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

Michel Rausch de Traubenberg and Marcus Slupinski

Vol. 19 (2022), No. 4, 165–181
Abstract

We consider structures on a Fano plane which allow a generalisation of Freudenthal’s construction of a norm and a bilinear multiplication law on an eight-dimensional vector space 𝕆 canonically associated to . We first determine necessary and sufficient conditions in terms of the incidence geometry of 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.

Keywords
Fano plane, incidence geometry, octonions
Mathematical Subject Classification
Primary: 17Dxx, 51Axx, 51Exx
Milestones
Received: 25 February 2022
Revised: 26 July 2022
Accepted: 25 August 2022
Published: 1 December 2022
Authors
Michel Rausch de Traubenberg
IPHC
University of Strasbourg
Strasbourg
France
Marcus Slupinski
Institut de recherche mathématique avancée
CNRS
Strasbourg
France