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I am reading introductory lecture notes on Lie groups and Lie algebras. There it is stated as a fact without proof, that any compact semi-simple Lie group has finite center.

Here, semi-simple means, that the corresponding Lie algebra can be written as a direct sum of simple Lie algebras (having no non-trivial ideals).

Since this is not immediately obvious for me, I wonder if the proof is actually complicated or if I am just too ignorant to see it. If anyone can give an idea of where this fact comes from, I would be thankful.

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If $G$ is a semi-simple Lie group, then its Lie algebra $\frak g$ is semi-simple, which implies that it has trivial center. Therefore, the Lie algebra of the center $Z(G)$ of $G$ is trivial, which means that $Z(G)$ is discrete. Since $G$ is compact, $Z(G)$ is finite.

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  • $\begingroup$ And a semi-simple Lie algebra $\frak{g}$ has trivial center, since if the center was non-trivial then one of the centers of the simple Lie algebras $\frak{g_i}$ forming the decomposition would be non-trivial, which is a contradiction to $\frak{g_i}$ being simple, since the center is an ideal. Is that correct? $\endgroup$ – TheAbelian May 31 '16 at 13:54

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