The quantum loop algebra of $s{l}_2$ and $q$-Racah type bivariate functions

Abstract

A unified algebraic framework for two different six-parameter families of bivariate $q$-Racah type functions is given using the representation theory of the quantum loop algebra $\mathcal{L}{U}_{q}s{l}_2$ of $s{l}_2$. The starting point of the analysis is two left and right coideal subalgebras of $\mathcal{L}{U}_{q}s{l}_2$ and six commutative subalgebras, built from eight elements in $\mathcal{L}{U}_{q}s{l}_2\otimes \mathcal{L}{U}_{q}s{l}_2$. The eight elements depend on two scalars $a,b\in {\mathbb{C}}^{\ast }$, and are diagonalized on a finite-dimensional vector space. The bivariate $q$-Racah type functions are interpreted as the overlap coefficients relating six `distinguished' eigenbases parametrized by $a,b$ of the tensor product (evaluation) representations of $\mathcal{L}{U}_{q}s{l}_2$ labeled by the evaluation parameters $u_1,u_2$. Upon certain conditions on $a,b,u_1,u_2$, it is shown that a subset of pairs of elements act as tridiagonal pairs of type I (also called $q$-Racah type). For $u_1/u_2=1$, another subset of pairs of elements are conjectured to act as factorized Leonard pairs. Thus, in both cases corresponding overlap coefficients relating the various eigenbases associated with different pairs are obtained. Some of their properties are also discussed, as well as their relation with known bivariate polynomials of Tratnik type and the rank 2 Askey--Wilson algebra.