On behavior of solvable ideals of Lie algebras under outer derivations

TL;DR

This paper investigates when the sum of all solvable ideals in a Lie algebra remains characteristic under outer derivations, providing new conditions especially in positive characteristic fields.

Contribution

It establishes that the sum of all solvable ideals is characteristic in certain cases, including characteristic zero and when the derived length is bounded in positive characteristic.

Findings

Sum of solvable ideals is characteristic in characteristic zero.

Sum of solvable ideals is characteristic if derived length is less than log2 p in characteristic p.

Provides estimations for derived length of ideals under derivations in characteristic zero.

Abstract

Let $L$ be a finite dimensional Lie algebra over a field $F$. It is well known that the solvable radical $S(L)$ of the algebra $L$ is a characteristic ideal of $L$ if $\char F=0$ and there are counterexamples to this statement in case $\char F=p>0$. We prove that the sum $S(L)$ of all solvable ideals of a Lie algebra $L$ (not necessarily finite dimensional) is a characteristic ideal of $L$ in the following cases: 1) $\char F=0;$ 2) $S(L)$ is solvable and its derived length is less than $\log_{2}p.$ Some estimations (in characteristic 0) for the derived length of ideals $I+D(I)+... +D^{k}(I)$ are obtained where $I$ is a solvable ideal of $L$ and $D\in Der(L).$