compact Lie group


Group Theory


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

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


Basic concepts

Universal constructions

Extra stuff, structure, properties


Basic statements


Analysis Theorems

topological homotopy theory



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


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 (