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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 anabelian group, which generalize the $A,B,C,D$ series of trigonometric Liealgebras. Among the main results, we identify $\widehat{X}_{S}$ with what arecalled 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 Liealgebra. We then show that restricted $\widehat{X}_{S}$-modules of level $\ell$naturally correspond to equivariant quasi modules for affine vertex algebrasrelated to $\mathcal{L}_{S}$. Furthermore, for any finite cyclic group $S$, wecompletely determine the structures of these four families of Lie algebras,showing that they are essentially affine Kac-Moody Lie algebras of certaintypes.

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