I'm struggling to understand the definition of a semisimple Lie algebra. The definitions I'm using are:
Simple: "A Lie algebra $\mathfrak{g}$ is simple if it is non-abelian and contains no non-trivial ideals."
Semisimple: "A Lie algebra $\mathfrak{g}$ is semisimple if it contains no non-trivial abelian ideals."
So, a simple Lie algebra is one which is non-abelian and $I=\mathfrak{g}$ and $I=0$ are the only ideals.
Here's my interpretation of the definition of semisimple: A semisimple Lie algebra can be abelian or non-abelian, and it is allowed to have non-trivial ideals as long as those non-trivial ideals are non-abelian, and the only ideals that are allowed to be abelian are $I=\mathfrak{g}$ and $I=0$ (though $I=\mathfrak{g}$ is clearly non-abelian if $\mathfrak{g}$ is non-abelian). Is this correct?
I'm rather confused about what it means for a Lie algebra to be semisimple. I know the definition that a semisimple Lie algebra is one that is a direct sum of simple Lie algebras, but I'd like to find a statement that explains semisimple Lie algebras like the one I have in bold. Any help would be much appreciated.
Best Answer
Another equivalent definition is that $L$ is semisimple if and only if $$ L=L_1\oplus \cdots \oplus L_n, $$ where $L_1,\ldots ,L_n$ are simple ideals, i.e., ideals with the definition of "simple" in boldface.
For semisimple, this is what you said: $L$ is semisimple iff every abelian ideal is trivial.