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

Abstract

The Dirac-Dunkl operator on the 2-sphere associated to the ${\mathbb{Z}}_2^3$ reflection group is considered. Its symmetries are found and are shown to generate the Bannai-Ito algebra. Representations of the Bannai-Ito algebra are constructed using ladder operators. Eigenfunctions of the spherical Dirac-Dunkl operator are obtained using a Cauchy-Kovalevskaia extension theorem. These eigenfunctions, which correspond to Dunkl monogenics, are seen to support finite-dimensional irreducible representations of the Bannai-Ito algebra.

17 pages, 1 diagram

References in corpus (2)

• Bispectrality of the Complementary Bannai-Ito Polynomials
• The Bannai-Ito algebra and some applications

Cited by in corpus (18)

• A higher rank Racah algebra and the ${\mathbb{Z}}_2^n$ Laplace-Dunkl operator
• ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$ generalizations of $\mathcal{N}=2$ super Schrödinger algebras and their representations
• The higher rank $q$-deformed Bannai-Ito and Askey-Wilson algebra
• On the algebra of symmetries of Laplace and Dirac operators
• The Racah algebra as a commutant and Howe duality
• Finite-dimensional irreducible modules of the Bannai--Ito algebra at characteristic zero
• Centralizers of the superalgebra osp(1|2): the Brauer algebra as a quotient of the Bannai-Ito algebra
• 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
• The dual pair $Pin(2n)\times \mathfrak{osp}(1|2)$, the Dirac equation and the Bannai-Ito algebra
• An embedding of the Bannai-Ito algebra in $\mathcal{U}(\mathfrak{osp}(1,2))$ and $-1$ polynomials
• The $q$-Bannai-Ito algebra and multivariate $(-q)$-Racah and Bannai-Ito polynomials
• The Racah Algebra as a Subalgebra of the Bannai-Ito Algebra
• Bannai-Ito algebras and the universal R-matrix of osp(1|2)
• Bivariate Bannai-Ito polynomials
• Finite-dimensional representations of the symmetry algebra of the dihedral Dunkl--Dirac operator
• Dunkl jump processes: relaxation and a phase transition
• Planar harmonic and monogenic polynomials of type A