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

SIGMA 7 (2011), 093, 13 pages      arXiv:1108.1603      https://doi.org/10.3842/SIGMA.2011.093

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

Abstract
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.

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