nLab
compact Lie group

Context

Group Theory

Topology

topology (point-set topology, point-free topology)

see also differential topology, algebraic topology, functional analysis and topological homotopy theory

Introduction

Basic concepts

Universal constructions

Extra stuff, structure, properties

Examples

Basic statements

Theorems

Analysis Theorems

topological homotopy theory

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

See also at relation between compact Lie groups and reductive algebraic groups

Revised on January 10, 2017 11:02:13 by Urs Schreiber (46.183.103.8)