Classical double, R-operators, and negative flows of integrable hierarchies

Using the classical double G of a Lie algebra gequipped with the classical R-operator, we define two sets of functions commuting with respect to the initial Lie-Poisson bracket on g* and its extensions. We consider examples of Lie algebras g with the "Adler-Kostant-Symes" R-operators and the two corresponding sets of mutually commuting functions in detail. Using the constructed commutative Hamiltonian flows on different extensions of g, we obtain zero-curvature equations with g-valued U-V pairs. The so-called negative flows of soliton hierarchies are among such equations. We illustrate the proposed approach with examples of two-dimensional Abelian and non-Abelian Toda field equations.