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

SIGMA 19 (2023), 008, 35 pages      arXiv:2206.03986      https://doi.org/10.3842/SIGMA.2023.008

An Askey-Wilson Algebra of Rank 2

Wolter Groenevelt and Carel Wagenaar
Delft Institute of Applied Mathematics, Technische Universiteit Delft, PO Box 5031, 2600 GA Delft, The Netherlands

Received June 30, 2022, in final form February 15, 2023; Published online March 05, 2023

Abstract
An algebra is introduced which can be considered as a rank 2 extension of the Askey-Wilson algebra. Relations in this algebra are motivated by relations between coproducts of twisted primitive elements in the two-fold tensor product of the quantum algebra $\mathcal{U}_{q}(\mathfrak{sl}(2,\mathbb C))$. It is shown that bivariate $q$-Racah polynomials appear as overlap coefficients of eigenvectors of generators of the algebra. Furthermore, the corresponding $q$-difference operators are calculated using the defining relations of the algebra, showing that it encodes the bispectral properties of the bivariate $q$-Racah polynomials.

Key words: Askey-Wilson algebra; $q$-Racah polynomials.

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