---
tag: P43WKB4372
kind: statement
role: theorem
name: Lie's theorems on transformation groups
msc: ["22E60", 57S20, 22F05]
uses: [7JBRGCVJRB, 96B4E2GPKJ, 0299VCW1TH, 68FGQEBD6J, H3QX1ZFC8K, P6QNEK44RX]
about: [96B4E2GPKJ]
sources:
  - url: https://en.wikipedia.org/wiki/Lie%27s_third_theorem
---

## Statement

Let $M$ be a smooth manifold.
Let $G$ be a [[Lie group|7JBRGCVJRB]] with Lie algebra $\mathfrak{g}$ and [[exponential map|H3QX1ZFC8K]] $\exp$.
Let $G$ have a smooth left [[action|96B4E2GPKJ]] on $M$.
The *fundamental vector field* of $X \in \mathfrak{g}$ is the [[vector field|0299VCW1TH]] $X_M$ on $M$ with $X_M(p) = \frac{d}{dt}\big|_{t=0} \exp(tX) \cdot p$.

1. The map $X \mapsto X_M$ is linear and satisfies $[X_M, Y_M] = -[X, Y]_M$, with the [[Lie bracket of vector fields|68FGQEBD6J]] on the left.
2. Each finite-dimensional Lie algebra $L$ of complete vector fields on $M$ is the set of fundamental vector fields of some simply connected Lie group action.
3. Every finite-dimensional real Lie algebra is the Lie algebra of a Lie group.

## Remarks

By part 1, the fundamental vector fields form a Lie algebra of vector fields of dimension at most $\dim G$.
Part 1 follows by differentiating the identity $\exp(sX)\exp(tY)\exp(-sX) = \exp(t\operatorname{Ad}(\exp sX)Y)$ along the action; the minus sign comes from using a left action together with the bracket of left-invariant vector fields.
Lie's original theorems were local: they concerned local groups of transformations and the vector fields that generate them.
The global form of part 2 is due to Palais.
Completeness of the vector fields is needed, since an incomplete field such as $x^2 \frac{d}{dx}$ on $\R$ generates no global action of $\R$.
Part 3 is [[Lie's third theorem|P6QNEK44RX]], which in the modern form is proved by Ado's theorem or by Cartan's method.
Example: the rotation action of $SO(3)$ on $\R^3$ has as fundamental vector fields the three infinitesimal rotations $y\partial_z - z\partial_y$, $z\partial_x - x\partial_z$, $x\partial_y - y\partial_x$ and their linear combinations, a Lie algebra isomorphic to $\mathfrak{so}(3)$.

## Explanations

### L1 human

A group of transformations of a space, such as the rotations of space, can be studied through its infinitesimal transformations, the velocity fields of motions that start at the identity.
Lie's theorems say that these velocity fields form a finite-dimensional space closed under the bracket of vector fields.
Conversely, such a space of vector fields generates a group of transformations.
So the study of continuous symmetry reduces to the study of finitely many vector fields and their brackets.

### L2 human

These results are the origin of Lie theory: Lie used them to study symmetries of differential equations.
Part 1 turns a group action into a Lie algebra homomorphism into vector fields, up to a sign, which is the basic tool for computing symmetries.
Part 2 integrates infinitesimal symmetries to actual group actions; without completeness one only gets a local action.
Part 3 shows that no Lie algebra is missed, so the classification of finite-dimensional Lie algebras gives all the local types of transformation groups.