Is it true if $[LL] = L$ then $L$ is a semisimple Lie algebra?

Question

Let $L$ be a finite dimensional Lie algebra over $\mathbb{C}$. It is classical theorem that if $L$ is semisimple, then $[LL] = L$.

Is it true if $[LL] = L$ then $L$ is a semisimple Lie algebra? I've been looking for counterexamples, but didn't find one yet.

No. Same question has been asked on MathOverflow: http://mathoverflow.net/questions/60498/lie-algebra-semisimple — Rafael Mrden, Jun 30 '14 at 11:43

score 4 · Accepted Answer · answered Jun 30 '14 at 19:51

4

One counterexample is the semidirect product $\mathfrak{sl}_n(\mathbb{C}) \ltimes \mathbb{C}^n$, where $\mathfrak{sl}_n(\mathbb{C})$ acts naturally on $\mathbb{C}^n$.

answered Jun 30 '14 at 19:51

Mikko Korhonen

25,224

Is it true if $[LL] = L$ then $L$ is a semisimple Lie algebra?

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