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Several statements I like to know their True or False statements about the compactness of Lie group.

  1. Semi-simple Lie algebra: Every semi-simple Lie group generated by the semi-simple Lie algebra is compact.

  2. Non-semi-simple Lie algebra: Non-semi-simple Lie group generated by the non-semi-simple Lie algebra is compact, if and only if the non-semi-simple Lie group is the direct product of compact U(1)$^N$ Abelian group and other semi-simple Lie groups.

  3. In general, the Lie group is compact, if and only if, the Lie algebra can be written as the direct product of U(1)$^N$ Abelian Lie algebra and other semi-simple Lie algebra.

  4. In general, the Lie group is compact, if and only if, the Lie algebra can be written as the direct product of U(1)$^N$ Abelian Lie algebra and other compact semi-simple Lie algebra.

True or False? If True, please provide your reasoning. If False, please give counter examples.

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  • $\begingroup$ What examples did you consider before asking this? Notice that Qiaochu's examples involve nothing more elaborate that what is surely the simplest possible example! $\endgroup$ Commented Jan 28, 2015 at 19:34
  • $\begingroup$ Sorry, what I really should ask is the 4: In general, the Lie group is compact, if and only if, the Lie algebra can be written as the direct product of U(1)$^N$ Abelian Lie algebra and other compact semi-simple Lie algebra. $\endgroup$ Commented Jan 28, 2015 at 20:04
  • $\begingroup$ Do edit the question to reflect your intended question. $\endgroup$ Commented Jan 28, 2015 at 20:30

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  1. This is false. For example, $\mathfrak{sl}_2(\mathbb{R})$ is a semisimple Lie algebra, but $SL_2(\mathbb{R})$ isn't compact.

  2. This is also false, and a counterexample to 1 also provides a counterexample here. For example, $U(1) \times SL_2(\mathbb{R})$ isn't compact.

  3. This is also false, and a counterexample to 2 also provides a counterexample here.

The term you want to look up is compact Lie algebra. See also compact real form.

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  • $\begingroup$ Thanks +1. Sorry, what I really should ask is the 4: In general, the Lie group is compact, if and only if, the Lie algebra can be written as the direct product of U(1)$^N$ Abelian Lie algebra and other compact semi-simple Lie algebra. True or False? $\endgroup$ Commented Jan 28, 2015 at 20:04
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    $\begingroup$ @miss-tery: that's still a bit buggy. First of all, $U(1)$ is a Lie group, not a Lie algebra. If you mean the Lie algebra of $U(1)$, which is just $\mathbb{R}$, then of course $\mathbb{R}$ itself (the Lie group) is a counterexample. $\endgroup$ Commented Jan 28, 2015 at 20:18
  • $\begingroup$ Thanks +1. Sorry, what I really should ask is the 4: In general, the Lie group is compact, if and only if, the Lie group can be written as the direct product of U(1)$^N$ Abelian Lie group and other compact semi-simple Lie group. True or False? (I will take Lie group as the one generated from some Lie algebra, so pardon me using it interchangeably. I am not a mathematician, but an engineer.) $\endgroup$ Commented Jan 28, 2015 at 23:58
  • $\begingroup$ @miss-tery: still not quite true. In particular it's possible to start with a compact Lie group of that form and quotient it by a "diagonal" central subgroup. For example, start with $U(1) \times SU(2)$, and quotient it by the subgroup generated by $(-1, -1)$. $\endgroup$ Commented Jan 29, 2015 at 1:19

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