Moment Maps to Loop Algebras, Classical R-Matrix and Integrable Systems

A class of Poisson embeddings of reduced, finite dimensional symplectic vector spaces into the dual space $\Lg_R^*$ of a loop algebra, with Lie Poisson structure determined by the classical split $R$--matrix $R=P_+ - P_-$ is introduced. These may be viewed as equivariant moment maps inducing natural Hamiltonian actions of the ``dual'' group $\LG_R = \LGp \times \LGm$ of a loop group $\LG$ on the symplectic space. The $R$--matrix version of the Adler-Kostant-Symes theorem is used to induce commuting flows determined by isospectral equations of Lax type. The compatibility conditions determine finite dimensional classes of solutions to integrable systems of PDE's, which can be integrated using the standard Liouville-Arnold approach. This involves an appropriately chosen ``spectral Darboux'' (canonical) coordinate system in which there is a complete separation of variables. As an example, the method is applied to the determination of finite dimensional quasi-periodic solutions of the sine-Gordon equation.