Multiple Hamiltonian structure of Bogoyavlensky-Toda lattices

Abstract

This paper is mainly a review of the multi--Hamiltonian nature of Toda and generalized Toda lattices corresponding to the classical simple Lie groups but it includes also some new results. The areas investigated include master symmetries, recursion operators, higher Poisson brackets, invariants and group symmetries for the systems. In addition to the positive hierarchy we also consider the negative hierarchy which is crucial in establishing the bi--Hamiltonian structure for each particular simple Lie group. Finally, we include some results on point and Noether symmetries and an interesting connection with the exponents of simple Lie groups. The case of exceptional simple Lie groups is still an open problem.

arXiv comment65 pages, 67 references

References in corpus (4)

• Multiple Hamiltonian Structures for Toda-type systems
• From the Toda Lattice to the Volterra lattice and back
• Symmetries of Toda Equations
• The Atiyah--Hitchin bracket and the open Toda lattice

Cited by in corpus (8)

• Lagrangian Formalism for nonlinear second-order Riccati Systems: one-dimensional Integrability and two-dimensional Superintegrability
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• The Poisson bracket compatible with the classical reflection equation algebra
• A Gentle (without Chopping) Approach to the Full Kostant-Toda Lattice
• Integrable hierarchies and the modular class
• The last integrable case of kozlov-Treshchev Birkhoff integrable potentials
• A centerless representation of the Virasoro algebra associated with the unitary circular ensemble
• Real Hamiltonian Forms of Affine Toda Models Related to Exceptional Lie Algebras