Compact Lie group
A Lie group that is compact as a manifold (equivalently, as a topological group).
Definition. A Lie group is compact if its underlying topological space is compact.
Remarks
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 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.