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I would like to ask, how to deduce a Lie group action, from infinitesimal action of its Lie algebra (the so called Lie-Palais theorem). More precisely, given a differential manifold $M$ and a Lie group $G$ with Lie algebra $\mathcal{G}$. Suppose we have a Lie algebra homomorphism $$\rho : \mathcal{G}\rightarrow\mathfrak{X}(M)$$ (where $\mathfrak{X}(M)$ denotes the space of vector fields of $M$).

How to deduce, from $\rho$, a smooth action $$G\times M\rightarrow M\ ?$$ In particular, is there a nice and elementary proof of the Lie-Palais theorem?

Thanks for you help.

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I guess if one can do it for one-parametric families, then one can use various group decompositions to write groups element as product of 1-parametric elements, locally. –  Sasha Aug 10 '11 at 20:17
    
Thanks Sasha, for your comment! I am interested in the third Lie theorem using Levi-Malcev decomposition of a Lie algebra $\mathcal{G}$ as a semi-direct sum of a semi-simple sub algebra $S$ and its radical $\mathfrak{rad}$ (maximal solvable ideal). In particular if $$\mathcal{G}=SL(2)\ltimes H_3$$ where $H_3$ is isomorphic to the Heisenberg Lie algebra, then what is the connected and simply connected Lie group associated to $\mathcal{G}$? –  amine Aug 11 '11 at 10:37

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