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This is obvious since there is a more general result for semi-simple Lie algebras, but I would like to prove it directly.

So suppose that the Killing form $\kappa$ is degenerate and let $X\in L$ be such that $\kappa(X,Y)=0$ for all $Y \in L$.

I would now like to use $X$ to construct a non-trivial ideal of $L$, thereby showing that it is not simple.

The ideal $\mathrm{span}_{\mathbb{F}}\{[X,Y] : Y \in L\} $ seems promising, but I haven't been able to show that it is non-trivial. I'm not sure if I am on the right track.

UPDATE Dietrich Burde's answer shows that this is in fact not true if for example the field $\mathbb{F}$ has characteristic 3. So in this case I assume I am only considering fields of characteristic 0, and moreover one may assume that $\kappa$ does not vanish identically.

prt13463
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2 Answers2

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Suppose that the characteristic is zero. $I=\{X:K(X,.)=0\}$ is an ideal. To see this, note that $K([X,Y],Z)=K(X,[Y,Z])$. If $K(X,.)=0,$ for every $Y,Z$, $K([X,Y],Z)=K(X,[Y,Z])=0$. This implies that $[X,Y]\in I$. Suppose that $I$ is not $\{0\}$. Then, since the Lie algebra ${\cal G}$ is simple, $I={\cal G}$ and $K=0$. This implies that ${\cal G}$ is solvable by the Cartan criterion. Contradiction.

https://en.wikipedia.org/wiki/Cartan%27s_criterion#Cartan.27s_criterion_for_solvability

Elías Guisado Villalgordo
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Tsemo Aristide
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  • Thank you. Our course did not cover solvability, but we were given the extra assumption that $\kappa$ does not vanish identically, so that allows us to exclude the case $I=G$. – prt13463 Apr 23 '17 at 18:57
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Here is a counterexample. Consider the simple Lie algebra $\mathfrak{sl}(3)/Z(\mathfrak{sl}(3))$ over a field of characteristic $p=3$. Then the Killing form is given by $$ \begin{pmatrix} 12 & -6 & 0 & 0 & 0 & 0 & 0 & 0\cr -6 & 12 & 0 & 0 & 0 & 0 & 0 & 0\cr 0 & 0 & 0 & 0 & 6 & 0 & 0 & 0 \cr 0 & 0 & 0 & 0 & 0 & 0 & 6 & 0 \cr 0 & 0 & 6 & 0 & 0 & 0 & 0 & 0 \cr 0 & 0 & 0 & 0 & 0 & 0 & 0 & 6 \cr 0 & 0 & 0 & 6 & 0 & 0 & 0 & 0 \cr 0 & 0 & 0 & 0 & 0 & 6 & 0 & 0 \end{pmatrix} $$ which is identical zero for $3=0$.

Dietrich Burde
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  • I don't know what to say... this was an exam question: "Prove that, if g is simple, then $\kappa$ is necessarily non-degenerate. [Here you may assume that $\kappa$ does not vanish identically.]" – prt13463 Apr 23 '17 at 18:47
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    For the exam question, probably, there was an assumption on the field $\mathbb{F}$. What did it say about the field? – Dietrich Burde Apr 23 '17 at 18:48
  • There was no explicit mention of that, but the course only covered Lie algebras over fields of characteristic $0$. Actually, just $\mathbb{R}$ and $\mathbb{C}$. – prt13463 Apr 23 '17 at 18:50
  • I'm new to the subject. Could tell me why (or some hints) that $\mathfrak{sl}(3)/Z$ is simple (in my case s.s. will be enough), and if I'm right, $Z$ is all the scalar matrices, which is one dimensional, so the resulting quotient algebra will be 7 dimensional. But your matrix for the Killing form is 8 by 8. Did I miss something? – S. D. Z Feb 09 '18 at 01:28