---
abstract: 'Let $V$ be a simple vertex operator algebra and $G$ a finite automorphism group. Then there is a natural right $G$-action on the set of all inequivalent irreducible $V$-modules. Let $\S$ be a finite set of inequivalent irreducible $V$-modules which is closed under the action of $G.$ There is a finite dimensional semisimple associative algebra $A_{\alpha}(G,\S)$ for a suitable 2-cocycle naturally determined by the $G$-action on $\S$ such that $A_{\alpha}(G,\S)$ and the vertex operator algebra $V^G$ form a dual pair on the sum of $V$-modules in $\S$ in the sense of Howe. In particular, every irreducible $V$-module is completely reducible $V^G$-module.'
---

-0.0cm -1.3cm 0.0cm

\#1[\_[\_[\#1]{}]{}]{} §[S]{} Ø[[O]{}]{} ł Ł c PS. ’[’\_g]{} ø amssym.def amssym

\[section\] \[th\][Proposition]{} \[th\][Corollary]{} \[th\][Lemma]{} \[th\][Remark]{} \[th\][Definition]{} \[th\][Hypothesis]{}

[ ]{}\
Chongying Dong[^1] and Gaywalee Yamskulna[^2]\
Department of Mathematics, University of California, Santa Cruz, CA 95064

Introduction
============

Let $V$ be a simple vertex operator algebra (see \[B\], \[FLM\]), and $G$ a finite automorphism group. A major problem in orbifold conformal field theory is to understand the module category for the vertex operator algebra $V^G$ of $G$-invariants. In the case $V$ is holomorphic, this is related to the quantum double (see \[DPR\] and \[DM5\]). 