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

TL;DR

This paper explores the Dirac-Dunkl operator on the 2-sphere, revealing its symmetries form the Bannai-Ito algebra, and constructs eigenfunctions that support finite-dimensional representations of this algebra.

Contribution

It introduces the Dirac-Dunkl operator on $S^2$, identifies its symmetry algebra as Bannai-Ito, and constructs explicit eigenfunctions using a Cauchy-Kovalevskaia extension.

Findings

Symmetries generate the Bannai-Ito algebra

Eigenfunctions are constructed explicitly

Eigenfunctions support finite-dimensional irreducible representations

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.