Octonionic bimodule

TL;DR

This paper explores the structure of octonionic bimodules, revealing their tensor product nature, category isomorphism to real vector spaces, and introduces cyclic decomposition to understand submodules, advancing octonionic functional analysis.

Contribution

It provides a comprehensive structural analysis of octonionic bimodules, including their tensor product form, category equivalence, and a novel cyclic decomposition approach.

Findings

Octonionic bimodules are tensor products and category equivalent to real vector spaces.

The real part structure is determined solely by left multiplication.

A complete description of submodules generated by one element is achieved.

Abstract

The structure of octonionic bimodules is formulated in this paper. It turns out that every octonionic bimodule is a tensor product, the category of octonionic bimodules is isomorphic to the category of real vector spaces. We show that there is also a real part structure on octonionic bimodules similar to the quaternion case. Different from the quaternion setting , the octonionic bimodule sturcture is uniquely determined by its left module structure and hence the real part can be obtained only by left multiplication. The structure of octonionic submodules generated by one element is more involved, which leads to many obstacles to further development of the octonionic functional analysis. We introduce a notion of cyclic decomposition to deal with this difficulty. Using this concept, we give a complete description of the submodule generated by one element in octonionic bimodules. This paper clears the barrier of the structure of $\O$-modules for the later study of octonionic functional analysis.