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Mathematics > Rings and Algebras
Title: Solid lines in axial algebras of Jordan type $\tfrac{1}{2}$ and Jordan algebras
(Submitted on 27 Mar 2024 (v1), last revised 23 Apr 2024 (this version, v2))
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.
Submission history
From: Jari Desmet [view email][v1] Wed, 27 Mar 2024 17:53:35 GMT (26kb)
[v2] Tue, 23 Apr 2024 16:37:25 GMT (17kb)
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