Loop Algebra Moment Maps and Hamiltonian Models for the Painleve Transcendants

Abstract

The isomonodromic deformations underlying the Painlev\'e transcendants are interpreted as nonautonomous Hamiltonian systems in the dual $\gR^*$ of a loop algebra $\tilde\grg$ in the classical $R$-matrix framework. It is shown how canonical coordinates on symplectic vector spaces of dimensions four or six parametrize certain rational coadjoint orbits in $\gR^*$ via a moment map embedding. The Hamiltonians underlying the Painlev\'e transcendants are obtained by pulling back elements of the ring of spectral invariants. These are shown to determine simple Hamiltonian systems within the underlying symplectic vector space.