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arXiv · 1702.05705

The octonions as a twisted group algebra

Tathagata Basak
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Abstract

We show that the octonions can be defined as the $\mathbb{R}$-algebra with basis $\lbrace e^x \colon x \in \mathbb{F}_8 \rbrace$ and multiplication given by $e^x e^y = (-1)^{\varphi(x,y)}e^{x + y}$, where $\varphi(x,y) = \operatorname{tr}(y x^6)$. While it is well known that the octonions can be described as a twisted group algebra, our purpose is to point out that this is a useful description. We show how the basic properties of the octonions follow easily from our definition. We give a uniform description of the sixteen orders of integral octonions containing the Gravesian integers, and a computation-free proof of their existence.

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Tathagata Basak. 2017-02-19. The octonions as a twisted group algebra. https://arxiv.org/abs/1702.05705

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