  We carry out a Hamiltonian analysis of Poisson-Lie T-duality based on the
loop geometry of the underlying phases spaces of the dual sigma and WZW models.
Duality is fully characterized by the existence of equivariant momentum maps on
the phase spaces such that the reduced phase space of the WZW model and a pure
central extension coadjoint orbit work as a bridge linking both the sigma
models. These momentum maps are associated to Hamiltonian actions of the loop
group of the Drinfeld double on both spaces and the duality transformations are
explicitly constructed in terms of these actions. Compatible dynamics arise in
a general collective form and the resulting Hamiltonian description encodes all
known aspects of this duality and its generalizations.
