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A non-local Poisson bracket for Coxeter--Toda lattices

Volume 161 / 2020

Eber Chuño Colloquium Mathematicum 161 (2020), 67-88 MSC: Primary 37K10; Secondary 53D17. DOI: 10.4064/cm7700-4-2019 Published online: 28 February 2020

Abstract

We present a non-local Poisson bracket defined on the phase space $G^{u,v}\!/H$, where $G^{u,v}$ is a Coxeter double Bruhat cell of $\operatorname{GL} _n$ and $H$ is the subgroup of diagonal matrices. The non-local Poisson bracket is written in an appropriate set of coordinates of $G^{u,v}/H$ derived from a set of factorization parameters for $G^{u,v}$. We show that the non-local Poisson bracket corresponds to the Atiyah–Hitchin bracket under the Moser map. As a consequence, the non-local Poisson bracket is compatible with a quadratic Poisson bracket obtained by M. Gekhtman, M. Shapiro and A. Vainshtein (2011).

Authors

  • Eber ChuñoUnidade Acadêmica de Serra Talhada
    Universidade Federal Rural de Pernambuco
    56909-535 Serra Talhada, Pernambuco, Brazil
    e-mail

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