Why is $\mathbb{C}^{g}$ the universal cover of any connected compact complex Lie group of dimension g? 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
 A: The following steps lead to a solution. Let me first define standard notation. If $G$ is a Lie group and if $g\in G$, then the map $\phi_{g}:G\to G$ defined by the rule $\phi_g(x)=gxg^{-1}$ for $x\in G$ is referred to as conjugation by $g$. If $f:M\to N$ is a smooth map, then we denote by $T_{p}(f):T_{p}(M)\to T_{f(p)}(N)$ the differential of $f$ at $p\in M$.
Exercise 1: Let $G$ be a complex Lie group. The adjoint representation of $G$ is the map $\rho:G\to \text{GL}(\mathfrak{g})$ defined by the rule $\rho(g)=T_{e}(\phi_{g})\in \text{GL}(\mathfrak{g})$ where $e\in G$ is the identity element. Prove that $\rho:G\to \text{GL}(\mathfrak{g})$ is a complex (holomorphic) representation of $G$.
Exercise 2: In the context of Exercise 1, assume that $G$ is in addition compact and connected. Prove that the adjoint representation $\rho:G\to \text{GL}(\mathfrak{g})$ is trivial. (Hint: use the maximum modulus principle.) Conclude that $G$ is an abelian group.
We have now set the stage to prove that the exponential map $\text{exp}:\mathfrak{g}\to G$ is a covering map.
Exercise 3: Let $G$ be a Lie group. If $X,Y\in \mathfrak{g}$ commute, i.e., $[X,Y]=0$, then prove that $e^{X}e^{Y}=e^{X+Y}=e^{Y}e^{X}$. In the context of Exercise 2, conclude that $\text{exp}:\mathfrak{g}\to G$ is a homomorphism.
Exercise 4: Use the Hopf-Rinow theorem (or look it up if necessary) to prove that $\text{exp}:\mathfrak{g}\to G$ is surjective in the context of Exercise 2. 
Exercise 5: Prove that $\text{exp}:\mathfrak{g}\to G$ is a covering map in the context of Exercise 2. (Hint: use Exercise 3 and Exercise 4. Recall that $\text{exp}:\mathfrak{g}\to G$ is a local homeomorphism at the origin of $\mathfrak{g}$ because its differential at the origin of $\mathfrak{g}$ is the identity map.)
I hope this helps!
