Lie algebra action

Definition

Given a Lie algebra $\mathfrak{g}$, and a smooth manifold $M$, a Lie algebra action of $\mathfrak{g}$ on $M$ is a Lie algebra morphism

$$\begin{array}{rl}\alpha :\mathfrak{g}& ⟶\mathfrak{X}(M)\\ \xi & ⟼{\xi }_M\end{array}$$

where $\mathfrak{X}(M)$ denotes the space of all vector fields on $M$.

From Lie groups

Every Lie group action of $G$ on $M$ induces a Lie algebra action defined by

$(\alpha (\xi ){)}_x=\frac{d}{dt}{|}_{t=0}\exp (-t\xi )\cdot x\forall \xi \in \mathfrak{g}$

these are the fundamental vector fields on a manifold.

If an infinitesimal action is induced by an action of $G$ on $M$ then the vector fields are complete. So, not every Lie algebra action comes from a Lie group action, but we do have a connection between some:

Theorem (Lie-Palais): Given a manifold $M$ and a 1-connected Lie group $G$ with Lie algebra $\mathfrak{g}$, given the equation above for induced Lie algebra actions, we have a 1-1 correspondence

\alpha: \mathfrak{g} \to \mathfrak{X}(M)\end{Bmatrix} \leftrightarrow \begin{Bmatrix} \text{Lie group actions} \\ \mathcal{A}: G \to \text{Diff}(M)\end{Bmatrix}$$