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

Let A be a compact abelian Lie group with the identity component ${\displaystyle A_{0}}$. If ${\displaystyle A/A_{0}}$ is a cyclic group, then ${\displaystyle A}$ 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.
• Procesi, Claudio (2007). Lie Groups: an approach through invariants and representation. Springer. ISBN 978-0387260402.