An Askey-Wilson Algebra of Rank 2

TL;DR

This paper introduces a rank 2 extension of the Askey-Wilson algebra, linking it to bivariate q-Racah polynomials and their bispectral properties through algebraic relations and q-difference operators.

Contribution

It presents a new algebraic structure that generalizes the Askey-Wilson algebra and connects it to bivariate q-Racah polynomials and their bispectrality.

Findings

Bivariate q-Racah polynomials appear as eigenvector overlap coefficients.

The algebra encodes bispectral properties of these polynomials.

Explicit q-difference operators are derived from the algebra.

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.