Contents

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 $G$ 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.

## 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 $K$ a compact Lie group there is a unique connected complex Lie group $G$ such that

1. the Lie algebra of $G$ is the complexification of the Lie algebra of $K$:

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

This $G$ is called the complexification of $K$.

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