A classification of sharp tridiagonal pairs

Ito, Tatsuro; Nomura, Kazumasa; Terwilliger, Paul

doi:10.1016/j.laa.2011.03.032

Cited by 26 publications

(25 citation statements)

References 95 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 algebra O q is the "most general" example of a tridiagonal algebra [17, p. 70]. Applications of O q to tridiagonal pairs can be found in [4,[15][16][17][18]22,23,27]. The algebra O q has applications to quantum integrable models [1][2][3][4][5][6][7][8][9][10], reflection equation algebras [12], and coideal subalgebras [14,19,20].…”

Section: Introductionmentioning

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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“…Shortly thereafter they appeared in physics, in the study of statistical mechanical models [4,Section 2]. Up to the present, the representation theory of O q remains an active area of research in mathematics [19,21,22,28,29,30,31,32,33] and physics [3,4,5,6,7,8,9,10,11,12,14,15]. This theory involves a linear algebraic object called a tridiagonal pair [20].…”

Section: Introductionmentioning

confidence: 99%

The q-Onsager Algebra and the Universal Askey-Wilson Algebra

Terwilliger¹

2018

SIGMA

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Recently Pascal Baseilhac and Stefan Kolb obtained a PBW basis for the q-Onsager algebra O q . They defined the PBW basis elements recursively, and it is obscure how to express them in closed form. To mitigate the difficulty, we bring in the universal Askey-Wilson algebra ∆ q . There is a natural algebra homomorphism : O q → ∆ q . We apply to the above PBW basis, and express the images in closed form. Our results make heavy use of the Chebyshev polynomials of the second kind.

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A classification of sharp tridiagonal pairs

Cited by 26 publications

References 95 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 q-Onsager Algebra and the Universal Askey-Wilson Algebra

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