Trigonometric Lie algebras, affine Lie algebras, and vertex algebras

Abstract

In this paper, we explore natural connections among trigonometric Lie algebras, (general) affine Lie algebras, and vertex algebras. Among the main results, we obtain a realization of trigonometric Lie algebras as what were called the covariant algebras of the affine Lie algebra Aˆ of Lie algebra A=gl∞⊕gl∞ with respect to certain automorphism groups. We then prove that restricted modules of level ℓ for trigonometric Lie algebras naturally correspond to equivariant quasi modules for the affine vertex algebras VAˆ(ℓ,0) (or VAˆ(2ℓ,0)). Furthermore, we determine irreducible modules and equivariant quasi modules for the simple vertex algebra LAˆ(ℓ,0) with ℓ a positive integer. In particular, we prove that every quasi-finite unitary highest weight (irreducible) module of level ℓ for type A trigonometric Lie algebra gives rise to an irreducible equivariant quasi LAˆ(ℓ,0)-module.