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In geometry, an abelian Lie group is a Lie group that is an abelian group.
A connected abelian real Lie group is isomorphic to R k × h {\displaystyle \mathbb {R} ^{k}\times ^{h}}. In particular, a connected abelian compact Lie group is a torus; i.e., a Lie group isomorphic to h {\displaystyle ^{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 C n {\displaystyle \mathbb {\mathbb {C} } ^{n}} by a lattice.
Let A be a compact abelian Lie group with the identity component A 0 {\displaystyle A_{0}}. If A / A 0 {\displaystyle A/A_{0}} is a cyclic group, then A {\displaystyle A} is topologically cyclic; i.e., has an element that generates a dense subgroup.