Symmetry, Integrability and Geometry: Methods and Applications (Nov 2011)
From sl_q(2) to a Parabosonic Hopf Algebra
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 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.
