Homomorphic images of solvable Lie algebras (Q1084487)

Let L be a finite dimensional solvable Lie algebra over a field F. The author defines L to be minimal if L is not nilpotent but every proper homomorphic image of L is nilpotent. Clearly, any non-nilpotent L can be mapped onto a minimal solvable Lie algebra. If F is algebraically closed and of characteristic zero then the two-dimensional solvable Lie algebra \(L_ 0=Fx+Fy\), where \([y,x]=x\), is the only minimal solvable Lie algebra [see \textit{W. Borho}, \textit{P. Gabriel} and \textit{R. Rentschler}, Primideale in Einhüllenden auflösbarer Lie-Algebren (Lect. Notes Math. 357) (1973; Zbl 0293.17005) Lemma 6.12], but the author gives an example to show that this is not the case if F is not algebraically closed. It is shown that over any field, any finite dimensional completely solvable non-nilpotent Lie algebra can be mapped onto \(L_ 0\). A description is given of the minimal solvable Lie algebras over a field of characteristic zero, and a partial description is given in the case of non-zero characteristic. The author states that these results can be used to prove that if L is a finite dimensional solvable non-nilpotent Lie algebra over a field of characteristic zero, then its universal enveloping algebra has an infinite link component.