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arXiv · 2403.18808

Solid lines in axial algebras of Jordan type $\tfrac{1}{2}$ and Jordan algebras

Jari Desmet
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Abstract

We show that a primitive axial algebra of Jordan type $\eta = \tfrac{1}{2}$ is a Jordan algebra if and only if every $2$-generated subalgebra is \emph{solid}, a notion introduced recently by Ilya Gorshkov, Sergey Shpectorov and Alexei Staroletov. As a byproduct, we show that a subalgebra generated by axes $a,b$ is solid if and only if the associator $[L_a,L_b]$ is a derivation. Moreover, we show that $2$-generated subalgebras that are not solid contain precisely $3$ axes.

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Jari Desmet. 2024-03-27. Solid lines in axial algebras of Jordan type $\tfrac{1}{2}$ and Jordan algebras. https://arxiv.org/abs/2403.18808

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