Poisson quasi-Nijenhuis structures with background

TL;DR

This paper introduces Poisson quasi-Nijenhuis structures with background on Lie algebroids, linking them to generalized complex structures, quasi-Lie bialgebroids, and known compatibility conditions between Poisson bivectors and 2-forms.

Contribution

It defines a new class of structures on Lie algebroids and establishes their connections to generalized complex structures and quasi-Lie bialgebroids, expanding the theoretical framework.

Findings

Any generalized complex structure on a Courant algebroid corresponds to a Poisson quasi-Nijenhuis structure with background.

Lie algebroids with such structures form quasi-Lie bialgebroids with their duals.

Pairs of Poisson bivectors and 2-forms induce these structures, recovering known compatibilities.

Abstract

We define the Poisson quasi-Nijenhuis structures with background on Lie algebroids and we prove that to any generalized complex structure on a Courant algebroid which is the double of a Lie algebroid is associated such a structure. We prove that any Lie algebroid with a Poisson quasi-Nijenhuis structure with background constitutes, with its dual, a quasi-Lie bialgebroid. We also prove that any pair $(\pi,\omega)$ of a Poisson bivector and a 2-form induces a Poisson quasi-Nijenhuis structure with background and we observe that particular cases correspond to already known compatibilities between $\pi$ and $\omega$.