  Various Hamiltonian actions of loop groups $\wt G$ and of the algebra
$\text{diff}_1$ of first order differential operators in one variable are
defined on the cotangent bundle $T^*\wt G$ of a Loop Group. The moment maps
generating the $\text{diff}_1$ actions are shown to factorize through those
generating the loop group actions, thereby defining commuting diagrams of
Poisson maps to the duals of the corresponding centrally extended algebras. The
maps are then used to derive a number of infinite commuting families of
Hamiltonian flows that are nonabelian generalizations of the dispersive water
wave hierarchies. As a further application, sets of pairs of generators of the
nonabelian mKdV hierarchies are shown to give a commuting hierarchy on $T^*\wt
G$ that contain the WZW system as its first element.
