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arXiv · 1502.05657

Jordan algebras and 3-transposition groups

Tom De Medts·Felix Rehren
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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.

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Tom De Medts, Felix Rehren. 2015-02-19. Jordan algebras and 3-transposition groups. https://arxiv.org/abs/1502.05657

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