On the conjecture of Shang about free alternative algebras
Vladimir Dotsenko
TL;DR
The paper refutes Shang's conjecture that the homology $H_k(\mathsf{ABG}(\mathrm{Alt}(V)), \mathbb{k})$ contains no trivial or adjoint $\mathfrak{sl}_3$-components for $k>1$, which would yield clean $GL_d$-module descriptions of free $d$-generated alternative algebras. It situates the ABG construction in an operadic framework, showing that $\mathsf{ABG}(\mathrm{Alt}(V))$ emerges as a twisted Lie algebra associated to the Alt operad and that the conjecture would imply a functorial characterization of these homologies; the paper then tests the conjecture via explicit counterexamples. It provides several obstructions: in particular, the conjecture fails in degree $k=17$ for the two-generator free alternative algebra, in degree $k=10$ for the one odd generator superalgebra, and in degree $k=7$ for the three-generator case, with additional issues in representation-theoretic (Schur-positivity) aspects for $\mathrm{Alt}(10)$. Nevertheless, up to degree $6$ the conjecture holds, and the analysis links to inner-derivation structures, showing that $\mathcal{B}(T(V)) \cong Inner(T(V))$ for tensor algebras, while leaving the general Alt case open for further refinement. Overall, the work clarifies the limits of the conjectural vanishing pattern and highlights the nuanced interaction between operadic, homological, and representation-theoretic features in 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 $GL_d$-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. 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 $GL_d$-module structure of the free $d$-generated alternative algebra. In this note, we explain why the conjecture of Shang is not true.