# nLab compact Lie group

group theory

### Cohomology and Extensions

#### Topology

topology

algebraic topology

# Contents

## Definition

A real Lie group is compact if its underlying topological group is a compact topological group.

## Properties

Compact Lie groups have a very well understood structure theory.

All maximal tori of a compact Lie group are conjugate by inner automorphisms. The dimension of a maximal torus $T$ of a compact Lie group is called a rank of $G$. The normalizer $N(T)$ of a maximal torus $T$ determines $G$. The Weyl group $W(G)=W(G,T)$ of $G$ with respect to a choice of a maximal torus $T$ is the group of automorphisms of $T$ which are restrictions of inner automorphisms of $G$. The maximal torus is of finite index in its normalizer; the quotient $N(T)/T$ is isomorphic to $W(G)$. The cardinality of $W(G)$ for a compact connected $G$, equals the Euler characteristic of the homogeneous space $G/T$ (“flag variety”).

Revised on June 19, 2014 21:03:27 by Urs Schreiber (82.136.246.44)