## Symmetry, Integrability and Geometry: Methods and Applications From slq(2) to a Parabosonic Hopf Algebra (2011)

### BibTeX

@MISC{Tsujimoto11symmetry,integrability,

author = {Satoshi Tsujimoto and Luc Vinet and Alexei Zhedanov},

title = {Symmetry, Integrability and Geometry: Methods and Applications From slq(2) to a Parabosonic Hopf Algebra},

year = {2011}

}

### OpenURL

### Abstract

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 slq(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. Key words: parabosonic algebra; dual Hahn polynomials; Clebsch–Gordan coefficients 2010 Mathematics Subject Classification: 17B37; 17B80; 33C45