title: Abelian Lie group
text: # 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 $\mathbb {\mathbb {C} } ^{n}$ 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.)
  
  