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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).