Multicomplex Ideals, Modules and Hilbert Spaces

Abstract

In this article we study some algebraic aspects of multicomplex numbers ${\mathbb{M}}_n$. For $n\ge 2$ 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 ${\Lambda }_n:={†}_1\cdots {†}_n$, 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.

arXiv comment27 pages

References in corpus (8)

• Relativistic quantum physics with hyperbolic numbers
• A second order differential equation for the relativistic description of electrons and photons
• Bicomplex quantum mechanics: I. The generalized Schrödinger equation
• Bicomplex Quantum Mechanics: II. The Hilbert Space
• Considerations on the hyperbolic complex Klein-Gordon equation
• Finite-Dimensional Bicomplex Hilbert Spaces
• On the Klein-Gordon equation and hyperbolic pseudoanalytic function theory
• The bicomplex quantum Coulomb potential problem

Cited by in corpus (2)

• Determinant, Characteristic Polynomial, and Inverse in Commutative Analogues of Clifford Algebras
• On Commutative Analogues of Clifford Algebras and Their Decompositions