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 (5)

• The Universal Askey-Wilson Algebra
• The Relationship between Zhedanov's Algebra AW(3) and the Double Affine Hecke Algebra in the Rank One Case
• Zhedanov's Algebra AW(3) and the Double Affine Hecke Algebra in the Rank One Case. II. The Spherical Subalgebra
• Representations of the rank two Racah algebra and orthogonal multivariate polynomials
• Higher Rank Askey-Wilson Algebras as Skein Algebras

Cited by in corpus (2)

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