Differential operator realization of braid group action on $\imath$quantum groups

TL;DR

This paper constructs a unique braid group action on certain quantum algebraic structures and provides a realization of this action on quasi-split $ extit{i}$quantum groups of type AIII, including a compatible action on polynomial rings.

Contribution

It introduces a novel braid group action on modified $q$-Weyl algebras and realizes it on $ extit{i}$quantum groups and polynomial rings, advancing understanding of their symmetries.

Findings

Braid group action on modified $q$-Weyl algebra constructed.

Realization of braid group action on quasi-split $ extit{i}$quantum groups of type AIII.

Compatible braid group action on polynomial rings established.

Abstract

We construct a unique braid group action on modified $q$-Weyl algebra $\mathbf A_q(S)$. Under this action, we give a realization of the braid group action on quasi-split $\imath$quantum groups $^{\imath}\mathbf U(S)$ of type $\mathrm{AIII}$. Furthermore, we directly construct a unique braid group action on polynomial ring $\mathbb P$ which is compatible with the braid group action on $\mathbf A_q(S)$ and $^{\imath}\mathbf U(S)$.