Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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.

share|cite|improve this question
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

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Browse other questions tagged or ask your own question.