---
tag: SWW2993CVJ
kind: concept
name: Universal covering group
msc: ["22E15", 57M10]
specializes: [Y1YPE6PM52]
sources:
  - url: https://en.wikipedia.org/wiki/Covering_group
---

## Summary

The universal covering group of a connected [[Lie group|7JBRGCVJRB]] $G$ is its [[universal cover|Y1YPE6PM52]] $\tilde G$ with a Lie group structure.
This structure makes the covering map $\tilde G \to G$ a homomorphism.
It is simply connected, and $G$ is the quotient of $\tilde G$ by a discrete central subgroup isomorphic to the [[fundamental group|S1XK3HXHFX]] of $G$.
All connected Lie groups with a given Lie algebra are quotients of the same universal covering group.

## Explanations

### L1 human

Some Lie groups have loops that cannot be shrunk, like the circle.
The universal covering group unwinds all such loops while keeping a group structure.
For the circle of unit complex numbers, it is the real line, mapped onto the circle by $t \mapsto e^{2\pi i t}$; adding real numbers matches multiplying points of the circle.
For the rotations of space $SO(3)$, it is the group $SU(2)$, and each rotation is covered by exactly two elements.