Quiver varieties and Hall algebras

TL;DR

This paper constructs quantum groups geometrically via Nakajima categories, linking algebraic Hall algebra realizations with geometric approaches, thereby deepening understanding of quantum group structures.

Contribution

It provides a geometric construction of quantum groups using Nakajima categories, connecting algebraic and geometric realizations of Hall algebras.

Findings

Established a direct link between algebraic and geometric realizations of quantum groups.

Provided a geometric construction of quantum groups via Nakajima categories.

Connected Bridgeland's Hall algebra approach with Qin's geometric approach.

Abstract

In this paper we give a geometric construction of the quantum group Ut(G) using Nakajima categories, which were developed in [29]. Our methods allow us to establish a direct connection between the algebraic realization of the quantum group as Hall algebra by Bridgeland [1] and its geometric counterpart by Qin [24].