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I have seen a statement that if $\mathfrak{g}$ is a simple Lie algebra, then there are only finitely many Lie groups with Lie algebra $\mathfrak{g}$. Equivalently, the simply connected group with Lie algebra $\mathfrak{g}$ has finite center. Equivalently, the fundamental group of a Lie group with Lie algebra $\mathfrak{g}$ is finite.

Can anyone provide a proof or reference for any of these equivalent statements?

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  • $\begingroup$ See this question and the linked questions and references, which together give the result. $\endgroup$ – Dietrich Burde Jan 17 at 15:41
  • $\begingroup$ Thanks for pointing me there! But the first answer is only for compact Lie groups, and I certainly hope I don’t need to use Ricci curvature like in the second answer. (And the links in the comments are broken.) $\endgroup$ – user320832 Jan 18 at 5:35

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