Poisson Nijenhuis Structures Research Articles

A generalization of Poisson–Nijenhuis structures

We generalize Poisson–Nijenhuis structures. We prove that on a manifold endowed with a Nijenhuis tensor and a Jacobi structure which are compatible, there is a hierarchy of pairwise compatible Jacobi structures. Furthermore, we study the homogeneous Poisson–Nijenhuis structures and their relations with Jacobi structures.

Jacobi—Nijenhuis manifolds and compatible Jacobi structures

We propose a definition of Jacobi—Nijenhuis structures, that includes the Poisson—Nijenhuis structures as a particular case. The existence of a hierarchy of compatible Jacobi structures on a Jacobi—Nijenhuis manifold is also obtained.

Lie algebroids and Poisson-Nijenhuis structures

Poisson-Nijenhuis structures for an arbitrary Lie algebroid are defined and studied by means of complete lifts of tensor fields.

The lie bialgebroid of a Poisson-Nijenhuis manifold

We describe a new class of Lie bialgebroids associated with Poisson-Nijenhuis structures.

A novel hierarchy of integrable lattices

In the framework of the reduction technique for Poisson-Nijenhuis structures, we derive a new hierarchy of integrable lattices, whose continuum limit is the AKNS hierarchy. In contrast to other differential-difference versions of the AKNS system, our hierarchy is endowed with a canonical Poisson structure and, moreover, it admits a vector generalization. We also solve the associated spectral problem and explicitly construct action-angle variables through the r-matrix approach.