Uniform product of Ag,n(V) for an orbifold model V and G-twisted Zhu algebra

Let V be a vertex operator algebra and G a finite automorphism group of V. For each g G and nonnegative rational number , an associative algebra Ag,n(V) plays an important role in the theory of vertex operator algebras, but the given product in Ag,n(V) depends on the eigenspaces of g. We show that if V has no negative weights then there is a uniform definition of products on V and we introduce a G-twisted Zhu algebra AG,n(V) which covers all Ag,n(V). Let V be a simple vertex operator algebra with no negative weights and let be a finite set of inequivalent irreducible twisted V-modules which is closed under the action of G. There is a finite dimensional semisimple associative algebra for a suitable 2-cocycle naturally determined by the G-action on . We show that a duality theorem of Schur–Weyl type holds for the actions of and VG on the direct sum of twisted V-modules in as an application of the theory of AG,n(V). It follows as a natural consequence of the result that for any g G every irreducible g-twisted V-module is a completely reducible VG-module.