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.
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