Abstract
We introduce quadratic Poisson structures on Lie groups associated with a class of solutions of the modified Yang-Baxter equation and apply them to the Hamiltonian description of Lax systems. The formal analog of these brackets on associative algebras provides second structures for certain integrable equations. In particular, the integrals of the Toda flow on generic orbits are shown to satisfy recursion relations. Finally, we exhibit a third order Poisson bracket for which ther-matrix approach is feasible.
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Li, LC., Parmentier, S. Nonlinear Poisson structures andr-matrices. Commun.Math. Phys. 125, 545–563 (1989). https://doi.org/10.1007/BF01228340
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DOI: https://doi.org/10.1007/BF01228340