𝔬𝔰𝔭(1,2) and generalized Bannai–Ito algebras

Genest, Vincent; Lapointe, Luc; Vinet, Luc

doi:10.1090/tran/7733

$\mathfrak {osp}(1,2)$ and generalized Bannai–Ito algebras
HTML articles powered by AMS MathViewer

by Vincent X. Genest, Luc Lapointe and Luc Vinet PDF

Trans. Amer. Math. Soc. 372 (2019), 4127-4148 Request permission

Abstract:

Generalizations of the (rank-$1$) Bannai–Ito algebra are obtained from a refinement of the grade involution of the Lie superalgebra $\mathfrak {osp}(1,2)$. A hyperoctahedral extension is derived by using a realization of $\mathfrak {osp}(1,2)$ in terms of Dunkl operators associated with the Weyl group $B_3$.

References

Satoshi Tsujimoto, Luc Vinet, and Alexei Zhedanov, Dunkl shift operators and Bannai-Ito polynomials, Adv. Math. 229 (2012), no. 4, 2123–2158. MR 2880217, DOI 10.1016/j.aim.2011.12.020
Eiichi Bannai and Tatsuro Ito, Algebraic combinatorics. I, The Benjamin/Cummings Publishing Co., Inc., Menlo Park, CA, 1984. Association schemes. MR 882540
Vincent X. Genest, Luc Vinet, and Alexei Zhedanov, The Bannai-Ito polynomials as Racah coefficients of the $sl_{-1}(2)$ algebra, Proc. Amer. Math. Soc. 142 (2014), no. 5, 1545–1560. MR 3168462, DOI 10.1090/S0002-9939-2014-11970-8
H. De Bie, V. X. Genest, S. Tsujimoto, L. Vinet, and A. Zhedanov, The Bannai-Ito algebra and some applications, J. Phys. Conf. Ser. 597 (2015), 012001.
H. De Bie, V. X. Genest, W. van de Vijver, and L. Vinet, Bannai-Ito algebras and the $osp(1; 2)$ superalgebra, Physical and mathematical aspects of symmetries, edited by S. Duarte, J. P. Gazeau, S. Faci, T. Micklitz, R. Scherer, and F. Toppan, Springer, Cham, 2017.
Vincent X. Genest, Luc Vinet, and Alexei Zhedanov, The non-symmetric Wilson polynomials are the Bannai-Ito polynomials, Proc. Amer. Math. Soc. 144 (2016), no. 12, 5217–5226. MR 3556266, DOI 10.1090/proc/13141
Vincent X. Genest, Luc Vinet, and Alexei Zhedanov, A Laplace-Dunkl equation on $S^2$ and the Bannai-Ito algebra, Comm. Math. Phys. 336 (2015), no. 1, 243–259. MR 3322373, DOI 10.1007/s00220-014-2241-4
Vincent X. Genest, Luc Vinet, and Alexei Zhedanov, The Bannai-Ito algebra and a superintegrable system with reflections on the two-sphere, J. Phys. A 47 (2014), no. 20, 205202, 13. MR 3200527, DOI 10.1088/1751-8113/47/20/205202
Hendrik De Bie, Vincent X. Genest, and Luc Vinet, A Dirac-Dunkl equation on $S^2$ and the Bannai-Ito algebra, Comm. Math. Phys. 344 (2016), no. 2, 447–464. MR 3500245, DOI 10.1007/s00220-016-2648-1
Hendrik De Bie, Vincent X. Genest, and Luc Vinet, The $\Bbb {Z}_2^n$ Dirac-Dunkl operator and a higher rank Bannai-Ito algebra, Adv. Math. 303 (2016), 390–414. MR 3552530, DOI 10.1016/j.aim.2016.08.007
Hendrik De Bie, Roy Oste, and Joris Van der Jeugt, On the algebra of symmetries of Laplace and Dirac operators, Lett. Math. Phys. 108 (2018), no. 8, 1905–1953. MR 3814732, DOI 10.1007/s11005-018-1065-0
Charles F. Dunkl, Differential-difference operators associated to reflection groups, Trans. Amer. Math. Soc. 311 (1989), no. 1, 167–183. MR 951883, DOI 10.1090/S0002-9947-1989-0951883-8
Charles F. Dunkl, Symmetric functions and $B_N$-invariant spherical harmonics, J. Phys. A 35 (2002), no. 48, 10391–10408. MR 1947314, DOI 10.1088/0305-4470/35/48/312
Akinori Nishino, Hideaki Ujino, and Miki Wadati, Rodrigues formula for the nonsymmetric multivariable Laguerre polynomial, J. Phys. Soc. Japan 68 (1999), no. 3, 797–802. MR 1697908, DOI 10.1143/JPSJ.68.797
Gerrit J. Heckman, A remark on the Dunkl differential-difference operators, Harmonic analysis on reductive groups (Brunswick, ME, 1989) Progr. Math., vol. 101, Birkhäuser Boston, Boston, MA, 1991, pp. 181–191. MR 1168482
H. De Bie, B. Ørsted, P. Somberg, and V. Souček, Dunkl operators and a family of realizations of $\mathfrak {osp}(1|2)$, Trans. Amer. Math. Soc. 364 (2012), no. 7, 3875–3902. MR 2901238, DOI 10.1090/S0002-9947-2012-05608-X
Misha Feigin and Tigran Hakobyan, On Dunkl angular momenta algebra, J. High Energy Phys. 11 (2015), 107, front matter+22. MR 3454988, DOI 10.1007/JHEP11(2015)107
A.-A. A. Jucys, Symmetric polynomials and the center of the symmetric group ring, Rep. Mathematical Phys. 5 (1974), no. 1, 107–112. MR 419576, DOI 10.1016/0034-4877(74)90019-6
G. E. Murphy, A new construction of Young’s seminormal representation of the symmetric groups, J. Algebra 69 (1981), no. 2, 287–297. MR 617079, DOI 10.1016/0021-8693(81)90205-2
Arun Ram, Seminormal representations of Weyl groups and Iwahori-Hecke algebras, Proc. London Math. Soc. (3) 75 (1997), no. 1, 99–133. MR 1444315, DOI 10.1112/S0024611597000282
V. X. Genest, L. Lapointe, and L. Vinet, Supersymmetries of the rational $B_N$-Calogero model for non-identical particles (in preparation).