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Are compact & connected Lie Groups in correspondence with semi-simple Lie groups? I think there is a condition on the center (discrete?) but I'm not sure.

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Where does this leave tori? –  Mariano Suárez-Alvarez Oct 18 '10 at 18:22

3 Answers 3

The answer to your title is "no"; lots of semi-simple Lie groups are not compact (for example, $SL_2(\mathbb{R})$). You're getting this mixed up with the fact that a complex semi-simple Lie group has a unique compact real form, and that this is a bijection to semi-simple compact Lie groups. (Complex reductive groups are in bijection with general compact Lie groups; this allows torus factors on both sides).

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Do you happen to know a source for the bijection between (possibly disconnected) complex reductive groups and (possibly disconnected) compact Lie groups? –  Victor Sep 26 '13 at 4:49

To amplify Ben's answer, I'd point to an earlier post that has lots more detail: here. The subject of compact groups is old and well-studied, so there are many references to choose from, even Wikipedia perhaps. Anyway, it's good to browse older Lie group entries on MO first.

PS: This supplementary "answer" is really a suggestion that the question is too close to the earlier post I cited to qualify as a fresh question. Textbook material of this kind calls mainly for references rather than discussion.

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I would like to add the following:

I think the source of confusion is the fact that the Killing form is nondegenerate (for semi-simple Lie groups) and negative definite (stronger than non-degenerate) for compact Lie groups with trivial center

$SL(2)$ is semi-simple but not compact.

The torus $S^1$ is compact but not semi-simple (abelian).

Compact groups are reductive and semi-simple only when in the case of trivial center.

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Dear amine, "Trivial center" should perhaps read "finite centre". Regards, Matthew –  Emerton Sep 11 '11 at 3:00
    
@Emerton: Thanks Emerton! Yes, 'trivial center' for Lie algebra which means 'finite center' for Lie groups. –  amine Sep 11 '11 at 4:48

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