---
tag: 96B4E2GPKJ
kind: concept
name: Lie group action
msc: [57S20, 22F05, "22E15"]
specializes: [0WCY42EV9V]
sources:
  - url: https://en.wikipedia.org/wiki/Lie_group_action
same_as:
  wikidata: Q17098482
  wikipedia: Lie group action
---

## Summary

A Lie group action is a [[group action|0WCY42EV9V]] of a [[Lie group|7JBRGCVJRB]] on a [[smooth manifold|F45VQ47GPV]] such that the action map is smooth.
It is the precise form of continuous symmetry of a geometric space.
Orbits of such actions are immersed submanifolds, and stabilisers are closed subgroups.
Differentiating the action turns elements of the Lie algebra into vector fields on the manifold.

## Explanations

### L1 human

A Lie group action lets each element of a Lie group move the points of a smooth space.
The motion depends smoothly on both the element and the point.
Rotations of the plane about the origin are an example: rotating by $\theta$ moves each point smoothly, and rotating twice adds the angles.
The orbits are the circles centred at the origin, together with the origin itself, which is fixed by every rotation.