Multicomplex Ideals, Modules and Hilbert Spaces

IF 1.1 2区数学 Q2 MATHEMATICS, APPLIED

Advances in Applied Clifford Algebras Pub Date : 2025-01-17 DOI:10.1007/s00006-025-01373-y

Derek Courchesne, Sébastien Tremblay

引用次数: 0

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 \(\Lambda _n:=\dagger _1\cdots \dagger _n\), 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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多复理想、模与希尔伯特空间

在这篇文章中，我们研究了多重复数的一些代数方面\({\mathbb {M}}_n\)。对于\(n\ge 2\)，规范表示是根据\(n-1\)幂等元素的乘法定义的。这种表示简化了该代数的计算，并使引入广义共轭\(\Lambda _n\)成为可能，即n个多复共轭\(\Lambda _n:=\dagger _1\cdots \dagger _n\)的组合，以及多复范数。然后详细研究了多复数环的理想，考虑了自由\({\mathbb {M}}_n\) -模及其线性算子，最后在多复数代数上建立了Hilbert空间。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Advances in Applied Clifford Algebras 数学-物理：数学物理

CiteScore

2.20

自引率

13.30%

发文量

审稿时长

3 months

期刊介绍： Advances in Applied Clifford Algebras (AACA) publishes high-quality peer-reviewed research papers as well as expository and survey articles in the area of Clifford algebras and their applications to other branches of mathematics, physics, engineering, and related fields. The journal ensures rapid publication and is organized in six sections: Analysis, Differential Geometry and Dirac Operators, Mathematical Structures, Theoretical and Mathematical Physics, Applications, and Book Reviews.