Geodesic
An Askey–Wilson Algebra of Rank $2$
Symmetry, integrability and geometry: methods and applications, Tome 19 (2023) Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

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 $\mathcal{U}_q(\mathfrak{sl}(2,\mathbb{C}))$. It is shown that bivariate $q$-Racah polynomials appear as overlap coefficients of eigenvectors of generators of the algebra. Furthermore, the corresponding $q$-difference operators are calculated using the defining relations of the algebra, showing that it encodes the bispectral properties of the bivariate $q$-Racah polynomials.
Keywords: Askey–Wilson algebra
Mots-clés : $q$-Racah polynomials. 
@article{SIGMA_2023_19_a7,
     author = {Wolter Groenevelt and Carel Wagenaar},
     title = {An {Askey{\textendash}Wilson} {Algebra} of {Rank~}$2$},
     journal = {Symmetry, integrability and geometry: methods and applications},
     year = {2023},
     volume = {19},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SIGMA_2023_19_a7/}
}
TY  - JOUR
AU  - Wolter Groenevelt
AU  - Carel Wagenaar
TI  - An Askey–Wilson Algebra of Rank $2$
JO  - Symmetry, integrability and geometry: methods and applications
PY  - 2023
VL  - 19
UR  - http://geodesic.mathdoc.fr/item/SIGMA_2023_19_a7/
LA  - en
ID  - SIGMA_2023_19_a7
ER  - 
%0 Journal Article
%A Wolter Groenevelt
%A Carel Wagenaar
%T An Askey–Wilson Algebra of Rank $2$
%J Symmetry, integrability and geometry: methods and applications
%D 2023
%V 19
%U http://geodesic.mathdoc.fr/item/SIGMA_2023_19_a7/
%G en
%F SIGMA_2023_19_a7
Wolter Groenevelt; Carel Wagenaar. An Askey–Wilson Algebra of Rank $2$. Symmetry, integrability and geometry: methods and applications, Tome 19 (2023). http://geodesic.mathdoc.fr/item/SIGMA_2023_19_a7/

[1] Askey R., Wilson J., Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials, Mem. Amer. Math. Soc., 54, 1985, iv+55 pp. | DOI | MR

[2] Baseilhac P., Vinet L., Zhedanov A., “The $q$-Onsager algebra and multivariable $q$-special functions”, J. Phys. A, 50 (2017), 395201, 22 pp., arXiv: 1611.09250 | DOI | MR

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

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

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

[7] 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

[8] 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

[9] 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

[10] Gasper G., Rahman M., Basic hypergeometric series, Encyclopedia Math. Appl., 96, 2nd ed., Cambridge University Press, Cambridge, 2004 | DOI | MR

[11] Gasper G., Rahman M., “Some systems of multivariable orthogonal Askey–Wilson polynomials”, Theory and Applications of Special Functions, Dev. Math., 13, Springer, New York, 2005, 209–219, arXiv: math.CA/0410249 | DOI | MR

[12] Gasper G., Rahman M., “Some systems of multivariable orthogonal $q$-Racah polynomials”, Ramanujan J., 13 (2007), 389–405, arXiv: math.CA/0410250 | DOI | MR

[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

[14] Granovskii Y.I., Zhedanov A.S., “Hidden symmetry of the Racah and Clebsch–Gordan problems for the quantum algebra $\mathfrak{sl}_q(2)$”, arXiv: hep-th/9304138 | MR

[15] Granovskii Y.I., Zhedanov A.S., “Linear covariance algebra for ${\rm SL}_q(2)$”, J. Phys. A, 26 (1993), L357–L359 | DOI | MR

[16] Groenevelt W., “Bilinear summation formulas from quantum algebra representations”, Ramanujan J., 8 (2004), 383–416, arXiv: math.QA/0201272 | DOI | MR

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

[18] 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

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

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

[21] Koekoek R., Lesky P.A., Swarttouw R.F., Hypergeometric orthogonal polynomials and their $q$-analogues, Springer Monogr. Math., Springer, Berlin, 2010 | DOI | MR

[22] Koelink E., “Eight lectures on quantum groups and $q$-special functions”, Rev. Colombiana Mat., 30 (1996), 93–180 | MR

[23] Koelink H.T., Van Der Jeugt J., “Convolutions for orthogonal polynomials from Lie and quantum algebra representations”, SIAM J. Math. Anal., 29 (1998), 794–822, arXiv: q-alg/9607010 | DOI | MR

[24] Koornwinder T.H., “Askey–Wilson polynomials as zonal spherical functions on the ${\rm SU}(2)$ quantum group”, SIAM J. Math. Anal., 24 (1993), 795–813 | DOI | MR

[25] Koornwinder T.H., “The relationship between Zhedanov's algebra ${\rm AW}(3)$ and the double affine Hecke algebra in the rank one case”, SIGMA, 3 (2007), 063, 15 pp., arXiv: math.QA/0612730 | DOI | MR

[26] Koornwinder T.H., “Zhedanov's algebra $\rm AW(3)$ and the double affine Hecke algebra in the rank one case. II The spherical subalgebra”, SIGMA, 4 (2008), 052, 17 pp., arXiv: 0711.2320 | DOI | MR

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

[28] Terwilliger P., “Leonard pairs from 24 points of view”, Rocky Mountain J. Math., 32 (2002), 827–888, arXiv: math.RA/0406577 | DOI | MR

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

[30] Terwilliger P., “The universal Askey–Wilson algebra and the equitable presentation of $U_q(\mathfrak{sl}_2)$”, SIGMA, 7 (2011), 099, 26 pp., arXiv: 1107.3544 | DOI | MR

[31] Terwilliger P., “The universal Askey–Wilson algebra and DAHA of type $(C^\vee_1,C_1)$”, SIGMA, 9 (2013), 047, 40 pp., arXiv: 1202.4673 | DOI | MR

[32] Terwilliger P., “The $q$-Onsager algebra and the universal Askey–Wilson algebra”, SIGMA, 14 (2018), 044, 18 pp., arXiv: 1801.06083 | DOI | MR

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

[34] Vidũnas R., “Normalized Leonard pairs and Askey–Wilson relations”, Linear Algebra Appl., 422 (2007), 39–57, arXiv: math.RA/0505041 | DOI | MR

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