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

We continue a previous study on $\Gamma$-vertex algebras and their quasimodules. In this paper we refine certain known results and we prove that for any $\Z$-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 $V^{\otimes N}$. 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 that twisted affine Lie algebras are related to vertex algebras and their twisted modules.