Dirac-Nijenhuis structures

In this paper we study the concept of Dirac-Nijenhuis structures. We consider them to be a pair (D,N) where D is a Dirac structure, defined with respect to a Lie bialgebroid (A, A*), and N is a Nijenhuis operator which defines a deformation of the Lie algebroid structure of D in a compatible way. The transformation can be considered to deform also the double of the Lie bialgebroid, which leads to the concept of Dirac-Nijenhuis structures of type I, or to not affect it, leading to Dirac-Nijenhuis structures of type II. We prove that the concept contains Poisson-Nijenhuis structures as a particular case and provides another example for the case of Kaehler manifolds