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

We continue a previous study on ΓΓ-vertex algebras and their quasimodules. In this paper we refine certain known results and we prove that for any ZZ-graded vertex algebra VV and a positive integer NN, the category of VV-modules is naturally isomorphic to the category of quasimodules of a certain type for V⊗NV⊗N. 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.