Slice regularity and harmonicity on Clifford algebras

TL;DR

This paper explores the mathematical relationships between various differential operators and slice-regular functions on Clifford algebras, extending function theory in hypercomplex analysis and deriving results like the Fueter-Sce Theorem.

Contribution

It introduces new formulas linking the Cauchy-Riemann operator, spherical Dirac operator, and slice regularity, advancing the understanding of hypercomplex function theory on Clifford algebras.

Findings

Derived formulas relating key differential operators and slice-regular functions.

Computed Laplacian of the spherical derivative, leading to Fueter-Sce Theorem.

Connected results to zonal harmonics and Poisson kernel in specific cases.

Abstract

We present some new relations between the Cauchy-Riemann operator on the real Clifford algebra $\mathbb R_n$ of signature $(0,n)$ and slice-regular functions on $\mathbb R_n$. The class of slice-regular functions, which comprises all polynomials with coefficients on one side, is the base of a recent function theory in several hypercomplex settings, including quaternions and Clifford algebras. In this paper we present formulas, relating the Cauchy-Riemann operator, the spherical Dirac operator, the differential operator characterizing slice regularity, and the {spherical derivative} of a slice function. The computation of the Laplacian of the spherical derivative of a slice regular function gives a result which implies, in particular, the Fueter-Sce Theorem. In the two four-dimensional cases represented by the paravectors of $\mathbb R_3$ and by the space of quaternions, these results are related to zonal harmonics on the three-dimensional sphere and to the Poisson kernel of the unit ball of $\mathbb R^4$.