# 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} }

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…

## 45 Citations

A Dirac–Dunkl Equation on S2 and the Bannai–Ito Algebra

- Physics, Mathematics
- 2013

The Dirac–Dunkl operator on the two-sphere associated to the $${{\mathbb{Z}_{2}^{3}}}$$Z23 reflection group is considered. Its symmetries are found and are shown to generate the Bannai–Ito algebra.…

The Z(2)(n) Dirac-Dunkl operator and a higher rank Bannai-Ito algebra

- Mathematics
- 2016

Abstract The kernel of the Z 2 n Dirac–Dunkl operator is examined. The symmetry algebra A n of the associated Dirac–Dunkl equation on S n − 1 is determined and is seen to correspond to a higher rank…

A higher rank Racah algebra and the $\mathbb{Z}_2^{n}$ Laplace-Dunkl operator

- Physics, Mathematics
- 2016

A higher rank generalization of the (rank one) Racah algebra is obtained as the symmetry algebra of the Laplace-Dunkl operator associated to the $\mathbb{Z}_2^n$ root system. This algebra is also the…

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 dual pair Pin(2n)×osp(1|2), the Dirac equation and the Bannai–Ito algebra

- Physics, MathematicsNuclear Physics B
- 2018

Abstract The Bannai–Ito algebra can be defined as the centralizer of the coproduct embedding of osp ( 1 | 2 ) in osp ( 1 | 2 ) ⊗ n . It will be shown that it is also the commutant of a maximal…

Embeddings of the Racah algebra into the Bannai-Ito algebra

- Physics, 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 q‐Bannai–Ito algebra and multivariate (−q)‐Racah and Bannai–Ito polynomials

- Mathematics, Physics
- 2019

The Gasper and Rahman multivariate $(-q)$-Racah polynomials appear as connection coefficients between bases diagonalizing different abelian subalgebras of the recently defined higher rank…

On the algebra of symmetries of Laplace and Dirac operators

- Physics, Mathematics
- 2017

We consider a generalization of the classical Laplace operator, which includes the Laplace–Dunkl operator defined in terms of the differential-difference operators associated with finite reflection…

The Higher Rank q-Deformed Bannai-Ito and Askey-Wilson Algebra

- Mathematics, PhysicsCommunications in Mathematical Physics
- 2019

The $q$-deformed Bannai-Ito algebra was recently constructed in the threefold tensor product of the quantum superalgebra $\mathfrak{osp}_q(1\vert 2)$. It turned out to be isomorphic to the…

The $\mathbb{Z}_2^n$ Dirac-Dunkl operator and a higher rank Bannai-Ito algebra

- Physics, Mathematics
- 2015

The kernel of the $\mathbb{Z}_2^{n}$ Dirac-Dunkl operator is examined. The symmetry algebra $\mathcal{A}_{n}$ of the associated Dirac-Dunkl equation on $\mathbb{S}^{n-1}$ is determined and is seen to…

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