Hello!

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.