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 conider 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 manifolds.
See also
Last revised on July 17, 2021 at 16:50:26. See the history of this page for a list of all contributions to it.