Multi-Hamiltonian structures for r-matrix systems

TL;DR

This paper explores the connections between different Poisson space levels associated with r-matrix systems, revealing integrable Hamiltonian systems and Nijenhuis coordinates across these levels.

Contribution

It establishes links between three levels of Poisson spaces for r-matrix systems and introduces compatible Poisson structures and Nijenhuis coordinates at each level.

Findings

Identifies Poisson structures on matrix-valued functions, spectral curves, and surface symmetric products.

Shows Hamiltonian systems are integrable across these Poisson structures.

Provides examples including well-known integrable systems.

Abstract

For the rational, elliptic and trigonometric r-matrices, we exhibit the links between three "levels" of Poisson spaces: (a) Some finite-dimensional spaces of matrix-valued holomorphic functions on the complex line; (b) Spaces of spectral curves and sheaves supported on them; (c) Symmetric products of a surface. We have, at each level, a linear space of compatible Poisson structures, and the maps relating the levels are Poisson. This leads in a natural way to Nijenhuis coordinates for these spaces. At level (b), there are Hamiltonian systems on these spaces which are integrable for each Poisson structure in the family, and which are such that the Lagrangian leaves are the intersections of the symplective leaves over the Poisson structures in the family. Specific examples include many of the well-known integrable systems.