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The Higher-Rank Askey–Wilson Algebra and Its Braid Group Automorphisms
Symmetry, integrability and geometry: methods and applications, Tome 19 (2023) Cet article a éte moissonné depuis la source Math-Net.Ru

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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.
Keywords: Askey–Wilson algebra, braid group. 
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     title = {The {Higher-Rank} {Askey{\textendash}Wilson} {Algebra} and {Its} {Braid} {Group} {Automorphisms}},
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}
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Nicolas Crampé; Luc Frappat; Loïc Poulain d'Andecy; Eric Ragoucy. The Higher-Rank Askey–Wilson Algebra and Its Braid Group Automorphisms. Symmetry, integrability and geometry: methods and applications, Tome 19 (2023). http://geodesic.mathdoc.fr/item/SIGMA_2023_19_a76/

[1] Bannai E., Ito T., Algebraic combinatorics, v. I, Association schemes, The Benjamin/Cummings Publishing Co., Inc., Menlo Park, CA, 1984 | MR | Zbl

[2] Chen H., On skein algebras of planar surfaces, arXiv: 2206.07856

[3] Cooke J., Lacabanne A., Higher rank Askey–Wilson algebras as skein algebras, arXiv: 2205.04414

[4] Crampé N., Frappat L., Gaboriaud J., Poulain d'Andecy L., Ragoucy E., Vinet L., “The Askey–Wilson algebra and its avatars”, J. Phys. A, 54 (2021), 063001, 32 pp., arXiv: 2009.14815 | DOI | MR | Zbl

[5] Crampé N., Frappat L., Ragoucy E., “Representations of the rank two Racah algebra and orthogonal multivariate polynomials”, Linear Algebra Appl., 664 (2023), 165–215, arXiv: 2206.01031 | DOI | MR

[6] Crampé N., Gaboriaud J., Poulain d'Andecy L., Vinet L., “Racah algebras, the centralizer $Z_n(\mathfrak{sl}_2)$ and its Hilbert–Poincaré series”, Ann. Henri Poincaré, 23 (2022), 2657–2682, arXiv: 2105.01086 | DOI | MR | Zbl

[7] Crampé N., Gaboriaud J., Vinet L., Zaimi M., “Revisiting the Askey–Wilson algebra with the universal $R$-matrix of ${\rm U}_q(\mathfrak{sl}_2)$”, J. Phys. A, 53 (2020), 05LT01, 10 pp., arXiv: 1908.04806 | DOI | MR | Zbl

[8] Crampé N., Poulain d'Andecy L., Vinet L., Zaimi M., “Askey–Wilson braid algebra and centralizer of ${\rm U}_q(\mathfrak{sl}_2)$”, Ann. Henri Poincaré, 24 (2023), 1897–1922, arXiv: 2206.11150 | DOI | MR

[9] Crampé N., Vinet L., Zaimi M., “Temperley–Lieb, Birman–Murakami–Wenzl and Askey–Wilson algebras and other centralizers of ${\rm U}_q(\mathfrak{sl}_2)$”, Ann. Henri Poincaré, 22 (2021), 3499–3528, arXiv: 2008.04905 | DOI | MR

[10] De Bie H., De Clercq H., “The $q$-Bannai–Ito algebra and multivariate $(-q)$-Racah and Bannai–Ito polynomials”, J. Lond. Math. Soc., 103 (2021), 71–126, arXiv: 1902.07883 | DOI | MR | Zbl

[11] De Bie H., De Clercq H., van de Vijver W., “The higher rank $q$-deformed Bannai–Ito and Askey–Wilson algebra”, Comm. Math. Phys., 374 (2020), 277–316, arXiv: 1805.06642 | DOI | MR | Zbl

[12] De Clercq H., “Higher rank relations for the Askey–Wilson and $q$-Bannai–Ito algebra”, SIGMA, 15 (2019), 099, 32 pp., arXiv: 1908.11654 | DOI | MR | Zbl

[13] Genest V.X., Iliev P., Vinet L., “Coupling coefficients of $\mathfrak{su}_q(1,1)$ and multivariate $q$-Racah polynomials”, Nuclear Phys. B, 927 (2018), 97–123, arXiv: 1702.04626 | DOI | MR | Zbl

[14] Genest V.X., Vinet L., Zhedanov A., “Superintegrability in two dimensions and the Racah–Wilson algebra”, Lett. Math. Phys., 104 (2014), 931–952, arXiv: 1307.5539 | DOI | MR | Zbl

[15] Geronimo J.S., Iliev P., “Multivariable Askey–Wilson function and bispectrality”, Ramanujan J., 24 (2011), 273–287 | DOI | MR | Zbl

[16] Granovskii Y.I., Zhedanov A.S., “Nature of the symmetry group of the $6j$-symbol”, J. Exp. Theor. Phys., 67 (1988), 1982–1985 | MR

[17] Granovskii Y.I., Zhedanov A.S., “Hidden symmetry of the Racah and Clebsch–Gordan problems for the quantum algebra $\mathfrak{sl}_q(2)$”, J. Group Theoret. Methods Phys., 1 (1993), 161–171, arXiv: hep-th/9304138

[18] Groenevelt W., “A quantum algebra approach to multivariate Askey–Wilson polynomials”, Int. Math. Res. Not., 2021 (2021), 3224–3266, arXiv: 1809.04327 | DOI | MR | Zbl

[19] Groenevelt W., Wagenaar C., “An Askey–Wilson algebra of rank 2”, SIGMA, 19 (2023), 008, 35 pp., arXiv: 2206.03986 | DOI | MR

[20] Huang H.-W., “Finite-dimensional irreducible modules of the universal Askey–Wilson algebra”, Comm. Math. Phys., 340 (2015), 959–984, arXiv: 1210.1740 | DOI | MR | Zbl

[21] Huang H.-W., “An embedding of the universal Askey–Wilson algebra into ${\rm U}_q(\mathfrak{sl}_2)\otimes {\rm U}_q(\mathfrak{sl}_2)\otimes {\rm U}_q(\mathfrak{sl}_2)$”, Nuclear Phys. B, 922 (2017), 401–434, arXiv: 1611.02130 | DOI | MR | Zbl

[22] Iliev P., “Bispectral commuting difference operators for multivariable Askey–Wilson polynomials”, Trans. Amer. Math. Soc., 363 (2011), 1577–1598, arXiv: 0801.4939 | DOI | MR | Zbl

[23] Kalnins E.G., Kress J.M., Miller Jr. W., “Second-order superintegrable systems in conformally flat spaces. I Two-dimensional classical structure theory”, J. Math. Phys., 46 (2005), 053509, 28 pp. | DOI | MR | Zbl

[24] Kassel C., Turaev V., Braid groups, Grad. Texts in Math., 247, Springer, New York, 2008 | DOI | MR | Zbl

[25] Leonard D.A., “Orthogonal polynomials, duality and association schemes”, SIAM J. Math. Anal., 13 (1982), 656–663 | DOI | MR | Zbl

[26] Post S., “Models of quadratic algebras generated by superintegrable systems in 2D”, SIGMA, 7 (2011), 036, 20 pp., arXiv: 1104.0734 | DOI | MR | Zbl

[27] Post S., Walter A., A higher rank extension of the Askey–Wilson algebra, arXiv: 1705.01860

[28] Terwilliger P., “The universal Askey–Wilson algebra”, SIGMA, 7 (2011), 069, 24 pp., arXiv: 1104.2813 | DOI | MR | Zbl

[29] Terwilliger P., Vidunas R., “Leonard pairs and the Askey–Wilson relations”, J. Algebra Appl., 3 (2004), 411–426, arXiv: math.QA/0305356 | DOI | MR | Zbl

[30] Vermaseren J.A.M., New features of FORM, , arXiv: https://github.com/vermaseren/formmath-ph/0010025

[31] Zhedanov A.S., ““Hidden symmetry” of Askey–Wilson polynomials”, Theoret. and Math. Phys., 89 (1991), 1146–1157 | DOI | MR | Zbl