Compact Lie group

Summary

A compact Lie group is a Lie group whose underlying manifold is compact. Examples are the circle, the tori, the orthogonal groups $\mathrm{O}(n)$ and the unitary groups $\mathrm{U}(n)$. These groups carry a finite invariant measure, so averaging over the group works as for finite groups. As a result their representation theory is as clean as that of finite groups. Their structure is described by maximal tori, root systems and Weyl groups.