Links among characteristically nilpotent, \(C\)-graded and derived filiform Lie algebras (Q812485)

The paper under review surveys a number of theorems on the mutual relationships between the three types of complex finite-dimensional Lie algebras referred to in the title: the characteristically nilpotent Lie algebras, the \(c\)-graded, and the derived ones. The main new result of the paper is the fact that a complex filiform Lie algebra \({\mathfrak g}\) of dimension \(\geq4\) is characteristically nilpotent (i.e., every derivation of \({\mathfrak g}\) is nilpotent) if and only if \({\mathfrak g}\) is not a derived Lie algebra, in the sense that there exists no Lie algebra \({\mathfrak h}\) such that \([{\mathfrak h},{\mathfrak h}]={\mathfrak g}\). One also determines the two nilpotent Lie algebras of smallest dimension (namely, of dimension~7) which are neither characteristically nilpotent nor derived. The paper concludes by describing an algorithmic procedure that allows one to see whether a given complex filiform Lie algebra possesses any of the properties referred to in the title.