A Laplace-Dunkl equation on $S^2$ and the Bannai-Ito algebra

Abstract

The analysis of the ${\mathbb{Z}}_2^3$ Laplace-Dunkl equation on the $2$-sphere is cast in the framework of the Racah problem for the Hopf algebra $s{l}_{-1}(2)$. 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 $s{l}_{-1}(2)$-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 (BI) algebra. Functions on $S^2$ spanning irreducible modules of the BI algebra are constructed and given explicitly in terms of Jacobi polynomials. The BI polynomials occur as expansion coefficients between two such bases composed of functions separated in different coordinate systems.

17 pages

References in corpus (1)

• The algebra of dual -1 Hahn polynomials and the Clebsch-Gordan problem of sl_{-1}(2)

Cited by in corpus (19)

• A higher rank Racah algebra and the ${\mathbb{Z}}_2^n$ Laplace-Dunkl operator
• A Dirac-Dunkl equation on $S^2$ and the Bannai-Ito algebra
• Embeddings of the Racah Algebra into the Bannai-Ito Algebra
• The higher rank $q$-deformed Bannai-Ito and Askey-Wilson algebra
• Superintegrability on the Dunkl oscillator model in three-Dimensional spaces of constant curvature
• On the algebra of symmetries of Laplace and Dirac operators
• A superintegrable model with reflections on $S^{n-1}$ and the higher rank Bannai-Ito algebra
• Finite-dimensional irreducible modules of the Bannai--Ito algebra at characteristic zero
• The total angular momentum algebra related to the ${\mathrm{S}}_3$ Dunkl Dirac equation
• The quantum superalgebra ${\mathfrak{osp}}_q(1|2)$ and a $q$-generalization of the Bannai-Ito polynomials
• An embedding of the Bannai-Ito algebra in $\mathcal{U}(\mathfrak{osp}(1,2))$ and $-1$ polynomials
• The dual pair $Pin(2n)\times \mathfrak{osp}(1|2)$, the Dirac equation and the Bannai-Ito algebra
• Generating functions for the osp(1|2) Clebsch-Gordan coefficients
• The Racah Algebra as a Subalgebra of the Bannai-Ito Algebra
• The equitable presentation of ${\mathfrak{osp}}_q(1|2)$ and a $q$-analog of the Bannai-Ito algebra
• Bivariate Bannai-Ito polynomials
• Superintegrability and the dual $-1$ Hahn algebra in superconformal quantum mechanics
• A Howe correspondence for the algebra of the $\mathfrak{osp}(1|2)$ Clebsch-Gordan coefficients
• Convolution identities for Dunkl orthogonal polynomials from the $\mathfrak{osp}(1|2)$ Lie superalgebra