The R-matrix theory and the reduction of Poisson manifolds

From the MR review by W.Oevel: "Three different constructions of multi-Hamiltonian structures associated with classical R-matrices are reviewed. The equivalence between the tensor product formulation and the algebraic operator approach is discussed. The main point of the work is the discussion of the reduction properties of the general Poisson brackets defined on the underlying algebra. In order to obtain the multi-Hamiltonian structure for a specific integrable system, one has to restrict or reduce the brackets to an orbit of suitable Lax matrices defining a submanifold of the entire algebra. A variety of examples are used to illustrate the possible complications arising in the reduction process. For the Gelfand-Dikii hierarchy and the finite gl(n)-Toda lattice the author reviews how the corresponding R-matrices successfully lead to the multi-Hamiltonian formulations. However, for the restriction of the Toda lattice to Lax matrices in the subalgebra sl(n) no systematic construction of the higher Poisson brackets is available, as this restriction is not compatible with the R-matrix approach to the higher brackets. Similar problems arise for the relativistic Toda hierarchy. Thus, it is explicitly shown that the reduction of the Poisson structures associated with R-matrices may become an obstruction in constructing the bi-Hamiltonian structures of a specific hierarchy of integrable equations."