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On the semi-regular module and vertex operator algebras
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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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