# Hom-Poisson-Nijenhuis structures on Hom-Lie algebroids and Hom-Dirac structures on Hom-Courant algebroids

## TL;DR

This paper extends the theory of Hom-Lie and Hom-Courant algebroids by introducing Hom-Poisson, Hom-Nijenhuis, and Hom-Dirac structures, establishing their properties, hierarchies, and relations to Maurer-Cartan equations.

## Contribution

It introduces new Hom-structures on Lie and Courant algebroids and explores their properties, hierarchies, and correspondences, advancing the understanding of Hom-algebroid geometry.

## Key findings

- Existence of hierarchy of Hom-Poisson-Nijenhuis structures
- Relation between Hom-Dirac structures and Maurer-Cartan equations
- One-to-one correspondence between Poisson structures and Poisson isomorphisms

## Abstract

In this paper, we develop the theory of Hom-Lie algebroids, Hom-Lie bialgebroids and Hom-Courant algebroids introduced by Cai, Liu and Sheng. Specifically, we introduce the notions of Hom-Poisson, Hom-Nijenhuis and Hom-Poisson-Nijenhuis structures on a Hom-Lie algebroid and the notion of Hom-Dirac structures on a Hom-Courant algebroid. We show that these structures satisfy similar properties to structures non "Hom-"version. For example, there exists the hierarchy of a Hom-Poisson-Nijenhuis structure and we have a relation between Hom-Dirac structures and Maurer-Cartan type equation. Moreover we show that there exists a one-to-one correspondence between the pairs consisting of a Poisson structure on $M$ and a Poisson isomorphism for it.