Multicomplex Ideals, Modules and Hilbert Spaces

TL;DR

This paper explores algebraic structures of multicomplex numbers, including ideals, modules, and Hilbert spaces, introducing new representations, conjugacy, and norms to facilitate their analysis.

Contribution

It introduces a canonical representation, generalized conjugacy, and a multicomplex norm, advancing the understanding of multicomplex algebraic structures and their Hilbert space formulation.

Findings

Defined a canonical representation for multicomplex numbers.

Developed a generalized conjugacy and multicomplex norm.

Studied ideals, modules, and Hilbert spaces within multicomplex algebra.

Abstract

In this article we study some algebraic aspects of multicomplex numbers $\mathbb M_n$. For $n\geq 2$ a canonical representation is defined in terms of the multiplication of $n-1$ idempotent elements. This representation facilitates computations in this algebra and makes it possible to introduce a generalized conjugacy $\Lambda_n$, i.e. a composition of the $n$ multicomplex conjugates $\Lambda_n:=\dagger_1\cdots \dagger_n$, as well as a multicomplex norm. The ideals of the ring of multicomplex numbers are then studied in details, free $\mathbb M_n$-modules and their linear operators are considered and, finally, we develop Hilbert spaces on the multicomplex algebra.