Covering groups of compact connected Lie groups is compact.

Question

Hi: I have a question as follows:

Is it true that the covering group of a compact, connected Lie group is also compact?

Thanks very much!

Answer 1

As Adam noted, this is wrong in general, but in the case of a compact and semisimple Lie group, it is true:

Theorem (Weyl): If $G$ is a connected semisimple Lie group with compact Lie algebra, then $G$ is compact and $Z(G)$ is finite.

This is proved for example in

Hilgert, Neeb: Structure and Geometry of Lie groups, Theorem 12.1.17.

Here a Lie algebra is compact if it is the Lie algebra of a compact Lie group (Proposition 12.1.4 in loc.cit.). In particular, Weyl's theorem applies to $G := \widetilde{K}$ the universal covering group of a compact semisimple Lie group $K$, proving that it is compact, too.

Answer 2

No, take $\Bbb R$ which is the universal cover of the circle group, $\Bbb R/\Bbb Z$.