nLab
complex Lie group

Contents

Context

Group Theory

Complex geometry

Contents

Idea

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 GG equipped with a group structure such that both the multiplication map G×GGG \times G \to G as well as the inverse map GGG \to G are holomorphic functions.

Properties

Relation to almost complex structure

One can also conider groups in almost complex manifolds. But every almost complex Lie group is automatically also a complex Lie group.

Complexification of compact Lie groups

For KK a compact Lie group there is a unique connected complex Lie group GG such that

  1. the Lie algebra of GG is the complexification of the Lie algebra of KK:

    Lie(G)Lie(K) , Lie(G) \simeq Lie(K) \otimes_{\mathbb{R}} \mathbb{C} \,,
  2. KK is a maximal compact subgroup of GG.

This GG is called the complexification of KK.

Oka property

Proposition

(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.

(review in Forstnerič & Lárusson 11, p. 9, Forstnerič 2013, Thm. 2.6)

References

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.