A conjecture concerning the $q$-Onsager algebra

Abstract

The $q$-Onsager algebra ${\mathcal{O}}_q$ is defined by two generators $W_0,W_1$ and two relations called the $q$-Dolan/Grady relations. Recently Baseilhac and Kolb obtained a PBW basis for ${\mathcal{O}}_q$ with elements denoted $\{B_{n\delta +{\alpha }_0}{\}}_{n=0}^{\infty },\{B_{n\delta +{\alpha }_1}{\}}_{n=0}^{\infty },\{B_{n\delta }{\}}_{n=1}^{\infty }$. In their recent study of a current algebra ${\mathcal{A}}_q$, Baseilhac and Belliard conjecture that there exist elements $\{W_{-k}{\}}_{k=0}^{\infty },\{W_{k+1}{\}}_{k=0}^{\infty },\{G_{k+1}{\}}_{k=0}^{\infty },\{{\tilde{G}}_{k+1}{\}}_{k=0}^{\infty }$ in ${\mathcal{O}}_q$ that satisfy the defining relations for ${\mathcal{A}}_q$. In order to establish this conjecture, it is desirable to know how the elements in the second list above are related to the elements in the first list above. In the present paper, we conjecture the precise relationship and give some supporting evidence. This evidence consists of some computer checks on SageMath due to Travis Scrimshaw, and a proof of our conjecture for a homomorphic image of ${\mathcal{O}}_q$ called the universal Askey-Wilson algebra.

25 pages, updated Intro, references added

References in corpus (5)

• The Universal Askey-Wilson Algebra
• Some algebra related to $P$-and $Q$-polynomial association schemes
• Non-Abelian symmetries of the half-infinite XXZ spin chain
• The $q$-Onsager algebra and the positive part of $U_q({\widehat{\mathfrak{sl}}}_2)$
• $ı$Hall algebra of the projective line and $q$-Onsager algebra

Cited by in corpus (5)

• The alternating central extension of the $q$-Onsager algebra
• The alternating presentation of $U_q(\widehat{g{l}_2})$ from Freidel-Maillet algebras
• The $q$-Onsager algebra and its alternating central extension
• A generating function associated with the alternating elements in the positive part of $U_q({\widehat{\mathfrak{sl}}}_2)$
• A new PBW basis for the alternating central extension of the $q$-Onsager algebra