Tridiagonal pairs of q-Racah type

Ito, Tatsuro; Terwilliger, Paul

doi:10.1016/j.jalgebra.2009.04.008

Cited by 50 publications

(50 citation statements)

References 65 publications

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“…As an application, tridiagonal pairs of q−Racah type (see 'case I' below) over C have been classified for q not a root of unity. For an algebraically closed field and no restrictions on q, note that a classification of tridiagonal pairs is given in [76] (see also [77]). However, to our knowledge, the connection with the theory of orthogonal polynomials has remained, in general, an open problem.…”

Section: Bispectrality and The Relation With Tridiagonal Pairsmentioning

confidence: 99%

A bispectral q-hypergeometric basis for a class of quantum integrable models

Baseilhac

Martín

2018

Journal of Mathematical Physics

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Abstract. For the class of quantum integrable models generated from the q−Onsager algebra, a basis of bispectral multivariable q−orthogonal polynomials is exhibited. In a first part, it is shown that the multivariable Askey-Wilson polynomials with N variables and N + 3 parameters introduced by Gasper and Rahman [1] generate a family of infinite dimensional modules for the q−Onsager algebra, whose fundamental generators are realized in terms of the multivariable q−difference and difference operators proposed by Iliev [2]. Raising and lowering operators extending those of Sahi [3] are also constructed. In a second part, finite dimensional modules are constructed and studied for a certain class of parameters and if the N variables belong to a discrete support. In this case, the bispectral property finds a natural interpretation within the framework of tridiagonal pairs. In a third part, eigenfunctions of the q−Dolan-Grady hierarchy are considered in the polynomial basis. In particular, invariant subspaces are identified for certain conditions generalizing Nepomechie's relations. In a fourth part, the analysis is extended to the special case q = 1. This framework provides a q−hypergeometric formulation of quantum integrable models such as the open XXZ spin chain with generic integrable boundary conditions (q = 1).MSC: 81R50; 81R10; 81U15; 39A70; 33D50; 39A13.

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Section: Bispectrality and The Relation With Tridiagonal Pairsmentioning

confidence: 99%

A bispectral q-hypergeometric basis for a class of quantum integrable models

Baseilhac

Martín

2018

Journal of Mathematical Physics

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“…and this is zero by (14). We have shown (28 [15,17,18,24]. In what follows, we freely invoke the notation and theory of tridiagonal pairs.…”

Section: The Higher Order Q-dolan/grady Relationsmentioning

confidence: 99%

The Lusztig automorphism of the q-Onsager algebra

Terwilliger¹

2018

Journal of Algebra

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Pascal Baseilhac and Stefan Kolb recently introduced the Lusztig automorphism L of the q-Onsager algebra O q . In this paper, we express each of L, L −1 as a formal sum involving some quantum adjoints. In addition, (i) we give a computer-free proof that L exists; (ii) we establish the higher order q-Dolan/Grady relations previously conjectured by Baseilhac and Thao Vu; (iii) we obtain a Lusztig automorphism for the current algebra A q associated with O q ; (iv) we describe what happens when a finite-dimensional irreducible O q -module is twisted via L.

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“…Proof: These equations are reformulations of the first two relations in (8). They can also be obtained from Lemma 2.7, by identifying k = y and using Lemma 3.5.…”

Section: Proof: Use Definition 23 and (6) ✷mentioning

confidence: 99%

The algebra Uq(sl2) in disguise

Bockting-Conrad

Terwilliger

2014

Linear Algebra and its Applications

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We discuss a connection between the algebra U q (sl 2 ) and the tridiagonal pairs of q-Racah type. To describe the connection, let x, y ±1 , z denote the equitable generators for U q (sl 2 ). Let U ∨ q denote the subalgebra of U q (sl 2 ) generated by x, y −1 , z. Using a tridiagonal pair of q-Racah type we construct two finite-dimensional U ∨ q -modules. The constructions yield two nonstandard presentations of U ∨ q by generators and relations. These presentations are investigated in detail.

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Tridiagonal pairs of q-Racah type

Cited by 50 publications

References 65 publications

A bispectral q-hypergeometric basis for a class of quantum integrable models

A bispectral q-hypergeometric basis for a class of quantum integrable models

The Lusztig automorphism of the q-Onsager algebra

The algebra Uq(sl2) in disguise

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