TITLE: when can we lift an action of Lie algebra?
QUESTION [5 upvotes]: Suppose $G$ is a Lie group, $\mathfrak{g}$ its Lie algebra, if we have a smooth representation $(\pi,V)$, then it induces an action of $\mathfrak{g}$ on $V$. Now conversely, if we have a nice (with properties you may assume) action of $\mathfrak{g}$ on $V$, can we say such action arises from some unique smooth action of $G$?
Here we may assume $G$ to be simply connected if needed. Thank you.

REPLY [7 votes]: Let $\pi$ represent a finite dimensional real Lie algebra $\mathfrak g$ on a Hilbert space
$\mathcal H$ by skew-adjoint operators.  Then $\pi$ integrates to the connected simply connected Lie group $G$ with Lie algebra $\mathfrak g$ if, and only if, the elements of
$\pi(\mathfrak g)$ have a common invariant dense domain.  This is an old result of Moshe Flato, Daniel Sternheimer and others.  My apologies to the mathematical physicists whose names I have omitted.