Jordan algebras and 3-transposition groups

An idempotent in a Jordan algebra induces a Peirce decomposition of the algebra into subspaces whose pairwise multiplication satisfies a certain fusion rule Phi)(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 rule Phi(a) for some a. 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 = Sym(n) and G = 3(2) : 2. Along the way, we also obtain results about Jordan algebras associated to root systems. (C) 2017 Elsevier Inc. All rights reserved.