A family of tridiagonal pairs

Abstract

Let F denote a field, and let V denote a vector space of finite positive dimension over F. Let A, A * denote a tridiagonal pair on V of diameter d 3. Assume the eigenvalue and dual eigenvalue sequences of A, A * satisfy θ i = q 2i θ , θ * i = q 2d−2i θ * for some nonzero scalars θ , θ * , q ∈ F, where q is not a root of unity. Assume that not all eigenvalues of A and A * have multiplicity one. Let M and M * denote the subalgebras of End(V ) generated by A and A * , respectively, and assume that V = Mv * + M * v for some eigenvectors v * of A * associated with θ * 0 and v of A associated with θ d . We find a nice basis for V and describe the action of A, A * on this basis in terms of six parameters.

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

“…, 2, 2, 1. Mild TDPs are studied in [1], where an "attractive" basis for the underlying vector space V is constructed. We are also interested in another simplifying property for TDPs.…”

“…TDPs of q-Serre type have been studied in [1,5,6,7]. In [1] the authors described the action of a mild TDP of q-Serre type on its underlying vector space.…”

“…In [1] the action of A * on the basis of Lemma 4.2 was described. The action of L follows from this action of A * and Lemmas 3.2 and 4.3.…”

“…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.…”

Irreducible modules for the quantum affine algebra Uq(slˆ2) and its Borel subalgebra

“…A tridiagonal pair is a mild generalization of a Leonard pair [10, Definition 1.1]. See [1,2] for related topics.…”