Hall algebras and quantum symmetric pairs II: reflection functors

TL;DR

This paper develops reflection functors for $ extit{i}$Hall algebras linked to quantum symmetric pairs, establishing symmetries and automorphisms that relate to braid group relations and aid in constructing bases for $ extit{i}$quantum groups.

Contribution

It introduces BGP type reflection functors for $ extit{i}$Hall algebras, leading to new symmetries and automorphisms of $ extit{i}$quantum groups, and provides a framework for constructing bases in ADE types.

Findings

Reflection functors induce isomorphisms of $ extit{i}$Hall algebras.

Symmetries satisfy braid group relations.

Framework for $q$-root vectors and PBW bases.

Abstract

Recently the authors initiated an $\imath$Hall algebra approach to (universal) $\imath$quantum groups arising from quantum symmetric pairs. In this paper we construct and study BGP type reflection functors which lead to isomorphisms of the $\imath$Hall algebras associated to acyclic $\imath$quivers. For Dynkin quivers, these symmetries on $\imath$Hall algebras induce automorphisms of universal $\imath$quantum groups, which are shown to satisfy the braid group relations associated to the restricted Weyl group of a symmetric pair; conjecturally these continue to hold for acyclic quivers/Kac-Moody setting. This leads to a conceptual construction of $q$-root vectors and PBW bases for (universal) quasi-split $\imath$quantum groups of ADE type.