On the conjecture of Shang about free alternative algebras

Abstract

Kashuba and Mathieu proposed a conjecture on vanishing of some components of the homology of certain Lie algebras, implying a description of the $G{L}_d$-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. Recently, Shang used a functorial construction of Allison, Benkart and Gao that builds Lie algebras out of alternative algebras to propose another conjecture on vanishing of some components of the homology of certain Lie algebras, implying a description of the $G{L}_d$-module structure of the free $d$-generated alternative algebra. In this note, we explain why the conjecture of Shang is not true.

21 pages, includes SageMath code in the Appendix. Version 2 implements several major changes: the main result is stronger (in particular, the conjecture is disproved already for two generators), and the argument is restructured to make all the main computations verifiable by a human