Double affine Lie algebras and finite groups

Nicolas Guay, David Hernandez and Sergey Loktev

Vol. 243 (2009), No. 1, 1–41

DOI: 10.2140/pjm.2009.243.1

Abstract

We begin to study the Lie theoretical analogues of symplectic reflection algebras for a finite cyclic group Γ; we call these algebras “cyclic double affine Lie algebras”. We focus on type A: In the finite (respectively affine, double affine) case, we prove that these structures are finite (respectively affine, toroidal) type Lie algebras, but the gradings differ. The case that is essentially new is sl_n(ℂ[u,v] ⋊ Γ). 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 A₁ instead of the polynomial ring ℂ[u,v], and, more generally, a rank one rational Cherednik algebra. We study quasifinite highest weight representations of these Lie algebras.

Keywords

affine Lie algebras, toroidal Lie algebras, symplectic reflection algebras

Mathematical Subject Classification 2000

Primary: 17B67

Milestones

Received: 6 January 2009

Revised: 13 June 2009

Accepted: 19 June 2009

Published: 1 November 2009