ABSTRACT
The method of the classical r-matrix originated in the papers of Sklyanin and as a byproduct of the quantum inverse scattering method. The r-matrix method led to a generalization of the AKS approach and helped to connect several methodologies of the group theoretic study. One important application of the classical r-matrix occurs in relation to the factorization problem initially studied by Semenor-Tian-Shansky and Reiman. This is actually a variant of the Birkoff decomposition, well-known in the domain of usual Lie algebras. The chapter shows that the classical r-matrix can also be used to study the properties of multidimensional integrable systems such as the KP equation and its variants. An important aspect of classical r-matrices is the dressing transformation is that it does not preserve the Hamiltonian structure, so a very deep and exhaustive analysis is needed for the understanding of their properties.
