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 [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
- ↑ Procesi 2007, Ch. 4. § 2..
- ↑ Knapp 2001, Ch. IV, § 6, Lemma 4.20..
Works cited
- Knapp, Anthony W. (2001). Representation theory of semisimple groups. An overview based on examples. Princeton Landmarks in Mathematics. Princeton University Press. ISBN 0-691-09089-0. https://books.google.com/books?id=QCcW1h835pwC&q=%22Lie+algebra%22.
- Procesi, Claudio (2007). Lie Groups: an approach through invariants and representation. Springer. ISBN 978-0387260402.
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