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Let $G$ be a Lie group, there is a natural inclusion of $G$ in Diff($G$), by left traslation.

Is true that there is a splitting Diff($G$)$\cong$ $G$ $\times$ Diff($G,e$)? Where Diff($G,e$) is the subgroup of Diff($G$) that fixes the identity element of the group.

This question comes from the example 1 of https://en.wikipedia.org/wiki/Diffeomorphism#Diffeomorphism_group

Andrés
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  • Is it a splitting as topological groups, or as spaces? For the latter, I think you can just look at the image of $e$ under a diffeomorphism as the splitting map. I don't think that is a homomorphism though... – Elle Najt May 10 '17 at 22:22
  • Splitting in which category? – JJR May 10 '17 at 22:29
  • @JJR First of all, as a spaces. – Andrés May 10 '17 at 23:02
  • kind of looks like ${\rm Diff}(G) \to G\times {\rm Diff}(G,e)$, $f\mapsto (g,g^{-1}f)$ where $g=f(e)$ – JJR May 10 '17 at 23:15

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