From $sl_q(2)$ to a Parabosonic Hopf Algebra

TL;DR

This paper introduces a new Hopf algebra called $sl_{-1}(2)$, which generalizes $sl_q(2)$ at $q=-1$, and explores its algebraic structure, representations, and connection to parabosonic oscillators.

Contribution

The paper constructs the $sl_{-1}(2)$ algebra as a $q=-1$ limit of $sl_q(2)$, develops its representations, and derives its Clebsch-Gordan coefficients in terms of dual -1 Hahn polynomials.

Findings

$sl_{-1}(2)$ encompasses $osp(1|2)$ as a special case.

The algebra has a noncocommutative coproduct.

Explicit CGC formulas are obtained and expressed via dual -1 Hahn polynomials.

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.