Geodesic
Higher Rank Relations for the Askey–Wilson and $q$-Bannai–Ito Algebra
Symmetry, integrability and geometry: methods and applications, Tome 15 (2019) Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

The higher rank Askey–Wilson algebra was recently constructed in the $n$-fold tensor product of $U_q(\mathfrak{sl}_2)$. In this paper we prove a class of identities inside this algebra, which generalize the defining relations of the rank one Askey–Wilson algebra. We extend the known construction algorithm by several equivalent methods, using a novel coaction. These allow to simplify calculations significantly. At the same time, this provides a proof of the corresponding relations for the higher rank $q$-Bannai–Ito algebra.
Keywords: Askey–Wilson algebra
Mots-clés : Bannai–Ito algebra. 
@article{SIGMA_2019_15_a98,
     author = {Hadewijch De Clercq},
     title = {Higher {Rank} {Relations} for the {Askey{\textendash}Wilson} and $q${-Bannai{\textendash}Ito} {Algebra}},
     journal = {Symmetry, integrability and geometry: methods and applications},
     year = {2019},
     volume = {15},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SIGMA_2019_15_a98/}
}
TY  - JOUR
AU  - Hadewijch De Clercq
TI  - Higher Rank Relations for the Askey–Wilson and $q$-Bannai–Ito Algebra
JO  - Symmetry, integrability and geometry: methods and applications
PY  - 2019
VL  - 15
UR  - http://geodesic.mathdoc.fr/item/SIGMA_2019_15_a98/
LA  - en
ID  - SIGMA_2019_15_a98
ER  - 
%0 Journal Article
%A Hadewijch De Clercq
%T Higher Rank Relations for the Askey–Wilson and $q$-Bannai–Ito Algebra
%J Symmetry, integrability and geometry: methods and applications
%D 2019
%V 15
%U http://geodesic.mathdoc.fr/item/SIGMA_2019_15_a98/
%G en
%F SIGMA_2019_15_a98
Hadewijch De Clercq. Higher Rank Relations for the Askey–Wilson and $q$-Bannai–Ito Algebra. Symmetry, integrability and geometry: methods and applications, Tome 15 (2019). http://geodesic.mathdoc.fr/item/SIGMA_2019_15_a98/

[1] Abrams L., Weibel C., “Cotensor products of modules”, Trans. Amer. Math. Soc., 354 (2002), 2173–2185, arXiv: math.RA/9912211 | DOI | MR | Zbl

[2] Baseilhac P., “Deformed Dolan–Grady relations in quantum integrable models”, Nuclear Phys. B, 709 (2005), 491–521, arXiv: hep-th/0404149 | DOI | MR | Zbl

[3] Baseilhac P., “An integrable structure related with tridiagonal algebras”, Nuclear Phys. B, 705 (2005), 605–619, arXiv: math-ph/0408025 | DOI | MR | Zbl

[4] Baseilhac P., Martin X., Vinet L., Zhedanov A., “Little and big $q$-Jacobi polynomials and the Askey–Wilson algebra”, Ramanujan J. (to appear) , arXiv: 1806.02656 | DOI

[5] Cherednik I., “Double affine Hecke algebras and Macdonald's conjectures”, Ann. of Math., 141 (1995), 191–216 | DOI | MR | Zbl

[6] Crampé N., Gaboriaud J., Vinet L., Zaimi M., Revisiting the Askey–Wilson algebra with the universal $R$-matrix of $U_q(\mathfrak{sl}(2))$, arXiv: 1908.04806

[7] De Bie H., De Clercq H., The $q$-Bannai–Ito algebra and multivariate $(-q)$-Racah and Bannai–Ito polynomials, arXiv: 1902.07883

[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. (to appear) , arXiv: 1805.06642 | DOI

[9] De Bie H., Genest V. X., Lemay J.-M., Vinet L., “A superintegrable model with reflections on $S^{n-1}$ and the higher rank Bannai–Ito algebra”, J. Phys. A: Math. Theor., 50 (2017), 195202, 10 pp., arXiv: 1601.07642 | DOI | MR | Zbl

[10] De Bie H., Genest V. X., van de Vijver W., Vinet L., “A higher rank Racah algebra and the ${\mathbb Z}^n_2$ Laplace–Dunkl operator”, J. Phys. A: Math. Theor., 51 (2018), 025203, 20 pp., arXiv: 1610.02638 | DOI | MR | Zbl

[11] De Bie H., Genest V. X., Vinet L., “The ${\mathbb Z}_2^n$ Dirac–Dunkl operator and a higher rank Bannai–Ito algebra”, Adv. Math., 303 (2016), 390–414, arXiv: 1511.02177 | DOI | MR | Zbl

[12] Dunkl C. F., “Differential-difference operators associated to reflection groups”, Trans. Amer. Math. Soc., 311 (1989), 167–183 | DOI | MR | Zbl

[13] Frappat L., Gaboriaud J., Ragoucy E., Vinet L., “The $q$-Higgs and Askey–Wilson algebras”, Nuclear Phys. B, 944 (2019), 114632, 13 pp., arXiv: 1903.04616 | DOI | MR

[14] Frappat L., Gaboriaud J., Ragoucy E., Vinet L., The dual pair $(U_q(\mathfrak{su}(1,1)),\mathfrak{o}_{q^{1/2}}(2n))$, $q$-oscillators and Askey–Wilson algebras, arXiv: 1908.04277 | MR

[15] Genest V. X., Vinet L., Zhedanov A., “The quantum superalgebra ${\mathfrak{osp}}_q(1|2)$ and a $q$-generalization of the Bannai–Ito polynomials”, Comm. Math. Phys., 344 (2016), 465–481, arXiv: 1501.05602 | DOI | MR | Zbl

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

[17] Granovskii Ya.I., Zhedanov A. S., “Linear covariance algebra for ${SL}_q(2)$”, J. Phys. A: Math. Gen., 26 (1993), L357–L359 | 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 | Zbl

[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 | Zbl

[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 | Zbl

[21] Ito T., Terwilliger P., “Double affine Hecke algebras of rank 1 and the ${\mathbb Z}_3$-symmetric Askey–Wilson relations”, SIGMA, 6 (2010), 065, 9 pp., arXiv: 1001.2764 | DOI | MR | Zbl

[22] Koekoek R., Lesky P. A., Swarttouw R. F., Hypergeometric orthogonal polynomials and their $q$-analogues, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2010 | DOI | MR | Zbl

[23] 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 | Zbl

[24] Koornwinder T. H., Mazzocco M., “Dualities in the $q$-Askey scheme and degenerate DAHA”, Stud. Appl. Math., 141 (2018), 424–473, arXiv: 1803.02775 | DOI | MR | Zbl

[25] Nomura K., “Tridiagonal pairs and the Askey-Wilson relations”, Linear Algebra Appl., 397 (2005), 99–106 | DOI | MR | Zbl

[26] Terwilliger P., “Two linear transformations each tridiagonal with respect to an eigenbasis of the other”, Linear Algebra Appl., 330 (2001), 149–203, arXiv: math.RA/0406555 | DOI | MR | Zbl

[27] Terwilliger P., “Two relations that generalize the $q$-Serre relations and the Dolan–Grady relations”, Physics and Combinatorics (Nagoya, 1999), World Sci. Publ., River Edge, NJ, 2001, 377–398, arXiv: math.QA/0307016 | DOI | MR | Zbl

[28] Terwilliger P., “Leonard pairs and the $q$-Racah polynomials”, Linear Algebra Appl., 387 (2004), 235–276, arXiv: math.QA/0306301 | DOI | MR | Zbl

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

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

[31] 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

[32] Vidunas R., “Askey–Wilson relations and Leonard pairs”, Discrete Math., 308 (2008), 479–495, arXiv: math.QA/0511509 | DOI | MR | Zbl

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