Poisson reduction of the space of polygons

TL;DR

This paper introduces a family of Poisson structures on the space of twisted polygons, explores their reductions, and connects them to integrable systems like Toda hierarchies, revealing new compatible structures.

Contribution

It defines a novel family of Poisson structures parametrized by odd periodic functions and relates them to well-known integrable systems through reduction and compatibility analysis.

Findings

Identified Poisson structures include lattice Virasoro and Toda structures.

Established connections between Poisson structures and Toda hierarchies.

Proved compatibility of certain Poisson structures for generating integrable hierarchies.

Abstract

A family of Poisson structures, parametrised by an arbitrary odd periodic function $\phi$, is defined on the space $\cW$ of twisted polygons in $\RR^\nu$. Poisson reductions with respect to two Poisson group actions on $\cW$ are described. The $\nu=2$ and $\nu=3$ cases are discussed in detail and the general $\nu$ case in less detail. Amongst the Poisson structures arising in examples are to be found the lattice Virasoro structure, the second Toda lattice structure and some extended Toda lattice structures. A general result is proved showing that, for any $\nu$, to certain concrete choices of $\phi$ there correspond compatible Poisson structures which generate all the extended bigraded Toda hierarchies of a suitable size.