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

A connected abelian real Lie group is isomorphic to ${\displaystyle \mathbb {R} ^{k}\times (S^{1})^{h}}$.^[1] In particular, a connected abelian (real) compact Lie group is a torus; i.e., a Lie group isomorphic to ${\displaystyle (S^{1})^{h}}$. 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 ${\displaystyle \mathbb {\mathbb {C} } ^{n}}$ by a lattice.

Let A be a compact abelian Lie group with the identity component ${\displaystyle A_{0}}$. If ${\displaystyle A/A_{0}}$ is a cyclic group, then ${\displaystyle A}$ is topologically cyclic; i.e., has an element that generates a dense subgroup.^[2] (In particular, a torus is topologically cyclic.)