---
tag: E3EYXM7GJN
kind: concept
name: Lie pseudogroup
msc: [58H05, 58A20]
specializes: [3017J7P6JD]
sources:
  - url: https://en.wikipedia.org/wiki/Pseudogroup
---

## Summary

A Lie pseudogroup is a [[pseudogroup|3017J7P6JD]] of local diffeomorphisms of a manifold that is cut out by a system of partial differential equations.
Membership of a local diffeomorphism is decided by its [[jets|1QE0Z0756D]] up to a fixed order at each point.
Isometries, conformal maps, symplectomorphisms, holomorphic maps and volume-preserving maps all form Lie pseudogroups.
Lie pseudogroups were introduced by Lie and studied by Cartan as the infinite-dimensional analogues of Lie groups.

## Explanations

### L1 human

Many geometric structures are defined by the maps that preserve them.
Often those maps are the solutions of differential equations: a map of the plane preserves area when its Jacobian determinant equals $1$.
The set of all local solutions of such an equation forms a pseudogroup, and this is a Lie pseudogroup.
Unlike a Lie group, it can be infinite-dimensional, as in the area-preserving example, where there are as many solutions as there are functions of two variables.