The $q$-Onsager Algebra and the Universal Askey-Wilson Algebra

TL;DR

This paper establishes a closed-form expression for the PBW basis of the $q$-Onsager algebra by leveraging the universal Askey-Wilson algebra and Chebyshev polynomials, simplifying previous recursive definitions.

Contribution

It introduces a new approach to express the PBW basis of the $q$-Onsager algebra in closed form using the universal Askey-Wilson algebra and Chebyshev polynomials.

Findings

Closed-form expressions for PBW basis elements obtained

Algebra homomorphism from $ ext{O}_q$ to $ riangle_q$ applied

Heavy use of Chebyshev polynomials of the second kind

Abstract

Recently Pascal Baseilhac and Stefan Kolb obtained a PBW basis for the $q$-Onsager algebra $\mathcal O_q$. They defined the PBW basis elements recursively, and it is obscure how to express them in closed form. To mitigate the difficulty, we bring in the universal Askey-Wilson algebra $\Delta_q$. There is a natural algebra homomorphism $\natural \colon \mathcal O_q \to \Delta_q$. We apply $\natural $ to the above PBW basis, and express the images in closed form. Our results make heavy use of the Chebyshev polynomials of the second kind.