A representation formula for slice regular functions over slice-cones in several variables

TL;DR

This paper extends slice analysis to functions with codomain in real vector spaces of even dimension, establishing a representation formula and broadening applications to various real algebras.

Contribution

It introduces a new framework for slice regular functions over slice-cones in several variables, including a representation formula and applications to diverse real algebras.

Findings

Established a representation formula for slice regular functions.

Extended slice analysis to real vector spaces of even dimension.

Applied the theory to various real algebras including quaternions, octonions, and Clifford algebras.

Abstract

The aim of this paper is to extend the so called slice analysis to a general case in which the codomain is a real vector space of even dimension, i.e. is of the form $\mathbb{R}^{2n}$. We define a cone $\mathcal{W}_\mathcal{C}^d$ in $[End(\mathbb{R}^{2n})]^d$ and we extend the slice-topology $\tau_s$ to this cone. Slice regular functions can be defined on open sets in $\left(\tau_s,\mathcal{W}_\mathcal{C}^d\right)$ and a number of results can be proved in this framework, among which a representation formula. This theory can be applied to some real algebras, called left slice complex structure algebras. These algebras include quaternions, octonions, Clifford algebras and real alternative $*$-algebras but also left-alternative algebras and sedenions, thus providing brand new settings in slice analysis.