A non-local Poisson bracket for Coxeter--Toda lattices
Tom 161 / 2020
Colloquium Mathematicum 161 (2020), 67-88
MSC: Primary 37K10; Secondary 53D17.
DOI: 10.4064/cm7700-4-2019
Opublikowany online: 28 February 2020
Streszczenie
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).
