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

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