Multicomplex Ideals, Modules and Hilbert Spaces
Derek Courchesne, Sébastien Tremblay
TL;DR
This paper investigates the algebraic and analytic structure of the multicomplex algebras $\mathbb{M}_n$, focusing on their ideals, free modules, and Hilbert space frameworks. It introduces a canonical idempotent representation built from the $n$ conjugates and a multiperplex-valued norm defined by the involution $\Lambda_n$, enabling componentwise arithmetic and a direct-sum decomposition. It characterizes all multiperplex ideals via minimal and maximal constituents and shows how complexification relates $\mathbb{D}_n$ to $\mathbb{M}_n$, yielding a bijection between ideal classes and a clear description of quotients. It develops a finite-dimensional free $\mathbb{M}_n$-module theory, multicomplex matrices with blockwise determinants, and a multicomplex Hilbert space with a self-adjoint spectral theorem, extending classical results to settings with zero divisors. The work sets the foundation for applications in analysis and physics by translating familiar linear-algebraic concepts to the multicomplex setting while accounting for the algebra’s zero-divisor structure.
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 $Λ_n$, i.e. a composition of the $n$ multicomplex conjugates $Λ_n:=\dagger_1\cdots \dagger_n$, 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.