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 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.