Leonard pairs and quantum algebra Uq(sl2)

Abstract

Let K denote an algebraically closed field of characteristic zero. Let V denote a vector space over K with finite positive dimension. A Leonard pair on V is an ordered pair of linear transformations in End(V) such that for each of these transformations there exists a basis for V with respect to which the matrix representing that transformation is diagonal and the matrix representing the other transformation is irreducible tridiagonal. Fix a nonzero scalar q∈K which is not a root of unity. Consider the quantum algebra Uq(sl2) with equitable generators x±1, y, z. Let d denote a nonnegative integer and let Vd,1 denote an irreducible Uq(sl2)-module of dimension d+1 and of type 1. In this paper, we determine all linear transformations A in End(Vd,1) such that on Vd,1, the pair A,x−1, the pair A,y and the pair A,z are all Leonard pairs.