Abstract
The Racah problem for the quantum superalgebra \({\mathfrak{osp}_{q}(1|2)}\) is considered. The intermediate Casimir operators are shown to realize a q-deformation of the Bannai–Ito algebra. The Racah coefficients of \({\mathfrak{osp}_q(1|2)}\) are calculated explicitly in terms of basic orthogonal polynomials that q-generalize the Bannai–Ito polynomials. The relation between these q-deformed Bannai–Ito polynomials and the q-Racah/Askey–Wilson polynomials is discussed.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Bannai E., Ito T.: Algebraic Combinatorics I. Association Schemes. Benjamin/Cummings Publishing Company, London (1984)
Chung W.S., Kalnins E.G., Miller W.: Tensor products of q-superalgebras representations and q-series identities. J. Phys. A Math. Gen. 30(20), 7147–7166 (1997)
Daskaloyannis C., Kanakoglou K., Tsohantjis I.: Hopf algebraic structure of the parabosonic and parafermionic algebras and paraparticle generalization of the Jordan–Schwinger map. J. Math. Phys. 41(2), 652 (2000)
De Bie H., Genest V.X., Tsujimoto S., Vinet L., Zhedanov A.: The Bannai–Ito algebra and some applications. J. Phys. Conf. Ser. 591, 012001 (2015)
De Bie, H., Genest, V.X., Vinet, L.: A Dirac-Dunkl equation on S 2 and the Bannai–Ito algebra. Commun. Math. Phys. (2016). (to appear)
Dunkl C.: Differential-difference operators associated to reflection groups. Trans. Am. Math. Soc. 311(13), 167–183 (1989)
Floreanini R., Vinet L.: q-analogues of the parabose and parafermi oscillators and representations of quantum algebras. J. Phys. A Math. Gen. 23(19), L1019 (1990)
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(20), 205202 (2014)
Genest V.X., Vinet L., Zhedanov A.: The Bannai–Ito polynomials as Racah coefficients of the sl −1(2) algebra. Proc. Am. Math. Soc. 142(5), 1545–1560 (2014)
Genest V.X., Vinet L., Zhedanov A.: A Laplace-Dunkl equation on S 2 and the Bannai–Ito algebra. Commun. Math. Phys. 336, 243–259 (2015)
Granovskii Y.A., Zhedanov A.: Nature of the symmetry group of the 6j symbol. J. Exp. Theor. Phys. 94(10), 1982–1985 (1988)
Groenevelt W.: Quantum analogs of tensor product representations of \({\mathfrak{su}(1,1)}\). SIGMA 7, 77 (2011)
Koekoek R., Lesky, P.A., Swarttouw R.F.: Hypergeometric Orthogonal Polynomials and their q-Analogues. Springer, New York (2010)
Koornwinder T.: The relationship between Zhedanov’s algebra AW(3) and the double affine Hecke algebra in the rank one case. SIGMA 311, 63 (2007)
Kulish P.P., Yu Reshetikhin N.: Universal R-matrix of the quantum superalgebra osp(1|2). Lett. Math. Phys. 18(2), 143–149 (1989)
Lesniewski A.: A remark on the Casimir elements of Lie superalgebras and quantized Lie superalgebras. J. Math. Phys. 36(3), 1457–1461 (1995)
Minnaert P., Mozrzymas M.: Racah coefficients and 6 − j symbols for the quantum superalgebra \({U_{q}(\mathrm{osp}(1|2))}\). J. Math. Phys. 36(2), 907 (1995)
Munkunda N., Sudarshan E.C.G., Sharma J.K., Mehta C.L.: Representations and properties of paraBose oscillator operators. I. Energy position and momentum eigenstates. J. Math. Phys. 21(9), 2386 (1980)
Terwilliger P., Vidunas R.: Leonard pairs and the Askey–Wilson relations. J. Algebra Appl. 311(4), 411–426 (2004)
Tsujimoto S., Vinet L., Zhedanov A.: From sl q (2) to a parabosonic Hopf algebra. SIGMA 7, 93–105 (2011)
Tsujimoto S., Vinet L., Zhedanov A.: Dunkl shift operators and Bannai–Ito polynomials. Adv. Math. 229(4), 2123–2158 (2012)
Underwood R.G.: An Introduction to Hopf Algebras. Springer, New York (2011)
Van der Jeugt, J.: 3nj-Coefficients and orthogonal polynomials of hypergeometric type. In: Orthogonal Polynomials and Special Functions. Lecture Notes in Mathematics, vol. 1817, pp. 25–92 (2003)
Vilenkin, N.J., Klimyk, A.U.: Representation of Lie groups and special functions. In: Mathematics and its Applications (Soviet Series), vol. 72. Springer, New York (1991)
Zachos C.K.: Altering the symmetry of wavefunctions in quantum algebras and supersymmetry. Mod. Phys. Lett. A. 7(18), 1595 (1992)
Zhedanov A.: Hidden symmetry of Askey–Wilson polynomials. Theor. Math. Phys. 89(2), 1146–1157 (1991)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by N. Reshetikhin
Rights and permissions
About this article
Cite this article
Genest, V.X., Vinet, L. & Zhedanov, A. The Quantum Superalgebra \({\mathfrak{osp}_{q}(1|2)}\) and a q-Generalization of the Bannai–Ito Polynomials. Commun. Math. Phys. 344, 465–481 (2016). https://doi.org/10.1007/s00220-016-2647-2
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00220-016-2647-2