Simple vertex operator algebras are nondegenerate

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1 Journal of Algebra Simple vertex operator algebras are nondegenerate Haisheng Li a,b, a Department of Mathematical Sciences, Rutgers University, Camden, NJ 0802, USA b Department of Mathematics, Harbin Normal University, Harbin, China Received 23 April 2002 Communicated by Geoffrey Mason Abstract In this paper it is shown that every irreducible vertex algebra of countable dimension is nondegenerate in the sense of Etingof and Kazhdan and that every simple vertex operator algebra is nondegenerate Elsevier Inc. All rights reserved.. Introduction In one of their series papers on quantization of Lie bialgebras [6], Etingof and Kazhdan introduced and studied certain fundamental notions of braided vertex operator algebra and quantum vertex operator algebra. While studying the axiomatic properties of braided and quantum vertex operator algebras, they introduced a notion of nondegeneracy of a vertex operator algebra. A vertex operator algebra V is said to be nondegenerate if for every positive integer n the linear map Z n from V n Cz z n to V z z n defined by Z n v v n f = fy v,z Y v n,z n is injective. It was proved therein that in their definition of the notion of braided vertex operator algebra, if the classical limit vertex operator algebra is nondegenerate, two of address: hli@crab.rutgers.edu. Partially supported by NSF grant DMS and a grant from Rutgers Research Council /03/$ see front matter 2003 Elsevier Inc. All rights reserved. doi:0.06/s x

2 200 H.-S. Li / Journal of Algebra the main axioms the quantum Yang Baxter equation and the unitarity condition are automatically satisfied, and that in their notion of quantum vertex operator algebra, the main axiom the hexagon relation is equivalent to a certain natural associativity property. In view of this, nondegeneracy is very useful in the study of braided and quantum vertex operator algebras. Let g be a finite-dimensional Lie algebra equipped with a nondegenerate symmetric invariant bilinear form,. Denote by ĝ the associated affine Lie algebra without the degree operator added. Associated to the affine Lie algebra ĝ and a complex number l one has a vertex operator algebra Vĝl, 0, whose underlying vector space is a certain generalized Verma module or Weyl module for the affine Lie algebra ĝ of level l cf. [7, 0,6,7]. It was proved in [6] that if Vĝl, 0 is an irreducible ĝ-module, then Vĝl, 0 is nondegenerate. Notice that the irreducibility of Vĝl, 0 as a ĝ-module amounts to the irreducibility of Vĝl, 0 as a module for the vertex operator algebra Vĝl, 0. Wealso notice that for a general vertex operator algebra V equipped with a conformal vector the irreducibility of the adjoint module V amounts to the simplicity of the vertex operator algebra V. In view of this, one may conjecturethat general simple vertex operator algebras are nondegenerate. In this paper, we shall prove that this conjecture is indeed true. In the literature, there are certain results closely related to the injectivity of the maps Z n. In [4], among other results it was proved that if V is a simple vertex operator algebra and W is an irreducible V -module, then Yv,zw 0for0 v V, 0 w W.Furthermore, among other results it was proved in [5] that the linear map Y viewed as a map from V W to Wz is injective. In view of this, one might guess that ideas in [4,5] and [6] would be highly valuable for proving the conjecture. Indeed, part of our proof uses some of their ideas. Notice that in the notion of vertex operator algebra used in [6], no conformal vector and no Z-grading are assumed. A vertex operator algebra in the sense of [6] is often called a vertex algebra. In this paper we consider a general vertex algebra V.ForanyV -module W and any positive integer n we define a linear map Z W n from V n W Cz z n to Wz z n by Z W n v v n w f = fy v,z Y v n,z n w. It follows from [8] that V n+ Cz z n is a natural vertex algebra with V n W Cz z n as a module. We first prove that the vector subspace ker Zn W of V n W Cz z n is a submodule. To describe submodules of V n W Cz z n we slightly generalize the result of [8] on the irreducibility of tensor product modules for tensor product vertex operator algebras in the context of vertex algebras of countable dimension. Using this, we show that if V is of countable dimension and irreducible in the sense that V is an irreducible V -module and if W is a V -module, then any submodule of V n W Cz z n is of the form V n U, where U is a V Cz z n -submodule of W Cz z n. Then it follows that ker Zn W = 0. From this we show that every irreducible vertex algebra of countable dimension is nondegenerate. In particular, this implies that every simple vertex operator algebra in the sense of [9] and [8] is nondegenerate.

3 H.-S. Li / Journal of Algebra The main results We here recall the notion of nondegeneracy of a vertex algebra from [6] and prove that every irreducible vertex algebra of countable dimension is nondegenerate and that every simple vertex operator algebra is nondegenerate. In the course of proving our main results we also extend a result of Frenkel, Huang and Lepowsky on the irreducibility of tensor product modules for tensor product vertex operator algebras in the context of vertex algebras of countable dimension. Throughout this paper, vector spaces are considered over the field C of complex numbers. In this paper we use the standard formal variable notations and conventions see [8,9] and we use the following definition of the notion of vertex algebra [4]; cf. [, 4,8,9,6]: Definition 2.. A vertex algebra is a vector space V equipped with a linear map, called the vertex operator map, Y : V EndV [[ z, z ]], v Yv,z= n Z v n z n v n EndV 2. and equipped with a distinguished vector V, called the vacuum vector, such that the following axioms hold: For u, v V, for v V, u n v = 0 forn sufficiently large; 2.2 Y,z= ; 2.3 Yv,z V [[z]] and Yv,z z=0 = v = v; 2.4 and for u, v V, z0 δ z z 2 Yu,z Y v, z 2 z0 z δ z2 z 0 z 0 = z 2 δ z z 0 z 2 Y Yu,z 0 v, z 2 Yv,x 2 Y u, x 2.5 the Jacobi identity. Remark 2.2. In the literature, there are variant definitions of the notion of vertex algebra cf. [,4,3,6]. It was proved in [6] that the definition given in [] is equivalent to the current definition with ground field C. The definition used in [6] and [3] where the vacuum vector is denoted by Ω, whose defining axioms had been proved before to give rise to an equivalent definition see [4,8,,6], is also equivalent to the current definition.

4 202 H.-S. Li / Journal of Algebra For a vertex algebra V, the vertex operator map Y is a linear map from V to HomV, V z. ThemapY can be considered as a linear map from V V to V z. Following [6], we alternatively denote this map by Yz. Define a linear operator D EndV by Then cf. [4] We also have Dv = v 2 Yv,z d =. 2.6 dz z=0 [ ] d D,Yu,z = YDu, z = Yu,z, 2.7 dz Yu,zv= e zd Yv, zu for u, v V. 2.8 Yu,z = e zd u for u V. 2.9 It was proved [4,6], cf. [8,9] that the Jacobi identity is equivalent to the following weak commutativity and associativity:foranyu, v V, there exists a nonnegative integer k such that z z 2 k Yu,z Y v, z 2 = z z 2 k Yv,z 2 Y u, z ; 2.0 and for any u, v, w V there exists a nonnegative integer l such that z 0 + z 2 l Yu,z 0 + z 2 Y v, z 2 w = z 0 + z 2 l Y Yu,z 0 v, z 2 w. 2. A V -module [4,6] is a vector space W equipped with a linear map Y from V to EndW[[z, z ]] such that all the axioms defining the notion of vertex algebra that make sense hold. That is, the truncation condition 2.2, the vacuum property 2.3 and the Jacobi identity 2.5 hold. The notion of ideal is defined in the obvious way; an ideal of a vertex algebra V is a subspace U such that u m v, v m u U for all v V, u U. Every vertex algebra V has trivial ideals 0 and V. Definition 2.3. A vertex algebra V is said to be simple if there is no nontrivial ideal and V is said to be irreducible if V is an irreducible V -module. Clearly, an irreducible vertex algebra is simple, but simple vertex algebras are not necessarily irreducible. For example, the vertex algebra constructed in [] from the commutative associative algebra C[x] with the standard derivation is simple, but not irreducible. On the other hand, simple vertex operator algebras in the sense of [9] and [8] are always irreducible because any submodule of V is an ideal cf. [8].

5 H.-S. Li / Journal of Algebra Let V be a vertex algebra, fixed throughout this section. Following [6], for a positive integer n, we define a linear map In [6], Z n was defined as Z n : V n Cz z n V z z n, v v n f fy v,z Y v n,z n. 2.2 Z n = Yz Yz 2 n Yz n n. 2.3 The following notion is due to Etingof and Kazhdan [6]: Definition 2.4. A vertex algebra V is said to be nondegenerate if the linear maps Z n are injective for all positive integers n. Remark 2.5. Consider the case n =. For v V, f Cz,wehave Z v f= fyv,z = fe zd v = e zd fv. Let π be the natural embedding of V Cz into V z. ThenZ = e zd π. It follows immediately that Z is injective. Now, let W be a V -module. For n, we define a linear map Z W n : V n W Cz z n Wz z n, v v n w f fy v,z Y v n,z n w 2.4 for v,...,v n V, w W, f Cz z n. Notice that Cz z n is a unital commutative associative algebra which is in fact a field and that for any vector space U, Uz z n is a Cz z n -module. It is clear that Zn W is Cz z n -linear where V n W Cz z n is considered as a Cz z n -module in the obvious way. Let E n be the embedding of V n into V n+ defined by Then E n v v n = v v n. 2.5 Z n = Z V n E n. 2.6 In the literature, there are certain results which are closely related to the injectivity of linear maps Z W. The following result is due to Dong and Mason [5] while the particular case is due to Dong and Lepowsky [4]:

6 204 H.-S. Li / Journal of Algebra Proposition 2.6. Let V be a simple vertex operator algebra and let W be an irreducible V -module. Let v,...,v r V be nonzero vectors and let w,...,w r W be linearly independent vectors. Then In particular, r Y v i,z w i 0 in Wz. 2.7 i= Yv,zw 0 for 0 v V, 0 w W. 2.8 Remark 2.7. Proposition 2.6 exactly asserts that Y viewed as a linear map from V W to Wz is injective. From [8], for any positive integer n, V n+ has a natural vertex algebra structure and for V -modules W,...,W n+, W W n+ has a natural V n+ -module structure. For i n +, let π i be the embedding of V into V n+ : π i : V V n+, v i v n+ i. 2.9 Let U be any subspace of W W n+. It is clear cf. [8] that U is a submodule if and only if for i =,...,n+, Y π i v, z U Uz for all v V Note that any commutative associative algebra with identity is naturally a vertex algebra. Then from [8], V n+ Cz z n is a vertex algebra. Now we state our first and key result: Proposition 2.8. Let W be any V -module. For any positive integer n, subspace ker Zn W a V n+ Cz z n -submodule of V n W Cz z n. is For convenience we first prove the following simple fact: Lemma 2.9. Let U be a vector space and let fz,...,z n Uz z n. 2.2 Assume that there exist nonnegative integers k ij for i<j n such that z i z j k ij fz,...,z n = Then fz,...,z n = 0. i<j n

7 H.-S. Li / Journal of Algebra Proof. The key issue here is the cancellation law. From [9], for any three formal series A,B and C, ifabc, AB and BC all exist algebraically, then Set A = i<j n ABC = ABC = ABC z i z j k ij, B = i<j n z i z j k ij, C = fz,...,z n, where we use the usual binomial expansion convention cf. [9]. Then it follows immediately from 2.22 that fz,...,z n = 0. Proof of Proposition Since Zn W prove is that for i =,...,n+, is already Cz z n -linear, what we must Y π i v, z ker Z W n ker Z W n z for v V That is, we must prove that if X V n W Cz z n with Zn W for i =,...,n+, X = 0, then Z W n Y πi v, z X = 0 forall v V We shall prove this in four steps. Claim : For i = n +, 2.25 holds. Let X = u α u nα n w β f γ 2.26 a finite sum. We shall use this general X for the whole proof. Then f γ Y u α,z Y u nα n,z n w β = Zn W X = Let v V. By the weak commutativity there exists a nonnegative integer k such that z z i k Yv,zY u iα i,z i = z zi k Y u iα i,z i Yv,z 2.28 for all of the indices i and α i. Multiplying 2.27 by z z k z z n k Yv,z from left and then using 2.28 we get z z k z z n k f γ Y u α,z Y u nα n,z n Yv,zw β =

8 206 H.-S. Li / Journal of Algebra In view of Lemma 2.9 we have f γ Y u α,z Y u nα n,z n Yv,zw β = Thus Z W n Y π n+v, zx = 0. Claim 2. For i = n, 2.25 holds. From 2.27, using 2.28 we get n z z i k f γ Y u α,z Y u n α n,z n Yv,zY u nα n,z n w β = 0. i= 2.3 In view of Lemma 2.9 we have f γ Y u α,z Y u n α n,z n Yv,zY u nα n,z n w β = By the weak associativity, there exists a nonnegative integer l such that z 0 + z n l Yv,z 0 + z n Y u nα n,z n w β = z 0 + z n l Y Yv,z 0 u nα n,z n w β 2.33 for all of the indices α n and β, which are finitely many. Then z 0 + z n l f γ Y u α,z Y u n α n,z n Y Yv,z0 u nα n,z n w β = z 0 + z n l f γ Y u α,z Y u n α n,z n Yv,z0 + z n Y u nα n,z n w β = By Lemma 2.9 we get f γ Y u α,z Y u n α n,z n Y Yv,z0 u nα n,z n w β = That is, Zn W Y π nv, zx = 0. Claim 3. ker Zn W is stable under the natural action of S n. Note that the symmetric group S n acts naturally on V n W Cz z n, the domain of Zn W. It suffices to prove that ker Zn W is stable under the actions of transpositions σ ii+ for i =,...,n. Let X ker Zn W and write X as in By the weak commutativity there exists a nonnegative integer k such that z i z i+ k Y u iα i,z i Yu,zi+ = z i z i+ k Yu,z i+ Y u iα i,z i 2.36

9 H.-S. Li / Journal of Algebra for all of the indices i and α i.then z i z i+ k Z W n σ ii+x = z i z i+ k Z W n X = In view of Lemma 2.9, we have Zn W σ ii+x = 0. That is, ker Zn W action of σ ii+ for i =,...,n. Therefore ker Zn W is S n-stable. Claim 4. For i n, 2.25 holds. Let σ be the transposition that exchanges i with n. Thenwehave is stable under the Y π i v, z X = σ Y π n v, z σx for v V By Claim 3, σx ker Z W n. Furthermore, by Claim 2 we have Y π n v, z σx ker Z W n z. By Claim 3 again we get σ Y π n v, z σx ker Z W n z. Therefore Z W n Y πi v, z X = Z W n σ Y πn v, z σx = 0. This proves Claim 4, completing the proof. Let W be a nonzero V -module. Noticing that for 0 w W, 0 f Cz z n, Z W n n w f = fy,z Y,z n w = fw 0, 2.39 we have ker Z W n V n W Cz z n If we can prove that V n W Cz z n is an irreducible V n+ Cz z n -module, in view of Proposition 2.8 we will immediately have ker Z W n = 0. When V is a simple vertex operator algebra and W is an irreducible V -module, it follows from [8] that V n W is an irreducible V n+ -module. But it is not clear if this is still true if V is just an irreducible vertex algebra. Motivated by this we next determine all the submodules of V n W Cz z n for the vertex algebra V n+ Cz z n with V an irreducible vertex algebra and W a general not necessarily irreducible V -module. In this direction we first prove the following simple result for which we have no reference:

10 208 H.-S. Li / Journal of Algebra Lemma 2.0. Let A and A 2 be associative algebras with identity element and let U and U 2 be modules for A and A 2, respectively. Assume that U is irreducible and End A U = C. Then any A A 2 -submodule of U U 2 is of the form U U 2,whereU 2 is an A 2 -submodule of U 2.Furthermore,ifU 2 is irreducible, then U U 2 is irreducible. Proof. Fix a basis {u α α I} for U.LetU be a nonzero A A 2 -submodule of U U 2.Letu be any nonzero element of U.Then u = u α u 2 + +u αr u 2r, where u 2,...,u 2r are finitely many nonzero vectors in U 2.SinceU is irreducible and End A U = C, in view of the density theorem cf. [2], there exists a A such that Then au α 0 and au αi = 0 fori = 2,...,r. 0 au α u 2 = a u U. Since U is irreducible, we have A au α = U. Thus U u 2 = A a u U. Similarly, we have U u 2i U for i = 2,...,r.Set U 2 u = A 2u 2 + +A 2 u 2r U 2. Notice that u 2i s are uniquely determined by u. Then u U U 2 u U. Setting U 2 = 0 u U U 2 u U 2, 2.4 we get U = U U 2, completing the proof. Closely related to Lemma 2.0 is the following result which can be found in [3], or [2]: Lemma 2.. Let A be an associative algebra with identity and let U be an irreducible A-module of countable dimension. Then End A U = C. We now apply Lemma 2. to vertex algebras. Lemma 2.2. Let V be a vertex algebra of countable dimension and let W be an irreducible V -module. Then W is of countable dimension and End V W = C.

11 H.-S. Li / Journal of Algebra Proof. Let w be any nonzero element of W. It was proved in [5] and [5] that the linear span of vectors v m w for v V, m Z is a submodule of W. Consequently, W = span{v m w v V, m Z}. Then we see at once that W has countable dimension. Furthermore, let A be the subalgebra of End W, generated by all the operators v m for v V, m Z. ThenA acts irreducibly on W and End A W = End V W. In view of Lemma 2. we have End A W = C. Thus End V W = End A W = C. The following result slightly generalizes the corresponding result of [8]: Proposition 2.3. Let V,...,V r be vertex algebras of countable dimension and let W,...,W r be irreducible modules for V,...,V r, respectively. Then W W r is an irreducible V V r -module with End V V r W W r = C. Proof. In view of Lemma 2.2, we have End Vi W i = C for i =,...,r. In particular, Proposition holds for r =. For r = 2, first it follows from Lemma 2.0 that W W 2 is an irreducible V V 2 -module. Then by Lemma 2.2, End V V 2 W W 2 = C. Now, Proposition follows immediately from induction on r and the assertion for r = 2. Now we are ready to prove our main result: Theorem 2.4. Let V be an irreducible vertex algebra of countable dimension and let W be any nonzero V -module. Then for every positive integer n, the linear map Zn W is injective. Furthermore, every irreducible vertex algebra V of countable dimension is nondegenerate. Proof. In view of Proposition 2.8, for any positive integer n, kerzn W is a V n+ Cz z n -submodule of V n W Cz z n. It follows from Proposition2.3thatV n is an irreducible V n -module and that End V n V n = C.LetA be the subalgebra of EndV n, generated by all the operators u m for u V n,m Z. ThenA acts irreducibly on V n and End A V n = C. In view of Lemma 2.0, we have ker Z W n = V n U, where U is a V Cz z n -submodule of W Cz z n.let F = w f + +w r f r be a generic element of U, wherew,...,w r are linearly independent vectors in W and f i Cz z n.thenwehave n F ker Z W n,sothat f w + +f r w r = r f i Y,z Y,z n w i = Zn W n F = i=

12 20 H.-S. Li / Journal of Algebra Consequently, f i = 0 for i =,...,r. This proves that U = 0, so kerzn W = 0. Furthermore, since Z n = Zn V E n,wheree n is the embedding of V n into V n+, Z n must be injective. This proves that V is nondegenerate. It follows from the definition of the notion of vertex operator algebra [8,9] that any vertex operator algebra in the sense of [9] and [8] has countable dimension. We also know that a simple vertex operator algebra is irreducible. Then we immediately have Corollary 2.5. Let V be a simple vertex operator algebra in the sense of [9] and [8] and let W be a nonzero V -module. Then for every positive integer n, the linear map Zn W is injective. Furthermore, every simple vertex operator algebra V is nondegenerate. We also immediately have the following generalization of Proposition 2.6: Corollary 2.6. Let V be an irreducible vertex algebra of countable dimension and let W be any not necessarily irreducible nonzero V -module. Then for each positive integer n, the linear map defined by F W n : V n W Wz z n F W n v v n w = Y v,z Y v n,z n w 2.43 is injective. Remark 2.7. Results of this paper can be appropriately generalized in terms of intertwining operators [8]. In fact, the corresponding result of [4] was formulated in terms of intertwining operators in the more general context of generalized vertex algebras. Acknowledgment We would like to thank Martin Karel for reading the manuscript and giving me many valuable suggestions. References [] R.E. Borcherds, Vertex algebras, Kac Moody algebras, and the Monster, Proc. Natl. Acad. Sci. USA [2] N. Chriss, V. Ginzburg, Representation Theory and Complex Geometry, Birkhäuser, Boston Basel Berlin, 997. [3] J. Dixmier, Enveloping Algebras, in: Grad. Stud. in Math., Vol. II, American Mathematical Society, 996. [4] C. Dong, J. Lepowsky, Generalized Vertex Algebras and Relative Vertex Operators, in: Progress in Math., Vol. 2, Birkhäuser, Boston, 993.

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