A family of tridiagonal pairs related to the quantum affine algebra U_q(sl_2)

Abstract

Abstract.A type of tridiagonal pair is considered, said to be mild of q-Serre type. It is shown that these tridiagonal pairs induce the structure of a quantum affine algebra Uq( sl 2 )-module on their underlying vector space. This is done by presenting an explicit basis for the underlying vector space and describing the Uq( sl 2 )-action on that basis.Key words. Leonard pair, Tridiagonal pair, Quantum affine algebra.AMS subject classifications. 20G42, 15A04, 33D80, 05E35, 33C45, 33D45.1. Introduction. In [1] the authors study the mild tridiagonal pairs of q-Serre type-the main result is a description of the members of this family by their action on an "attractive" basis for the underlying vector space. In this paper we use this action to describe a U q ( sl 2 )-module structure on the underlying vector space of each mild tridiagonal pair of q-Serre type. We do so by constructing linear operators on this vector space which essentially satisfy the defining relations for U q ( sl 2 ) in the Chevalley presentation. To state our result precisely we recall some definitions. Throughout this paper, let F denote a field, and let V denote a vector space over

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

“…Theorem 3.3 extends some work of Al-Najjar and Curtin [4,5]. They give a U q ( sl 2 )-action for those tridiagonal pairs that satisfy the assumptions of Theorem 3.3 and for which the dimensions of the V i , V * i are all at most 2.…”

Tridiagonal pairs of q-Racah type

“…. , 1) are called Leonard pairs [55,Definition 1.1], and these are classified up to isomorphism [55,58]. This classification yields a correspondence between the Leonard pairs and a family of orthogonal polynomials consisting of the q-Racah polynomials and their relatives [2,59].…”

Generalized q-Onsager Algebras and Boundary Affine Toda Field Theories

“…Indeed, examples of algebra homomorphisms for the standard generators A 0 , A 1 have been proposed for ρ 0 = 0, ρ 1 = 0, and related finite dimensional representations studied in details. We refer the reader to [ITer1,Bas2,AlCu,ITer2] for details. In particular, the following realization immediately follows from [Bas2]:…”