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?
Best Answer
Nate's hint does the trick. Let $G$ be an abelian connected Lie group with Lie algebra $\mathfrak g$. The exponential map $\exp:\mathfrak g\to G$ is actually a homomorphism of abelian groups. The image is open in $G$, so $\exp$ is surjective because $G$ is connected. The fact that $\mathfrak g\to G$ is a local homeomorphism means that $\ker(\exp)$ is a discrete subgroup of $\mathfrak g$. It is known that such groups are of the form $\Lambda=\mathbf Z x_1+\cdots + \mathbf Z x_n$, for $x_1,\dots,x_n\in \mathfrak g$ linearly independent over $\mathbf R$. We can extend $x_1,\dots,x_n$ to a basis of $\mathfrak g$ to see that $$ G \simeq (S^1)^r \times \mathbf R^s $$ In other words, every connected abelian Lie group is a product of affine space and a torus.
For example, $\mathbf C_{\ne 0} = \mathbf C^\times$ is the product $\mathbf R\times S^1$, via $(r,\theta)\mapsto r e^{i\theta}$.