Poisson-Lie T-Duality and non trivial monodromies

We describe a general framework for studying duality between different phase spaces which share the same symmetry group $\mathrm{H}$. Solutions corresponding to collective dynamics become dual in the sense that they are generated by the same curve in $\mathrm{H}$. Explicit examples of phase spaces which are dual with respect to a common non trivial coadjoint orbit $\mathcal{O}_{c,0}(\mathbf{\alpha},1) \subset\mathfrak{h}^{\ast}$ are constructed on the cotangent bundles of the factors of a double Lie group $\mathrm{H}=\mathrm{N}\Join\mathrm{N}^{\ast}$. In the case $\mathrm{H}=LD$, the loop group of a Drinfeld double Lie group $D$, a hamiltonian description of Poisson-Lie T-duality for non trivial monodromies and its relation with non trivial coadjoint orbits is obtained.