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Suppose $G$ is a Lie group, $\mathcal{g}$ \mathfrak{g}$ its Lie algebra, if we have a smooth representation $(\pi,V)$, then it induces a an action of $\mathcal{g}$ \mathfrak{g}$ on $V$. Now conversely, if we have a nice (with properties you may assume) action of $\mathcal{g}$ \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. |
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when can we lift an action of Lie algebra?Suppose $G$ is a Lie group, $\mathcal{g}$ its Lie algebra, if we have a smooth representation $(\pi,V)$, then it induces a action of $\mathcal{g}$ on $V$. Now conversely, if we have a nice action of $\mathcal{g}$ 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.
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