Classical groups
Finite groups
Group schemes
Topological groups
Lie groups
Super-Lie groups
Higher groups
Cohomology and Extensions
Related concepts
geometry, complex numbers, complex line
$dim = 1$: Riemann surface, super Riemann surface
A complex Lie group is a Lie group that is a group object not just internal to smooth manifolds but in fact to complex manifolds. Hence it is a complex manifold $G$ equipped with a group structure such that both the multiplication map $G \times G \to G$ as well as the inverse map $G \to G$ are holomorphic functions.
One can also consider groups in almost complex manifolds. But every almost complex Lie group is automatically also a complex Lie group.
For $K$ a compact Lie group there is a unique connected complex Lie group $G$ such that
the Lie algebra of $G$ is the complexification of the Lie algebra of $K$:
$K$ is a maximal compact subgroup of $G$.
This $G$ is called the complexification of $K$.
(coset spaces of complex Lie groups are Oka manifolds)
Every complex Lie group and every coset space (homogeneous space) of complex Lie groups is an Oka manifold.
See also
Last revised on May 21, 2024 at 04:35:12. See the history of this page for a list of all contributions to it.