TY - JOUR T1 - Classical double, R-operators, and negative flows of integrable hierarchies JF - Theoretical and Mathematical Physics. Volume 172, Issue 1, July 2012, Pages 911-931 Y1 - 2012 A1 - Boris Dubrovin A1 - Taras V. Skrypnyk AB - Using 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. PB - SISSA UR - http://hdl.handle.net/1963/6468 U1 - 6413 U2 - Mathematics U4 - 1 U5 - MAT/07 FISICA MATEMATICA ER -