A new PBW basis for the alternating central extension of the $q$-Onsager algebra

Abstract

We establish a new PBW basis for $\mathcal A_q$, the alternating central extension of the $q$-Onsager algebra. Terwilliger showed that the alternating generators form a PBW basis in the block order $\mathcal G<\mathcal W^-<\mathcal W^+<\widetilde{\mathcal G}$. We prove that they also form a PBW basis in the different block order $\mathcal W^-<\mathcal G<\widetilde{\mathcal G}<\mathcal W^+$. Consequently, multiplication induces a vector-space isomorphism \[ \mathcal W^-\otimes\mathcal G\otimes\widetilde{\mathcal G}\otimes\mathcal W^+ \longrightarrow \mathcal A_q, \] thereby confirming a conjecture of Terwilliger.