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 ${ℝ}^k\times (S^1{)}^h$.Template:Sfn 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 ${ℂ}^n$ 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.Template:Sfn (In particular, a torus is topologically cyclic.)

See also

• Cartan subgroup

Works cited

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