01113nas a2200121   4500008004100000245008000041210006900121260001000190520071100200100002000911700002400931856003600955       2012                        en  d00aClassical double, R-operators, and negative flows of integrable hierarchies0 aClassical double Roperators and negative flows of integrable hie  bSISSA3 aUsing the classical double G of a Lie algebra g equipped 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.1 aDubrovin, Boris1 aSkrypnyk, Taras, V.  uhttp://hdl.handle.net/1963/6468