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

Abstract

In this paper, we study a family of infinite-dimensional Lie algebras XˆS, 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 XˆS with what are called the covariant algebras of the affine Lie algebra LSˆ with respect to some automorphism groups, where LS is an explicitly defined associative algebra viewed as a Lie algebra. We then show that restricted XˆS-modules of level ℓ naturally correspond to equivariant quasi modules for affine vertex algebras related to LS. 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.