Symplectic geometries on \(T^*\widetilde {G}\), Hamiltonian group actions and integrable systems. (English) Zbl 0829.53027

For many integrable systems the underlying phase space may be taken as the cotangent bundle \(T^* LG\) of a loop group \(LG\). The authors derive, using the moment maps generating certain infinitesimal Hamiltonian actions of the algebra \(\text{diff }_1\) of first-order differential operators in one variable on \(T^* LG\), some new, nonstandard Lax equations determining infinite commuting families of flows. Integrable systems associated with the dispersive water wave hierarchy are constructed. They provide the systematic study of the various Hamiltonian actions of the group \(\overline{G}\) of smooth maps \(g : R \to G\) \((G\) is a Lie group) on \(T^* \overline{G}\) and on the corresponding spaces \((\overline {g} \oplus \overline {g})^{\wedge *}\), \((\overline {g} + \overline {g}_A)^{\wedge *}\). The associated moments maps are derived and shown to form commuting triplets of Poisson maps into \((\overline {g} \oplus \overline {g})^{\wedge *}\) (or \((\overline {g} + \overline {g}_A)^{\wedge *})\).


53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)
37J35 Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
20N05 Loops, quasigroups
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