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

Let A be a compact abelian Lie group with the identity component [math]\displaystyle{ A_0 }[/math]. If [math]\displaystyle{ A/A_0 }[/math] is a cyclic group, then [math]\displaystyle{ A }[/math] is topologically cyclic; i.e., has an element that generates a dense subgroup.[2] (In particular, a torus is topologically cyclic.)

See also

Citations

  1. Procesi 2007, Ch. 4. § 2..
  2. Knapp 2001, Ch. IV, § 6, Lemma 4.20..

Works cited