Differential operator realization of braid group action on $ı$quantum groups

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 $ı$quantum groups ${}^{ı}\mathbf{U}(S)$ 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 ${}^{ı}\mathbf{U}(S)$.