Quasi-filiform Lie algebras of maximum length (Q5947462)

The aim of the paper under review is to classify the Lie algebras referred to in the title. (For other recent results on classification of ``almost filiform Lie algebras'' see e.g. [\textit{J. M. Cabezas, L. M. Camacho, J. R. Gómez} and \textit{R. M. Navarro}, Commun. Algebra 28, 4489-4499 (2000; Zbl 0970.17007)]). NEWLINENEWLINENEWLINEA complex Lie algebra \({\mathfrak g}\) of finite dimension \(n\) is said to be quasi-filiform if \({\mathcal C}^{n-3}{\mathfrak g}\neq\{0\}\) and \({\mathcal C}^{n-2}{\mathfrak g}=\{0\}\), where \(\{{\mathcal C}^i{\mathfrak g}\}_{i\geq 0}\) is the descending central series of \({\mathfrak g}\), that is, \({\mathcal C}^0{\mathfrak g}=\{\mathfrak g\}\) and \({\mathcal C}^i{\mathfrak g}=[{\mathfrak g},{\mathcal C}^{i-1}{\mathfrak g}]\) for \(i\geq 1\). The Lie algebra \({\mathfrak g}\) is said to have maximum length if it possesses a gradation \({\mathfrak g}={\mathfrak g}_{n_1}\oplus{\mathfrak g}_{n_1+1}\oplus\cdots\oplus {\mathfrak g}_{n_1+n-1}\), where \(n_1\in{\mathbb Z}\) and \({\mathfrak g}_{n_1+j}\neq\{0\}\) for \(j=0,1,\dots,n-1\). (Note that \({\mathfrak g}\) cannot have a gradation with more than \(n=\dim{\mathfrak g}\) nonzero terms, and the maximum length condition is equivalent to \(\dim{\mathfrak g}_{n_1+j}=1\) for \(j=0,1,\dots,n-1\).) NEWLINENEWLINENEWLINEThe classification of quasi-filiform Lie algebras of maximum length is stated in Theorem 3.8. The basic idea of the proof is to construct adequate homogeneous bases in the algebra under consideration. NEWLINENEWLINENEWLINEThe paper under review ends with a most interesting section concerning the use of the package Mathematica in Lie algebra classification problems.