Poisson-Nijenhuis structures on quiver path algebras

TL;DR

This paper introduces noncommutative Poisson-Nijenhuis structures on quiver path algebras, linking them to integrable systems like Calogero-Moser and Gibbons-Hermsen, and offers new insights into bihamiltonian reduction.

Contribution

It develops the theory of noncommutative Poisson-Nijenhuis structures on quiver algebras and applies it to integrable systems, providing new interpretations and frameworks.

Findings

New noncommutative Poisson-Nijenhuis structures introduced

Application to Calogero-Moser and Gibbons-Hermsen systems

Reinterpretation of bihamiltonian reduction

Abstract

We introduce a notion of noncommutative Poisson-Nijenhuis structure on the path algebra of a quiver. In particular, we focus on the case when the Poisson bracket arises from a noncommutative symplectic form. The formalism is then applied to the study of the Calogero-Moser and Gibbons-Hermsen integrable systems. In the former case, we give a new interpretation of the bihamiltonian reduction performed in [3].