On Levi extensions of nilpotent Lie algebras

Levi's theorem decomposes any arbitrary Lie algebra over a field of characteristic zero, as a direct sum of a semisimple Lie algebra (named Levi factor) and its solvable radical. Given a solvable Lie algebra $R$, a semisimple Lie algebra $S$ is said to be a Levi extension of $R$ in case a Lie structure can be defined on the vector space $S\oplus R$. The assertion is equivalent to $\rho(S)\subseteq \mathrm{Der}(R)$, where $\mathrm{Der}(R)$ is the derivation algebra of $R$, for some representation $\rho$ of $S$ onto $R$. Our goal in this paper, is to present some general structure results on nilpotent Lie algebras admitting Levi extensions based on free nilpotent Lie algebras and modules of semisimple Lie algebras. In low nilpotent index a complete classification will be given. The results are based on linear algebra methods and leads to computational algorithms.