A Schur-Weyl type duality for twisted weak modules over a vertex algebra

Let V V be a vertex algebra of countable dimension, G G a subgroup of A u t V AutV of finite order, V G VG the fixed point subalgebra of V V under the action of G G , and S S a finite G G -stable set of inequivalent irreducible twisted weak V V -modules associated with possibly different automorphisms in G G . We show a Schur–Weyl type duality for the actions of A α ( G , S ) Aα(G,S) and V G V^G on the direct sum of twisted weak V V -modules in S S where A α ( G , S ) Aα(G,S) is a finite dimensional semisimple associative algebra associated with G , S G,S , and a 2 2 -cocycle α α naturally determined by the G G -action on S S . It follows as a natural consequence of the result that for any g ∈ G g∈ G every irreducible g g -twisted weak V V -module is a completely reducible weak V G V^G -module.