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We define two algebra automorphisms Ͳ0 and Ͳ1 of the q-Onsager algebra $$ {\mathcal{B}}_c $$, 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 $$ {\mathcal{B}}_c $$. 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 $$ \hat{\mathfrak{s}{\mathfrak{l}}_2} $$.

BRAID GROUP ACTION AND ROOT VECTORS FOR THE q-ONSAGER ALGEBRA