nLab
compact Lie group
Context
Group Theory
group theory

Classical groups Finite groups Group schemes Topological groups Lie groups Super-Lie groups Higher groups Cohomology and Extensions
Topology
topology (point-set topology , point-free topology )

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

Introduction

Basic concepts

open subset , closed subset , neighbourhood

topological space , locale

base for the topology , neighbourhood base

finer/coarser topology

closure , interior , boundary

separation , sobriety

continuous function , homeomorphism

uniformly continuous function

embedding

open map , closed map

sequence , net , sub-net , filter

convergence

category Top

Universal constructions

Extra stuff, structure, properties

nice topological space

metric space , metric topology , metrisable space

Kolmogorov space , Hausdorff space , regular space , normal space

sober space

compact space , proper map

sequentially compact , countably compact , locally compact , sigma-compact , paracompact , countably paracompact , strongly compact

compactly generated space

second-countable space , first-countable space

contractible space , locally contractible space

connected space , locally connected space

simply-connected space , locally simply-connected space

cell complex , CW-complex

pointed space

topological vector space , Banach space , Hilbert space

topological group

topological vector bundle , topological K-theory

topological manifold

Examples

empty space , point space

discrete space , codiscrete space

Sierpinski space

order topology , specialization topology , Scott topology

Euclidean space

cylinder , cone

sphere , ball

circle , torus , annulus , Moebius strip

polytope , polyhedron

projective space (real , complex )

classifying space

configuration space

path , loop

mapping spaces : compact-open topology , topology of uniform convergence

Zariski topology

Cantor space , Mandelbrot space

Peano curve

line with two origins , long line , Sorgenfrey line

K-topology , Dowker space

Warsaw circle , Hawaiian earring space

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 automorphism s. The dimension of a maximal torus $T$ of a compact Lie group is called the 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 ”).

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

Last revised on January 10, 2017 at 11:02:13.
See the history of this page for a list of all contributions to it.