TITLE: The description of abelian Lie groups
QUESTION [14 upvotes]: Question. Classify all abelian connected Lie groups.

There is a problem in my problem sheet which asks me to describe all abelian connected Lie groups (moreover this is the first problem so it should be rather easy). I don't understand how this description should look. They mean description up to an isomorphism (of Lie groups), don't they?
I can list some abelian connected (real) Lie groups:

$\mathbb{R}^n$,
$\mathbb{C}_{\ne 0}$ (as a real group under multiplication),
$S^1$ (i.e. $\big\{z\in\mathbb{C}: |z|=1\big\}$),
also some different finite products.

Could you help me to classify them?

REPLY [4 votes]: If one knows the fundamental fact that a simply connected connected Lie group is completely determined by its Lie algebra, one can proceed as follows:
Let $G$ be an connected abelian Lie group of dimension $n$. The universal covering space $\tilde G$ is also a connected abelian Lie group of dimension $n$ which is, of course, simply connected; in particular, the Lie algebra of $\tilde G$ is of dimension $n$ and abelian. Since $\mathbb R^n$ is also a connected simply-connected Lie group with abelian Lie algebra of dimension $n$, we must have $\tilde G\cong\mathbb R^n$ as Lie groups. The point here is that there is exactly one^abelian Lie algebra of each dimension.
Now, the covering map $p:\tilde G\to G$ is a group homomorphism, so that $\ker p$ is a discrete subgroup of $\tilde G\cong\mathbb R^n$, so it is isomorphic to the subgroup generated by a linearly independent subset of $\mathbb R^n$. Using this it is easy to conclude what we want.