Tridiagonal pairs and the quantum affine algebra U_q( sl₂)

2003/10/03 by Tatsuro Ito, Paul Terwilliger, Ito, Tatsuro +1 · 1 citation
Computer Science · Mathematics · #Advanced Topics in Algebra #Algebraic structures and combinatorial models #Matrix Theory and Algorithms #math.CV #math.QA #msc:20G42

paper · pdf · doi:10.48550/arxiv.math/0310042

23 pages

arxiv created 2003/10/03 · arxiv updated 2009/12/01

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→ V and A^*:V→ 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( sl₂).

Cited by

• Every Q-polynomial distance-regular graph is sharp over ℝ

Related

• Two linear transformations each tridiagonal with respect to an eigenbasis of the other; an algebraic approach to the Askey scheme of orthogonal polynomials
• Leonard pairs and the q-Racah polynomials
• Some algebra related to P-and Q-polynomial association schemes
• Two linear transformations each tridiagonal with respect to an eigenbasis of the other
• Tridiagonal pairs of q-Racah type, the double lowering operator ψ, and the quantum algebra U_q(\mathfraksl₂)