nLab
compact Lie group

Context

Group Theory

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 TT of a compact Lie group is called a 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”).

Revised on February 2, 2012 11:51:56 by Urs Schreiber (82.169.65.155)