Leonard Pairs and the Askey–wilson Relations

Abstract

Let K denote a field and let V denote a vector space over K with finite positive dimension. We consider an ordered pair of linear transformations A : V → V and A * : V → V which satisfy the following two properties:(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 irreducible tridiagonal and the matrix representing A is diagonal.We call such a pair a Leonard pair on V . Referring to the above Leonard pair, we show there exists a sequence of scalars β, γ, γ * , ̺, ̺ * , ω, η, η * taken from K such that bothThe sequence is uniquely determined by the Leonard pair provided the dimension of V is at least 4. The equations above are called the Askey-Wilson relations.

The Bannai-Ito polynomials as Racah coefficients of the $sl_{-1}(2)$ algebra

“…Given a Leonard pair (K 1 , K 2 ), it is known [18,23,27] that K 1 , K 2 obey the so-called Askey-Wilson relations…”

“…Our approach consists in constructing the Jordan algebra of the intermediary Casimir operators that appear in the coproduct [6] of three sl −1 (2) algebras; this anticommutator algebra coincides with the Bannai-Ito algebra [22], a special case of the Askey-Wilson algebra introduced in [27]. The two Casimir operators are then shown to form a Leonard pair [3,5,11,16,17,18], an observation which allows to recover the recurrence relation of the Bannai-Ito polynomials for the overlap (Racah) coefficients.…”

A family of tridiagonal pairs and related symmetric functions

“…These are called Leonard pairs, classified in [6]. In particular, they satisfy (for details, see [7]) the Askey-Wilson (AW) relations (4) first introduced by Zhedanov [5]. Other examples of TD pairs are for instance the subset corresponding to ρ = ρ * = 0 which reduce (5) to the q−Serre relations of U q 1/2 ( sl 2 ).…”

“…In particular, their matrix representations of size 2j + 1 × 2j + 1 provide an example of Leonard pairs [7] with shape vector (1, 1, ..., 1). For the simplest case j = 1/2, W …”

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

“…Indeed, the original motivation for our work came from the study of tridiagonal pairs [10,11] and the closely related Leonard pairs [17][18][19][20][21][22][23]. A Leonard pair is a pair of semisimple linear transformations on a finite-dimensional vector space, each of which acts tridiagonally on an eigenbasis for the other [18, Definition 1.1].…”