Poisson Quasi-Nijenhuis Structures with Background

We define Poisson quasi-Nijenhuis structures with background on Lie algebroids and we prove that any generalized complex structure on a Courant algebroid which is the double of a Lie algebroid has an associated Poisson quasi-Nijenhuis structure with background. 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 (π,ω) 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 π and ω. This paper was presented as a poster in the conference “Poisson 2008”, EPFL, Lausanne, in July 2008.