Could someone give me a suggestion to solve this problem?
Let $\frak{g}$ a nilpotent Lie álgebra. Prove that there is an ideal $\frak{h}\subseteq\frak{g}$ such that $\dim(\frak{g}) = \dim(\frak{h}) + 1$
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- It's false for $\mathfrak{g}={0}$ which is nilpotent. 2) It's true for every non-perfect Lie algebra (and hence every nonzero nilpotent Lie algebra): any hyperplane containing $[\mathfrak{g},\mathfrak{g}]$ works (this is user466747's argument).
– YCor Sep 24 '17 at 20:26 -
I can't do "Lie álgebra" on my keyboard directly, looks good. – Dietrich Burde Aug 21 '18 at 20:55
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Hint: Consider the quotient $\mathfrak{g}/[\mathfrak{g}, \mathfrak{g}]$ and note that any codimension 1 subspace is an ideal since this Lie algebra is abelian. Also note that $\mathfrak{g} \neq [\mathfrak{g}, \mathfrak{g}]$ since $\mathfrak{g}$ is nilpotent.
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