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

@article{Genest2015ALE,
title={A Laplace-Dunkl Equation on S2 and the Bannai–Ito Algebra},
author={Vincent X. Genest and Luc Vinet and Alexei S. Zhedanov},
journal={Communications in Mathematical Physics},
year={2015},
volume={336},
pages={243-259}
}
• Published 20 December 2013
• Mathematics
• Communications in Mathematical Physics
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 shown to correspond to a quadratic expression in the total Casimir operator of the tensor product of three irreducible sl−1(2)-modules. The operators commuting with the Dunkl Laplacian are seen to coincide with the intermediate Casimir operators and to realize a central extension of the Bannai–Ito…
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## References

SHOWING 1-10 OF 40 REFERENCES
Dunkl shift operators and Bannai-Ito polynomials
• Mathematics
• 2011
We consider the most general Dunkl shift operator $L$ with the following properties: (i) $L$ is of first order in the shift operator and involves reflections; (ii) $L$ preserves the space of
The Bannai–Ito algebra and a superintegrable system with reflections on the two-sphere
• Physics, Mathematics
• 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 Bannai-Ito polynomials as Racah coefficients of the sl_{-1}(2) algebra
• Mathematics, Physics
• 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
Clifford-Gegenbauer polynomials related to the Dunkl Dirac operator
• Mathematics
• 2010
We introduce the so-called Clifford-Gegenbauer polynomials in the framework of Dunkl operators, as well on the unit ball B(1), as on the Euclidean space $R^m$. In both cases we obtain several
The Dunkl Oscillator in the Plane II: Representations of the Symmetry Algebra
• Mathematics, Physics
• 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
FROM slq(2) TO A PARABOSONIC HOPF ALGEBRA
• Mathematics, Physics
• 2011
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
The Howe Duality for the Dunkl Version of the Dirac Operator
• Mathematics
• 2009
Abstract.The classical Fischer decomposition of polynomials on Euclidean space makes it possible to express any polynomial as a sum of harmonic polynomials multiplied by powers of |x|2. A deformation
Bispectrality of the Complementary Bannai-Ito Polynomials
• Mathematics, Physics
• 2013
A one-parameter family of operators that have the complementary Bannai{ Ito (CBI) polynomials as eigenfunctions is obtained. The CBI polynomials are the kernel partners of the Bannai{Ito polynomials
The Dunkl oscillator in the plane I : superintegrability, separated wavefunctions and overlap coefficients
• Mathematics, Physics
• 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
The algebra of dual −1 Hahn polynomials and the Clebsch-Gordan problem of sl−1(2)
• Mathematics, Physics
• 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