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arXiv · 2405.04683

Multicomplex Ideals, Modules and Hilbert Spaces

Derek Courchesne·Sébastien Tremblay
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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.

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Derek Courchesne, Sébastien Tremblay. 2024-05-07. Multicomplex Ideals, Modules and Hilbert Spaces. https://doi.org/10.1007/s00006-025-01373-y

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