Recently there is some interest in the Poisson geometry of "cluster manifolds", which are varieties associated to cluster algebras. See for example the works of Gekhtman, Shapiro and Vainshtein. In this sprite Fock-Goncharov invented a "higer Teichmüller theory", a crucial point of which can be roughly described as follows:
Let $\mathcal{F}$ be the real flag manifold consisting of flags $\mathbb{RP}^m$, take $N\geq 3$ and let $U\subset\mathcal{F}^N$ be the set of $N$-tuples of flags in general position. So $\mathcal{X}=U/\mbox{PGL}_m(\mathbb{R})$ is a "configurations space of flags". Fock and Goncharov gave explicit construction of many birational maps from the algebraic torus of certain dimension to $\mathcal{X}$, which can be considered as coordinate systems of $\mathcal{X}$. They proved that the transitions between these coordinate systems are provided by certain explicit birational functions call "mutations". They also defined a natural Poisson structure on $\mathcal{X}$ independent of the choice of coordinates.
It would be useful for my purpose if there is an alternative construction of the Poisson structure on $\mathcal{X}$ without using the cluster coordinates, for example, does it comes from a "Poisson reduction"? To make it precise, we can ask:
Is there some natural Poisson structures on $U$ for which the action of the Poisson Lie group $PGL_m(\mathbb{R})$ is a Poisson action? Moreover, does it pass to the quotient to define a Poisson structure on $\mathcal{X}$?
Here I take the standard Poisson-Lie structure on $PGL_m(\mathbb{R})$, i.e., defined by a $r$-matrix whose symmetric part is the Casimir element. Note that the case $m=2$ is already non trivial. In this case $\mathcal{X}$ is the set of $N$-tuple of distinct points in $\mathbb{P}^1$.
