### 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 the r-matrix approach is feasible.

Original language | English (US) |
---|---|

Pages (from-to) | 545-563 |

Number of pages | 19 |

Journal | Communications in Mathematical Physics |

Volume | 125 |

Issue number | 4 |

DOIs | |

State | Published - Dec 1 1989 |

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### All Science Journal Classification (ASJC) codes

- Statistical and Nonlinear Physics
- Mathematical Physics

### Cite this

*Communications in Mathematical Physics*,

*125*(4), 545-563. https://doi.org/10.1007/BF01228340

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*Communications in Mathematical Physics*, vol. 125, no. 4, pp. 545-563. https://doi.org/10.1007/BF01228340

**Nonlinear Poisson structures and r-matrices.** / Li, Luen-chau; Parmentier, Serge.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Nonlinear Poisson structures and r-matrices

AU - Li, Luen-chau

AU - Parmentier, Serge

PY - 1989/12/1

Y1 - 1989/12/1

N2 - 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 the r-matrix approach is feasible.

AB - 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 the r-matrix approach is feasible.

UR - http://www.scopus.com/inward/record.url?scp=0001306018&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0001306018&partnerID=8YFLogxK

U2 - 10.1007/BF01228340

DO - 10.1007/BF01228340

M3 - Article

AN - SCOPUS:0001306018

VL - 125

SP - 545

EP - 563

JO - Communications in Mathematical Physics

JF - Communications in Mathematical Physics

SN - 0010-3616

IS - 4

ER -