Geodesic
Embeddings of the Racah Algebra into the Bannai–Ito Algebra
Symmetry, integrability and geometry: methods and applications, Tome 11 (2015) Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

Embeddings of the Racah algebra into the Bannai–Ito algebra are proposed in two realizations. First, quadratic combinations of the Bannai–Ito algebra generators in their standard realization on the space of polynomials are seen to generate a central extension of the Racah algebra. The result is also seen to hold independently of the realization. Second, the relationship between the realizations of the Bannai–Ito and Racah algebras by the intermediate Casimir operators of the $\mathfrak{osp}(1|2)$ and $\mathfrak{su}(1,1)$ Racah problems is established. Equivalently, this gives an embedding of the invariance algebra of the generic superintegrable system on the two-sphere into the invariance algebra of its extension with reflections, which are respectively isomorphic to the Racah and Bannai–Ito algebras.
Mots-clés : Bannai–Ito polynomials; Bannai–Ito algebra; Racah polynomials; Racah algebra. 
@article{SIGMA_2015_11_a49,
     author = {Vincent X. Genest and Luc Vinet and Alexei Zhedanov},
     title = {Embeddings of the {Racah} {Algebra} into the {Bannai{\textendash}Ito} {Algebra}},
     journal = {Symmetry, integrability and geometry: methods and applications},
     year = {2015},
     volume = {11},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SIGMA_2015_11_a49/}
}
TY  - JOUR
AU  - Vincent X. Genest
AU  - Luc Vinet
AU  - Alexei Zhedanov
TI  - Embeddings of the Racah Algebra into the Bannai–Ito Algebra
JO  - Symmetry, integrability and geometry: methods and applications
PY  - 2015
VL  - 11
UR  - http://geodesic.mathdoc.fr/item/SIGMA_2015_11_a49/
LA  - en
ID  - SIGMA_2015_11_a49
ER  - 
%0 Journal Article
%A Vincent X. Genest
%A Luc Vinet
%A Alexei Zhedanov
%T Embeddings of the Racah Algebra into the Bannai–Ito Algebra
%J Symmetry, integrability and geometry: methods and applications
%D 2015
%V 11
%U http://geodesic.mathdoc.fr/item/SIGMA_2015_11_a49/
%G en
%F SIGMA_2015_11_a49
Vincent X. Genest; Luc Vinet; Alexei Zhedanov. Embeddings of the Racah Algebra into the Bannai–Ito Algebra. Symmetry, integrability and geometry: methods and applications, Tome 11 (2015). http://geodesic.mathdoc.fr/item/SIGMA_2015_11_a49/

[1] Gao S., Wang Y., Hou B., “The classification of Leonard triples of Racah type”, Linear Algebra Appl., 439 (2013), 1834–1861 | DOI | MR | Zbl

[2] Genest V. X., Vinet L., Zhedanov A., “Bispectrality of the complementary Bannai–Ito polynomials”, SIGMA, 9 (2013), 018, 20 pp., arXiv: 1211.2461 | DOI | MR | Zbl

[3] Genest V. X., Vinet L., Zhedanov A., “The Bannai–Ito algebra and a superintegrable system with reflections on the two-sphere”, J. Phys. A: Math. Theor., 47 (2014), 205202, 13 pp., arXiv: 1401.1525 | DOI | MR | Zbl

[4] Genest V. X., Vinet L., Zhedanov A., “The Bannai–Ito polynomials as Racah coefficients of the $sl_{-1}(2)$ algebra”, Proc. Amer. Math. Soc., 142 (2014), 1545–1560, arXiv: 1205.4215 | DOI | MR | Zbl

[5] 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

[6] Genest V. X., Vinet L., Zhedanov A., “The equitable Racah algebra from three $\mathfrak{su}(1,1)$ algebras”, J. Phys. A: Math. Theor., 47 (2014), 025203, 12 pp., arXiv: 1309.3540 | DOI | MR | Zbl

[7] Genest V. X., Vinet L., Zhedanov A., “A Laplace–Dunkl equation on $S^2$ and the Bannai–Ito algebra”, Comm. Math. Phys., 336 (2015), 243–259, arXiv: 1312.6604 | DOI | MR | Zbl

[8] Granovskiĭ Ya. A., Zhedanov A. S., “Nature of the symmetry group of the $6j$-symbol”, Soviet Phys. JETP, 94 (1988), 1982–1985 | MR

[9] Kalnins E. G., Miller W. (Jr.), Pogosyan G. S., “Superintegrability and associated polynomial solutions: Euclidean space and the sphere in two dimensions”, J. Math. Phys., 37 (1996), 6439–6467 | DOI | MR | Zbl

[10] Kalnins E. G., Miller W. (Jr.), Pogosyan G. S., “Superintegrability on the two-dimensional hyperboloid”, J. Math. Phys., 38 (1997), 5416–5433 | DOI | MR | Zbl

[11] Kalnins E. G., Miller W. (Jr.), Post S., “Wilson polynomials and the generic superintegrable system on the 2-sphere”, J. Phys. A: Math. Theor., 40 (2007), 11525–11538 | DOI | MR | Zbl

[12] Kalnins E. G., Miller W. (Jr.), Post S., “Contractions of 2D 2nd order quantum superintegrable systems and the Askey scheme for hypergeometric orthogonal polynomials”, SIGMA, 9 (2013), 057, 28 pp., arXiv: 1212.4766 | DOI | MR | Zbl

[13] 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

[14] Lesniewski A., “A remark on the Casimir elements of Lie superalgebras and quantized Lie superalgebras”, J. Math. Phys., 36 (1995), 1457–1461 | DOI | MR | Zbl

[15] Lévy-Leblond J. M., Lévy-Nahas M., “Symmetrical coupling of three angular momenta”, J. Math. Phys., 6 (1965), 1372–1380 | DOI | MR | Zbl

[16] Miller W. (Jr.), Post S., Winternitz P., “Classical and quantum superintegrability with applications”, J. Phys. A: Math. Theor., 46 (2013), 423001, 97 pp., arXiv: 1309.2694 | DOI | MR | Zbl

[17] 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 | Zbl

[18] Tsujimoto S., Vinet L., Zhedanov A., “From $sl_{q}(2)$ to a parabosonic Hopf algebra”, SIGMA, 7 (2011), 093, 13 pp., arXiv: 1108.1603 | DOI | MR | Zbl

[19] Tsujimoto S., Vinet L., Zhedanov A., “Dunkl shift operators and Bannai–Ito polynomials”, Adv. Math., 229 (2012), 2123–2158, arXiv: 1106.3512 | DOI | MR | Zbl

[20] Vilenkin N. Ja., Klimyk A. U., Representation of Lie groups and special functions, v. 1, Mathematics and its Applications (Soviet Series), 72, Simplest Lie groups, special functions and integral transforms, Kluwer Academic Publishers Group, Dordrecht, 1991 | DOI | MR | Zbl