Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 7 (2011), 093, 13 pages      arXiv:1108.1603

From slq(2) to a Parabosonic Hopf Algebra

Satoshi Tsujimoto a, Luc Vinet b and Alexei Zhedanov c
a) Department of Applied Mathematics and Physics, Graduate School of Informatics, Kyoto University, Sakyo-ku, Kyoto 606-8501, Japan
b) Centre de Recherches Mathématiques, Université de Montréal, P.O. Box 6128, Centre-ville Station, Montréal (Québec), H3C 3J7 Canada
c) Donetsk Institute for Physics and Technology, Donetsk 83114, Ukraine

Received August 25, 2011; Published online October 07, 2011

A Hopf algebra with four generators among which an involution (reflection) operator, is introduced. The defining relations involve commutators and anticommutators. The discrete series representations are developed. Designated by sl−1(2), this algebra encompasses the Lie superalgebra osp(1|2). It is obtained as a q=−1 limit of the slq(2) algebra and seen to be equivalent to the parabosonic oscillator algebra in irreducible representations. It possesses a noncocommutative coproduct. The Clebsch-Gordan coefficients (CGC) of sl−1(2) are obtained and expressed in terms of the dual −1 Hahn polynomials. A generating function for the CGC is derived using a Bargmann realization.

Key words: parabosonic algebra; dual Hahn polynomials; Clebsch-Gordan coefficients.

pdf (368 Kb)   tex (16 Kb)


  1. 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 (2000), 652-660, math-ph/9902005.
  2. Dunkl C.F., Orthogonal polynomials of types A and B and related Calogero models, Comm. Math. Phys. 197 (1998), 451-487, q-alg/9710015.
  3. Floreanini R., Vinet L., Quantum algebras and q-special functions, Ann. Physics 221 (1993), 53-70.
  4. Frappat L., Sorba P., Sciarrino A., Dictionary on Lie superalgebras, hep-th/9607161.
  5. Horváthy P.A., Plyushchay M.S., Valenzuela M., Bosons, fermions and anyons in the plane, and supersymmetry, Ann. Physics 325 (2010), 1931-1975, arXiv:1001.0274.
  6. Jafarov E.I., Stoilova N.I., Van der Jeugt J., Finite oscillator models: the Hahn oscillator, J. Phys. A: Math. Theor. 44 (2011), 265203, 15 pages, arXiv:1101.5310.
  7. Jafarov E.I., Stoilova N.I., Van der Jeugt J., The su(2)α Hahn oscillator and a discrete Hahn-Fourier transform, J. Phys. A: Math. Theor. 44 (2011), 355205, 18 pages, arXiv:1106.1083.
  8. Kanakoglou K., Daskaloyannis C., Graded structure and Hopf structures in parabosonic algebra. An alternative approach to bosonisation, in New Techniques in Hopf Algebras and Graded Ring Theory, K. Vlaam. Acad. Belgie Wet. Kunsten (KVAB), Brussels, 2007, 105-116, arXiv:0706.2825.
  9. Koelink E., Koornwinder T.H., The Clebsch-Gordan coefficients for the quantum group SμU(2) and q-Hahn polynomials, Nederl. Akad. Wetensch. Indag. Math. 92 (1989), 443-456.
  10. Lapointe L., Vinet L., Exact operator solution of the Calogero-Sutherland model, Comm. Math. Phys. 178 (1996), 425-452, q-alg/9509003.
  11. Macfarlane A.J., Generalised oscillator systems and their parabosonic interpretation, in Proc. Inter. Workshop on Symmetry Methods in Physics, Editors A.N. Sissakian, G.S. Pogosyan and S.I. Vinitsky, JINR, Dubna, 1994, 319-325.
  12. Mukunda N., Sudarshan E.C.G., Sharma J.K., Mehta C.L., Representations and properties of para-Bose oscillator operators. I. Energy position and momentum eigenstates, J. Math. Phys. 21 (1980), 2386-2394.
  13. Plyushchay M.S., R-deformed Heisenberg algebra, Modern Phys. Lett. A 11 (1996), 2953-2964, hep-th/9701065.
  14. Plyushchay M.S., Deformed Heisenberg algebra with reflection, Nuclear Phys. B 491 (1997) 619-634, hep-th/9701091.
  15. Roche P., Arnaudon D., Irreducible representations of the quantum analogue of SU(2), Lett. Math. Phys. 17 (1989), 295-300.
  16. Rosenblum M., Generalized Hermite polynomials and the Bose-like oscillator calculus, in Nonselfadjoint Operators and Related Topics (Beer Sheva, 1992), Oper. Theory Adv. Appl., Vol. 73, Birkhäuser, Basel, 1994, 369-396, math.CA/9307224.
  17. Rösler M., Dunkl operators: theory and applications, in Orthogonal Polynomials and Special Functions (Leuven, 2002), Lecture Notes in Mathematics, Vol. 1817, Springer, Berlin, 2003, 93-135.
  18. Tsujimoto S., Vinet L., Zhedanov A., Dunkl shift operators and Bannai-Ito polynomials, arXiv:1106.3512.
  19. Tsujimoto S., Vinet L., Zhedanov A., Dual −1 Hahn polynomials: "classical" polynomials beyond the Leonard duality, Proc. Amer. Math. Soc., to appear, arXiv:1108.0132.
  20. Vasiliev M.A., Higher spin algebras and quantization on the sphere and hyperboloid, Internat. J. Modern Phys. A 6 (1991), 1115-1135.
  21. Vilenkin N.Ya., Klimyk A.U., Representation of Lie groups and special functions, Vol. 1, Simplest Lie groups, special functions and integral transforms, Kluwer, Dordrecht, 1991.
  22. Vinet L., Zhedanov A., A "missing" family of classical orthogonal polynomials, J. Phys. A: Math. Theor. 44 (2011), 085201, 16 pages, arXiv:1011.1669.
  23. Vinet L., Zhedanov A., A limit q=−1 for big q-Jacobi polynomials, Trans. Amer. Math. Soc., to appear, arXiv:1011.1429.
  24. Vinet L., Zhedanov A., A Bochner theorem for Dunkl polynomials, SIGMA 7 (2011), 020, 9 pages, arXiv:1011.1457.

Previous article   Next article   Contents of Volume 7 (2011)