An exceptional symmetry algebra for the 3D Dirac-Dunkl operator

TL;DR

This paper introduces a new symmetry algebra for the 3D Dirac-Dunkl operator linked to the exceptional G2 root system, providing both abstract and explicit constructions, and developing ladder operators for this algebra.

Contribution

It defines and realizes a novel symmetry algebra for the 3D Dirac-Dunkl operator related to G2, including ladder operators construction.

Findings

Defined the symmetry algebra for the 3D Dirac-Dunkl operator

Provided explicit realization of the algebra

Constructed ladder operators for the symmetry algebra

Abstract

We initiate the study of an algebra of symmetries for the 3D Dirac-Dunkl operator associated with the Weyl group of the exceptional root system $G_2$. For this symmetry algebra, we give both an abstract definition and an explicit realisation. We then construct ladder operators, using an intermediate result we prove for the Dirac-Dunkl symmetry algebra associated with arbitrary finite reflection group acting on a three-dimensional space.