (V) and O′ (V) coincide. In other n n,n n,n words, O′′ (V), O′′′ (V) are subspaces of O′ (V). We suspect that this is true in general. n,n n,n n,n That is, O (V) and its subspace O′ (V) are equal. It seems that this is a very difficult n,m n,m problem and we cannot find a proof for this in this paper. 10