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 ${\displaystyle {\mathbb{ℂ}}^n}$ 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.)

• Cartan subgroup

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