The R-matrix theory and the reduction of Poisson manifolds.

The relation between the R-matrix approach to the construction of bi-Hamiltonian structures and the reduction of Poisson manifolds is discussed. The crucial role of the reduction process is emphasized by showing, on the basis of specific examples, that the knowledge of a R matrix may not be sufficient. The reduction of the Poisson structures related with R from the entire algebra to the Lax submanifold may become an obstruction in constructing the bi-Hamiltonian structures of a specific hierarchy of integrable equations. [ABSTRACT FROM AUTHOR]