Compact Lie group

Definition. A Lie group $G$ is compact if its underlying topological space is compact.

Core structural features.

• Compactness implies the existence of a bi-invariant Haar measure, enabling averaging arguments throughout geometry and representation theory.
• Every compact Lie group admits a bi-invariant Riemannian metric ; see bi-invariant metrics on compact Lie groups .
• Maximal tori control much of the structure: every element lies in some maximal torus, and conjugacy classes intersect a fixed maximal torus in Weyl-group orbits (see the maximal torus theorem and the Weyl group ).

Representation-theoretic context. Finite-dimensional continuous representations of compact Lie groups are completely reducible , and the regular representation on $L^2(G)$ decomposes discretely (compare the Peter–Weyl theorem ).

For a global decomposition of compact connected groups into torus and semisimple parts, see structure of compact Lie groups .