Jordan algebras and 3-transposition groups

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 $G=\mathrm{Sym}(n)$ and $G=3^2:2$. Along the way, we also obtain results about Jordan algebras associated to root systems.

20 pages