Webs, Lenard schemes, and the local geometry of bihamiltonian Toda and Lax structures

TL;DR

This paper introduces a geometric criterion for bihamiltonian structures to have constant coefficient local coordinates, applies it to Toda lattices, and explores their local geometric decompositions and implications for integrable systems.

Contribution

The paper develops a geometric approach to analyze bihamiltonian structures, providing new criteria and decompositions for Toda lattices and related integrable systems.

Findings

Open Toda lattice is locally isomorphic to a Kronecker pair of brackets with constant coefficients.

Periodic Toda lattice decomposes into a product of two open Toda lattices.

Bihamiltonian structures with non-degenerate Lax structures are locally isomorphic to the open Toda lattice.

Abstract

We introduce a criterion that a given bihamiltonian structure allows a local coordinate system where both brackets have constant coefficients. This criterion is applied to the bihamiltonian open Toda lattice in a generic point, which is shown to be locally isomorphic to a Kronecker odd-dimensional pair of brackets with constant coefficients. This shows that the open Toda lattice cannot be locally represented as a product of two bihamiltonian structures. In a generic point the bihamiltonian periodic Toda lattice is shown to be isomorphic to a product of two open Toda lattices (one of which is a (trivial) structure of dimension 1). While the above results might be obtained by more traditional methods, we use an approach based on general results on geometry of webs. This demonstrates a possibility to apply a geometric language to problems on bihamiltonian integrable systems, such a possibility may be no less important than the particular results proven in this paper. Based on these geometric approaches, we conjecture that decompositions similar to the decomposition of the periodic Toda lattice exist in local geometry of the Volterra system, the complete Toda lattice, the multidimensional Euler top, and a regular bihamiltonian Lie coalgebra. We also state general conjectures about geometry of more general ``homogeneous'' finite-dimensional bihamiltonian structures. The class of homogeneous structures is shown to coincide with the class of system integrable by Lenard scheme. The bihamiltonian structures which allow a non-degenerate Lax structure are shown to be locally isomorphic to the open Toda lattice.