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arXiv · 2203.03261

Incidence geometry of the Fano plane and Freudenthal's ansatz for the construction of (split) octonions

Michel Rausch de Traubenberg·Marcus J. Slupinski
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Abstract

In this article we consider structures on a Fano plane ${\cal F}$ which allow a generalisation of Freudenthal's construction of a norm and a bilinear multiplication law on an eight-dimensional vector space ${\mathbb O_{\cal F}}$ canonically associated to ${\cal F}$. We first determine necessary and sufficient conditions in terms of the incidence geometry of ${\cal F}$ for these structures to give rise to division composition algebras, and classify the corresponding structures using a logarithmic version of the multiplication. We then show how these results can be used to deduce analogous results in the split composition algebra case.

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Michel Rausch de Traubenberg, Marcus J. Slupinski. 2022-03-07. Incidence geometry of the Fano plane and Freudenthal's ansatz for the construction of (split) octonions. https://arxiv.org/abs/2203.03261

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