On certain generalizations of twisted affine Lie algebras and quasimodules for Γ-vertex algebras

We continue a previous study on Γ-vertex algebras and their quasimodules. In this paper we refine certain known results and we prove that for any image-graded vertex algebra V and a positive integer N, the category of V-modules is naturally isomorphic to the category of quasimodules of a certain type for Vcircle times operatorN. We also study certain generalizations of twisted affine Lie algebras and we relate such Lie algebras to vertex algebras and their quasimodules, in a way similar to how twisted affine Lie algebras are related to vertex algebras and their twisted modules.