Matsuo algebras in characteristic 2

TL;DR

This paper extends Matsuo algebra theory to characteristic 2, introducing nilpotent generators instead of idempotents, and classifies when these algebras maintain a $Z/2Z$-graded fusion law, revealing a special case with an algebraic Miyamoto group.

Contribution

It develops a new framework for Matsuo algebras in characteristic 2 using nilpotent elements and classifies the conditions for a $Z/2Z$-graded fusion law.

Findings

Classification of when the fusion law is $Z/2Z$-graded

Identification of a special case with an algebraic Miyamoto group

Extension of Matsuo algebra theory to characteristic 2

Abstract

We extend the theory of Matsuo algebras, which are certain non-associative algebras related to 3-transposition groups, to characteristic 2. Instead of idempotent elements associated to points in the corresponding Fischer space, our algebras are now generated by nilpotent elements associated to lines. For many 3-transposition groups, this still gives rise to a $\mathbb{Z}/2\mathbb{Z}$-graded fusion law, and we provide a complete classification of when this occurs. In one particular small case, arising from the 3-transposition group $\operatorname{Sym}(4)$, the fusion law is even stronger, and the resulting Miyamoto group is an algebraic group $\mathbb{G}_a^2 \rtimes \mathbb{G}_m$.