Braid group action and root vectors 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 ${\widehat{\mathfrak{sl}}}_2$.

Substantial revision following referee comments; changed ordering of positive roots to unify (proof of) commutation relations; simplified proof of Proposition 5.10; minor changes to the introduction; final version; to appear in Transformation Groups; 24 pages