## 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

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3 |
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2 |
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Citation Context ...been found that these polynomials can be identified as a q → −1 limits of some q-orthogonal polynomials, the simplest among them being the little −1 Jacobi polynomials introduced in [22]. In [18] and =-=[19]-=- this approach was generalized to Dunkl shift operators. This provided a theoretical framework for the Bannai–Ito and the dual −1 Hahn polynomials.2 S. Tsujimoto, L. Vinet and A. Zhedanov With this p... |

2 |
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Citation Context ...e, that is of operators that are first order in the derivative and involve reflections. We have thus discovered certain families of “classical” orthogonal polynomials that had hitherto escaped notice =-=[23, 24]-=-. It has been found that these polynomials can be identified as a q → −1 limits of some q-orthogonal polynomials, the simplest among them being the little −1 Jacobi polynomials introduced in [22]. In ... |

1 |
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Citation Context ...1 The result is not surprising. We have seen that the sl−1(2) algebra is a q → −1 limit of the slq(2) algebra and for the latter algebra, the CGC are expressed in terms of the dual q-Hahn polynomials =-=[9]-=-. Also, when µ1 = µ2 = 0 the dual −1 Hahn polynomials coincide with the ordinary Krawtchouk polynomials. This result is also expected: the case µ1 = µ2 = 0 corresponds to the case when both sl−1(2) al... |