Jordan Matsuo algebras over fields of characteristic 3

TL;DR

This paper characterizes when Matsuo algebras over fields of characteristic 3 are Jordan algebras, linking this property to specific Fischer spaces like affine space of order 3 and symmetric groups.

Contribution

It provides a complete classification of Matsuo algebras as Jordan algebras in characteristic 3, based on Fischer space isomorphisms.

Findings

Matsuo algebra is Jordan iff Fischer space is affine space of order 3 or related to symmetric group

Characterization of affine spaces of order 3 used in the proof

Equivalence of Jordan identities over fields of characteristic 3 when algebra is spanned by idempotents

Abstract

The Matsuo algebra associated with a connected Fischer space is shown to be a Jordan algebra over a field of characteristic 3 if and only if the Fischer space is isomorphic to either the affine space of order 3 or the Fischer space associated with the symmetric group. The proof uses a characterization of the affine spaces of order 3 and equivalence of Jordan and linearized Jordan identities over a field of characteristic 3 in case the algebra is spanned by idempotents.