Jordan algebras and 3-transposition groups

TL;DR

This paper classifies Jordan algebras that are isomorphic to Matsuo algebras with specific fusion rules, linking algebraic structures to 3-transposition groups and root systems.

Contribution

It provides a classification of Jordan algebras isomorphic to Matsuo algebras with fusion rules rac{1}{2}, identifying specific groups involved.

Findings

G = Sym(n) and G = 3^2:2 are the only such groups.

Results connect Jordan algebras to root systems.

Characterization of automorphism groups of these Jordan algebras.

Abstract

An idempotent in a Jordan algebra induces a Peirce decomposition of the algebra into subspaces whose pairwise multiplication satisfies certain fusion rules $\Phi(\frac{1}{2})$. On the other hand, $3$-transposition groups $(G,D)$ can be algebraically characterised as Matsuo algebras $M_\alpha(G,D)$ with idempotents satisfying the fusion rules $\Phi(\alpha)$ for some $\alpha$. We classify the Jordan algebras $J$ which are isomorphic to a Matsuo algebra $M_{1/2}(G,D)$, in which case $(G,D)$ is a subgroup of the (algebraic) automorphism group of $J$; the only possibilities are $G = \operatorname{Sym}(n)$ and $G = 3^2:2$. Along the way, we also obtain results about Jordan algebras associated to root systems.