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 $\mathcal{G}<{\mathcal{W}}^{-}<{\mathcal{W}}^{+}<\tilde{\mathcal{G}}$. We prove that they also form a PBW basis in the different block order ${\mathcal{W}}^{-}<\mathcal{G}<\tilde{\mathcal{G}}<{\mathcal{W}}^{+}$. 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.

14 pages

References in corpus (4)

• Universal K-matrices for quantum Kac-Moody algebras
• A conjecture concerning the $q$-Onsager algebra
• The $q$-Onsager algebra and its alternating central extension
• Universal TT- and TQ-relations via centrally extended q-Onsager algebra