Poisson Quasi-Nijenhuis Manifolds

Abstract

We introduce the notion of Poisson quasi-Nijenhuis manifolds generalizing the Poisson-Nijenhuis manifolds of Magri-Morosi. We also investigate the integration problem of Poisson quasi-Nijenhuis manifolds. In particular, we prove that, under some topological assumption, Poisson (quasi)-Nijenhuis manifolds are in one-one correspondence with symplectic (quasi)-Nijenhuis groupoids. As an application, we study generalized complex structures in terms of Poisson quasi-Nijenhuis manifolds. We prove that a generalized complex manifold corresponds to a special class of Poisson quasi-Nijenhuis structures. As a consequence, we show that a generalized complex structure integrates to a symplectic quasi-Nijenhuis groupoid recovering a theorem of Crainic.

18 pages, title changed, introduction rewritten, order of sections changed, references added, minor changes to body text, to appear in Comm. Math. Phys

Cited by in corpus (24)

• Holomorphic Poisson Manifolds and Holomorphic Lie Algebroids
• Reduction of Generalized Complex Structures
• Lie theory of multiplicative tensors
• Integration of Holomorphic Lie Algebroids
• Generalized Contact Bundles
• Poisson quasi-Nijenhuis structures with background
• Deformations of Lie 2-algebras
• Holomorphic Jacobi Manifolds and Holomorphic Contact Groupoids
• Infinitesimal Automorphisms of VB-groupoids and algebroids
• E-Courant algebroids
• Dirac structures and Nijenhuis operators
• Nijenhuis and Compatible Tensors on Lie and Courant algebroids
• On Jacobi quasi-Nijenhuis algebroids and Courant-Jacobi algebroid morphisms
• Lie-Nijenhuis bialgebroids
• Poisson-Nijenhuis groupoids
• Local formulas for multiplicative forms
• Poisson quasi-Nijenhuis manifolds and the Toda system
• Jacobi quasi-Nijenhuis Algebroids
• Pseudo-Poisson Nijenhuis manifolds
• Quasi-Lie bialgebroids, Dirac structures and deformations of Poisson quasi-Nijenhuis manifolds
• Integrating Nijenhuis Structures
• Reduction and construction of Poisson quasi-Nijenhuis manifolds with background
• Some remarks on invariant Poisson quasi-Nijenhuis structures on Lie groups
• Courant-Nijenhuis algebroids