From sl_q(2) to a Parabosonic Hopf Algebra

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...

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Bibliographic Details
Main Authors: Satoshi Tsujimoto, Luc Vinet, Alexei Zhedanov
Format: Article
Language:English
Published: National Academy of Science of Ukraine 2011-11-01
Series:Symmetry, Integrability and Geometry: Methods and Applications
Subjects:
Online Access:http://dx.doi.org/10.3842/SIGMA.2011.093
Description
Summary: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 sl_q(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.
ISSN:1815-0659