Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 14 (2018), 044, 18 pages      arXiv:1801.06083
Contribution to the Special Issue on Orthogonal Polynomials, Special Functions and Applications (OPSFA14)

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

Paul Terwilliger
Department of Mathematics, University of Wisconsin, Madison, WI 53706-1388, USA

Received January 25, 2018, in final form May 01, 2018; Published online May 07, 2018

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.

Key words: $q$-Onsager algebra; universal Askey-Wilson algebra; Chebyshev polynomial.

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