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

Research paper by **Tatsuro Ito, Paul Terwilliger**

Indexed on: **03 Oct '03**Published on: **03 Oct '03**Published in: **Mathematics - Quantum Algebra**

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#### 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
$U_q({\hat {sl}}_2)$.