Why semi-simple Lie algebra cannot have a abelian ideals?

Question

I am trying to understand why semi-simple Lie algebra cannot have a abelian ideal. More specifically I search for an argument as direct as possible saying if an algebra is a direct some of simple Lie algebras then it cannot have abelian ideals, and possibly an argument for the converse. I would prefer an argument which does not involve solvable algebras (i did not understand them well enough yet..) Thanks you.

Answer 1

A semi-simple Lie algebra is a sum $S_1\oplus...\oplus S_n$ of simple ideals $S_i$, by definition, $S_i$ is simple if and only if it is non abelian and has no non zero proper ideal. Let $I$ be an abelian ideal of $S$. Let $p:I\rightarrow S_i$ such that $p(u)=u_i$, where $u=u_1+...+u_n, u_i\in S_i$,$p_i(I)$ is an ideal of $S_i$: for every $s\in S_i, [s,u]\in I, [s,u]=[s,u_1+..+u_n]=[s,u_i]$ since $[S_i,S_j]=0$ for $i\neq j$. Since $S_i$ is simple, $p_i(I)=0$ and $I=0$.

Answer 2

All abelian ideals of any Lie Algebra are solvable and since, you are taking your Lie Algebra say, $L$ to be semi-simple. So, Radical of $L$ or maximal solvable ideal has to be zero. Hence, all abelian ideals are zero.