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 ℝk×(S1)h.[1] In particular, a connected abelian (real) compact Lie group is a torus; i.e., a Lie group isomorphic to (S1)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 ℂn by a lattice.

Let A be a compact abelian Lie group with the identity component A0. If A/A0 is a cyclic group, then A is topologically cyclic; i.e., has an element that generates a dense subgroup.[2] (In particular, a torus is topologically cyclic.)

## See also

* Cartan subgroup

## Citations

1. ↑ Procesi 2007, Ch. 4. § 2..
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="Lie+algebra".
* Procesi, Claudio (2007). Lie Groups: an approach through invariants and representation. Springer. ISBN 978-0387260402.

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