Double affine Lie algebras and finite groups

Abstract

We introduce and begin to study Lie theoretical analogs of symplectic reflection algebras for a finite cyclic group, which we call "cyclic double affine Lie algebra". We focus on type A : in the finite (resp. affine, double affine) case, we prove that these structures are finite (resp. affine, toroidal) type Lie algebras, but the gradings differ. The case which is essentially new involves $\mathbb{C}[u,v]$. We describe its universal central extensions and start the study of its representation theory, in particular of its highest weight integrable modules and Weyl modules. We also consider the first Weyl algebra $A_1$ instead of the polynomial ring $\mathbb{C}[u,v]$, and, more generally, a rank one rational Cherednik algebra. We study quasi-finite highest weight representations of these Lie algebras.