Courant-Nijenhuis algebroids

Abstract

We introduce Courant 1-derivations, which describe a compatibility between Courant algebroids and linear (1,1)-tensor fields and lead to the notion of Courant-Nijenhuis algebroids. We provide examples of Courant 1-derivations on exact Courant algebroids and show that holomorphic Courant algebroids can be viewed as special types of Courant-Nijenhuis algebroids. By considering Dirac structures, one recovers the Dirac-Nijenhuis structures previously studied by the authors (in the special case of the standard Courant algebroid) and obtains an equivalent description of the Lie-Nijenhuis bialgebroids introduced by the second author via Manin triples.