Английская Википедия:Abelian Lie group

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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 <math>\mathbb{R}^k \times (S^1)^h</math>.Шаблон:Sfn In particular, a connected abelian (real) compact Lie group is a torus; i.e., a Lie group isomorphic to <math>(S^1)^h</math>. 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 <math>\mathbb{\Complex}^n</math> by a lattice.

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

See also

• Cartan subgroup

Works cited

• Шаблон:Cite book
• Шаблон:Cite book

Шаблон:Geometry-stub