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A symplectic principal bundle is a principal bundle $(X,B, G)$ with projection map $q:X\to B$ such that $X$ and $B$ are symplectic manifolds and the right action of $G$ preserves the symplectic structure of $X$.

1)Does this imply that $G$ admit a symplectic structure?

2)Under what conditions the following map preserves the standard Poisson Lie structures ?

$$ T:C^{\infty} (X) \to C^{\infty} (B)\\ T(f)(b)= \int_{q^{-1}(b)} fd\mu $$

Here $\mu$ is the natural normalized Haar measure of the fibers of $X$.

Remark: Considering the natural principal bundle structure $q:S^3 \to S^2$, the above construction provides a Lie bracket structure on $C^{\infty}(S^3)$ with $[f,g]=[T(f),T(g)]\circ q$ where right hand bracket is the natural Lie bracket on $C^{\infty}(S^2)$. (But does it satisfy the Leibniz rule, hence giving a poisson structure on $S^3$?)

With a modification of the same construction one can give an alternative Lie bracket structure on $\chi^{\infty}(S^3)$.

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    $\begingroup$ The answer of 1. is yes see section 1.1 of François Lalonde, Dusa McDuff , Symplectic structures on fiber bundles sciencedirect.com/science/article/pii/S0040938301000209 $\endgroup$
    – user21574
    Commented Oct 23, 2017 at 13:28
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    $\begingroup$ @HassanJolany Thank you for your very helpful comment and very interesting link. $\endgroup$
    – Ali Taghavi
    Commented Oct 24, 2017 at 6:09
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    $\begingroup$ $S^2$ is coadjoint orbit, hence it has Kirillov-Kostant-Souriau symplectic structure , and every symplectic manifold has Poisson structure, see staff.www.ltu.se/~norbert/home_journal/electronic/93art5.pdf . Also $S^3\cong SU(2)$, you can find in p.13 a Poisson structure on $SU(2)$, mat.univie.ac.at/~michor/lie-pois.pdf $\endgroup$
    – user21574
    Commented Oct 25, 2017 at 8:42
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    $\begingroup$ @HassanJolany Thanks for your comment. Regarding the first of your comment, I think there is an easier Poison structure for $S^2$ since it is a symplectic manifold with its volume form. But I am not sure that this is equivalent to the poison structure which you explained. I was not aware of such structure, I will read the linked and references you provided. But I wonder whether the Lie bracket on S^3 which I suggested in the question, gives an alternative poison structure on S^3. $\endgroup$
    – Ali Taghavi
    Commented Oct 25, 2017 at 9:10
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    $\begingroup$ I think(if I remember correcctly) you may try for such Poisson structure $\pi=\sum_{i,j,k,l}X_i\frac{\partial}{\partial X_j}\wedge \frac{\partial}{\partial X_k}\wedge \frac{\partial}{\partial X_l}$ where $X_i$ means rotation in direction $X_i$-axis on $S^3$, (I remember the proof of such Poisson structure took 3 lecture of Ctirad Klimcik) $\endgroup$
    – user21574
    Commented Oct 25, 2017 at 9:25

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