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arXiv · 2507.00437

On the conjecture of Kashuba and Mathieu about free Jordan algebras

Vladimir Dotsenko·Irvin Roy Hentzel
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Abstract

Kashuba and Mathieu proposed a conjecture on vanishing of Lie algebra homology, implying a description of the $GL_d$-module structure of the free $d$-generated Jordan algebra. Their conjecture relies on a functorial version of the Tits--Kantor--Koecher construction that builds Lie algebras out of Jordan algebras. In this note, we summarize new intricate computational data concerning free Jordan algebras and explain why, despite a lot of overwhelmingly positive evidence, the conjecture of Kashuba and Mathieu is not true.

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Vladimir Dotsenko, Irvin Roy Hentzel. 2025-07-01. On the conjecture of Kashuba and Mathieu about free Jordan algebras. https://arxiv.org/abs/2507.00437

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