Reduction of Poisson-Nijenhuis Lie algebroids to symplectic-Nijenhuis Lie algebroids with nondegenerate Nijenhuis tensor

TL;DR

This paper presents a method to reduce Poisson-Nijenhuis Lie algebroids to symplectic-Nijenhuis Lie algebroids with nondegenerate Nijenhuis tensors, generalizing previous manifold reduction techniques for applications in bi-Hamiltonian systems.

Contribution

It generalizes the reduction process from Poisson-Nijenhuis manifolds to Lie algebroids, enabling new geometric insights into bi-Hamiltonian systems.

Findings

Established reduction conditions for Lie algebroids.

Provided an explicit example of the reduction process.

Extended Magri and Morosi's work to a broader framework.

Abstract

We show how to reduce, under certain regularities conditions, a Poisson-Nijenhuis Lie algebroid to a symplectic-Nijenhuis Lie algebroid with nondegenerate Nijenhuis tensor. We generalize the work done by Magri and Morosi for the reduction of Poisson-Nijenhuis manifolds. The choice of the more general framework of Lie algebroids is motivated by the geometrical study of some reduced bi-Hamiltonian systems. An explicit example of reduction of a Poisson-Nijenhuis Lie algebroid is also provided.