Nijenhuis structures on Courant algebroids

TL;DR

This paper investigates Nijenhuis structures on Courant algebroids using Poisson brackets, establishing conditions for skew-symmetry and deformations, with specific results on Lie bialgebroid doubles.

Contribution

It provides new criteria for Nijenhuis torsion skew-symmetry and deformation conditions on Courant algebroids, especially in the context of Lie bialgebroid doubles.

Findings

Skew-symmetric Nijenhuis torsion when N^2 is proportional to identity

Necessary and sufficient conditions for deformed Courant structures

Torsion of N on a double equals sum of torsions of n and its transpose

Abstract

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.