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 ${\displaystyle \mathbb {R} ^{k}\times (S^{1})^{h}}$. 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. (In particular, a torus is topologically cyclic.)

See also

• Cartan subgroup

Citations

Works cited

1. Poincaré group
2. Non-abelian group
3. Lie algebra
4. Abelian group
5. Simple Lie group
6. Circle group
7. Table of Lie groups
8. Lie group–Lie algebra correspondence
9. Elementary abelian group
10. Lie group
11. Nilpotent group
12. Algebraic group
13. Group of Lie type
14. Nilmanifold
15. Torsion-free abelian group
16. Finite group
17. Locally compact abelian group
18. List of group theory topics
19. General linear group
20. Simple group