The Bannai-Ito polynomials as Racah coefficients of the 𝑠𝑙₋₁(2) algebra

Genest, Vincent; Vinet, Luc; Zhedanov, Alexei

doi:10.1090/S0002-9939-2014-11970-8

The Bannai-Ito polynomials as Racah coefficients of the $sl_{-1}(2)$ algebra
HTML articles powered by AMS MathViewer

by Vincent X. Genest, Luc Vinet and Alexei Zhedanov

Proc. Amer. Math. Soc. 142 (2014), 1545-1560

DOI: https://doi.org/10.1090/S0002-9939-2014-11970-8

Published electronically: February 18, 2014

PDF | Request permission

Abstract:

The Bannai-Ito polynomials are shown to arise as Racah coefficients for $sl_{-1}(2)$. This Hopf algebra has four generators, including an involution, and is defined with both commutation and anticommutation relations. It is also equivalent to the parabosonic oscillator algebra. The coproduct is used to show that the Bannai-Ito algebra acts as the hidden symmetry algebra of the Racah problem for $sl_{-1}(2)$. The Racah coefficients are recovered from a related Leonard pair.

References

M. Arik and U. Kayserilioglu, The anticommutator spin algebra, its representations and quantum group invariance, Internat. J. Modern Phys. A 18 (2003), no. 27, 5039–5045. MR 2020304, DOI 10.1142/S0217751X03015933
Eiichi Bannai and Tatsuro Ito, Algebraic combinatorics. I, The Benjamin/Cummings Publishing Co., Inc., Menlo Park, CA, 1984. Association schemes. MR 882540
George M. F. Brown, Totally bipartite/abipartite Leonard pairs and Leonard triples of Bannai/Ito type, Electron. J. Linear Algebra 26 (2013), 258–299. MR 3084644, DOI 10.13001/1081-3810.1654
T. S. Chihara, An introduction to orthogonal polynomials, Mathematics and its Applications, Vol. 13, Gordon and Breach Science Publishers, New York-London-Paris, 1978. MR 0481884
Brian Curtin, Modular Leonard triples, Linear Algebra Appl. 424 (2007), no. 2-3, 510–539. MR 2329491, DOI 10.1016/j.laa.2007.02.024
C. Daskaloyannis, K. Kanakoglou, and I. Tsohantjis, Hopf algebraic structure of the parabosonic and parafermionic algebras and paraparticle generalization of the Jordan-Schwinger map, J. Math. Phys. 41 (2000), no. 2, 652–660. MR 1737022, DOI 10.1063/1.533157
Charles F. Dunkl, Orthogonal polynomials of types $A$ and $B$ and related Calogero models, Comm. Math. Phys. 197 (1998), no. 2, 451–487. MR 1652751, DOI 10.1007/s002200050460
M. F. Gorodniĭ and G. B. Podkolzin, Irreducible representations of graded Lie algebras, Spectral theory of operators and infinite-dimensional analysis, Akad. Nauk Ukrain. SSR, Inst. Mat., Kiev, 1984, pp. 66–77, iii (Russian). MR 817219
Y. I. Granovskii and A. S. Zhedanov. Hidden symmetry of the Racah and Clebsch-Gordan problems for the quantum algebra $sl_q(2)$. arXiv:hep-th/9304138, 1993.
H. S. Green, A generalized method of field quantization, Phys. Rev. (2) 90 (1953), 270–273. MR 55215
Hau-wen Huang, The classification of Leonard triples of QRacah type, Linear Algebra Appl. 436 (2012), no. 5, 1442–1472. MR 2890930, DOI 10.1016/j.laa.2011.08.033
Roelof Koekoek, Peter A. Lesky, and René F. Swarttouw, Hypergeometric orthogonal polynomials and their $q$-analogues, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2010. With a foreword by Tom H. Koornwinder. MR 2656096, DOI 10.1007/978-3-642-05014-5
N. Mukunda, E. C. G. Sudarshan, J. K. Sharma, and C. L. Mehta, Representations and properties of para-Bose oscillator operators. I. Energy position and momentum eigenstates, J. Math. Phys. 21 (1980), no. 9, 2386–2394. MR 585590, DOI 10.1063/1.524695
V. L. Ostrovskiĭ and S. D. Sil′vestrov, Representations of real forms of the graded analogue of a Lie algebra, Ukraïn. Mat. Zh. 44 (1992), no. 11, 1518–1524 (Russian, with Russian and Ukrainian summaries); English transl., Ukrainian Math. J. 44 (1992), no. 11, 1395–1401 (1993). MR 1213895, DOI 10.1007/BF01071514
Marvin Rosenblum, Generalized Hermite polynomials and the Bose-like oscillator calculus, Nonselfadjoint operators and related topics (Beer Sheva, 1992) Oper. Theory Adv. Appl., vol. 73, Birkhäuser, Basel, 1994, pp. 369–396. MR 1320555
Paul Terwilliger, Two linear transformations each tridiagonal with respect to an eigenbasis of the other, Linear Algebra Appl. 330 (2001), no. 1-3, 149–203. MR 1826654, DOI 10.1016/S0024-3795(01)00242-7
Paul Terwilliger, Two relations that generalize the $q$-Serre relations and the Dolan-Grady relations, Physics and combinatorics 1999 (Nagoya), World Sci. Publ., River Edge, NJ, 2001, pp. 377–398. MR 1865045, DOI 10.1142/9789812810199_{0}013
Paul Terwilliger and Raimundas Vidunas, Leonard pairs and the Askey-Wilson relations, J. Algebra Appl. 3 (2004), no. 4, 411–426. MR 2114417, DOI 10.1142/S0219498804000940
Satoshi Tsujimoto, Luc Vinet, and Alexei Zhedanov, Dual $-1$ Hahn polynomials: “classical” polynomials beyond the Leonard duality, Proc. Amer. Math. Soc. 141 (2013), no. 3, 959–970. MR 3003688, DOI 10.1090/S0002-9939-2012-11469-8
Satoshi Tsujimoto, Luc Vinet, and Alexei Zhedanov, From $sl_q(2)$ to a parabosonic Hopf algebra, SIGMA Symmetry Integrability Geom. Methods Appl. 7 (2011), Paper 093, 13. MR 2861183, DOI 10.3842/SIGMA.2011.093
Satoshi Tsujimoto, Luc Vinet, and Alexei Zhedanov, Jordan algebras and orthogonal polynomials, J. Math. Phys. 52 (2011), no. 10, 103512, 8. MR 2894610, DOI 10.1063/1.3653482
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
Raimundas Vidūnas, Askey-Wilson relations and Leonard pairs, Discrete Math. 308 (2008), no. 4, 479–495. MR 2376472, DOI 10.1016/j.disc.2007.03.037
Luc Vinet and Alexei Zhedanov, A Bochner theorem for Dunkl polynomials, SIGMA Symmetry Integrability Geom. Methods Appl. 7 (2011), Paper 020, 9. MR 2804576, DOI 10.3842/SIGMA.2011.020
Luc Vinet and Alexei Zhedanov, A limit $q=-1$ for the big $q$-Jacobi polynomials, Trans. Amer. Math. Soc. 364 (2012), no. 10, 5491–5507. MR 2931336, DOI 10.1090/S0002-9947-2012-05539-5
Luc Vinet and Alexei Zhedanov, A ‘missing’ family of classical orthogonal polynomials, J. Phys. A 44 (2011), no. 8, 085201, 16. MR 2770369, DOI 10.1088/1751-8113/44/8/085201
A. S. Zhedanov, “Hidden symmetry” of Askey-Wilson polynomials, Teoret. Mat. Fiz. 89 (1991), no. 2, 190–204 (Russian, with English summary); English transl., Theoret. and Math. Phys. 89 (1991), no. 2, 1146–1157 (1992). MR 1151381, DOI 10.1007/BF01015906