Dirac structures and Nijenhuis operators

TL;DR

This paper introduces a compatibility concept between Dirac structures and Nijenhuis operators, exploring their properties, geometric implications, hierarchies, and integration to Lie groupoids, extending Poisson-Nijenhuis theory.

Contribution

It extends the theory of Poisson-Nijenhuis structures by defining Dirac-Nijenhuis structures and analyzing their geometric and integrative properties.

Findings

Established compatibility conditions for Dirac and Nijenhuis structures

Connected Dirac-Nijenhuis structures with holomorphic Dirac structures

Explored integration of these structures to Lie groupoids

Abstract

We introduce a notion of compatibility between (almost) Dirac structures and (1,1)-tensor fields extending that of Poisson-Nijenhuis structures. We study several properties of the "Dirac-Nijenhuis" structures thus obtained, including their connection with holomorphic Dirac structures, the geometry of their leaves and quotients, as well as the presence of hierarchies. We also consider their integration to Lie groupoids, which includes the integration of holomorphic Dirac structures as a special case.