  In this paper we introduce a notion of quantum Hamiltonian (co)action of Hopf
algebras endowed with Drinfel'd twist structure (resp., 2-cocycles). First, we
define a classical Hamiltonian action in the setting of Poisson Lie groups
compatible with the 2-cocycle stucture and we discuss a concrete example. This
allows us to construct, out of the classical momentum map, a quantum momentum
map in the setting of Hopf coactions and to quantize it by using Drinfel'd
approach.
