# FROM slq(2) TO A PARABOSONIC HOPF ALGEBRA

@article{Tsujimoto2011FROMST, title={FROM slq(2) TO A PARABOSONIC HOPF ALGEBRA}, author={Satoshi Tsujimoto and Luc Vinet and Alexei S. Zhedanov}, journal={Symmetry Integrability and Geometry-methods and Applications}, year={2011}, volume={7}, pages={093} }

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 en- compasses the Lie superalgebra osp(1j2). 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 repre- sentations. It possesses a noncocommutative coproduct. The…

## 32 Citations

The Bannai-Ito polynomials as Racah coefficients of the sl_{-1}(2) algebra

- Mathematics
- 2012

The Bannai-Ito polynomials are shown to arise as Racah coefficients for sl_{-1}(2). This Hopf algebra has four generators including an involution and is defined with both commutation and…

The Bannai–Ito algebra and a superintegrable system with reflections on the two-sphere

- Mathematics, Physics
- 2014

A quantum superintegrable model with reflections on the two-sphere is introduced. Its two algebraically independent constants of motion generate a central extension of the Bannai–Ito algebra. The…

The algebra of dual −1 Hahn polynomials and the Clebsch-Gordan problem of sl−1(2)

- Mathematics
- 2013

The algebra H of the dual −1 Hahn polynomials is derived and shown to arise in the Clebsch-Gordan problem of sl−1(2). The dual −1 Hahn polynomials are the bispectral polynomials of a discrete…

Deformed su(1;1) Algebra as a Model for Quantum Oscillators

- Mathematics
- 2012

The Lie algebra su(1; 1) can be deformed by a reflection operator, in such a way that the positive discrete series representations of su(1; 1) can be extended to representa- tions of this deformed…

A Laplace-Dunkl Equation on S2 and the Bannai–Ito Algebra

- Mathematics
- 2015

The analysis of the $${\mathbb{Z}_2^{3}}$$Z23 Laplace-Dunkl equation on the 2-sphere is cast in the framework of the Racah problem for the Hopf algebra sl−1(2). The related Dunkl-Laplace operator is…

The Hahn superalgebra and supersymmetric Dunkl oscillator models

- Mathematics
- 2013

A supersymmetric extension of the Hahn algebra is introduced. This quadratic superalgebra, which we call the Hahn superalgebra, is constructed using the realization provided by the Dunkl oscillator…

Embeddings of the Racah algebra into the Bannai-Ito algebra

- Mathematics
- 2015

Embeddings of the Racah algebra into the Bannai-Ito algebra are proposed in two realiza- tions. First, quadratic combinations of the Bannai-Ito algebra generators in their standard realization on the…

The Dunkl Oscillator in the Plane II: Representations of the Symmetry Algebra

- Mathematics
- 2013

The superintegrability, wavefunctions and overlap coefficients of the Dunkl oscillator model in the plane were considered in the first part. Here finite-dimensional representations of the symmetry…

An infinite family of superintegrable Hamiltonians with reflection in the plane

- Mathematics, Physics
- 2011

We introduce a new infinite class of superintegrable quantum systems in the plane. Their Hamiltonians involve reflection operators. The associated Schrodinger equations admit the separation of…

The Dunkl oscillator in the plane I : superintegrability, separated wavefunctions and overlap coefficients

- Mathematics
- 2013

The isotropic Dunkl oscillator model in the plane is investigated. The model is defined by a Hamiltonian constructed from the combination of two independent parabosonic oscillators. The system is…

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