# Braid group action and root vectors for the $q$-Onsager algebra

## TL;DR

This paper introduces automorphisms and root vectors for the $q$-Onsager algebra, establishing a PBW basis and $q$-commutation relations inspired by quantum group theory.

## Contribution

It defines braid group automorphisms for the $q$-Onsager algebra and constructs root vectors that form a PBW basis with $q$-commutation relations.

## Key findings

- Automorphisms $T_0$, $T_1$ mimic braid group actions.
- Root vectors satisfy $q$-analogues of Onsager relations.
- Provides a PBW basis for the $q$-Onsager algebra.

## Abstract

We define two algebra automorphisms $T_0$ and $T_1$ of the $q$-Onsager algebra $B_c$, which provide an analog of G. Lusztig's braid group action for quantum groups. These automorphisms are used to define root vectors which give rise to a PBW basis for $B_c$. We show that the root vectors satisfy $q$-analogs of Onsager's original commutation relations. The paper is much inspired by I. Damiani's construction and investigation of root vectors for the quantized enveloping algebra of $\widehat{\mathfrak{sl}}_2$.