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

TL;DR

This paper explores Hamiltonian group actions on the cotangent bundle of loop groups, revealing new integrable systems and hierarchies that generalize known dispersive water wave and mKdV systems.

Contribution

It introduces novel Hamiltonian actions of loop groups and differential operators, and constructs new integrable hierarchies on the cotangent bundle of loop groups.

Findings

Derived infinite commuting families of Hamiltonian flows.

Established factorization of moment maps through loop group actions.

Connected the hierarchies to known systems like WZW and mKdV.

Abstract

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.