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I have some background in Poisson geometry, in particular Poisson-Lie groups and I would like to initiate myself to dressing transformations.

If $(G,\pi)$ is a Poisson-Lie group, then its Lie algebra $\mathcal{G}$ carries a bialgebra structure. Its dual $\mathcal{G}^\star$ is also a Lie bialgebra which can be integrated to a 'dual' Poisson-Lie group $G^*$.

1) How to define the action of $G^*$ on $G$, to identify the orbits of the action, with symplectic leafs of $(G,\pi)$?

2) Are left dressing vector fields hamiltonians?

Thanks for your help.

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Please do not crosspost:… Pick a site you believe to be appropriate and start there. – Qiaochu Yuan Aug 10 '11 at 13:48
Amine: the two sites are not clones. However here in MO it is considered bad etiquette to crosspost as the aim of both sites is very different: MO is for research-level mathematics. So if you post the question to both sites, you give the impression that the question is not sufficiently high level for MO. You can always ask in "meta" whether a question is deemed suitable for MO. In my opinion, the present question is borderline. Asking for tutorials on topics is frowned upon, but you could ask instead for references on this subject. – José Figueroa-O'Farrill Aug 10 '11 at 14:54
It's considered bad etiquette to crosspost in general because of the possibility of duplication of effort: if multiple people spend time giving you answers on different sites where they can't see that other people have done the same, they may feel like they have wasted their time. – Qiaochu Yuan Aug 10 '11 at 16:52
Why is mathoverflow becoming a place with so many rules and etiquettes? I mean I understand that a lot of mathematicians are really like systems. But can't we chill out a little bit? Do we really want MO to turn into my grandma's dinner table? Like a guy just posted a question on two sites--- doesn't strike me as a situation that needs a rule or policy. There are lots of questions that people have a feeling are borderline research level and in that case posting on both sites seems ok. Would people feel better if someone put "ALSO POSTED ON STACK" with a link? – Daniel Pomerleano Aug 10 '11 at 18:29
There is lot of material which gets lost if not organized properly. Explaining mathematics is time-consuming and if it is wasted into unordered material on several sites it is not serving the purpose. While I dislike rules in general, it is good to have strong organizing principles here. – Zoran Skoda Aug 24 '11 at 16:21

If memory serves me, I think that your questions are answered in the paper Dressing symmetries by Babelon and Bernard.

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Thanks José, for the reference. The paper by Lu and Weinstein is also a good reference – amine Aug 10 '11 at 16:39

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