---
tag: 7JBRGCVJRB
kind: concept
name: Lie group
msc: ["22E15", "22E10"]
specializes: [92700QAB5Y]
sources:
  - url: https://en.wikipedia.org/wiki/Lie_group
same_as:
  wikidata: Q622679
  wikipedia: Lie group
---

## Summary

A Lie group is a group that is also a smooth manifold, in such a way that multiplication and inversion are smooth maps.
It is the natural setting for continuous symmetry: rotations, translations, matrix groups and the symmetry groups of physics are all Lie groups.
Its structure near the identity is captured by a finite-dimensional Lie algebra, which turns many questions about the group into linear algebra.

## Explanations

### L1 human

A Lie group is a family of symmetries that can be varied smoothly, like the rotations of a plane through any angle.
Each element is a point of a smooth space, and you can move from one element to a nearby one by a small change.
The circle of complex numbers of absolute value $1$, with multiplication, is a simple example: it is a smooth curve, and multiplying two points of it just adds their angles.
The invertible $n \times n$ real matrices form another example, a group that is an open subset of the space of all matrices.