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.