Higher Rank Askey-Wilson Algebras as Skein Algebras

Cooke, Juliet; Lacabanne, Abel

doi:10.48550/arxiv.2205.04414

Cited by 3 publications

(42 citation statements)

References 32 publications

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“…= 2 3 ψ 00 n,p + µ (12) n 2 + µ (12) n µ (1) + µ (2) + µ (3) + µ (123) p − µ (12) n − ψ 00 n,p − 2 3 ψ 00 n,p + ξ 12 µ (12) n + ξ 23 ψ 00 n,p + ξ 0 ,…”

Section: Special Racah Algebra Sr(3)mentioning

confidence: 99%

“…The following theorem ensures that they are also sufficient. Hence we get representations of sR(4) labeled by the multiplet j (1) , j (2) , j (3) , j (4) , j (0) , a 12 , a 123 .…”

Section: It Remains To Determine the Explicit Expressions Of [C 234 ]mentioning

confidence: 99%

“…provide a finite-dimensional real symmetric irreducible representation of sR( 4) with ψ, ρ and ϕ given in appendix B. This representation is labeled by the multiplet of real parameters j (1) , j (2) , j (3) , j (4) , j (0) subject to the constraint N = j (1) + j (2) − j (3) + j (4) + j (0) ∈ Z >0 and inequalities (3.55).…”

Section: Finite-dimensional Symmetric Irreducible Representationsmentioning

confidence: 99%

“…The section 3 was devoted in constructing one finite-dimensional representation of sR(4) labeled by the multiplet j (1) , j (2) , j (3) , j (4) , j (0) . The isomorphisms of sR(4) proved in proposition 2.1, composed with this representation, allows us to construct new representations.…”

Section: Construction Of Other Representationsmentioning

confidence: 99%

“…In this paper, we consider the rank two Racah algebra R(4), which is the first generalization of the usual Racah algebra R (3). Their definitions are recalled in details in section 2.1.…”

Section: Introductionmentioning

confidence: 99%

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Representations of the rank two Racah algebra and orthogonal multivariate polynomials

Crampé¹,

Frappat²,

Ragoucy³

2022

Preprint

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The algebraic structure of the rank two Racah algebra is studied in detail. We provide an automorphism group of this algebra, which is isomorphic to the permutation group of five elements. This group can be geometrically interpreted as the symmetry of a folded icosidodecahedron. It allows us to study a class of equivalent irreducible representations of this Racah algebra. They can be chosen symmetric so that their transition matrices are orthogonal. We show that their entries can be expressed in terms of Racah polynomials. This construction gives an alternative proof of the recurrence, difference and orthogonal relations satisfied by the Tratnik polynomials, as well as their expressions as a product of two monovariate Racah polynomials. Our construction provides a generalization of these bivariate polynomials together with their properties.

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“…= 2 3 ψ 00 n,p + µ (12) n 2 + µ (12) n µ (1) + µ (2) + µ (3) + µ (123) p − µ (12) n − ψ 00 n,p − 2 3 ψ 00 n,p + ξ 12 µ (12) n + ξ 23 ψ 00 n,p + ξ 0 ,…”

Section: Special Racah Algebra Sr(3)mentioning

confidence: 99%

“…The following theorem ensures that they are also sufficient. Hence we get representations of sR(4) labeled by the multiplet j (1) , j (2) , j (3) , j (4) , j (0) , a 12 , a 123 .…”

Section: It Remains To Determine the Explicit Expressions Of [C 234 ]mentioning

confidence: 99%

Section: Finite-dimensional Symmetric Irreducible Representationsmentioning

confidence: 99%

Section: Construction Of Other Representationsmentioning

confidence: 99%

“…In this paper, we consider the rank two Racah algebra R(4), which is the first generalization of the usual Racah algebra R (3). Their definitions are recalled in details in section 2.1.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Representations of the rank two Racah algebra and orthogonal multivariate polynomials

Crampé¹,

Frappat²,

Ragoucy³

2022

Preprint

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An Askey-Wilson Algebra of Rank 2

Groenevelt¹,

Carel²

2023

SIGMA

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An algebra is introduced which can be considered as a rank 2 extension of the Askey-Wilson algebra. Relations in this algebra are motivated by relations between coproducts of twisted primitive elements in the two-fold tensor product of the quantum algebra U q (sl(2, C)). It is shown that bivariate q-Racah polynomials appear as overlap coefficients of eigenvectors of generators of the algebra. Furthermore, the corresponding qdifference operators are calculated using the defining relations of the algebra, showing that it encodes the bispectral properties of the bivariate q-Racah polynomials.

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The Higher-Rank Askey-Wilson Algebra and Its Braid Group Automorphisms

Crampé,

Frappat,

Poulain d'Andecy

et al. 2023

SIGMA

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We propose a definition by generators and relations of the rank $n-2$ Askey-Wilson algebra $\mathfrak{aw}(n)$ for any integer $n$, generalising the known presentation for the usual case $n=3$. The generators are indexed by connected subsets of $\{1,\dots,n\}$ and the simple and rather small set of defining relations is directly inspired from the known case of $n=3$. Our first main result is to prove the existence of automorphisms of $\mathfrak{aw}(n)$ satisfying the relations of the braid group on $n+1$ strands. We also show the existence of coproduct maps relating the algebras for different values of $n$. An immediate consequence of our approach is that the Askey-Wilson algebra defined here surjects onto the algebra generated by the intermediate Casimir elements in the $n$-fold tensor product of the quantum group ${\rm U}_q(\mathfrak{sl}_2)$ or, equivalently, onto the Kauffman bracket skein algebra of the $(n+1)$-punctured sphere. We also obtain a family of central elements of the Askey-Wilson algebras which are shown, as a direct by-product of our construction, to be sent to $0$ in the realisation in the $n$-fold tensor product of ${\rm U}_q(\mathfrak{sl}_2)$, thereby producing a large number of relations for the algebra generated by the intermediate Casimir elements.

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Higher Rank Askey-Wilson Algebras as Skein Algebras

Cited by 3 publications

References 32 publications

Representations of the rank two Racah algebra and orthogonal multivariate polynomials

Representations of the rank two Racah algebra and orthogonal multivariate polynomials

An Askey-Wilson Algebra of Rank 2

The Higher-Rank Askey-Wilson Algebra and Its Braid Group Automorphisms

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