On nilpotent Lie algebras having a torus of derivations
Let \({\mathfrak g}\) be a complex nilpotent Lie algebra of dimension \(n\). For \(x \in {\mathfrak g} - [{\mathfrak g}, {\mathfrak g}]\) let \(c(x)\) be the set of dimensions of Jordan blocks of ad \(x\) in decreasing order. Let \(c({\mathfrak g}) = \text{Sup} (c(x),x \in {\mathfrak g} - [{\mathfrak g}, {\mathfrak g}])\) (relative to the lexicographical order). Then \({\mathfrak g}\) is said to be filiform if \(c ({\mathfrak g})\) is maximal, i.e., \((n - 1,1)\). The authors classify the filiform Lie algebras admitting a nonzero diagonalizable derivation. Then, in a section on characteristically nilpotent algebras (i.e., algebras with all derivations nilpotent), they prove that for dimensions greater than 8 any algebraic irreducible component of the variety of complex nilpotent filiform laws of Lie algebras contains an open set whose elements are characteristically nilpotent laws.
scientific article; zbMATH DE number 701157
scientific article