On the semi-regular module and vertex operator algebras.pdf

On the semi-regular module and vertex operator algebras.pdf

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On the semi-regular module and vertex operator algebras

a r X i v : 0 7 1 1 .3 0 7 4 v 2 [ m a t h .R T ] 3 D e c 2 0 0 7 ON THE SEMI-REGULAR MODULE AND VERTEX OPERATOR ALGEBRAS MINXIAN ZHU 1. Introduction The aim of this paper is to give a proof of a conjecture stated in a previous paper by the author ([Z1]). Let g be a simple complex Lie algebra, g? be the affine Lie algebra and h∨ be the dual Coxeter number of g. Let Ag,k be the vertex algebroid associated to g and a complex number k, according to [GMS1], we can construct a vertex algebra UAg,k, called the enveloping algebra of Ag,k. Set V = UAg,k. It is shown in [AG] and [GMS2] that not only V is a g?-representation of level k, it is also a g?-representation of the dual level k? = ?2h∨ ? k. Moreover the two copies of g?-actions commute with each other, i.e. V is a g?k ⊕ g?k?-representation. When k /∈ Q, the vertex operator algebra V decomposes into ⊕λ∈P+Vλ,k ? Vλ?,k? as a g?k ⊕ g?k?-module (see [FS], [Z1]). Here P + is the set of dominant integral weights of g, Vλ,k is the Weyl module induced from Vλ, the irreducible representation of g with highest weight λ, in level k, and Vλ?,k? is induced from V ? λ in the dual level k?. In fact the vertex operators can be constructed using intertwining operators and Knizhnik-Zamolodchikov equations (see [Z1]). In the case where k ∈ Q, the g?k ⊕ g?k?-module structure of V is much more com- plicated. In the present paper, we prove a result about the existence of canonical filtrations of V conjectured at the end of [Z1]. More precisely we will prove the following. Theorem 1. Let k ∈ Q, k ?h∨. The vertex operator algebra V admits an in- creasing (resp. a decreasing) filtration of g?k ⊕ g?k?-submodules with factors isomorphic to Vλ,k ? V c λ,k? (resp. V c λ,k ? Vλ,k?), λ ∈ P +, where V c λ,k? is the contragredient module of Vλ,k? defined by the anti-involution: x(n) 7→ ?x(?n), c 7→ c of g?. We need two ingredients to prove the theorem: one is the semi-regular module; the other is the regular representation of the correspondi

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