Poisson–Lie TT-duality and non-trivial monodromies

We describe a general framework for studying duality among different phase spaces which share the same symmetry group H. Solutions corresponding to collective dynamics become dual in the sense that they are generated by the same curve in H. Explicit examples of phase spaces which are dual with respect to a common non-trivial coadjoint orbit Oc,0(α,1)⊂h∗ are constructed on the cotangent bundles of the factors of a double Lie group H=N⋈N∗. In the case H=LD, the loop group of a Drinfeld double Lie group DD, we built up a hamiltonian description of Poisson–Lie TT-duality for non-trivial monodromies and its relation with non-trivial coadjoint orbits is obtained.