A Lie-theoretic construction of spherical symplectic reflection algebras

TL;DR

This paper introduces a Lie-theoretic method to construct spherical symplectic reflection algebras of any rank, utilizing quantum Hamiltonian reduction of universal enveloping algebras, and provides a way to build their finite-dimensional representations.

Contribution

It presents a novel Lie-theoretic construction of spherical symplectic reflection algebras for arbitrary rank using quantum Hamiltonian reduction.

Findings

Construction from universal enveloping algebras via Hamiltonian reduction

Explicit method for finite-dimensional representations

Applicable to star-shaped affine Dynkin diagrams

Abstract

We propose a construction of the spherical subalgebra of a symplectic reflection algebra of an arbitrary rank corresponding to a star-shaped affine Dynkin diagram. Namely, it is obtained from the universal enveloping algebra of a certain semi-simple Lie algebra by the process of quantum Hamiltonian reduction. As an application, we propose a construction of finite-dimensional representations of the spherical subalgebra.