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We defined a Lie algebra to be simple, if it has no proper Lie ideals and is not $k$ (the ground field). We have the proposition that $\mathfrak{sl}_n(\mathbb{R})$ and $\mathfrak{sl}_n(\mathbb{C})$ are simple $\mathbb{R}$-Lie algebras, and $\mathfrak{sl}_n(\mathbb{C})$ is also simple as a $\mathbb{C}$-Lie algebra. We have shown the proposition for the case $n=2$ using an $\mathfrak{sl}_2$-triple.

How would I show the statement for general $n \in \mathbb{N}$?

I suspect there's a way to use induction, but don't see how that would work. Also, I have no idea how I would generalize and apply the idea with the $\mathfrak{sl}_2$-triple to higher dimensions. I can't even just do the case $n=3$. How do I generalize?

user302234
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1 Answers1

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There is a direct proof that $\mathfrak{sl}(n,K)$ is a simple Lie algebra for any field $K$ of characteristic zero, which just uses Lie brackets of traceless matrices to show that a nontrivial ideal $J$ must be $\mathfrak{sl}(n,K)$ itself, see 6.4 in the book "Naive Lie Theory". This works uniformly for all $n\ge 2$. For a proof using a bit more theory, see Lemma $1.3$ here.

Dietrich Burde
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  • Thank you very much for the references. I looked up the elementary proof and it is rather straight-forward. I am a bit confused though: We never really make use of bilinearity (over $K$) of the Lie bracket. Does this mean the proof given in "Naive Lie Theory" works for $\mathfrak{sl}_n(\mathbb{C})$ as both $\mathbb{R}$- and $\mathbb{C}$-Lie algebra? – user302234 Jan 04 '16 at 13:09
  • For this question see "real forms", or "complexification" and "realification" of simple Lie algebras, see $2.2.1.9$ here. You do not need an extra proof for $\mathbb{sl}(n)$, this follows from general principles. – Dietrich Burde Jan 04 '16 at 13:26
  • I'll have a look at your reference as soon as possible. What exactly do you mean by $\mathbb{sl}(n)$? Thanks a lot! – user302234 Jan 04 '16 at 13:36
  • @user302234 It means $\mathfrak{sl}_n(K)$, see the last link. Did you had time to look at the reference in the meantime? – Dietrich Burde Oct 23 '21 at 08:16