New York Journal of Mathematics

New York J. Math. 14 (2008) 261–284.

Vertex tensor category structure on a category of Kazhdan–Lusztig

Lin Zhang

Abstract. We incorporate a category considered by Kazhdan and Lusztig of certain modules (of a fixed level ℓ, not a positive integer) for an affine Lie algebra, into the representation theory of vertex operator algebras. We do this using the logarithmic tensor product theory for generalized modules for a vertex operator algebra developed by Huang, Lepowsky and the author; we prove that the conditions for applying this general logarithmic tensor product theory hold. As a consequence, we prove that this category has a natural vertex tensor category structure, and in particular we obtain a new, vertex-algebraic, construction of the natural associativity isomorphisms and proof of their properties.

Contents

1. Introduction 261 2. Kazhdan–Lusztig’s tensor product and the category O_{κ} 263 3. Tensor product for modules for a conformal vertex algebra 269 4. Strong lower truncation condition 273 5. Generalized modules 278 References 283

1. Introduction

In a series of papers [KL2]–[KL5] (see also [KL1]), Kazhdan and Lusztig constructed a braided tensor category structure on a category O_{κ} of certain modules for an affine Lie algebra of a fixed level κ − h, h the dual Coxeter number of the Lie algebra and κ a complex number not in ℚ_{≥0}, the set of nonnegative rational numbers, and showed that this braided tensor

Received April 22, 2008. Mathematics Subject Classification. 17B69; 17B67; 81T40; 18D10. Key words and phrases. Vertex operator algebra, generalized module, logarithmic ten-

sor product theory, vertex tensor category, braided tensor category.

Partial support from NSF grant DMS-0070800 is gratefully acknowledged.

261^{ISSN 1076-9803/08}

category is equivalent to a tensor category of modules for a quantum group constructed from the same Lie algebra. The most interesting cases are those of the allowable negative levels. Their construction of the tensor product uses ideas in the important work [MS] by Moore and Seiberg. While the category in [MS] consists essentially of the integrable highest weight modules for an affine Lie algebra of a fixed positive integral level rather than modules of a fixed negative level, for example, it was first discovered by Moore and Seiberg in that paper that this positive-level category has a braided tensor category structure; however, this was based on the strong assumption of the axioms for conformal field theory and in particular, the assumption of the existence of the “operator product expansion” for “intertwining operators.”

Suitable modules for affine Lie algebras give rise to an important family of vertex operator algebras and their modules. In [HL1]–[HL4] and [H2], Huang and Lepowsky developed a substantial tensor product theory for modules for a “rational” vertex operator algebra, under certain conditions. As one of the applications of this theory, they proved in [HL5] that the conditions required for this theory are satisfied for the module category of the vertex operator algebra constructed from the category of integrable highest weight modules for an affine Lie algebra of a fixed positive integral level. As a result they directly constructed a braided tensor category structure, and further, a “vertex tensor category” structure, on this category.

It has been expected that the category O_{κ} considered by Kazhdan and Lusztig should also be covered by a suitable generalization of the tensor product theory developed by Huang and Lepowsky, even though κ − h cannot be positive integral. However, this category is very different from the one associated with the positive integral level case. For example, the objects of this category in general are only direct sums of generalized eigenspaces, rather than eigenspaces, for the operator L_{0} that defines the conformal weights. Therefore, such a generalization, if it exists, should include categories of these more general modules. Recently, such a generalization has been achieved in [HLZ1] and [HLZ2] by Huang, Lepowsky and the author. A fundamental subtlety in this generalization is that the corresponding intertwining operators involve logarithms of the variables. The questions are now whether the category of Kazhdan and Lusztig satisfies the required conditions for the generalized tensor product theory of [HLZ1] and [HLZ2], and if the answer is affirmative, whether the resulting tensor product construction is equivalent to the original construction given by Kazhdan and Lusztig.

In this paper, we prove that category O_{κ} indeed satisfies all necessary conditions in [HLZ1] and [HLZ2]. We establish an equivalent condition for the subtle “compatibility condition” in the construction of the tensor product and use it to prove that the two constructions of the tensor product functor are identical. We then use the methods in [H2] and [H3] and their generalizations in [HLZ1] and [HLZ2] to obtain a new construction, very different from the original one by Kazhdan and Lusztig, of the natural associativity isomorphisms. As a result, we incorporate the tensor category theory of O_{κ} into the theory of vertex operator algebras and more importantly, prove that O_{κ} has a natural vertex tensor category structure.

The contents of this paper are as follows: In Section 2 we recall the construction of the tensor product by Kazhdan and Lusztig and some of their results on the category O_{κ}. In Section 3 we recall from [HLZ1] and [HLZ2] the construction of tensor product of generalized modules for a vertex operator algebra. In Section 4 we prove an equivalent condition for the “compatibility condition.” Then in Section 5 we first apply the general theory to the case of O_{κ} and prove the equivalence of the two constructions of the tensor product functor; then we show that the objects of the category O_{κ} are C_{1}-cofinite and quasi-finite-dimensional. These results imply that the conditions for applying the results in [HLZ1] to the category O_{κ} are satisfied, and thus O_{κ} has a vertex tensor category structure. This paper is heavily based on the generalized tensor product theory developed in [HLZ1] and [HLZ2].

In this paper, ℂ, ℕ and ℤ_{+} are the complex numbers, the nonnegative integers and the positive integers, respectively.

Acknowledgments. This paper is a revised version of part of the author’s Ph.D. thesis [Z]. The author would like to thank Professor James Lepowsky and Professor Yi-Zhi Huang for their long time encouragement and support, both spiritually and financially, and for numerous discussions and suggestions on this and related works; he would also like to thank Professor Haisheng Li for his encouragement and discussions from the very beginning. Part of the contents of this paper was presented in seminar talks given at Stony Brook and at Rutgers by the author in April and November, 2003. The author would like to thank Bin Zhang for inviting him to present this work at Stony Brook.

2. Kazhdan–Lusztig’s tensor product and the category O_{κ}

In this section we recall the “double dual” construction of tensor product of certain modules for an affine Lie algebra of a fixed level, given by Kazhdan and Lusztig in [KL2] (see also [KL1]). We also recall from their papers the category O_{κ} for a complex number κ ∉ ℚ_{≥0} and the result on closedness of tensor product on this category (see also [Y]).

Let 𝔤 be a complex semisimple finite-dimensional Lie algebra equipped with a nondegenerate invariant symmetric bilinear form (·, ·). The (untwisted) affine Lie algebra associated with 𝔤 is the vector space

𝔤̂ = 𝔤 ⊗ ℂ[t, t^{−1}] ⊕ ℂk,

equipped with the bilinear bracket operations

[a ⊗ t^{m}, b ⊗ t^{n}] = [a, b] ⊗ t^{m+n} + m(a, b)δ_{m+n,0}k, (2.1) [·, k] = 0 = [k, ·] (2.2)

for a, b ∈ 𝔤 and m, n ∈ ℤ. We will also need its algebraic completion 𝔤 ⊗ ℂ((t)) ⊕ ℂk, which satisfies the same bracket relations (2.1) and (2.2), and more generally, for a, b ∈ 𝔤 and g_{1}, g_{2} ∈ ℂ((t)),

(2.3)  [a ⊗ g_{1}, b ⊗ g_{2}] = [a, b] ⊗ g_{1}g_{2} + {g_{1}, g_{2}}(a, b)k

where \{g_{1}, g_{2}\} = Res g_{2} \frac{d}{dt}g_{1}, the coefficient of t^{−1} in the formal Laurent series g_{2} \frac{d}{dt}g_{1}.

Equipped with the ℤ-grading

𝔤̂ = ∐_{n∈ℤ} 𝔤̂_{(n)},

where

𝔤̂_{(0)} = 𝔤 ⊕ ℂk and 𝔤̂_{(n)} = 𝔤 ⊗ t^{−n} for n ≠ 0,

𝔤̂ becomes a ℤ-graded Lie algebra. We have the following graded subalgebras of 𝔤̂:

𝔤̂_{(±)} = ∐_{n>0} 𝔤 ⊗ t^{∓n},
𝔤̂_{(≤0)} = 𝔤̂_{(−)} ⊕ 𝔤 ⊕ ℂk.

A 𝔤̂-module W is said to be of level ℓ ∈ ℂ if k acts on W as scalar ℓ. A module W for 𝔤̂ or 𝔤̂_{(≤0)} is said to be restricted if for any a ∈ 𝔤 and w ∈ W, (a ⊗ t^{n})w = 0 for n sufficiently large. Note that a restricted module W for 𝔤̂ is naturally a module for 𝔤 ⊗ ℂ((t)) ⊕ ℂk by letting a ⊗ ∑_{n∈ℤ} c_{n}t^{n}, a ∈ 𝔤, c_{n} ∈ ℂ, act on w ∈ W as ∑_{n∈ℤ} c_{n}(a ⊗ t^{n})w.

A 𝔤̂-module W is said to be smooth if for any w ∈ W, there is N ∈ ℕ such that for any a_{1}, …, a_{N} ∈ 𝔤, (a_{1} ⊗ t) ⋯ (a_{N} ⊗ t)w = 0. By (2.1) and the fact that 𝔤 = [𝔤, 𝔤] it is clear that a smooth 𝔤̂-module must be restricted.

Let m be a positive integer. First recall that the direct sum 𝔤^{⊕m} of m copies of 𝔤 is a semisimple Lie algebra with nondegenerate invariant symmetric bilinear form (·, ·) given by

((a_{1}, ⋯, a_{m}), (b_{1}, ⋯, b_{m})) = (a_{1}, b_{1}) + ⋯ + (a_{m}, b_{m})

for a_{1}, …, a_{m}, b_{1}, …, b_{m} ∈ 𝔤.

Given m restricted 𝔤̂-modules of a fixed level ℓ ∈ ℂ, the goal is to produce a “tensor product” of these modules that is also a 𝔤̂-module of the same level. (Note that the usual tensor product for Lie algebra modules is a module of level mℓ.) As we recall from [KL1] and [KL2] below, this can be defined in terms of a Riemann sphere with m + 1 distinct points and local coordinates at these points.

Let p_{0}, p_{1}, …, p_{m} be distinct points, or punctures (see [H1]), on the Riemann sphere C = ℂP^{1} and let φ_{s} : C → ℂP^{1} be isomorphisms such that φ_{s}(p_{s}) = 0 for each of s = 0, 1, …, m, that is, φ_{s} is the local coordinate around p_{s} for each s. We will still use C for the Riemann sphere equipped with these punctures and local coordinates.

Let R denote the algebra of regular functions on C\{p_{0}, p_{1}, …, p_{m}}. Define {f_{1}, f_{2}} = Res_{p_{0}}f_{2}df_{1}, i.e., the residue of the meromorphic 1-form f_{2}df_{1} on C at the point p_{0}. Then {·, ·} : R × R → ℂ is a bilinear form satisfying

{f_{1}, f_{2}} + {f_{2}, f_{1}} = 0, {f_{1}f_{2}, f_{3}} + {f_{2}f_{3}, f_{1}} + {f_{3}f_{1}, f_{2}} = 0

for all f_{1}, f_{2}, f_{3} ∈ R. As a result the Lie algebra 𝔤 ⊗ R has a natural central extension Γ_{R} = (𝔤 ⊗ R) ⊕ ℂk with central element k and bracket relations

(2.4)  [a ⊗ f_{1}, b ⊗ f_{2}] = [a, b] ⊗ f_{1}f_{2} + {f_{1}, f_{2}}(a, b)k,

for a, b ∈ 𝔤 and f_{1}, f_{2} ∈ R.

Remark 2.1. In [KL2], C is allowed to be a smooth curve with k connected components each of which is isomorphic to ℂP^{1}, and the construction would give a 𝔤̂^{⊕k}-module as the tensor product. For the purpose of this paper, however, we need only the k = 1 case.

For each s = 0, 1, …, m, denote by ι_{p_{s}} : R → ℂ((t)) the linear map which sends f ∈ R to the power series expansion of f ∘ φ_{s}^{−1} around 0. Then we have Lie algebra homomorphisms

Γ_{R} → 𝔤 ⊗ ℂ((t)) ⊕ ℂk, (2.5)

a ⊗ f ↦ a ⊗ ι_{p_{0}}f, k ↦ k

for a ∈ 𝔤, f ∈ R, and

Γ_{R} → (𝔤 ⊗ ℂ((t)))^{⊕m} ⊕ ℂk, (2.6)

a ⊗ f ↦ (a ⊗ ι_{p_{1}}f, ⋯, a ⊗ ι_{p_{m}}f), k ↦ −k

for a ∈ 𝔤 and f ∈ R (cf. Remark 2.2 below). Here we see

(𝔤 ⊗ ℂ((t)))^{⊕m} ⊕ ℂk = 𝔤^{⊕m} ⊗ ℂ((t)) ⊕ ℂk

as the algebraic completion of the affine Lie algebra

𝔤̂^{⊕m} = 𝔤^{⊕m} ⊗ ℂ[t, t^{−1}] ⊕ ℂk

associated with the semisimple Lie algebra 𝔤^{⊕m}; so in particular,

(2.7) [(a_{1} ⊗ g_{1}, ⋯, a_{m} ⊗ g_{m}), (a′_{1} ⊗ g′_{1}, ⋯, a′_{m} ⊗ g′_{m})]

= ([a_{1}, a′_{1}] ⊗ g_{1}g′_{1}, ⋯, [a_{m}, a′_{m}] ⊗ g_{m}g′_{m}) + ∑_{i=1}^{m}(a_{i}, a′_{i}){g_{i}, g′_{i}}k

for a_{1}, …, a_{m}, a′_{1}, …, a′_{m} ∈ 𝔤 and g_{1}, …, g_{m}, g′_{1}, …, g′_{m} ∈ ℂ((t)).

Remark 2.2. That (2.5) is a Lie algebra homomorphism is straightforward by (2.4) and (2.3). Formula (2.6) gives a Lie algebra homomorphism due to (2.4), (2.7) and the fact that for any f_{1}, f_{2} ∈ R,

{ι_{p_{1}}f_{1}, ι_{p_{1}}f_{2}} + ⋯ + {ι_{p_{m}}f_{1}, ι_{p_{m}}f_{2}} = −{ι_{p_{0}}f_{1}, ι_{p_{0}}f_{2}},

by the residue theorem.

Now let W_{1}, W_{2}, …, W_{m} be 𝔤̂-modules of level ℓ. The vector space tensor product W_{1}⊗⋯⊗W_{m} is naturally a module for the affine Lie algebra 𝔤̂^{⊕m} = 𝔤^{⊕m} ⊗ ℂ[t, t^{−1}] ⊕ ℂk = (𝔤 ⊗ ℂ[t, t^{−1}])^{⊕m} ⊕ ℂk by k acting as scalar ℓ and

(a_{1} ⊗ t^{n_{1}}, ⋯, a_{m} ⊗ t^{n_{m}})(w_{(1)} ⊗ ⋯ ⊗ w_{(m)}) (2.8)

= (a_{1} ⊗ t^{n_{1}})w_{(1)} ⊗ w_{(2)} ⊗ ⋯ ⊗ w_{(m)} + ⋯
+ w_{(1)} ⊗ w_{(2)} ⊗ ⋯ ⊗ (a_{m} ⊗ t^{n_{m}})w_{(m)}

for a_{1}, …, a_{m} ∈ 𝔤, n_{1}, …, n_{m} ∈ ℤ and w_{(i)} ∈ W_{i}, i = 1, …, m. This follows from the bracket relations (2.7) where g_{i} and g′_{i}’s are in ℂ[t, t^{−1}].

If each 𝔤̂-module W_{i} is restricted, i = 1, …, m, then it is clear from (2.8) that W_{1}⊗⋯⊗W_{m} is a restricted 𝔤̂^{⊕m}-module, and hence naturally a module for 𝔤^{⊕m} ⊗ ℂ((t)) ⊕ ℂk, satisfying

(a_{1} ⊗ g_{1}, ⋯, a_{m} ⊗ g_{m})(w_{(1)} ⊗ ⋯ ⊗ w_{(m)}) (2.9)

= (a_{1} ⊗ g_{1})w_{(1)} ⊗ w_{(2)} ⊗ ⋯ ⊗ w_{(m)} + ⋯
+ w_{(1)} ⊗ w_{(2)} ⊗ ⋯ ⊗ (a_{m} ⊗ g_{m})w_{(m)}.

for a_{1}, …, a_{m} ∈ 𝔤, g_{1}, …, g_{m} ∈ ℂ((t)) and w_{(i)} ∈ W_{i}, i = 1, …, m. Thus by (2.6) we have a Γ_{R}-module structure on W_{1} ⊗ ⋯ ⊗ W_{m} where k acts as the scalar −ℓ. The dual vector space (W_{1}⊗⋯⊗W_{m})^{∗} = Hom(W_{1}⊗⋯⊗W_{m}, ℂ) has an induced natural Γ_{R}-module structure by

(2.10)  ⟨ξ(λ), w⟩ = −⟨λ, ξ(w)⟩

for all ξ ∈ Γ_{R}, λ ∈ (W_{1} ⊗ ⋯ ⊗ W_{m})^{∗} and w ∈ W_{1} ⊗ ⋯ ⊗ W_{m}. Here and below we use ⟨·, ·⟩ to denote the natural pairing between a vector space and its dual.

Let N be a positive integer. Let G_{N} be the subspace of U(Γ_{R}) spanned by all products (a_{1} ⊗ f_{1}) ⋯ (a_{N} ⊗ f_{N}) with a_{1}, …, a_{N} ∈ 𝔤 and f_{1}, …, f_{N} ∈ R satisfying ι_{p_{0}}f_{i} ∈ tℂ[[t]] for i = 1, …, N. Here and below we use notation U(L) for the univeral enveloping algebra of a Lie algebra L. Define Z^{N} ⊂ (W_{1} ⊗ ⋯ ⊗ W_{m})^{∗} to be the annihilator of G_{N}(W_{1} ⊗ ⋯ ⊗ W_{m}). Then

Z^{N} = {λ ∈ (W_{1} ⊗ ⋯ ⊗ W_{m})^{∗} | G_{N}λ = 0},

and we have an increasing sequence Z^{1} ⊂ Z^{2} ⊂ · · ·. Let Z^{∞} = ∪_{N∈ℤ_{+}}Z^{N}. It is clear that Z^{∞} is a Γ_{R}-submodule of (W_{1} ⊗ ⋯ ⊗ W_{m})^{∗}.

Define a 𝔤 ⊗ ℂ((t)) ⊕ ℂk-module structure on Z^{∞} as follows: k acts as scalar ℓ; and for λ ∈ Z^{∞}, a ∈ 𝔤 and g ∈ ℂ((t)), choose N ∈ ℕ such that λ ∈ Z^{N}, choose f ∈ R such that ι_{p_{0}}f − g ∈ t^{N}ℂ[[t]], and define

(2.11)  (a ⊗ g)λ = (a ⊗ f)λ.

It is easy to verify that this is independent of the choice of f and gives a 𝔤⊗ℂ((t))⊕ℂk-module structure on Z^{∞} with k acting as scalar ℓ. Restricted to Lie subalgebra 𝔤̂ of 𝔤 ⊗ ℂ((t)) ⊕ ℂk, we have on Z^{∞} a structure of 𝔤̂- module of level ℓ. We will denote this 𝔤̂-module by ∘_{C}(W_{1}, W_{2}, ⋯, W_{m}), and when m = 2, simply by W_{1} ∘_{C} W_{2}.

Finally we need:

Definition 2.3. Given a 𝔤̂-module W, consider Hom(W, ℂ) as a 𝔤̂-module with the actions given by

(2.12)  ((a ⊗ t^{n})λ)(w) = −λ((a ⊗ (−t)^{−n})w), (k · λ)(w) = λ(kw)

for v ∈ 𝔞, λ ∈ W^{∗} and w ∈ W. The contragredient module D(W) of W is defined by

D(W) = {λ ∈ Hom(W, ℂ) | there is N ∈ ℕ such that for any a_{1}, …, a_{N} ∈ 𝔤, (a_{1} ⊗ t) ⋯ (a_{N} ⊗ t)λ = 0},

a 𝔤̂-submodule of Hom(W, ℂ).

The 𝔤̂-module D(∘_{C}(W_{1}, W_{2}, ⋯, W_{m})) is defined to be the desired tensor product of W_{1}, …, W_{m}.

Now we recall the category O_{κ} for a complex number κ ∉ ℚ_{≥0} from [KL2]. Let M be a module for 𝔤̂_{(≤0)} satisfying the condition that k acts as a

scalar ℓ. Then the induced 𝔤̂-module

(2.13)  U(𝔤̂) ⊗_{U(𝔤̂_{(≤0)})} M

is a 𝔤̂-module of level ℓ. In case M is finite-dimensional, restricted, and the subalgebra 𝔤_{(−)} acts nilpotently, the corresponding induced module is called a generalized Weyl module.

Definition 2.4. Given complex number κ ∉ ℚ_{≥0}, the category O_{κ} is defined to be the full subcategory of the category of 𝔤̂-modules whose objects are quotients of some generalized Weyl module of level ℓ = κ − h, where h is the dual Coxeter number of 𝔤.

For an object W of O_{κ}, define the Segal–Sugawara operator L_{k} : W → W by

(2.14)  L_{k}(w) = \frac{1}{2κ} ∑_{j≥−k/2} ∑_{p}(c_{p} ⊗ t^{−j})(c_{p} ⊗ t^{j+k})w

+ \frac{1}{2κ} ∑_{j<−k/2} ∑_{p}(c_{p} ⊗ t^{j+k})(c_{p} ⊗ t^{−j})w

where {c_{p}} is an orthonormal basis of 𝔤. For any n ∈ ℂ denote by W_{[n]} the generalized eigenspace for L_{0} with respect to the eigenvalue n. Then it is shown in [KL2] that W = ∐_{n∈ℂ} W_{[n]} and dim W_{[n]} < ∞.

The following result is proved in [KL2] Theorem 2.5. For W_{1} and W_{2} in O_{κ}, W_{1} ∘_{C} W_{2} is again an object of O_{κ}; the functor D(·) is closed in O_{κ}, and furthermore, as a vector space

D(W) = ∐_{n∈ℂ}(W_{[n]})^{∗}.

Let z be a nonzero complex number. Consider the Riemann sphere C with punctures p_{0} = z, p_{1} = ∞ and p_{2} = 0 and local coordinates given by φ_{0}(ϵ) = ϵ − z, φ_{1}(ϵ) = 1/ϵ and φ_{2}(ϵ) = ϵ at p_{0}, p_{1} and p_{2}, respectively. A Riemann sphere equipped with these punctures and local coordinates is denoted by Q(z) as in [HL2].

By (2.6) and (2.9), the action of Γ_{R} on W_{1} ⊗ W_{2} associated with Q(z) is given by k acting as −ℓ and

(2.15)  (a ⊗ f)(w_{(1)} ⊗ w_{(2)}) = (a ⊗ ι_{∞}f)(w_{(1)}) ⊗ w_{(2)}
+ w_{(1)} ⊗ (a ⊗ ι_{0}f)(w_{(2)})

for a ∈ 𝔤, f ∈ R, w_{(1)} ∈ W_{1} and w_{(2)} ∈ W_{2}. Hence by (2.10) the action of Γ_{R} on (W_{1} ⊗ W_{2})^{∗} is given by k acting as ℓ and

⟨(a ⊗ f)(λ), w_{(1)} ⊗ w_{(2)}⟩ = −⟨λ, (a ⊗ f)(w_{(1)} ⊗ w_{(2)})⟩ = −⟨λ, (a ⊗ ι_{∞}f)(w_{(1)}) ⊗ w_{(2)} + w_{(1)} ⊗ (a ⊗ ι_{0}f)(w_{(2)})⟩

for a ∈ 𝔤, f ∈ R, λ ∈ (W_{1} ⊗ W_{2})^{∗}, w_{(1)} ∈ W_{1} and w_{(2)} ∈ W_{2}. This gives an action of the Lie algebra 𝔤 ⊗ ℂ[t, t^{−1}, (z + t)^{−1}] ⊕ ℂk. (Note that ℂ[t, t^{−1}, (z + t)^{−1}] = ι_{z}R.) In particular, for f = t^{n}, n ∈ ℤ, we have ι_{z}t^{n} = (z + t)^{n}, and

⟨(a ⊗ (z + t)^{n})(λ), w_{(1)} ⊗ w_{(2)}⟩ (2.16)

= −⟨λ, (a ⊗ ι_{∞}t^{n})(w_{(1)}) ⊗ w_{(2)} + w_{(1)} ⊗ (a ⊗ ι_{0}t^{n})(w_{(2)})⟩
= −⟨λ, (a ⊗ t^{−n})(w_{(1)}) ⊗ w_{(2)} + w_{(1)} ⊗ (a ⊗ t^{n})(w_{(2)})⟩

for any a ∈ 𝔤, λ ∈ (W_{1} ⊗ W_{2})^{∗}, w_{(1)} ∈ W_{1} and w_{(2)} ∈ W_{2}; and in case f = (t − z)^{n}, n ∈ ℤ, we have ι_{z}(t − z)^{n} = t^{n}, and

(2.17) ⟨(a ⊗ t^{n})(λ), w_{(1)}⊗w_{(2)}⟩

= −⟨λ, (a ⊗ ι_{∞}(t − z)^{n})(w_{(1)}) ⊗ w_{(2)}

+ w_{(1)} ⊗ (a ⊗ ι_{0}(t − z)^{n})(w_{(2)})⟩

= −⟨λ, (a ⊗ ι_{+}(t^{−1} − z)^{n})(w_{(1)}) ⊗ w_{(2)}

+ w_{(1)} ⊗ (a ⊗ ι_{+}(t − z)^{n})(w_{(2)})⟩

= −⟨λ, ∑_{i∈ℕ} \binom{n}{i}(−z)^{i}(a ⊗ t^{i−n})(w_{(1)}) ⊗ w_{(2)}+

+ w_{(1)} ⊗ ∑_{i∈ℕ} \binom{n}{i}(−z)^{n−i}(a ⊗ t^{i})(w_{(2)})⟩

for any a ∈ 𝔤, λ ∈ (W_{1} ⊗ W_{2})^{∗}, w_{(1)} ∈ W_{1} and w_{(2)} ∈ W_{2}.

By the general construction we now have

W_{1} ∘_{Q(z)} W_{2} (2.18)

= {λ ∈ (W_{1} ⊗ W_{2})^{∗} | for some N ∈ ℕ, ξ_{1}ξ_{2} ⋯ ξ_{N}λ = 0
for any ξ_{1}, ξ_{2}, …, ξ_{N} ∈ 𝔤 ⊗ tℂ[t, (z + t)^{−1}]}.

The tensor product is then defined as

D(W_{1} ∘_{Q(z)} W_{2}).

3. Tensor product for modules for a conformal vertex algebra

In this section we recall the construction and some results in the tensor product theory for suitable module categories for a conformal vertex algebra from [HL2]–[HL4], [H2] and [HLZ1], [HLZ2].

We assume the reader is familiar with the material in [FLM] and [FHL], such as the language of formal calculus, the notion of vertex operator algebra and their modules, etc. Results from [HL2]–[HL4], [H2] and [HLZ1], [HLZ2] will be recalled without proof. We refer the reader to these papers for details.

We will focus on the “Q(z)-tensor product” in this section, due to the fact that tensor product constructed in [KL1] corresponds to Q(1).

In [HLZ1] and [HLZ2], for an abelian group A and an abelian group Ã containing A as a subgroup, the notions of strongly A-graded conformal vertex algebra and strongly Ã-graded generalized modules were introduced. The vertex operator algebras and their (ordinary) modules are exactly the conformal vertex algebras and their (ordinary) modules that are strongly graded with respect to the trivial group (see Remark 2.4 in [HLZ1] and Remarks 2.24 and 2.27 in [HLZ2]). In the present paper, we shall work in the special case of [HLZ1] and [HLZ2] in which A and Ã are trivial.

Let V be a vertex operator algebra, that is, as a special case of the general theory developed in [HLZ1] and [HLZ2], a strongly A-graded conformal vertex algebra with trivial A. A generalized V -module is a strongly Ã-graded generalized module in the sense of [HLZ1] and [HLZ2] with trivial Ã. It can also be defined directly in the same way as a V -module except that instead of being the direct sum of eigenspaces for the operator L(0), it is assumed to be the direct sum of generalized eigenspaces for L(0).

Recall from Definition 2.5 in [HLZ1] or Definition 3.32 in [HLZ2] that given a generalized V -module (W, Y_{W}) with

W = ∐_{n∈ℂ}W_{[n]}

where W_{[n]} is the generalized eigenspace for L(0) with respect to the eigenvalue n, its contragredient module is the vector space

W′ = ∐_{n∈ℂ}(W_{[n]})^{∗}

equipped with the vertex operator map Y′ defined by

⟨Y′(v, x)w′, w⟩ = ⟨w′, Y_{W}^{o} (v, x)w⟩,

for any v ∈ V, w′ ∈ W′ and w ∈ W, where

Y_{W}^{o} (v, x) = Y_{W} (e^{xL(1)}(−x^{−2})^{L(0)}v, x^{−1}),

for any v ∈ V, is the opposite vertex operator (cf. [FHL]). We will use the standard notation

\overline{W} = ∏_{n∈ℂ}W_{[n]},

the formal completion of W with respect to the ℂ-grading.

Fix a nonzero complex number z. The concept of Q(z)-intertwining map is defined as follows (see Definition 4.17 of [HLZ1] or Definition 4.32 of [HLZ2]):

Definition 3.1. Let W_{1}, W_{2} and W_{3} be generalized modules for a vertex operator algebra V. A Q(z)-intertwining map of type \binom{W_{3}}{W_{1} W_{2}} is a linear map I : W_{1} ⊗ W_{2} → \overline{W}_{3} such that the following conditions are satisfied: the lower truncation condition: for any elements w_{(1)} ∈ W_{1}, w_{(2)} ∈ W_{2}, and any n ∈ ℂ,

(3.1)  π_{n−m}I(w_{(1)} ⊗ w_{(2)}) = 0 for m ∈ ℕ sufficiently large;

and the Jacobi identity:

z^{−1}δ (\frac{x_{1} − x_{0}}{z} ) Y_{3}^{o}(v, x_{0})I(w_{(1)} ⊗ w_{(2)}) (3.2)

= x_{0}^{−1}δ (\frac{x_{1} − z}{x_{0}} ) I(Y_{1}^{o}(v, x_{1})w_{(1)} ⊗ w_{(2)})

− x_{0}^{−1}δ (\frac{z − x_{1}}{−x_{0}} ) I(w_{(1)} ⊗ Y_{2}(v, x_{1})w_{(2)})

for v ∈ V, w_{(1)} ∈ W_{1} and w_{(2)} ∈ W_{2} (note that the left-hand side of (3.2) is meaningful because any infinite linear combination of v_{n} of the form ∑_{n<N} a_{n}v_{n} (a_{n} ∈ ℂ) acts on any I(w_{(1)} ⊗ w_{(2)}), due to (3.1)).

Given generalized V -modules W_{1} and W_{2}, we first have the notion of a Q(z)-product, as follows (see Definition 4.39 of [HLZ2]):

Definition 3.2. Let W_{1} and W_{2} be generalized V -modules. A Q(z)-product of W_{1} and W_{2} is a generalized V -module (W_{3}, Y_{3}) together with a Q(z)- intertwining map I_{3} of type \binom{W_{3}}{W_{1}W_{2}}, We denote it by (W_{3}, Y_{3}; I_{3}) or simply by (W_{3}, I_{3}). Let (W_{4}, Y_{4}; I_{4}) be another Q(z)-product of W_{1} and W_{2}. A morphism from (W_{3}, Y_{3}; I_{3}) to (W_{4}, Y_{4}; I_{4}) is a module map η from W_{3} to W_{4} such that

I_{4} = \overline{η} ∘ I_{3}.

where \overline{η} is the natural map from \overline{W}_{3} to \overline{W}_{4} uniquely extending η.

Let C be a full subcategory of the category of generalized V -modules. The notion of Q(z)-tensor product of W_{1} and W_{2} in C is defined in term of a universal property as follows (see Definition 4.40 of [HLZ2]):

Definition 3.3. For W_{1}, W_{2} ∈ ob C, a Q(z)-tensor product of W_{1} and W_{2} in C is a Q(z)-product (W_{0}, Y_{0}; I_{0}) with W_{0} ∈ ob C such that for any Q(z)-product (W, Y; I) with W ∈ ob C, there is a unique morphism from (W_{0}, Y_{0}; I_{0}) to (W, Y; I). Clearly, a Q(z)-tensor product of W_{1} and W_{2} in C, if it exists, is unique up to a unique isomorphism. In this case we will denote it as (W_{1} ⊠_{Q(z)} W_{2}, Y_{Q(z)}; ⊠_{Q(z)}) and call the object (W_{1} ⊠_{Q(z)} W_{2}, Y_{Q(z)}) the Q(z)-tensor product module of W_{1} and W_{2} in C. We will skip the term “in C” if the category C under consideration is clear in context.

Now we recall the construction of the Q(z)-tensor product from Section 5.3 of [HLZ2], which generalizes that in [HL2]–[HL4]. Let W_{1} and W_{2} be generalized V -modules. We first have the following linear action τ_{Q(z)} of the space

V ⊗ ι_{+}ℂ[t, t^{−1}, (z + t)^{−1}]

on (W_{1} ⊗ W_{2})^{∗}:

(τ_{Q(z)} (z^{−1}δ (\frac{x_{1} − x_{0}}{z} ) Y_{t}(v, x_{0})) λ) (w_{(1)} ⊗ w_{(2)}) (3.3)

= x_{0}^{−1}δ (\frac{x_{1} − z}{x_{0}} ) λ(Y_{1}^{o}(v, x_{1})w_{(1)} ⊗ w_{(2)})

− x_{0}^{−1}δ (\frac{z − x_{1}}{−x_{0}} ) λ(w_{(1)} ⊗ Y_{2}(v, x_{1})w_{(2)}).

for v ∈ V, λ ∈ (W_{1} ⊗ W_{2})^{∗}, w_{(1)} ∈ W_{1}, w_{(2)} ∈ W_{2}, where

(3.4)  Y_{t}(v, x) = v ⊗ t^{−1}δ(\frac{t}{x}).

This includes an action Y′_{Q(z)} of V ⊗ ℂ[t, t^{−1}] defined by

(3.5)  Y′_{Q(z)}(v, x) = τ_{Q(z)}(v ⊗ t^{−1}δ(\frac{t}{x})),

that is, by taking Res_{x_{1}} in (3.3),

(Y′_{Q(z)}(v, x_{0})λ)(w_{(1)} ⊗ w_{(2)}) (3.6)

= λ(Y_{1}^{o}(v, x_{0} + z)w_{(1)} ⊗ w_{(2)})

− Res_{x_{1}}x_{0}^{−1}δ (\frac{z − x_{1}}{−x_{0}} ) λ(w_{(1)} ⊗ Y_{2}(v, x_{1})w_{(2)}).

We also have the operators L′_{Q(z)}(n), n ∈ ℤ defined by

Y′_{Q(z)}(ω, x) = ∑_{n∈ℤ}L′_{Q(z)}(n)x^{−n−2}.

We have the following construction of W_{1} _{Q(z)}W_{2}, a subspace of (W_{1} ⊗ W_{2})^{∗} (see Definition 5.60 and Theorem 5.74 of [HLZ2]):

Definition 3.4. Given W_{1} and W_{2} as above, the vector space W_{1} _{Q(z)}W_{2} consists of all the elements λ ∈ (W_{1} ⊗ W_{2})^{∗} satisfying the following two conditions:

(1) (Q(z)-compatibility condition)

(a) (Lower truncation condition) For all v ∈ V, the formal Laurent series Y′_{Q(z)}(v, x)λ involves only finitely many negative powers of x.

(b) The following formula holds:

τ_{Q(z)} (z^{−1}δ (\frac{x_{1} − x_{0}}{z} ) Y_{t}(v, x_{0})) λ (3.7)

= z^{−1}δ (\frac{x_{1} − x_{0}}{z} ) Y′_{Q(z)}(v, x_{0})λ for all v ∈ V.

(2) (Q(z)-local grading restriction condition)

(a) (Grading condition) λ is a (finite) sum of generalized eigenvectors of (W_{1} ⊗ W_{2})^{∗} for the operator L′_{Q(z)}(0).

(b) The smallest subspace W_{λ} of (W_{1} ⊗ W_{2})^{∗} containing λ and stable under the component operators τ_{Q(z)}(v ⊗ t^{n}) of the operators Y′_{Q(z)}(v, x) for v ∈ V, n ∈ ℤ, have the properties:

dim(W_{λ})_{[n]} < ∞ (3.8) (3.9) (W_{λ})_{[n+k]} = 0 for k ∈ ℤ sufficiently negative;

for any n ∈ ℂ, where the subscripts denote the ℂ-grading by L′_{Q(z)}(0)-eigenvalues.

The importance of the space W_{1} _{Q(z)}W_{2} is given by the following theorem from [HL2] and its generalization in [HLZ2] (see Theorems 5.70, 5.71, 5.72, 5.73 and 5.74 of [HLZ2]):

Theorem 3.5. The vector space W_{1} _{Q(z)}W_{2} is closed under the action Y′_{Q(z)} of V and the Jacobi identity holds on W_{1} _{Q(z)}W_{2}. Furthermore, the Q(z)- tensor product of W_{1} and W_{2} in C exists if and only if W_{1} _{Q(z)}W_{2} equipped with Y′_{Q(z)} is an object of C, and in this case, this Q(z)-tensor product is the contragredient module of (W_{1} _{Q(z)}W_{2}, Y′_{Q(z)}).

4. Strong lower truncation condition

In this section we define what we shall call the strong lower truncation condition and prove its equivalence to the compatibility condition.

By using (3.4), (3.5) and the delta function identity

z^{−1}δ (\frac{x_{1} − x_{0}}{z} ) = x_{1}^{−1}δ (\frac{z + x_{0}}{x_{1}} ) ,

formula (3.7) in the compatibility condition can be written as

τ_{Q(z)} (x_{1}^{−1}δ (\frac{z + x_{0}}{x_{1}} ) Y_{t}(v, x_{0})) λ = x_{1}^{−1}δ (\frac{z + x_{0}}{x_{1}} ) τ_{Q(z)}(Y_{t}(v, x_{0}))λ.

Taking the coefficient of x_{1}^{n} for n ∈ ℤ of both sides we get

τ_{Q(z)}((z + x_{0})^{−n−1}Y_{t}(v, x_{0}))λ = (z + x_{0})^{−n−1}Y′_{Q(z)}(v, x_{0})λ.

or, by using (3.4) and the property of the δ-function

t^{−1}δ(\frac{t}{x_{0}} )f(x_{0}) = t^{−1}δ(\frac{t}{x_{0}} )f(t)

for formal series f(x_{0}) we have

(4.1)  τ_{Q(z)}((z + t)^{−n−1}Y_{t}(v, x_{0}))λ = (z + x_{0})^{−n−1}Y′_{Q(z)}(v, x_{0})λ.

Further taking the coefficient of x_{0}^{−m−1} for m ∈ ℤ this becomes

(4.2) τ_{Q(z)}(v ⊗ (z + t)^{−n−1}t^{m})λ = ∑_{i∈ℕ} \binom{−n − 1}{i}z^{−n−1−i}τ_{Q(z)}(v ⊗ t^{m+i})λ.

We have:

Proposition 4.1. Let W_{1} and W_{2} be modules for V as a vertex algebra, λ ∈ (W_{1} ⊗ W_{2})^{∗}, v ∈ V and n ∈ ℕ. Then

τ_{Q(z)}((z + t)^{−n−1}Y_{t}(v, x_{0}))λ

is lower truncated in x_{0} if and only if Y′_{Q(z)}(v, x_{0})λ is lower truncated in x_{0} and (4.1) holds.

Proof. The “if” part is obvious. For the “only if” part, suppose that

τ_{Q(z)}((z + t)^{−n−1}Y_{t}(v, x_{0}))λ = τ_{Q(z)}((z + x_{0})^{−n−1}Y_{t}(v, x_{0}))λ

is lower truncated in x_{0}, then so is

(z + x_{0})^{n+1}τ_{Q(z)}((z + x_{0})^{−n−1}Y_{t}(v, x_{0}))λ (4.3)

= τ_{Q(z)}((z + x_{0})^{n+1}(z + x_{0})^{−n−1}Y_{t}(v, x_{0}))λ
= τ_{Q(z)}(Y_{t}(v, x_{0}))λ
= Y′_{Q(z)}(v, x_{0})λ.

(Note that here we need the existence of the triple product

(z + x_{0})^{n+1}(z + x_{0})^{−n−1}Y_{t}(v, x_{0})

which can be seen by, for example, observing that the coefficient of each power of t exists.) That is, Y′_{Q(z)}(v, x_{0})λ is lower truncated in x_{0}.

By (4.3) we also have

(z + x_{0})^{n+1} (4.4)

· (τ_{Q(z)}((z + x_{0})^{−n−1}Y_{t}(v, x_{0}))λ − (z + x_{0})^{−n−1}Y′_{Q(z)}(v, x_{0})λ)
= (z + x_{0})^{n+1}τ_{Q(z)}((z + x_{0})^{−n−1}Y_{t}(v, x_{0}))λ − Y′_{Q(z)}(v, x_{0})λ
= Y′_{Q(z)}(v, x_{0})λ − Y′_{Q(z)}(v, x_{0})λ
= 0.

Since both terms of

(4.5)  τ_{Q(z)}((z + x_{0})^{−n−1}Y_{t}(v, x_{0}))λ − (z + x_{0})^{−n−1}Y′_{Q(z)}(v, x_{0})λ

are lower truncated in x_{0}, (4.4) implies that (4.5) is equal to 0, as desired. □

Now we define the strong lower truncation condition:

Definition 4.2. Let v ∈ V. An element λ in (W_{1} ⊗ W_{2})^{∗} is said to satisfy the strong lower truncation condition with respect to v ∈ V if there exists N ∈ ℕ depending on v and λ such that

(4.6)  (τ_{Q(z)}(v ⊗ t^{m}(z + t)^{−n−1}))λ = 0

for all m ≥ N and n ∈ ℤ. We say that λ satisfies the strong lower truncation condition if it satisfies the strong lower truncation condition with respect to every vector in V.

As a consequence of Proposition 4.1 we have the following equivalent condition for the Q(x)-compatibility condition:

Proposition 4.3. Let λ ∈ (W_{1}⊗W_{2})^{∗}. Then λ satisfies the Q(z)-compatibil-ity condition if and only if λ satisfies the strong lower truncation condition.

Proof. Suppose that λ satisfies the Q(z)-compatibility condition. Then (4.2) holds for any v ∈ V and m, n ∈ ℤ. But then by part (a) of the compatibility condition we see that the right-hand side of (4.2) is 0 when m is large enough, independent of n. This proves half of the statement. The other half follows directly from Proposition 4.1. □

We will need:

Lemma 4.4. Let N ∈ ℕ. For v ∈ V and λ ∈ (W_{1} ⊗ W_{2})^{∗}, (4.6) holds for any m ≥ N and n ∈ ℤ if and only if for any w_{(1)} ∈ W_{1} and w_{(2)} ∈ W_{2},

(x_{1} − z)^{N}λ(Y_{1}^{o}(v, x_{1})w_{(1)} ⊗ w_{(2)} − w_{(1)} ⊗ Y_{2}(v, x_{1})w_{(2)}) = 0.

Proof. By definition, (4.6) holds for any m ≥ N and n ∈ ℤ if and only if all powers of x_{0} with nonzero coefficients in

τ_{Q(z)} (z^{−1}δ (\frac{x_{1} − x_{0}}{z} ) Y_{t}(v, x_{0})) λ

are at least −N. By definition of the action τ_{Q(z)} in (3.3) this in turn is equivalent to the condition that Res_{x_{0}}x_{0}^{m} of the right-hand side of (3.3) is 0 for any m ≥ N, w_{(1)} ∈ W_{1} and w_{(2)} ∈ W_{2}. The statement now follows from this. □

The following result in formal calculus will be handy for us:

Lemma 4.5. Let x and y be formal variables, ξ be either a formal variable or a complex number. Let K ∈ ℕ and let f_{k}(x, ξ), k ∈ ℤ be a sequence of formal series with coefficients in some vector space satisfying the condition that f_{k}(x, ξ) = 0 for any k ≥ K. Suppose that for N_{1}, N_{2} ∈ ℕ we have

(4.7)  (x + ξ)^{N_{1}}(x + y + ξ)^{N_{2}} ∑_{n∈ℤ}f_{n}(x, ξ)y^{−n−1} = 0.

Then

(4.8)  (x + ξ)^{N_{1}+N_{2}+s}f_{K−1−s}(x, ξ) = 0

for any s ∈ ℕ.

Proof. By taking coefficient of powers y, we see that (4.7) is equivalent to

(4.9)  ∑_{i=0}^{N_{2}} \binom{N_{2}}{i}(x + ξ)^{N_{1}+N_{2}−i}f_{i+k}(x, ξ) = 0

for any k ∈ ℤ. Since f_{k}(x, ξ) = 0 for any k ≥ K, setting k = K−1 in (4.9) we obtain (x + ξ)^{N_{1}+N_{2}}f_{K−1}(x, ξ) = 0. Now (4.8) follows by induction on s, as follows: If the case 0, 1, …, s−1 holds, then by setting k = K −1−s in (4.9) and multiplying both sides by (x+ξ)^{s} we see that only the i = 0 term remains and must equal the right-hand side 0, that is, (x+ξ)^{N_{1}+N_{2}+s}f_{K−1−s}(x, ξ) = 0 also holds. □

We now have:

Proposition 4.6. Let u, v ∈ V and λ ∈ (W_{1} ⊗ W_{2})^{∗}. Suppose that λ satisfies the strong lower truncation condition with respect to both u and v. Then for any k ∈ ℤ, λ satisfies the strong lower truncation condition with respect to u_{k}v. More precisely, let N_{1} be the integer such that (4.6) holds with v replaced by u for any m ≥ N_{1} and n ∈ ℤ and let N_{2} be the corresponding number for v, then (4.6) holds with v replaced by u_{k}v for any m ≥ N_{1} + N_{2} + K − 1 − k and n ∈ ℤ, where K ∈ ℕ is such that u_{n}v = 0 for any n ≥ K.

276 Lin Zhang Proof. By assumption and Lemma 4.4 we have

(x_{1} − z)^{N_{1}}λ(Y_{1}^{o}(u, x_{1})w_{(1)} ⊗ w_{(2)} − w_{(1)} ⊗ Y_{2}(u, x_{1})w_{(2)}) = 0. and

(x_{1} − z)^{N_{2}}λ(Y_{1}^{o}(v, x_{1})w_{(1)} ⊗ w_{(2)} − w_{(1)} ⊗ Y_{2}(v, x_{1})w_{(2)}) = 0.

for any w_{(1)} ∈ W_{1} and w_{(2)} ∈ W_{2}. For k ∈ ℤ, we need a formula similar to either of these, with u or v replaced by u_{k}v. We derive as follows: First, for formal variables x − 1, y_{0} and y_{1}, using the above identities we have:

(x_{1} − z)^{N_{2}}(x_{1} + y_{0} − z)^{N_{1}}λ(y_{0}^{−1}δ(\frac{y_{1} − x_{1}}{y_{0}} )Y_{1}^{o}(v, x_{1})Y_{1}^{o}(u, y_{1})w_{(1)} ⊗ w_{(2)})

= y_{0}^{−1}δ (\frac{y_{1} − x_{1}}{y_{0}} ) (x_{1} − z)^{N_{2}}(x_{1} + y_{0} − z)^{N_{1}}

· λ(Y_{1}^{o}(v, x_{1})Y_{1}^{o}(u, y_{1})w_{(1)} ⊗ w_{(2)})

= y_{0}^{−1}δ (\frac{y_{1} − x_{1}}{y_{0}} ) (x_{1} − z)^{N_{2}}(x_{1} + y_{0} − z)^{N_{1}}

· λ(Y_{1}^{o}(u, y_{1})w_{(1)} ⊗ Y_{2}(v, x_{1})w_{(2)})

= y_{0}^{−1}δ (\frac{y_{1} − x_{1}}{y_{0}} ) (x_{1} − z)^{N_{2}}(y_{1} − z)^{N_{1}}λ(Y_{1}^{o}(u, y_{1})w_{(1)} ⊗ Y_{2}(v, x_{1})w_{(2)})

= y_{0}^{−1}δ (\frac{y_{1} − x_{1}}{y_{0}} ) (x_{1} − z)^{N_{2}}(y_{1} − z)^{N_{1}}λ(w_{(1)} ⊗ Y_{2}(u, y_{1})Y_{2}(v, x_{1})w_{(2)})

= y_{0}^{−1}δ (\frac{y_{1} − x_{1}}{y_{0}} ) (x_{1} − z)^{N_{2}}(x_{1} + y_{0} − z)^{N_{1}}

· λ(w_{(1)} ⊗ Y_{2}(u, y_{1})Y_{2}(v, x_{1})w_{(2)}) = (x_{1} − z)^{N_{2}}(x_{1} + y_{0} − z)^{N_{1}}

· λ (w_{(1)} ⊗ y_{0}^{−1}δ (\frac{y_{1} − x_{1}}{y_{0}} ) Y_{2}(u, y_{1})Y_{2}(v, x_{1})w_{(2)}) .

On the other hand,

(x_{1} − z)^{N_{2}}(x_{1} + y_{0} − z)^{N_{1}}λ(y_{0}^{−1}δ(\frac{x_{1} − y_{1}}{−y_{0}} )Y_{1}^{o}(u, y_{1})Y_{1}^{o}(v, x_{1})w_{(1)} ⊗ w_{(2)})

= y_{0}^{−1}δ (\frac{x_{1} − y_{1}}{−y_{0}} ) (x_{1} − z)^{N_{2}}(x_{1} + y_{0} − z)^{N_{1}}

· λ(Y_{1}^{o}(u, y_{1})Y_{1}^{o}(v, x_{1})w_{(1)} ⊗ w_{(2)})

= y_{0}^{−1}δ (\frac{x_{1} − y_{1}}{−y_{0}} ) (x_{1} − z)^{N_{2}}(y_{1} − z)^{N_{1}}λ(Y_{1}^{o}(u, y_{1})Y_{1}^{o}(v, x_{1})w_{(1)} ⊗ w_{(2)})

= y_{0}^{−1}δ (\frac{x_{1} − y_{1}}{−y_{0}} ) (x_{1} − z)^{N_{2}}(y_{1} − z)^{N_{1}}λ(Y_{1}^{o}(v, x_{1})w_{(1)} ⊗ Y_{2}(u, y_{1})w_{(2)})

= y_{0}^{−1}δ (\frac{x_{1} − y_{1}}{−y_{0}} ) (x_{1} − z)^{N_{2}}(x_{1} + y_{0} − z)^{N_{1}}

· λ(Y_{1}^{o}(v, x_{1})w_{(1)} ⊗ Y_{2}(u, y_{1})w_{(2)})

= y_{0}^{−1}δ (\frac{x_{1} − y_{1}}{−y_{0}} ) (x_{1} − z)^{N_{2}}(x_{1} + y_{0} − z)^{N_{1}}

· λ(w_{(1)} ⊗ Y_{2}(v, x_{1})Y_{2}(u, y_{1})w_{(2)})

= (x_{1} − z)^{N_{2}}(x_{1} + y_{0} − z)^{N_{1}}

· λ (w_{(1)} ⊗ y_{0}^{−1}δ (\frac{x_{1} − y_{1}}{−y_{0}} ) Y_{2}(v, x_{1})Y_{2}(u, y_{1})w_{(2)}) .

Taking difference of these two equalities, using the Jacobi identity and the opposite Jacobi identity we have

(x_{1} − z)^{N_{2}}(x_{1} + y_{0} − z)^{N_{1}}λ(x_{1}^{−1}δ(\frac{y_{1} − y_{0}}{x_{1}} ) Y_{1}^{o}(Y (u, y_{0})v, x_{1})w_{(1)} ⊗ w_{(2)}) =

(x_{1} − z)^{N_{2}}(x_{1} + y_{0} − z)^{N_{1}}λ(w_{(1)} ⊗ x_{1}^{−1}δ (\frac{y_{1} − y_{0}}{x_{1}} ) Y_{2}(Y (u, y_{0})v, x_{1})w_{(2)}).

Applying Res_{y_{1}} we have

(x_{1} − z)^{N_{2}}(x_{1} + y_{0} − z)^{N_{1}}· λ(Y_{1}^{o}(Y (u, y_{0})v, x_{1})w_{(1)} ⊗ w_{(2)} − w_{(1)} ⊗ Y_{2}(Y (u, y_{0})v, x_{1})w_{(2)}) = 0.

Let K be a number such that u_{k}v = 0 for all k ≥ K. Then Lemma 4.5 applies with ξ being −z and f_{k}(x, ξ) being

λ(Y_{1}^{o}(u_{k}v, x_{1})w_{(1)} ⊗ w_{(2)} − w_{(1)} ⊗ Y_{2}(u_{k}v, x_{1})w_{(2)}).

As a result we have

(x_{1}−z)^{N_{2}+N_{2}+s}λ(Y_{1}^{o}(u_{K−1−s}v, x_{1})w_{(1)}⊗w_{(2)}−w_{(1)}⊗Y_{2}(u_{K−1−s}v, x_{1})w_{(2)}) = 0,

for any s ∈ ℕ. This is exactly what we need. □

Combining Proposition 4.3 and Proposition 4.6 we obtain:

Theorem 4.7. Let V be a vertex algebra and S a generating set for V in the sense that

V =

span{a_{n_{k}}^{(k)} ⋯ a_{n_{2}}^{(2)}a_{n_{1}}^{(1)}a^{(0)} | a^{(0)}, a^{(1)}, …, a^{(k)} ∈ S, n_{1}, n_{1}, …, n_{k} ∈ ℤ, k ∈ ℕ}.

Let W_{1} and W_{2} be generalized V -modules. Then λ ∈ (W_{1} ⊗ W_{2})^{∗} satisfies the Q(z)-compatibility condition if and only if λ satisfies the strong lower truncation condition with respect to all elements of S.

5. Generalized modules for vertex operator algebras associated to 𝔤̂

In this section we recall the vertex operator algebra constructed from suitable modules for the affine Lie algebra 𝔤̂. Then we show that the construction of the tensor products in Section 2 and Section 3 are equivalent, for modules in O_{κ}. In particular, this shows that the tensor product constructed by Kazhdan and Lusztig satisfies the universal property in Definition 3.3. Then we show that the objects of the category O_{κ} are C_{1}-cofinite and quasi-finite-dimensional. These results together with a result in [H3] imply that the conditions needed for applying the results obtained in [HLZ1] and [HLZ2] are satisfied. Hence we prove the existence of the associativity isomorphisms and we obtain the braided tensor category structure.

Recall the complex semisimple Lie algebra 𝔤, the nondegenerate invariant bilinear form (·, ·) on 𝔤 and the affine Lie algebra 𝔤̂ in Section 2.

Given any 𝔤-module U and any complex number ℓ, consider U as a 𝔤̂_{(≤)}- module with 𝔤̂_{(−)} acting trivially and k acting as the scalar ℓ. Then the induced 𝔤̂-module

Ind𝔤̂ ; 𝔤(U) = U(𝔤̂) ⊗_{U(𝔤̂_{(≤)})} U

is restricted and is of level ℓ. When U = ℂ is the trivial 𝔤-module, we will write

V𝔤̂(ℓ, 0) = Ind𝔤̂ ; 𝔤(ℂ).

In particular, fix a Cartan subalgebra 𝔥 ⊂ 𝔤 and a set of positive roots, let 𝔤 = 𝔫_{+} ⊕𝔥⊕𝔫_{−} be the corresponding triangular decomposition. Let U = L(λ) be the irreducible highest weight 𝔤-module with highest weight λ ∈ 𝔥^{∗}. That is, L(λ) is the quotient of V (λ) by its maximal proper submodule, where V (λ) is the 𝔤-module induced by the (𝔥 ⊕ 𝔫_{+})-module ℂv_{λ} with

hv_{λ} = λ(h)v_{λ} for h ∈ 𝔥

and 𝔫_{+} acts on v_{λ} as 0. L(λ) is finite dimensional if and only if λ is dominant integral in the sense that

λ(h_{α}) = \frac{2(λ, α)}{(α, α)} ∈ ℕ, for α ∈ Δ_{+}.

In this case,

L(λ) = U(𝔫_{−})/U(𝔫_{−})α_{i}^{λ(h_{α_{i}})+1}

and we will denote the induced 𝔤̂-module by M(ℓ, λ), called the Weyl module for 𝔤̂ with respect to λ. Let J(ℓ, λ) be the maximal proper 𝔤̂-submodule of M(ℓ, λ). Then L(ℓ, λ) = M(ℓ, λ)/J(ℓ, λ) is an irreducible 𝔤̂-module of level ℓ.

We have (see [KL2]):

Proposition 5.1. (1) The operator L(0) acts semisimply on M(ℓ, λ).

(2) The category O_{κ} consists of all the 𝔤̂-modules of level ℓ having a finite composition series all of whose irreducible subquotients are of the form L(ℓ, λ) for various λ.

Note that V𝔤̂(ℓ, 0) is spanned by the elements of the form

a^{(1)}(−n_{1}) ⋯ a^{(r)}(−n_{r})1,

where a^{(1)}, …, a^{(r)} ∈ 𝔤 and n_{1}, …, n_{r} ∈ ℤ_{+}; here and below we use a(−n) to denote the representation image of a ⊗ t^{−n} for a ∈ 𝔤 and n ∈ ℤ.

The following theorem is well-known:

Theorem 5.2 ([FZ]; cf. [LL]). There is a unique vertex algebra structure (V𝔤̂(ℓ, 0), Y, 1) on V𝔤̂(ℓ, 0) such that 1 = 1 ∈ ℂ is the vacuum vector and such that

Y (a(−1)1, x) = ∑_{n∈ℤ}a(n)x^{−n−1}

for a ∈ 𝔤. We have

Y (a^{(0)}(−n_{0})a^{(1)}(−n_{1}) ⋯ a^{(r)}(−n_{r})1, x)

= Res_{x_{1}}(x_{1} − x)^{−n_{0}}Y (a^{(0)}(−1)1, x_{1})Y (a^{(1)}(−n_{1}) ⋯ a^{(r)}(−n_{r})1, x)

− Res_{x_{1}}(−x + x_{1})^{−n_{0}}Y (a^{(1)}(−n_{1}) ⋯ a^{(r)}(−n_{r})1, x)Y (a^{(0)}(−1)1, x_{1}).

Any restricted 𝔤̂-module W of level ℓ has a unique V𝔤̂(ℓ, 0)-module structure with the same action as above. Furthermore, in case ℓ ≠ −h,

ω = \frac{1}{2(ℓ + h)} ∑_{i=1}^{dim 𝔤} g^{i}(−1)^{2}1

is a conformal element, where {g^{i}}_{i=1,...,dim 𝔤} is an orthonormal basis of 𝔤 with respect to the form (·, ·), and the quadruple (V𝔤̂(ℓ, 0), Y, 1, ω) is a vertex operator algebra.

Since every object of O_{κ} is a restricted 𝔤̂-module of level κ − h, it is a module for the vertex algebra V𝔤̂(κ−h, 0), and when κ ≠ 0, it is a generalized module for V𝔤̂(κ − h, 0) viewed as a vertex operator algebra.

For any element λ ∈ W_{1} ∘_{Q(z)} W_{2} (recall (2.18)), by Corollary 4.3 and the fact that 𝔤 ⊗ t^{N}ℂ[t, (z + t)^{−1}] ⊂ (𝔤 ⊗ tℂ[t, (z + t)^{−1}])^{N} in U(𝔤 ⊗ ℂ((t))) we see that λ satisfies the Q(z)-compatibility condition. On the other hand, the closedness of tensor product in O_{κ} in Theorem 2.5 shows that (W_{1} ∘_{Q(z)} W_{2})′, and hence W_{1} ∘_{Q(z)} W_{2} itself, is an object of O_{κ}. So all elements of W_{1} ∘_{Q(z)} W_{2} also satisfy the Q(z)-local grading restriction condition. Hence W_{1} ∘_{Q(z)} W_{2} ⊆ W_{1} _{Q(z)}W_{2}.

Conversely, if λ ∈ W_{1} _{Q(z)}W_{2}, then from the Q(z)-local grading restriction condition we see that λ generates a generalized V -module. But since for any a ∈ 𝔤, a ⊗ t = (a(−1)1)_{1} as an operator on W_{1} _{Q(z)}W_{2} reduces generalized weights by 1, we see that when N is large enough we have ξ_{1} ⋯ ξ_{N}λ = 0 for all ξ_{1}, …, ξ_{N} ∈ 𝔤 ⊗ tℂ[t, (z + t)^{−1}]. Hence W_{1} _{Q(z)}W_{2} ⊆ W_{1} ∘_{Q(z)} W_{2}.

We have proved:

Theorem 5.3. For W_{1} and W_{2} in O_{κ}, the two subspaces W_{1} ∘_{Q(z)} W_{2} and W_{1} _{Q(z)}W_{2} of (W_{1} ⊗ W_{2})^{∗} are equal to each other. In particular, the tensor product of two modules constructed in [KL2] with respect to Q(z) satisfies the universal property in Definition 3.3.

Now we proceed to the existence and construction of the associativity isomorphism for this tensor product. For this, we now work in the setting of tensor product associated with another type of Riemann spheres with punctures and local coordinates, namely, the spheres with punctures and local coordinates of type P (z); recall from [H1] that for a nonzero complex number z, P (z) denotes the Riemann sphere with ordered punctures ∞, z, 0 and local coordinates 1/w, w − z, w around these punctures.

Remark 5.4. The reason that we use P (z) here is because it is most convenient for the formulation of the associativity isomorphisms, due to the fact that spheres with punctures and local coordinates of type P (z) are closed under sewing and subsequently decomposing. The corresponding associativity isomorphisms for other types of tensor products can be constructed from those for the type P (z) by natural transformations associated to certain parallel transport over the moduli space of spheres with punctures and local coordinates. The P (z)-tensor product of W_{1} and W_{2} in O_{κ} exists if and only if their Q(z)-tensor product exists; the details are given in [HLZ2].

We need the following notions from [HLZ1] and [HLZ2], which generalize the corresponding notions in [H2] to the logarithmic case:

Definition 5.5. Let V be a vertex operator algebra and C be a category of generalized V -modules. We say that products of intertwining operators in C satisfy the convergence and extension property if for any objects W_{1}, W_{2}, W_{3}, W_{4} and M_{1} of C, and intertwining operator Y_{1} and Y_{2} of types \binom{W_{4}}{W_{1} M_{1}} and \binom{M_{1}}{W_{2} W_{3}}, respectively, there exists an integer N depending only on Y_{1} and Y_{2}, and for any w_{(1)} ∈ W_{1}, w_{(2)} ∈ W_{2}, w_{(3)} ∈ W_{3}, w′_{(4)} ∈ W′_{4}, there exist M ∈ ℕ, r_{k}, s_{k} ∈ ℝ, i_{k}, j_{k} ∈ ℕ, k = 1, …, M and analytic functions f_{i_{k}j_{k}}(z) on |z| < 1, k = 1, …, M, satisfying

wt w_{(1)} + wt w_{(2)} + s_{k} > N, k = 1, …, M,

such that

⟨w′_{(4)}, Y_{1}(w_{(1)}, x_{2})Y_{2}(w_{(2)}, x_{2})w_{(3)}⟩_{W_{4}} |_{x_{1}=z_{1}, x_{2}=z_{2}}

is convergent when |z_{1}| > |z_{2}| > 0 and can be analytically extended to the multi-valued analytic function

∑_{k=1}^{M} z_{2}^{r_{k}}(z_{1} − z_{2})^{s_{k}}(log z_{2})^{i_{k}}(log(z_{1} − z_{2}))^{j_{k}}f_{i_{k}j_{k}} (\frac{z_{1} − z_{2}}{z_{2}} )

in the region |z_{2}| > |z_{1} − z_{2}| > 0.

Definition 5.6. Let V be a vertex operator algebra and C be a category of generalized V -modules. If for any n ∈ ℤ_{+}, any generalized V -modules W_{i}, i = 0, …, n + 1 and W̃_{i}, i = 1, …, n − 1 in C, and intertwining operators

Y_{1}, Y_{2}, …, Y_{n−1}, Y_{n}, of types \binom{W′_{0}}{W_{1}W̃_{1}}, \binom{W̃_{1}}{W_{2}W̃_{2}}, …, \binom{W̃_{n−2}}{W_{n−1}W̃_{n−1}}, \binom{W̃_{n−1}}{W_{n}W_{n+1}}, respectively, the series

(5.1)  ⟨w_{0}, Y_{1}(w_{1}, z_{1}) ⋯ Y_{n}(w_{n}, z_{n})w_{n+1}⟩

is absolutely convergent in the region |z_{1}| > ⋯ > |z_{n}| > 0, then we say that products of arbitrary number of intertwining operators among objects of C are convergent.

The following theorem was proved in [HLZ1] and [HLZ2]:

Theorem 5.7. Let V be a vertex operator algebra of central charge c ∈ ℂ and C a category of generalized V -modules closed under the contragredient functor (·)′ and under taking direct sums and quotients. Assume that the convergence and extension property for products of intertwining operators holds in C and that products of arbitrary number of intertwining operators among objects of C are convergent. Further assume that every finitely generated lower truncated generalized V -module is an object of C and, for any generalized V -modules W_{1} and W_{2}, W_{1} _{P (z)}W_{2} is an object of C. Then the category C has a natural structure of vertex tensor category (see [HL1]) of central charge equal to the central charge c of V such that for each z ∈ ℂ^{×}, the tensor product bifunctor ⊠_{ψ(P (z))} associated to ψ(P (z)) ∈ K̃^{c}(2) is equal to ⊠_{P (z)}. In particular, the category C has a braided tensor category structure.

By definition it is clear that O_{κ} is closed under taking direct sums and quotients. By Theorems 2.5 and 5.3 and Remark 5.4, we have that O_{κ} is closed under the contragredient functor and the Q(z)- and P (z)-tensor product functors. We now prove:

Proposition 5.8. A lower truncated, finitely generated generalized V𝔤̂(ℓ, 0)- module is an object of O_{κ}.

Proof. Let W be a lower truncated generalized V -module generated by a finite set S of elements. That is, we have

W = U(𝔤̂)S = U(𝔤̂_{(>)})U(𝔤̂_{(≤)})S.

Let N = U(𝔤_{(≤)})S, the U(𝔤_{(≤)})-submodule of W generated by S. Then since W is lower truncated and S is finite, N is finite-dimensional. Let N^{κ} be the 𝔤̂-module induced by N as in (2.13). Then N^{κ} is a generalized Weyl module and there is a unique 𝔤̂-homomorphism from N^{κ} to W fixing N. It is clear that this is a surjection and thus W is a quotient module of generalized Weyl module N^{κ}. Hence W is in O_{κ}. □

Remark 5.9. Note that in general a generalized Weyl module may not be generated by its lowest weight subspace, even if it is indecomposable.

Let V be a vertex operator algebra. A generalized V -module W is C_{1}- cofinite if W/C_{1}(W) is finite-dimensional, where C_{1}(W) is the subspace of W spanned by elements of the form u_{−1}w for u ∈ V_{+} = ∐_{n∈ℕ} V_{(n)} and w ∈ W.

Proposition 5.10. The objects of O_{κ} are C_{1}-cofinite as generalized V𝔤̂(ℓ, 0)- modules.

Proof. Let W be an object of O_{κ}. Then W is a quotient of a generalized Weyl module U(𝔤̂) ⊗_{U(𝔤̂_{(≤0)})} M. Let M̃ be the image of M under projection from the generalized Weyl module to W. Then W is spanned by elements of M̃ together with elements of the form a(−n)w for w ∈ W and n ∈ ℤ_{+}. By the L(−1)-derivative property, we have

a(−n) = (a(−1)1)_{−n}

= \frac{(L(−1)a(−1)1)_{−n+1}}{n − 1}

when n ≠ 1. Thus we see that W is spanned by elements of M̃ together with elements of the form u_{−1}w for u ∈ (V𝔤̂(ℓ, 0))_{+} and w ∈ W. Since M̃ is finite-dimensional, we see that W/C_{1}(W) is also finite-dimensional. □

Let V be a vertex operator algebra. A generalized V -module W is quasi-finite-dimensional if for any real number N, ∐_{ℜ(n)<N} W_{[n]} is finitedimensional.

We have the following:

Proposition 5.11. The objects of O_{κ} are quasi-finite-dimensional as generalized V𝔤̂(ℓ, 0)-modules.

Proof. Since generalized Weyl modules are obviously quasi-finite-dimen-sional, the objects of O_{κ}, as quotients of generalized Weyl modules, are quasi-finite-dimensional. □

By Theorems 2.5 and 5.3, Remark 5.4, Corollary 5.10, Proposition 5.11 and Theorem 7.2 in [HLZ1], we obtain the main conclusion of the present paper:

Theorem 5.12. The category O_{κ} has a natural structure of vertex tensor category and in particular, a natural structure of braided tensor category.

Vertex tensor category structure 283

References

[FHL] Frenkel, Igor B.; Huang, Yi-Zhi; Lepowsky, James. On axiomatic approaches to vertex operator algebras and modules. Memoirs Amer. Math. Soc. 104 (1993), no. 494. MR1142494 (94a:17007), Zbl 0789.17022.

[FLM] Frenkel, Igor; Lepowsky, James; Meurman, Arne. Vertex operator algebras and the monster. Pure and Appl. Math., 134. Academic Press, Boston, 1988. ISBN: 0-12-267065-5 MR0996026 (90h:17026), Zbl 0674.17001.

[FZ] Frenkel, Igor B.; Zhu, Yongchang. Vertex operator algebras associated to representations of affine and Virasoro algebras. Duke Math. J. 66 (1992) 123–168. MR1159433 (93g:17045), Zbl 0848.17032.

[H1] Huang, Yi-Zhi. Two-dimensional conformal field theory and vertex operator algebras. Progress in Math., 148. Birkhäuser, Boston, 1997.

[H2] Huang, Yi-Zhi. A theory of tensor products for module categories for a vertex operator algebra. IV. J. Pure Appl. Alg. 100 (1995) 173–216. MR1344849 (98a:17050), Zbl 0841.17015.

[H3] Huang, Yi-Zhi. Differential equations and intertwining operators. Commun. Contemp. Math. 7 (2005) 375–400. MR2151865 (2006e:17037), Zbl 1070.17012.

[HL1] Huang, Yi-Zhi; Lepowsky, James. Tensor products of modules for a vertex operator algebras and vertex tensor categories. Lie Theory and Geometry. Progr. Math., 123. Birkhäuser, Boston, 1994, 349–383. MR1327541 (96e:17061).

[HL2] Huang, Yi-Zhi; Lepowsky, James. A theory of tensor products for module categories for a vertex operator algebra. I. Selecta Math. (New Series) 1 (1995) 699–756. MR1383584 (98a:17047), Zbl 0854.17032.

[HL3] Huang, Yi-Zhi; Lepowsky, James. A theory of tensor products for module categories for a vertex operator algebra. II. Selecta Math. (New Series) 1 (1995) 757–786. MR1383584 (98a:17047), Zbl 0854.17033.

[HL4] Huang, Yi-Zhi; Lepowsky, James. A theory of tensor products for module categories for a vertex operator algebra. III. J. Pure Appl. Alg. 100 (1995) 141– 171. MR1344848 (98a:17049), Zbl 0841.17014.

[HL5] Huang, Yi-Zhi; Lepowsky, James. Intertwining operator algebras and vertex tensor categories for affine Lie algebras. Duke Math. J. 99 (1999) 113–134. MR1700743 (2001j:17046), Zbl 0953.17016.

[HL6] Huang, Yi-Zhi; Lepowsky, James. A theory of tensor products for module categories for a vertex operator algebra. V. To appear.

[HLZ1] Huang, Yi-Zhi; Lepowsky, James; Zhang, Lin. A logarithmic generalization of tensor product theory for modules for a vertex operator algebra. Internat. J. Math. 17 (2006), 975-1012. MR2261644 (2007i:17034).

[HLZ2] Huang, Yi-Zhi; Lepowsky, James; Zhang, Lin. Logarithmic tensor product theory for generalized modules for a conformal vertex algebra. To appear: arXiv:0710.2687.

[KL1] Kazhdan, David; Lusztig, George. Affine Lie algebras and quantum groups. Internat. Math. Res. Notices (1991), no. 2, 21–29. MR1104840 (92g:17015), Zbl 0726.17015.

[KL2] Kazhdan, David; Lusztig, George. Tensor structures arising from affine Lie algebras. I. J. Amer. Math. Soc. 6 (1993) 905–947. MR1186962 (93m:17014), Zbl 0786.17017.

[KL3] Kazhdan, David; Lusztig, George. Tensor structures arising from affine Lie algebras. II. J. Amer. Math. Soc. 6 (1993) 949–1011. MR1186962 (93m:17014), Zbl 0786.17017.

284 Lin Zhang

[KL4] Kazhdan, David; Lusztig, George. Tensor structures arising from affine Lie algebras. III. J. Amer. Math. Soc. 7 (1994) 335–381. MR1239506 (94g:17048), Zbl 0802.17007.

[KL5] Kazhdan, David; Lusztig, George. Tensor structures arising from affine Lie algebras. IV. J. Amer. Math. Soc. 7 (1994) 383–453. MR1239507 (94g:17049), Zbl 0802.17008.

[LL] Lepowsky, James; Li, Haisheng. Introduction to vertex operator algebras and their representations. Progress in Math., 227. Birkhäuser, Boston, 2003. MR2023933 (2004k:17050), Zbl 1055.17001.

[MS] Moore, Gregory; Seiberg, Nathan. Classical and quantum conformal field theory. Commun. Math. Phys. 123 (1989) 177–254. MR1002038 (90e:81216), Zbl 0694.53074.

[Y] Yakimov, Milen. Categories of modules over an affine Kac–Moody algebra and the Kazhdan–Lusztig tensor product. arXiv:math.RT/0210180.

[Z] Zhang, Lin. Vertex operator algebras and Kazhdan-Lusztig’s tensor category. Ph.D. Thesis, Rutgers University, 2004.

Department of Mathematics, Rutgers University, Piscataway, NJ 08854 linzhang@math.rutgers.edu

This paper is available via http://nyjm.albany.edu/j/2008/14-12.html.