New properties of filiform Lie algebras and its computational processing (Q1855723)

A nilpotent Lie algebra \(\mathfrak{g}\) of dimension \(n\) is called filiform if its nilindex is \(n-1.\) The authors analyze these algebras when they have the supplementary property of being characteristically nilpotent (i.e. \(Der(\mathfrak{g})\) is nilpotent) [\textit{J. Dixmier} and \textit{W. G. Lister}, Proc. Am. Math. Soc. 8, 155-158 (1957; Zbl 0079.04802)]. It is proven that a filiform Lie algebra \(\mathfrak{g}\) is characteristically nilpotent if and only if is not isomorphic to a derived ideal of a Lie algebra (this result being false for non-filiform nilpotent algebras, as shown in \textit{E. M. Luks} [Proc. Am. Math. Soc. 56, 42-44 (1976; Zbl 0353.17006)]). Further, three polynomial time algorithms for the analysis of filiform algebras are discussed. One of these algorithms can also be used to check the characteristic nilpotence.