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.

Last revised on June 19, 2014 at 20:49:01. See the history of this page for a list of all contributions to it.