Tridiagonal pairs of q-Racah type, the Bockting operator ψ, and L-operators for U_q(L(sl_2))

Terwilliger, Paul

doi:10.26493/1855-3974.1283.456

Cited by 6 publications

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References 20 publications

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“…{θ * i } d i=0 ) in Lemma 2.4 (resp. Lemma 2.7), we see that this tridiagonal pair has q-Racah type in the sense of [38,Section 1].…”

Section: Introductionmentioning

confidence: 89%

Twisting finite-dimensional modules for the $q$-Onsager algebra $\mathcal O_q$ via the Lusztig automorphism

Terwilliger¹

2020

Preprint

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The q-Onsager algebra O q is defined by two generators A, A * and two relations, called the q-Dolan/Grady relations. Recently P. Baseilhac and S. Kolb found an automorphism L of O q , that fixes A and sends A * to a linear combination of A * , A 2 A * , AA * A, A * A 2 . Let V denote an irreducible O q -module of finite dimension at least two, on which each of A, A * is diagonalizable. It is known that A, A * act on V as a tridiagonal pair of q-Racah type, giving access to four familiar elements K, B, K ↓ , B ↓ in End(V ) that are used to compare the eigenspace decompositions for A, A * on V . We display an invertiblewhere a is a certain scalar that is used to describe the eigenvalues of A on V . We use the conjugation results to compare the eigenspace decompositions for A, A * , L ±1 (A * ) on V . In this comparison we use the notion of an equitable triple; this is a 3-tuple of elements in End(V ) such that any two satisfy a q-Weyl relation. Our comparison involves eight equitable triples. One of them is aA − a 2 K, M −1 , K where M = (aK − a −1 B)(a − a −1 ) −1 . The map M appears in earlier work of S. Bockting-Conrad concerning the double lowering operator ψ of a tridiagonal pair.

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“…{θ * i } d i=0 ) in Lemma 2.4 (resp. Lemma 2.7), we see that this tridiagonal pair has q-Racah type in the sense of [38,Section 1].…”

Section: Introductionmentioning

confidence: 89%

Twisting finite-dimensional modules for the $q$-Onsager algebra $\mathcal O_q$ via the Lusztig automorphism

Terwilliger¹

2020

Preprint

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“…The paper [121] shows howĤ q is related to Leonard pairs. Additional results in the literature link tridiagonal pairs and Leonard pairs with the Lie algebra sl 2 (see [3, 7-9, 26, 81, 120]), the quantum algebras U q (sl 2 ) (see [2,29,30,88,147,162]), U q ( sl 2 ) (see [5,24,41,75,76,84,152]), the tetrahedron Lie algebra (see [25,56,79,96]), and its q-deformation q (see [42,73,77,78,80,164,165]).…”

Section: Casementioning

confidence: 99%

Totally bipartite tridiagonal pairs

Nomura

Terwilliger²

2021

ELA

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There is a concept in linear algebra called a tridiagonal pair. The concept was motivated by the theory of $Q$-polynomial distance-regular graphs. We give a tutorial introduction to tridiagonal pairs, working with a special case as a concrete example. The special case is called totally bipartite, or totally bipartite (TB). Starting from first principles, we give an elementary but comprehensive account of TB tridiagonal pairs. The following topics are discussed: (i) the notion of a TB tridiagonal system; (ii) the eigenvalue array; (iii) the standard basis and matrix representations; (iv) the intersection numbers; (v) the Askey--Wilson relations; (vi) a recurrence involving the eigenvalue array; (vii) the classification of TB tridiagonal systems; (viii) self-dual TB tridiagonal pairs and systems; (ix) the $\mathbb{Z}_3$-symmetric Askey--Wilson relations; (x) some automorphisms and antiautomorphisms associated with a TB tridiagonal pair; and (xi) an action of the modular group ${\rm PSL}_2(\mathbb{Z})$ associated with a TB tridiagonal pair.

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“…The paper [121] shows how Ĥq is related to Leonard pairs. Additional results in the literature link tridiagonal pairs and Leonard pairs with the Lie algebra sl 2 (see [3, 7-9, 26, 81, 120]), the quantum algebras U q (sl 2 ) (see [2,29,30,88,147,162]), U q ( sl 2 ) (see [5,24,41,75,76,84,152]), the tetrahedron Lie algebra ⊠ (see [25,56,79,96]) and its q-deformation ⊠ q (see [42,73,77,78,80,164,165]).…”

Section: Casementioning

confidence: 99%

Totally bipartite tridiagonal pairs

Nomura

Terwilliger²

2017

Preprint

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There is a concept in linear algebra called a tridiagonal pair. The concept was motivated by the theory of Q-polynomial distance-regular graphs. We give a tutorial introduction to tridiagonal pairs, working with a special case as a concrete example. The special case is called totally bipartite, or TB. Starting from first principles, we give an elementary but comprehensive account of TB tridiagonal pairs. The following topics are discussed: (i) the notion of a TB tridiagonal system; (ii) the eigenvalue array; (iii) the standard basis and matrix representations; (iv) the intersection numbers; (v) the Askey-Wilson relations; (vi) a recurrence involving the eigenvalue array; (vii) the classification of TB tridiagonal systems; (viii) self-dual TB tridiagonal pairs and systems; (ix) the Z 3symmetric Askey-Wilson relations; (x) some automorphisms and antiautomorphisms associated with a TB tridiagonal pair; (xi) an action of the modular group PSL 2 (Z) associated with a TB tridiagonal pair.

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Tridiagonal pairs of q-Racah type, the Bockting operator ψ, and L-operators for U_q(L(sl_2))

Abstract: We describe the Bockting operator ψ for a tridiagonal pair of q-Racah type, in terms of a certain L-operator for the quantum loop algebra U q (L(sl 2 )).

Cited by 6 publications

References 20 publications

Twisting finite-dimensional modules for the $q$-Onsager algebra $\mathcal O_q$ via the Lusztig automorphism

Twisting finite-dimensional modules for the $q$-Onsager algebra $\mathcal O_q$ via the Lusztig automorphism

Totally bipartite tridiagonal pairs

Totally bipartite tridiagonal pairs

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