Vertex operator algebras, generalized doubles and dual pairs

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 \({\cal S}\) 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 \(A_{\alpha}(G,{\cal S})\) for a suitable 2-cocycle \(\alpha\) naturally determined by the G-action on \({\cal S}\) such that \(A_{\alpha}(G,{\cal S})\) and the vertex operator algebra \(V^G\) form a dual pair on the sum of V-modules in \({\cal S}\) in the sense of Howe. In particular, every irreducible V-module is completely reducible \(V^G\)-module.