Double Affine Hecke Algebra of Rank 1 and Orthogonal Polynomials on the Unit Circle

Tsujimoto, Satoshi; Vinet, Luc; Zhedanov, Alexei

doi:10.1007/s00365-019-09468-z

Cited by 5 publications

(3 citation statements)

References 38 publications

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“…In a different line of development, the paper introduced analogues of Askey–Wilson polynomials that are orthogonal on the unit circle, and constructed a DAHA associated with them.…”

Section: Summary Of Other Related Work and Further Perspectivementioning

confidence: 99%

Dualities in the q‐Askey Scheme and Degenerate DAHA

Koornwinder

Mazzocco

2018

Stud Appl Math

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The Askey–Wilson polynomials are a four‐parameter family of orthogonal symmetric Laurent polynomials Rnfalse[zfalse] that are eigenfunctions of a second‐order q‐difference operator L, and of a second‐order difference operator in the variable n with eigenvalue z+z−1=2x. Then, L and multiplication by z+z−1 generate the Askey–Wilson (Zhedanov) algebra. A nice property of the Askey–Wilson polynomials is that the variables z and n occur in the explicit expression in a similar and to some extent exchangeable way. This property is called duality. It returns in the nonsymmetric case and in the underlying algebraic structures: the Askey–Wilson algebra and the double affine Hecke algebra (DAHA). In this paper, we follow the degeneration of the Askey–Wilson polynomials until two arrows down and in four different situations: for the orthogonal polynomials themselves, for the degenerate Askey–Wilson algebras, for the nonsymmetric polynomials, and for the (degenerate) DAHA and its representations.

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“…In a different line of development, the paper introduced analogues of Askey–Wilson polynomials that are orthogonal on the unit circle, and constructed a DAHA associated with them.…”

Section: Summary Of Other Related Work and Further Perspectivementioning

confidence: 99%

Dualities in the q‐Askey Scheme and Degenerate DAHA

Koornwinder

Mazzocco

2018

Stud Appl Math

View full text Add to dashboard Cite

show abstract

“…Askey-Wilson algebras have moreover found their way in the general framework of knot theory through their identification with the Kauffman bracket skein algebras (KBSA) of the four-punctured sphere Sk iq 1/2 (Σ 0,4 ) and other elementary surfaces [37][38][39]. This is also closely connected to double affine Hecke algebras (DAHA) as the Askey-Wilson algebra is related to the spherical subalgebra of the DAHA of type (C ∨ 1 , C 1 ) [20,28,[40][41][42][43][44][45][46]. This overview of the relevance of Askey-Wilson algebras in different domains motivates the present topical report.…”

Section: Introductionmentioning

confidence: 99%

The Askey–Wilson algebra and its avatars

Crampé¹,

Frappat²,

Gaboriaud³

et al. 2021

J. Phys. A: Math. Theor.

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The original Askey–Wilson algebra introduced by Zhedanov encodes the bispectrality properties of the eponym polynomials. The name Askey–Wilson algebra is currently used to refer to a variety of related structures that appear in a large number of contexts. We review these versions, sort them out and establish the relations between them. We focus on two specific avatars. The first is a quotient of the original Zhedanov algebra; it is shown to be invariant under the Weyl group of type D 4 and to have a reflection algebra presentation. The second is a universal analogue of the first one; it is isomorphic to the Kauffman bracket skein algebra (KBSA) of the four-punctured sphere and to a subalgebra of the universal double affine Hecke algebra ( C 1 ∨ , C 1 ) . This second algebra emerges from the Racah problem of U q ( s l 2 ) and is related via an injective homomorphism to the centralizer of U q ( s l 2 ) in its threefold tensor product. How the Artin braid group acts on the incarnations of this second avatar through conjugation by R-matrices (in the Racah problem) or half Dehn twists (in the diagrammatic KBSA picture) is also highlighted. Attempts at defining higher rank Askey–Wilson algebras are briefly discussed and summarized in a diagrammatic fashion.

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“…Askey-Wilson algebras have moreover found their way in the general framework of knot theory through their identification with the Kauffman bracket skein algebras of the four-punctured sphere Sk iq 1/2 (Σ 0,4 ) and other elementary surfaces [37][38][39]. This is also closely connected to double affine Hecke algebras (DAHA) as the Askey-Wilson algebra is related to the spherical subalgebra of the DAHA of type (C ∨ 1 , C 1 ) [20,28,[40][41][42][43][44][45][46]. This overview of the relevance of Askey-Wilson algebras in different domains motivates the present topical report.…”

Section: Introductionmentioning

confidence: 99%

The Askey-Wilson algebra and its avatars

Crampé,

Frappat,

Gaboriaud

et al. 2020

Preprint

Self Cite

View full text Add to dashboard Cite

The original Askey-Wilson algebra introduced by Zhedanov encodes the bispectrality properties of the eponym polynomials. The name Askey-Wilson algebra is currently used to refer to a variety of related structures that appear in a large number of contexts. We review these versions, sort them out and establish the relations between them. We focus on two specific avatars. The first is a quotient of the original Zhedanov algebra; it is shown to be invariant under the Weyl group of type D 4 and to have a reflection algebra presentation. The second is a universal analogue of the first one; it is isomorphic to the Kauffman bracket skein algebra (KBSA) of the four-punctured sphere and to a subalgebra of the universal double affine Hecke algebra (C ∨ 1 , C 1 ). This second algebra emerges from the Racah problem of U q (sl 2 ) and is related via an injective homomorphism to the centralizer of U q (sl 2 ) in its threefold tensor product. How the Artin braid group acts on the incarnations of this second avatar through conjugation by R-matrices (in the Racah problem) or half Dehn twists (in the diagrammatic KBSA picture) is also highlighted. Attempts at defining higher rank Askey-Wilson algebras are briefly discussed and summarized in a diagrammatic fashion.Keywords: Askey-Wilson algebra, Kauffman bracket skein algebra, U q (sl 2 ) algebra, double affine Hecke algebra, centralizer, universal R-matrix, W (D 4 ) Weyl group, half Dehn twist.

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Double Affine Hecke Algebra of Rank 1 and Orthogonal Polynomials on the Unit Circle

Cited by 5 publications

References 38 publications

Dualities in the q‐Askey Scheme and Degenerate DAHA

Dualities in the q‐Askey Scheme and Degenerate DAHA

The Askey–Wilson algebra and its avatars

The Askey-Wilson algebra and its avatars

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