---
tag: 7JA7WH92K0
kind: concept
name: Representation of a Lie group
msc: ["22E45", "22E47", "22E15"]
specializes: [WYGRXCG2RN]
sources:
  - url: https://en.wikipedia.org/wiki/Representation_of_a_Lie_group
same_as:
  wikidata: Q763735
  wikipedia: Representation of a Lie group
---

## Summary

A representation of a [[Lie group|7JBRGCVJRB]] $G$ is a continuous homomorphism from $G$ to the group of invertible linear maps of a vector space.
It lets the group act on vectors by linear maps, so that group theory can be studied with linear algebra.
Each finite-dimensional representation has a derivative, a representation of the Lie algebra, which often determines it.
Representations of Lie groups are central in harmonic analysis, number theory and quantum physics.

## Explanations

### L1 human

A representation turns each element of a group into a matrix, so that composing elements becomes multiplying matrices.
The rotation group of space acts on vectors of $\R^3$ by rotation matrices, and this is a representation.
It also acts on functions on the sphere, by rotating the function's graph, and this is a larger representation.
The two must be continuous: rotations by nearby angles give nearby matrices.
Breaking a representation into irreducible pieces is like the Fourier series of a function.