An Askey–Wilson Algebra of Rank $2$

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.