PBW bases and centralisers for the $q$-Onsager algebra

Abstract

We settle five conjectures concerning PBW bases and centralisers for the $q$-Onsager algebra. We first prove that the Baseilhac--Kolb root vectors give a PBW basis in every linear order whenever $q$ is not a root of unity, removing the previous transcendence hypothesis. We next establish twelve PBW bases in the alternating generators and show that they persist under arbitrary scalar central specialisation of the alternating central extension. This proves conjectures of Terwilliger and of Baseilhac and Belliard. Finally, we determine the centralisers of the negative and imaginary alternating subalgebras and show that all four single-family alternating polynomial subalgebras are maximal commutative. The proofs combine explicit straightening for Damiani root vectors, the degeneration $\mathrm{gr} O_q \simeq U_q^+(\widehat{\mathfrak{sl}}_2)$, and large-index triangular crossing arguments.