A proof of Hozo's conjecture on the homology of a class of nilpotent Lie algebras (Q1355608)

A rooted tree with root \(r\) and vertex set \(\{1,2,\dots, n\}\) is considered as a poset with vertex set \(\{1,2,\dots,n\}\). Defining \(Z(P)\) as the span of all \(n\times n\) matrices \(z_{uv}\) such that \(u<_P v\), where \(z_{uv}\) is the \(n\times n\) matrix with a 1 in the \(u,v\) entry and 0's elsewhere, we obtain a nilpotent subalgebra of \(gl_n(\mathbb{C})\) whose algebra homology is the aim of the author's investigations. On the other hand, one way to compute the homology of any complex is the use of the Laplacian [\textit{B. Konstant}, Ann. Math., II. Ser. 74, 329-387 (1961; Zbl 0134.03501)]. In the present paper the author proves a conjecture due to \textit{Iztok Hozo} [Electron. J. Comb. 2, Paper R14 (1995; Zbl 0827.58059)], which identifies a second class of posets for which the Laplacians of the associated Lie algebras have integer eigenvalues.