Combinatorial structures and Lie algebras of upper triangular matrices (Q427661)

Inspired by \textit{A. Carriazo} et al. [Linear Algebra Appl. 389, 43--61 (2004; Zbl 1053.05059)] and also by \textit{L. M. Fernández} and \textit{L. Martín-Martínez} [Linear Algebra Appl. 407, 43--63 (2005; Zbl 1159.17302)], the authors associate the solvable Lie algebra \(\mathfrak{h}_n\) (\(n \in \mathbb{N}\)) consisting of all \(n\times n\) upper triangular matrices over \(\mathbb{C}\), with a combinatorial structure. The classification of these structures and some of its properties are presented.NEWLINENEWLINEThe last section of the paper provides a method for computing the minimal \(n\) for which a given finite-dimensional solvable Lie algebra \(\mathfrak{L}\) over \(\mathbb{C}\) can be represented as a subalgebra in the matrix algebra \(\mathfrak{h}_n\). The method given here is new and stresses the compatibility of combinatorial structures associated with \(\mathfrak{h}_n\) and \(\mathfrak{L}\), based on the study of maximal abelian dimension and the maximal abelian subalgebras. As application, the authors compute the minimal matrix representation of the \(r\)-dimensional solvable Lie algebra \(\mathfrak{L}_r\) (\(r\geq 3\)) with the law: \([e_k, e_r] = e_k\), (\(k < r\)).