In this article, we provide a complete classification of left octonionic modules (finite or infinite dimensions) in terms of new notions such as associative elements and conjugate associative elements. We give a simple approach to determine the irreducible left \(\mathbb {O}\)-modules by utilizing the Clifford algebra \(C\ell _{7}\). We find that every left \(\mathbb {O}\)-module has a basis in some sense. This means that every left \(\mathbb {O}\)-module is a “free” module.

中文翻译：

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左八键模块的分类

在本文中，我们根据新概念（例如，关联元素和共轭关联元素）提供了一个完整的左张回音模块分类（有限或无限尺寸）。我们给出一种简单的方法，通过利用Clifford代数\（C \ ell _ {7} \）来确定不可约的左\（\ mathbb {O} \） -模。我们发现每个左\（\ mathbb {O} \）-模块在某种意义上都有一个基础。这意味着每个左\（\ mathbb {O} \）-模块都是一个“免费”模块。