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

Abstract

AbstractThe 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 $${\mathbb {R}}^{2n}$$ R 2 n . This is a new setting which contains and encompasses in a nontrivial way other cases already studied in the literature and which requires new tools. To this end, we define a cone $${\mathcal {W}}_{\mathcal {C}}^d$$ W C d in $$[{\text {End}}({\mathbb {R}}^{2n})]^d$$ [ End ( R 2 n ) ] d and we extend the slice topology $$\tau _s$$ τ s to this cone. Slice regular functions can be defined on open sets in $$\left( \tau _s,{\mathcal {W}}_{\mathcal {C}}^d\right) $$ τ s , W C d 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.