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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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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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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