Theoretical and Mathematical Physics

, Volume 172, Issue 1, pp 911–931 | Cite as

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

  • B. A. DubrovinEmail author
  • T. V. Skrypnyk


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.


classical R-operator integrable hierarchy 


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Copyright information

© Pleiades Publishing, Ltd. 2012

Authors and Affiliations

  1. 1.International School for Advanced StudiesTriesteItaly
  2. 2.Lomonosov Moscow State UniversityMoscowRussia
  3. 3.Universita di Milano BicoccaMilanItaly
  4. 4.Bogolyubov Institute for Theoretical PhysicsKievUkraine

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