Title: 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 $\widehat{\mathfrak{sl}}_2$.