A family of the Poisson brackets compatible with the Sklyanin bracket

@article{Tsiganov2006AFO,
  title={A family of the Poisson brackets compatible with the Sklyanin bracket},
  author={A. V. Tsiganov},
  journal={arXiv: Exactly Solvable and Integrable Systems},
  year={2006}
}
  • A. Tsiganov
  • Published 12 December 2006
  • Mathematics, Physics
  • arXiv: Exactly Solvable and Integrable Systems
We introduce a family of compatible Poisson brackets on the space of $2\times 2$ polynomial matrices, which contains the Sklyanin bracket, and use it to derive a multi-Hamiltonian structure for a set of integrable systems that includes $XXX$ Heisenberg magnet, the open and periodic Toda lattices, the discrete self-trapping model and the Goryachev-Chaplygin gyrostat. 
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