compact Lie group



Group Theory


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

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


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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

Last revised on August 26, 2019 at 07:48:31. See the history of this page for a list of all contributions to it.