Jacobi manifolds, Dirac structures and Nijenhuis operators

Abstract

In a recent paper [2], we studied the concept of Dirac-Nijenhuis structures. We de ned them as deformations of the canonical Lie algebroid structure of a Dirac bundle D de ned in the double of a Lie bialgebroid (A;A¤) which satisfy certain properties. In this paper, we introduce the concept of generalized Dirac- Nijenhuis structures as the natural analogue when we replace the double of the Lie bialgebroid by the double of a generalized Lie bialgebroid. We also show the usefulness of the concept by proving that a Jacobi-Nijenhuis manifold has an associated generalized Dirac-Nijenhuis structure of a certain type.