Abelian Lie group explained
In geometry, an abelian Lie group is a Lie group that is an abelian group.
A connected abelian real Lie group is isomorphic to
. In particular, a connected abelian (real) compact Lie group is a
torus; i.e., a Lie group isomorphic to
. 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
by a lattice.
Let A be a compact abelian Lie group with the identity component
. If
is a
cyclic group, then
is topologically cyclic; i.e., has an element that generates a dense subgroup. (In particular, a torus is topologically cyclic.)
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