Dirac–Nijenhuis structures

Abstract

In this paper we study the concept of Dirac–Nijenhuis structures. We consider them to be a pair where D is a Dirac structure, defined with respect to a Lie bialgebroid (A, A*), and 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 Kahler manifolds.