Let $M$ be a smooth manifold. A local flow is a (suitable restriction of) map $\theta: \mathbb R \times M \to M$, such that $\theta(t,\theta(s,p)) = \theta(t+s,p)$ whenever both sides are defined, and $\theta(0,p)=p$. For a local flow $\theta$, we can associate the infinitesimal generator $V\in \mathfrak X(M)$, defined as
$$V_p = \left. \frac{d}{dt} \right|_{t=0} \theta(t,p).$$
Fundamental theorem of flow guarantees the one-to-one correspondence between a maximal flow $\theta$ and its infinitesimal generator $V$.
Question: Can we generalize this notion by replacing $\mathbb R$ to a Lie group $G$?
In this case, we should define $\theta: G \times M \to M$, with the group-like axioms as above. Also, infinitesimal generator could be a Lie algebra $\mathfrak g$-valued vector field, i.e., for each $X\in \mathfrak g$, $V^X \in \mathfrak X(M)$ is defined as
$$V_p^X = \left. \frac{d}{dt} \right|_{t=0} \theta(e^{tX},p).$$
I expect $V^{[X,Y]} = [V^X, V^Y]$. Furthermore, generalizing the fundamental theorem of flow, I also expect that a maximal $G$-valued flow $\theta$ is uniquely associated with the infinitesimal generator $V: \mathfrak g \to \mathfrak X(M)$, which is a Lie algebra homomorphism.
Is my conjecture correct? Is there any references or concepts that deals on this matter?
Best Answer
You are essentially correct, though "Lie group-valued flow" and "Lie algebra-valued vector field" are non-standard (and, in my opinion, anti-idiomatic) terminology. The former is known as a Lie group action, and it is typically defined as a group action $\theta:G\times M\to M$ that is also a smooth map. The latter is often called the "infinitesimal generator" of the action, which as you say is a Lie algebra homomorphism $\hat\theta:\mathfrak{g}\to\mathfrak{X}M$ corresponding to $\theta$. One standard reference for the topic is Lee's Introuction to Smooth Manifolds.