Theoretical and Mathematical Physics

, Volume 146, Issue 1, pp 73–84 | Cite as

Hamiltonian Structures of Fermionic Two-Dimensional Toda Lattice Hierarchies

  • V. V. Gribanov
  • V. G. Kadyshevsky
  • A. S. Sorin


By exhibiting the corresponding Lax-pair representations, we propose a wide class of integrable two-dimensional (2D) fermionic Toda lattice (TL) hierarchies, which includes the 2D N=(2|2) and N=(0|2) supersymmetric TL hierarchies as particular cases. We develop the generalized graded R-matrix formalism using the generalized graded bracket on the space of graded operators with involution generalizing the graded commutator in superalgebras, which allows describing these hierarchies in the framework of the Hamiltonian formalism and constructing their first two Hamiltonian structures. We obtain the first Hamiltonian structure for both bosonic and fermionic Lax operators and the second Hamiltonian structure only for bosonic Lax operators.


integrable systems Toda lattices R-matrix Yang-Baxter equation 


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Copyright information

© Springer Science+Business Media, Inc. 2006

Authors and Affiliations

  • V. V. Gribanov
    • 1
  • V. G. Kadyshevsky
    • 1
  • A. S. Sorin
    • 1
  1. 1.Bogoliubov Laboratory of Theoretical PhysicsJoint Institute for Nuclear ResearchDubna, Moscow OblastRussia

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