Classification of left octonion modules

TL;DR

This paper classifies left octonion modules, revealing new phenomena and introducing concepts like associative elements, ultimately showing all finite-dimensional modules are direct sums of two distinct structures.

Contribution

It provides a complete classification of left octonion modules and introduces new notions such as associative and conjugate associative elements.

Findings

Submodules generated by one element can be the whole module or not.

Octonion modules have two distinct structures.

All finite-dimensional octonion modules are direct sums of these structures.

Abstract

It is natural to study octonion Hilbert spaces as the recently swift development of the theory of quaternion Hilbert spaces. In order to do this, it is important to study first its algebraic structure, namely, octonion modules. In this article, we provide complete classification of left octonion modules. In contrast to the quaternionic setting, we encounter some new phenomena. That is, a submodule generated by one element $m$ may be the whole module and may be not in the form $\O m$. This motivates us to introduce some new notions such as associative elements, conjugate associative elements, cyclic elements. We can characterize octonion modules in terms of these notions. It turns out that octonions admit two distinct structures of octonion modules, and moreover, the direct sum of their several copies exhaust all octonion modules with finite dimensions.