Marcus J. Slupinski | Searcharxiv

arXiv2022

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

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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arXiv2012

The geometry of special symplectic representations

We show there is a class of symplectic Lie algebra representations over any field of characteristic not 2 or 3 that have many of the exceptional algebraic and geometric properties of both symmetric three forms in two dimensions and alternating three forms in six dimensions. All nonzero orbits are coisotropic and the covariants satisfy relations generalising classical identities of Eisenstein and Mathews. The main algebraic result is that suitably generic elements of these representation spaces can be uniquely written as the sum of two elements of a naturally defined Lagrangian subvariety. We give universal explicit formulae for the summands and show how they lead to the existence of geometric structure on appropriate subsets of the representation space. Over the reals this structure reduces to either a conic, special pseudo-K\" ahler metric or a conic, special para-K\" ahler metric.

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