Tridiagonal pairs and the quantum affine algebra $U_q({\hat {sl}}_2)$

Abstract

Let $K$ denote a field and let $V$ denote a vector space over $K$ with finite positive dimension. By definition a Leonard pair on $V$ is a pair of linear transformations $A:V\to V$ and $A^*:V\to V$ that satisfy the following two conditions: (i) There exists a basis for $V$ with respect to which the matrix representing $A$ is irreducible tridiagonal and the matrix representing $A^*$ is diagonal. (ii) There exists a basis for $V$ with respect to which the matrix representing $A$ is diagonal and the matrix representing $A^*$ is irreducible tridiagonal. There is a correspondence between Leonard pairs and a family of orthogonal polynomials consisting of the $q$-Racah and some related polynomials of the Askey scheme. In this paper we discuss a mild generalization of a Leonard pair which we call a tridiagonal pair. We will show how certain tridiagonal pairs are associated with finite dimensional modules for the quantum affine algebra $U_q({\hat {sl}}_2)$.