# On the algebra of symmetries of Laplace and Dirac operators

## TL;DR

This paper explores the algebraic structure of symmetries related to generalized Laplace and Dirac operators involving Dunkl operators, revealing new algebraic relations and connections to known symmetry algebras.

## Contribution

It introduces a generalized symmetry algebra for Laplace-Dunkl operators and defines a Dirac operator with explicit symmetry operators, extending classical angular momentum algebra.

## Key findings

- Identified symmetry operators commuting with the generalized Laplace operator.
- Derived algebraic relations for the symmetry algebra, generalizing angular momentum.
- Connected the symmetry algebra to the higher rank Bannai-Ito algebra.

## Abstract

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 groups called Dunkl operators. For this Laplace-like operator, we determine a set of symmetries commuting with it, which are generalized angular momentum operators, and we present the algebraic relations for the symmetry algebra. In this context, the generalized Dirac operator is then defined as a square root of our Laplace-like operator. We explicitly determine a family of graded operators which commute or anti-commute with our Dirac-like operator depending on their degree. The algebra generated by these symmetry operators is shown to be a generalization of the standard angular momentum algebra and the recently defined higher rank Bannai-Ito algebra.