Nijenhuis structures on Courant algebroids

Details Nijenhuis structures on Courant algebroids Nijenhuis structures on Courant algebroids

2011-02-07

arxiv.org/abs/1102.1410v2

Mathematics

Differential Geometry

25 pages; a footnote and an example added; to appear in Bull. Braz. Math. Soc

Scientific paper

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 skew-symmetric if the square of N 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 skew-symmetric 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 of N is the sum of the torsion of n and that of the transpose of n.

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