Trigonometric Lie algebras, affine Kac-Moody Lie algebras, and equivariant quasi modules for vertex algebras

In this paper, we study a family of infinite-dimensional Lie algebras ${\widehat{X}}_S$, where $X$ stands for the type: $A,B,C,D$, and $S$ is an abelian group, which generalize the $A,B,C,D$ series of trigonometric Lie algebras. Among the main results, we identify ${\widehat{X}}_S$ with what are called the covariant algebras of the affine Lie algebra $\widehat{{\mathcal{L}}_S}$ with respect to some automorphism groups, where ${\mathcal{L}}_S$ is an explicitly defined associative algebra viewed as a Lie algebra. We then show that restricted ${\widehat{X}}_S$-modules of level $\mathrm{\ell }$ naturally correspond to equivariant quasi modules for affine vertex algebras related to ${\mathcal{L}}_S$. Furthermore, for any finite cyclic group $S$, we completely determine the structures of these four families of Lie algebras, showing that they are essentially affine Kac-Moody Lie algebras of certain types.