This question came up while I was studying the book "Complex Abelian Varieties" by Lange/Birkenhake.
More precisely, the authors prove in Lemma 1.1 that every connected compact complex Lie group of dimension $g$, $g$ a positive integer, is a complex torus. At one point they are using the fact that the universal cover of any such Lie group is a complex vector space of dimension $g$. They even give a reference (Theorem 18.4.1, "The Structure of Lie Groups" by Gerhard Hochschild). However, Theorem 18.4.1 (which is instead listed as Proposition 18.4.1 in the cited book) reads
"If a semisimple analytic group has a faithful finite-dimensional continuous representation then its center is finite"
I really don't have a clue how this should apply to my problem, because a complex torus is abelian, hence in particular not semisimple. On the other hand my knowledge of Lie groups is very very rudimentary so I might overlook something.
Can anyone show me how Proposition 18.4.1 in Hochschild's book helps me, or instead give me a reference to a proof of the statement in the title of my question? Of course any outline of a proof would also be appreciated, however I'd rather favor a reference. A reference to a direct proof of the above cited Lemma 1.1 would be very helpful too.
Thank you in advance