Nijenhuis structures on Courant algebroids

We study Nijenhuis structures on Courant algebroids in terms of the canonical Poisson bracket on their symplectic realizations. We prove that the Nijenhuis torsion of a skew-symmetric endomorphism N of a Courant algebroid is skewsymmetric if N (2) is proportional to the identity, and only in this case when the Courant algebroid is irreducible. We derive a necessary and sufficient condition for a skewsymmetric endomorphism to give rise to a deformed Courant structure. In the case of the double of a Lie bialgebroid (A, A*), given an endomorphism N of A that defines a skew-symmetric endomorphism N of the double of A, we prove that the torsion ofN is the sum of the torsion of N and that of the transpose of N.