Abstract
An embedding of the Bannai–Ito algebra in the universal enveloping algebra of \(\mathfrak {osp}(1,2)\) is provided. A connection with the characterization of the little \(-1\) Jacobi polynomials is found in the holomorphic realization of \(\mathfrak {osp}(1,2)\). An integral expression for the Bannai–Ito polynomials is derived as a corollary.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Bannai, E., Ito, T.: Algebraic combinatorics I: association schemes. Benjamin & Cummings, San Francisco (1984)
Baseilhac, P., Gainutdinov, A.M., Vu, T.T.: Cyclic tridiagonal pairs, higher order Onsager algebras and orthogonal polynomials. Linear Algebra Appl. 522, 71–110 (2017)
De Bie, H., Genest, V.X., Vinet, L.: A Dirac–Dunkl equation on \(S^2\) and the Bannai–Ito algebra. Commun. Math. Phys. 344, 447–464 (2016)
Genest, V.X., Ismail, M., Vinet, L., Zhedanov, A.: Tridiagonalization of the hypergeometric operator and the Racah–Wilson algebra. Proc. Am. Math. Soc. 144, 4441–4454 (2016)
Genest, V.X., Vinet, L., Zhedanov, A.: Bispectrality of the complementary Bannai–Ito polynomials. SIGMA Symmetry Integr. Geom. Methods Appl. 9, 18–37 (2013)
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, 1545–1560 (2014)
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, 205202 (2014)
Genest, V.X., Vinet, L., Zhedanov, A.: The equitable racah algebra from three \(\mathfrak{su}(1,1)\) algebras. J. Phys. A: Math. Theor. 47, 025203 (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)
Genest, V.X., Vinet, L., Zhedanov, A.: The equitable presentation of \(\mathfrak{osp}_{q}(1|2)\) and a \(q\)-analog of the Bannai–Ito algebra. Lett. Math. Phys. 105, 1725–1734 (2015)
Genest, V.X., Vinet, L., Zhedanov, A.: The non-symmetric Wilson polynomials are the Bannai–Ito polynomials. Proc. Am. Math. Soc. 144, 5217–5226 (2016)
Granovskii, Y.I., Zhedanov, A.: Linear covariance algebra for \(SL_q(2)\). J. Phys. A: Math. Gen. 26, L357 (1993)
Huang, H.: Finite-dimensional irreducible modules of the universal Askey–Wilson algebra. Commun. Math. Phys. 340, 959–964 (2016)
Huang, H.: An embedding of the universal Askey-Wilson algebra into \(\fancyscript {U}_q(sl(2)) \otimes \fancyscript {U} _q(sl(2)) \otimes \fancyscript {U}_q(sl(2))\). Nucl. Phys. B 922, 401–434 (2017)
Huang, H.: Center of the universal Askey–Wilson algebra at roots of unity. Nucl. Phys. B 909, 260–296 (2016)
Ismail, M., Koelink, E.: Spectral properties of operators using tridiagonalisation. Anal. Appl. 10, 327 (2012)
Koornwinder, T.H.: Special orthogonal polynomial systems mapped onto each other by the Fourier–Jacobi transform. In: Lecture Notes in Mathematics. Springer, Berlin, pp 174–183 (1984)
Terwilliger, P.: Leonard pairs and dual polynomial sequences. Preprint (1987)
Terwilliger, P.: Two linear transformations each tridiagonal with respect to an eigenbasis of the other. Linear Algebra Appl. 330, 149–203 (2001)
Terwilliger, P., Vidunas, R.: Leonard pairs and the Askey–Wilson relations. J. Algebra Appl. 3, 411–426 (2004)
Vidunas, R.: Normalized Leonard pairs and Askey–Wilson relations. Linear Algebra Appl. 422, 39–57 (2007)
Tsujimoto, S., Vinet, L., Zhedanov, A.: Dunkl shift operators and Bannai–Ito polynomials. Adv. Math. 229, 2123–2158 (2012)
Vinet, L., Zhedanov, A.: A ‘missing’ family of classical orthogonal polynomials. J. Phys. A: Math. Theor. 44, 085201 (2011)
Zhedanov, A.S.: “Hidden symmetry” of Askey–Wilson polynomials. Theor. Math. Phys. 89, 1146–1157 (1991)
Acknowledgements
PB, VXG and AZ acknowledge the hospitality of the CRM and LV that of the Université de Tours where parts of the reported research have been realized. PB is supported by C.N.R.S. VXG holds a postdoctoral fellowship from the Natural Science and Engineering Research Council (NSERC) of Canada. LV is grateful to NSERC for support through a discovery grant.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Baseilhac, P., Genest, V.X., Vinet, L. et al. An embedding of the Bannai–Ito algebra in \(\mathscr {U}(\mathfrak {osp}(1,2))\) and \(-1\) polynomials. Lett Math Phys 108, 1623–1634 (2018). https://doi.org/10.1007/s11005-017-1041-0
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11005-017-1041-0