---
tag: 9TFC6ADC1C
kind: concept
name: Compact Lie group
msc: ["22E15", 22C05]
specializes: [7JBRGCVJRB, EVHCE5VR6Y]
sources:
  - url: https://en.wikipedia.org/wiki/Compact_group
---

## Summary

A compact Lie group is a [[Lie group|7JBRGCVJRB]] 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|3JVGZVVTB0]], root systems and Weyl groups.

## Explanations

### L1 human

A compact Lie group is a group of symmetries that is also a smooth, closed and bounded shape.
The rotations of the plane form a circle, and the circle is the simplest example.
The rotations of space form the group $\mathrm{SO}(3)$, a compact space of dimension $3$.
Because the group is bounded, one can take the average of a function over all its elements.
This averaging trick makes many questions about these groups as easy as for finite groups.