True or False statements about compactness of Lie group

Question

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.

Answer 1

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.