On the conjecture of Kashuba and Mathieu about free Jordan algebras

2025/07/01 by Dotsenko, Vladimir, Hentzel, Irvin Roy
#FOS: Mathematics #K-Theory and Homology (math.KT) #Quantum Algebra (math.QA) #Rings and Algebras (math.RA)

paper · doi:10.48550/arxiv.2507.00437

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.

Citations

• On the conjecture of Shang about free alternative algebras
• Allison-Benkart-Gao functor and the cyclicity of free alternative functors
• An effective criterion for Nielsen-Schreier varieties
• Structure theory for the group algebra of the symmetric group, with applications to polynomial identities for the octonions
• Categories of Jordan Structures and Graded Lie Algebras
• The On-Line Encyclopedia of Integer Sequences

Related

• On the conjecture of Shang about free alternative algebras
• On the Free Jordan Algebras
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