# Thread: Semisimple Lie Algebra, if L=[L,L] is it true L is semisimple?

1. ## Semisimple Lie Algebra, if L=[L,L] is it true L is semisimple?

I have went through a couple of books, none of which mention the validity of this. In my class we just proved the converse and we were asked to think about this.

For the past couple of weeks we have been talking about representations of sl(n,F) and just now started talking about Killing form. I don't see how to apply what we've done thus far to find a counter example, as I tried the few lie algebras I am familiar with. So I am thinking it might be true.

I do know L is semi-simple if and only if it is a direct sum of simple ideals.

2. Originally Posted by twilighttulip
I have went through a couple of books, none of which mention the validity of this. In my class we just proved the converse and we were asked to think about this.

For the past couple of weeks we have been talking about representations of sl(n,F) and just now started talking about Killing form. I don't see how to apply what we've done thus far to find a counter example, as I tried the few lie algebras I am familiar with. So I am thinking it might be true.

I do know L is semi-simple if and only if it is a direct sum of simple ideals.
I am new to Lie stuff and so my answer is probably naive. I think in Fulton/Harris L=[L,L] is called "perfect", and no, perfect does not imply semi-simple: a counterexample is given in Excercise 9.5
I don't understand it so I can't explain it
But what I wonder is if perfect and semi-simple is the same in finite dim?