Hall algebras and quantum symmetric pairs III: Quiver varieties

TL;DR

This paper develops a geometric framework for universal $ extit{ extbf{i}}$-quantum groups using Nakajima-Keller-Scherotzke quiver varieties, extending previous realizations of quantum groups to the setting of quantum symmetric pairs.

Contribution

It introduces a geometric construction of universal $ extit{ extbf{i}}$-quantum groups and their dual canonical bases via quiver varieties, generalizing Qin's work for type ADE.

Findings

Dynkin $ extit{ extbf{i}}$-quiver algebras as Nakajima-Keller-Scherotzke categories

Construction of dual canonical bases with positivity

Extension of geometric realization to quantum symmetric pairs

Abstract

The $\imath$quiver algebras were introduced recently by the authors to provide a Hall algebra realization of universal $\imath$quantum groups, which is a generalization of Bridgeland's Hall algebra construction for (Drinfeld doubles of) quantum groups; here an $\imath$quantum group and a corresponding Drinfeld-Jimbo quantum group form a quantum symmetric pair. In this paper, the Dynkin $\imath$quiver algebras are shown to arise as new examples of singular Nakajima-Keller-Scherotzke categories. Then we provide a geometric construction of the universal $\imath$quantum groups and their ``dual canonical bases" with positivity, via the quantum Grothendieck rings of Nakajima-Keller-Scherotzke quiver varieties, generalizing Qin's geometric realization of quantum groups of type ADE.