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

TL;DR

This paper introduces a family of infinite-dimensional Lie algebras generalizing trigonometric Lie algebras, identifies their structure with covariant algebras of affine Lie algebras, and relates their modules to equivariant quasi modules for vertex algebras.

Contribution

It establishes the structure of these Lie algebras as affine Kac-Moody types and connects their modules to equivariant quasi modules for affine vertex algebras.

Findings

Identification of $\u2208{X}_S$ with covariant algebras of affine Lie algebras

Classification of these Lie algebras as affine Kac-Moody types for finite cyclic groups

Correspondence between restricted modules and equivariant quasi modules

Abstract

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 $\ell$ 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.