PBW bases for $\imath$quantum groups

TL;DR

This paper constructs PBW type bases for $ ext{ } ext{ extit{i}}$quantum groups of finite type, providing explicit formulas and demonstrating their role as integral bases, thus advancing the understanding of their algebraic structure.

Contribution

It introduces PBW type bases for $ ext{ extit{i}}$quantum groups of arbitrary finite type using braid group symmetries, with explicit root vector formulas and integral basis properties.

Findings

Established PBW bases for all finite type $ ext{ extit{i}}$quantum groups.

Provided explicit formulas for root vectors in rank 1 cases.

Demonstrated these bases form integral bases for modified $ ext{ extit{i}}$quantum groups.

Abstract

We establish PBW type bases for $\imath$quantum groups of arbitrary finite type, using the relative braid group symmetries. Explicit formulas for root vectors are provided for $\imath$quantum groups of each rank 1 type. We show that our PBW type bases give rise to integral bases for the modified $\imath$quantum groups. The leading terms of our bases can be identified with the usual PBW bases in the theory of quantum groups.