Is the one-dimensional Lie algebra L=C semisimple?

Question

I might be wrong, but I think it is not simple since $[x,x]=0$ and thus abelian which means it cant be simple? Any clarification would be appreciated!

Answer 1

It is not semi-simple. There are two definitions of semi-simple (it is not trivial that they are equivalent).

The first is 'a direct sum of simple Lie algebras'. This one is ruled out by what you write.

In principle this is enough, but I will say something about the other definition of semi-simple: 'the algebra has no non-zero solvable ideals'

If your intuition tricks you into thinking that an ideal is always a proper subspace it seems that the Lie algebra in fact satisfies this. BUT in reality the Lie algebra itself also counts as an ideal in itself and since it is abelian it is solvable. So again we see that the Lie algebra is not semi-simple.

The lie algebra is reductive however. Reductive Lie algebras are a class of Lie algebras that behave very similar to semi-simple ones but allow some not too wild abelian ideals. The idea of reductive Lie algebras is that it includes not only all semi-simple ones (in particular the $\mathfrak{sl}_n$) but also the slightly bigger and closely related $\mathfrak{gl}_n$ which of course are of interest in many contexts.

More info on Wikipedia: https://en.wikipedia.org/wiki/Reductive_Lie_algebra

(Ok there reductive Lie algebras are defined as the direct sum of a semi-simple Lie algbera and an abelian Lie algebra. I insist that the semi-simple part can also be the Lie algebra $\{0\}$ so that the Lie algebra is abelian, but maybe different authors have different opinions on that. Update: on closer inspection Wikipedia also agrees with me here.)