The hierarchy of Poisson brackets for the open Toda lattice and its spectral curves

Abstract

We establish a new representation of the infinite hierarchy of Poisson brackets for the open Toda lattice in terms of its spectral curve. For the Poisson brackets we give a representation in the form of a contour integral of some special Abelian differential (meromorphic one-form) on the spectral curve. All higher brackets of the infinite hierarchy are obtained by multiplication of the one-form by a power of the spectral parameter.A universal algebraic–geometrical formula of the infinite hierarchy of Poisson bracket for a classical integrable system was conjectured by the author in Vaninsky (2017). This paper is a confirmation of this conjecture for the finite open Toda lattice.