In geometry, an abelian Lie group is a
Lie group
In mathematics, a Lie group (pronounced ) is a group (mathematics), group that is also a differentiable manifold, such that group multiplication and taking inverses are both differentiable.
A manifold is a space that locally resembles Eucli ...
that is an
abelian group
In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is commu ...
.
A connected abelian real Lie group is isomorphic to
. In particular, a connected abelian (real)
compact Lie group
In mathematics, a compact (topological) group is a topological group whose topology realizes it as a compact space, compact topological space (when an element of the group is operated on, the result is also within the group). Compact groups are ...
is a
torus
In geometry, a torus (: tori or toruses) is a surface of revolution generated by revolving a circle in three-dimensional space one full revolution about an axis that is coplanarity, coplanar with the circle. The main types of toruses inclu ...
; i.e., a Lie group isomorphic to
. A connected
complex Lie group
In geometry, a complex Lie group is a Lie group over the complex numbers; i.e., it is a complex-analytic manifold that is also a group in such a way G \times G \to G, (x, y) \mapsto x y^ is holomorphic. Basic examples are \operatorname_n(\math ...
that is a compact group is abelian and a connected compact complex Lie group is a
complex torus
In mathematics, a complex torus is a particular kind of complex manifold ''M'' whose underlying smooth manifold is a torus in the usual sense (i.e. the cartesian product of some number ''N'' circles). Here ''N'' must be the even number 2''n'', whe ...
; 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
In abstract algebra, a cyclic group or monogenous group is a Group (mathematics), group, denoted C_n (also frequently \Z_n or Z_n, not to be confused with the commutative ring of P-adic number, -adic numbers), that is Generating set of a group, ge ...
, then
is topologically cyclic; i.e., has an element that generates a dense subgroup. (In particular, a torus is topologically cyclic.)
See also
*
Cartan subgroup
In the theory of algebraic groups, a Cartan subgroup of a connected linear algebraic group G over a (not necessarily algebraically closed) field k is the centralizer of a maximal torus. Cartan subgroups are smooth (equivalently reduced), connec ...
Citations
Works cited
*
*
Abelian group theory
Geometry
Lie groups
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