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 V^{G} the fixed point subalgebra of V V under the action of G G , and S \mathscr {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 ) \mathscr {A}_{\alpha }(G,\mathscr {S}) and V G V^G on the direct sum of twisted weak V V -modules in S \mathscr {S} where A α ( G , S ) \mathscr {A}_{\alpha }(G,\mathscr {S}) is a finite dimensional semisimple associative algebra associated with G , S G,\mathscr {S} , and a 2 2 -cocycle α \alpha naturally determined by the G G -action on S \mathscr {S} . It follows as a natural consequence of the result that for any g ∈ G g\in G every irreducible g g -twisted weak V V -module is a completely reducible weak V G V^G -module.