Abelian Lie group

In geometry, an abelian Lie group is a Lie group that is an abelian group.

A connected abelian real Lie group is isomorphic to $$\mathbb{R}^k \times (S^1)^h$$. In particular, a connected abelian (real) compact Lie group is a torus; i.e., a Lie group isomorphic to $$(S^1)^h$$. A connected complex Lie group that is a compact group is abelian and a connected compact complex Lie group is a complex torus; i.e., a quotient of $$\mathbb{\Complex}^n$$ by a lattice.

Let A be a compact abelian Lie group with the identity component $$A_0$$. If $$A/A_0$$ is a cyclic group, then $$A$$ is topologically cyclic; i.e., has an element that generates a dense subgroup. (In particular, a torus is topologically cyclic.)