An Askey-Wilson Algebra of Rank 2

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.

References in corpus (15)

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• Bivariate $P$-polynomial association schemes
• A Generalized Dynamic Asymmetric Exclusion Process: Orthogonal Dualities and Degenerations