On the nilpotent Lie algebras of dimension \(\leq 7)\) (Q1076782)

The author considers the embedding problem of a nilpotent Lie algebra into a stratified one and constructs all nilpotent Lie algebras of dimension \(\leq 7\) having a fixed Lie algebra of codimension 1, and gets among other results, a new classification of 6-dimensional nilpotent Lie algebras (in the paper Morosov's classification of 6-dimensional nilpotent Lie algebras is given). Here, the author uses the following method: If \({\mathfrak g}\) is a nilpotent Lie algebra, and \(\delta\) is a nilpotent derivative of \({\mathfrak g}\), then \({\mathbb{R}}\delta \oplus {\mathfrak g}\) is a nilpotent Lie algebra, where \(\oplus\) is a semi-direct product. The study has applications in the analysis of deformations of the internal space of 11-dimensional Kaluza-Klein theories (non-abelian theories) and in 11-dimensional supergravity.