Letters in Mathematical Physics

, Volume 105, Issue 12, pp 1725–1734 | Cite as

The Equitable Presentation of \({\mathfrak{osp}_q(1|2)}\) and a q-Analog of the Bannai–Ito Algebra

  • Vincent X. GenestEmail author
  • Luc Vinet
  • Alexei Zhedanov


The equitable presentation of the quantum superalgebra \({\mathfrak{osp}_q(1|2)}\), in which all generators appear on an equal footing, is exhibited. It is observed that in their equitable presentations, the quantum algebras \({\mathfrak{osp}_q(1|2)}\) and \({\mathfrak{sl}_q(2)}\) are related to one another by the formal transformation \({q\rightarrow -q}\). A q-analog of the Bannai–Ito algebra is shown to arise as the covariance algebra of \({\mathfrak{osp}_q(1|2)}\).


quantum superalgebra \({\mathfrak{osp}_{q}(1|2)}\) Bannai–Ito algebra 

Mathematics Subject Classification

17B37 20G42 81R50 


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  1. 1.
    De Bie, H., Genest, V.X., Vinet, L.: A Dirac-Dunkl equation on S 2 and the Bannai–Ito algebra. (2015). arXiv:1501.03108
  2. 2.
    Chakrabarti A.: On the coupling of 3 angular momenta. Annales de l’Institut Henri Poincaré, Section A 1(3), 301–327 (1964)zbMATHGoogle Scholar
  3. 3.
    Edmonds A.R.: Angular momentum in quantum mechanics. Investigations in physics. Princeton University Press, Princeton (1996)Google Scholar
  4. 4.
    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)zbMATHMathSciNetCrossRefADSGoogle Scholar
  5. 5.
    Gao S., Wang Y., Hou B.: The classification of Leonard triples of Racah type. Linear Algebra Appl. 439(7), 1834–1861 (2013)zbMATHMathSciNetCrossRefGoogle Scholar
  6. 6.
    Genest V.X., Vinet L., Zhedanov A.: Superintegrability in two dimensions and the Racah–Wilson algebra. Lett. Math. Phys. 104(8), 931–952 (2014)zbMATHMathSciNetCrossRefADSGoogle Scholar
  7. 7.
    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)MathSciNetCrossRefADSGoogle Scholar
  8. 8.
    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)zbMATHMathSciNetCrossRefGoogle Scholar
  9. 9.
    Genest V.X., Vinet L., Zhedanov A.: The equitable Racah algebra from three \({\mathfrak{su}(1,1)}\) algebras. J. Phys. A Math. Theor. 47(2), 025203 (2014)MathSciNetCrossRefADSGoogle Scholar
  10. 10.
    Genest V.X., Vinet L., Zhedanov A.: A Laplace-Dunkl equation on S 2 and the Bannai–Ito algebra. Commun. Math. Phys. 336(1), 243–259 (2015)zbMATHMathSciNetCrossRefADSGoogle Scholar
  11. 11.
    Genest, V.X., Vinet, L., Zhedanov, A.: The quantum superalgebra \({\mathfrak{osp}_q(1|2)}\) and a q-generalization of the Bannai–Ito polynomials. (2015). arXiv:1501.05602
  12. 12.
    Granovskii Y.A., Zhedanov A.: Nature of the symmetry group of the 6j symbol. J. Exp. Theor. Phys. 94(10), 1982–1985 (1988)MathSciNetGoogle Scholar
  13. 13.
    Granovskii Y.A., Zhedanov A.: Hidden symmetry of the Racah and Clebsch–Gordan problems for the quantum algebra sl q(2). J. Gr. Theory Phys. 1, 161–171 (1993)Google Scholar
  14. 14.
    Granovskii Y.A., Zhedanov A.: Linear covariance algebra for SL q(2). J. Phys. A Math. Gen. 26(7), L357–L359 (1993)MathSciNetCrossRefADSGoogle Scholar
  15. 15.
    Hartwig B., Terwilliger P.: The Tetrahedron algebra, the Onsager algebra, and the \({\mathfrak{sl}_2}\) loop algebra. J. Algebra 308(2), 840–863 (2007)zbMATHMathSciNetCrossRefGoogle Scholar
  16. 16.
    Ito T., Terwilliger P., Weng C.: The quantum algebra \({U_{q}(\mathfrak{sl}_2)}\) and its equitable presentation. J. Algebra 298(1–2), 284–301 (2006)zbMATHMathSciNetCrossRefGoogle Scholar
  17. 17.
    Kalnins E.G., Miller W., Post S.: Wilson polynomials and the generic superintegrable system on the 2-sphere. J. Phys. A Math. Theor. 40(38), 11525 (2007)zbMATHMathSciNetCrossRefADSGoogle Scholar
  18. 18.
    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)MathSciNetGoogle Scholar
  19. 19.
    Kulish P.P., Yu Reshetikhin N.: Universal R-Matrix of the quantum superalgebra osp(1|2). Lett. Math. Phys. 18(2), 143–149 (1989)zbMATHMathSciNetCrossRefADSGoogle Scholar
  20. 20.
    Lesniewski A.: A remark on the Casimir elements of Lie superalgebras and quantized Lie superalgebras. J. Math. Phys. 36(3), 1457–1461 (1995)zbMATHMathSciNetCrossRefADSGoogle Scholar
  21. 21.
    Lévy-Leblond J.M., Lévy-Nahas M.: Symmetrical coupling of three angular momenta. J. Math. Phys. 6(9), 1372 (1965)zbMATHCrossRefADSGoogle Scholar
  22. 22.
    Terwilliger P.: The equitable presentation for the quantum group \({U_q(\mathfrak{g})}\) associated with a symmetrizable Kac–Moody algebra \({\mathfrak{g}}\). J. Algebra 298(1–2), 302–319 (2006)zbMATHMathSciNetCrossRefGoogle Scholar
  23. 23.
    Terwilliger P.: The universal Askey–Wilson algebra and the equitable presentation of \({U_{q}(\mathfrak{sl}_2)}\). SIGMA 7, 99 (2011)MathSciNetGoogle Scholar
  24. 24.
    Tsujimoto S., Vinet L., Zhedanov A.: From sl q(2) to a parabosonic Hopf algebra. SIGMA 7, 93–105 (2011)MathSciNetGoogle Scholar
  25. 25.
    Tsujimoto S., Vinet L., Zhedanov A.: Dunkl shift operators and Bannai–Ito polynomials. Adv. Math. 229(4), 2123–2158 (2012)zbMATHMathSciNetCrossRefGoogle Scholar
  26. 26.
    Underwood, R.G.: An introduction to Hopf algebras. Springer (2011)Google Scholar
  27. 27.
    Zhedanov A.: “Hidden” symmetry of Askey–Wilson polynomials. Theor. Math. Phys. 89(2), 1146–1157 (1991)zbMATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2015

Authors and Affiliations

  • Vincent X. Genest
    • 1
    Email author
  • Luc Vinet
    • 1
  • Alexei Zhedanov
    • 2
  1. 1.Centre de recherches mathématiquesUniversité de MontréalMontrealCanada
  2. 2.Donetsk Institute for Physics and TechnologyDonetskUkraine

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