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

Pascal Baseilhac (1) , Stefan Kolb (2)

1 IDP - Institut Denis Poisson
2 School of Mathematics, Statistics and Physics

We define two algebra automorphisms $T_0$ and $T_1$ of the q-Onsager algebra Bc, 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 Bc. 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 $\widehat{sl_2}$.

Mots clés: Quantum groups; q-Onsager algebra; Braid group

Domaines: Algèbres quantiques [math.QA]; Physique mathématique [math-ph]; Théorie des représentations [math.RT]

Pascal Baseilhac, Stefan Kolb. Braid group action and root vectors for the q-Onsager algebra. Transformation Groups, 2020, 25 (June), pp.363-389. ⟨10.1007/s00031-020-09555-7⟩. ⟨hal-01549066⟩