# Pseudo-Poisson Nijenhuis manifolds

## TL;DR

This paper introduces pseudo-Poisson Nijenhuis manifolds, generalizing Poisson Nijenhuis manifolds, and explores their associated quasi-Lie bialgebroids and Courant algebroids, providing new examples and structural insights.

## Contribution

It defines pseudo-Poisson Nijenhuis manifolds, links them to quasi-Lie bialgebroids, and offers methods to construct Courant algebroids with numerous examples.

## Key findings

- Pseudo-Poisson Nijenhuis manifolds generalize Poisson Nijenhuis manifolds.
- Associated quasi-Lie bialgebroids can be used to construct Courant algebroids.
- Nondegenerate cases allow reduction of conditions and generation of examples.

## Abstract

We introduce the notion of pseudo-Poisson Nijenhuis manifolds. These manifolds are generalizations of Poisson Nijenhuis manifolds by Magri and Morosi \cite{MM}. We show that any pseudo-Poisson Nijenhuis manifold has an associated quasi-Lie bialgebroid as in the case of Poisson quasi-Nijenhuis manifolds by Sti$\acute{\mathrm{e}}$non and Xu \cite{SX}. Hence, since a quasi-Lie bialgebroid has an associated Courant algebroid, we have new materials to construct Courant algebroids. In the "nondegenerate" case, we show that the conditions of pseudo-Poisson Nijenhuis structures can be reduced. Therefore we can provide lots of non-trivial examples of pseudo-Poisson Nijenhuis manifolds.