On behavior of solvable ideals of Lie algebras under outer derivations

Details On behavior of solvable ideals of Lie algebras under outer derivations On behavior of solvable ideals of Lie algebras under outer derivations

2008-08-24

arxiv.org/abs/0808.3262v1

Mathematics

Rings and Algebras

Scientific paper

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 $D\in Der(L).$

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