Hall algebras and quantum symmetric pairs II: reflection functors

Abstract

Recently the authors initiated an $ı$Hall algebra approach to (universal) $ı$quantum groups arising from quantum symmetric pairs. In this paper we construct and study BGP type reflection functors which lead to isomorphisms of the $ı$Hall algebras associated to acyclic $ı$quivers. For Dynkin quivers, these symmetries on $ı$Hall algebras induce automorphisms of universal $ı$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 $ı$quantum groups of ADE type.

v2, 55 pages, labels of Theorems in Introduction modified, Section 3 expanded, accepted by CMP

References in corpus (3)

• Canonical bases for tensor products and super Kazhdan-Lusztig theory
• Symmetries and invariants of twisted quantum algebras and associated Poisson algebras
• Teichmüller Theory of Bordered Surfaces

Cited by in corpus (3)

• A Drinfeld type presentation of affine $ı$quantum groups II: split BCFG type
• Relative braid group symmetries on $ı$quantum groups of Kac-Moody type
• Braid group action and quasi-split affine $ı$quantum groups I