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

TL;DR

This paper explores how the incidence geometry of the Fano plane can be used to generalize Freudenthal's construction of octonions and split octonions, providing conditions for division algebra structures.

Contribution

It establishes geometric conditions on the Fano plane for constructing (split) octonions and classifies these structures using a novel logarithmic multiplication approach.

Findings

Identified conditions for Fano plane structures to produce division composition algebras.

Classified algebraic structures via a logarithmic multiplication method.

Extended results to split composition algebras.

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.