Symplectic Geometries on $T^*\widetilde{G}$, Hamiltonian Group Actions and Integrable Systems

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.