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For a simple Lie algebra $\mathfrak{g}$ who contains a subalgebra isomorphic to $\mathfrak{sl}(2,\mathbb{R})$, I’m trying to show that a nontrivial irreducible representation $\pi:\mathfrak{g}\to\mathrm{End}(V)$ is faithful.

I tried to apply the fact that $\pi$ is isomorphic to $\vee^k\mathbb{C}^2$ for any irreducible complex representation of $\mathfrak{sl}(2,\mathbb{R})$ but failed. Can someone please give me some hints to guide me?

itkyitfbku
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1 Answers1

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The kernel of $\pi$ is an ideal of $\mathfrak g$. Since $\mathfrak g$ is simple, the ideal $\ker\pi$ is either $0$ or $\mathfrak g$. Since $\pi$ is non-trivial, $\ker\pi\ne\mathfrak g$, hence $\ker\pi=0$. This means $\pi$ is faithful.

Kenta S
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