Quivers with relations for symmetrizable Cartan matrices III: Convolution algebras

TL;DR

This paper constructs a convolution algebra of constructible functions on module varieties of specific Gorenstein algebras to realize the enveloping algebra of the positive part of a semisimple Lie algebra.

Contribution

It introduces a novel realization of Lie algebra enveloping algebras via convolution algebras on module varieties of Gorenstein algebras.

Findings

Realization of the enveloping algebra as a convolution algebra

Connection between Lie algebras and Gorenstein algebra module varieties

New algebraic structures for Lie algebra representations

Abstract

We realize the enveloping algebra of the positive part of a semisimple complex Lie algebra as a convolution algebra of constructible functions on module varieties of some Iwanaga-Gorenstein algebras of dimension 1.