Related papers: Dirac structures and Nijenhuis operators

We study higher-order analogues of Dirac structures, extending the multisymplectic structures that arise in field theory. We define higher Dirac structures as involutive subbundles of $TM+\wedge^k TM^*$ satisfying a weak version of the…

We generalize Poisson-Nijenhuis structures. We prove that on a manifold endowed with a Nijenhuis tensor and a Jacobi structure which are compatible, there is a hierarchy of pairwise compatible Jacobi structures. Furthermore, we study the…

Poisson homogeneous spaces for Poisson groupoids are classfied in terms of Dirac structures for the corresponding Lie bialgebroids. Applications include Drinfel'd's classification in the case of Poisson groups and a description of leaf…

In this paper, we introduce right-invariant Poisson-Nijenhuis Structures on Lie groupoids and their infinitesimal counterparts as called (Poisson bivector, Nijenhuis operator) structures. Also, we present a one-to-one correspondence between…

We study the integrability of Poisson and Dirac structures that arise from quotient constructions. From our results we deduce several classical results as well as new applications. We also give explicit constructions of Lie groupoids…

Geometric structures underlying commutative and non commutative integrable dynamics are analyzed. They lead to a new characterization of noncommutative integrability in terms of spectral properties and of Nijenhuis torsion of an invariant…

In this paper we introduce multiplicative Dirac structures on Lie groupoids, providing a unified framework to study both multiplicative Poisson bivectors (i.e., Poisson group(oid)s) and multiplicative closed 2-forms (e.g., symplectic…

In this paper, we develop the theory of Hom-Lie algebroids, Hom-Lie bialgebroids and Hom-Courant algebroids introduced by Cai, Liu and Sheng. Specifically, we introduce the notions of Hom-Poisson, Hom-Nijenhuis and Hom-Poisson-Nijenhuis…

We give simple characterizations of contact 1-forms in terms of Dirac structures. We also relate normal almost contact structures to the theory of Dirac structures.

We introduce Morita equivalence for Nijenhuis groupoids and for their infinitesimal counterparts, establishing a global-to-infinitesimal correspondence under the Lie functor. A special case is that of holomorphic Lie groupoids and…

We extend to the context of Courant algebroids several hierarchies that can be constructed on Poisson-Nijenhuis manifolds. More precisely, we introduce several notions (Poisson-Nijenhuis, deformation-Nijenhuis and Nijenhuis pairs) that…

We investigate Nijenhuis deformations of $L_\infty$-algebras, a notion that unifies several Nijenhuis deformations, namely those of Lie algebras, Lie algebroids, Poisson structures and Courant structures. Additional examples, linked to Lie…

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…

Introducing Nijenhuis forms on Lie-infinity algebras gives a general frame to understand deformations of the latter. We give here a Nijenhuis interpretation of a deformation of an arbitrary Lie algebroid into a Lie-infinity algebra. Then we…

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…

We show that well known structures on Lie algebroids can be viewed as Nijenhuis tensors or pairs of compatible tensors on Courant algebroids. We study compatibility and construct hierarchies of these structures.

We define hypersymplectic structures on Lie algebroids recovering, as particular cases, all the classical results and examples of hypersymplectic structures on manifolds. We prove a 1-1 correspondence theorem between hypersymplectic…

We extend the definition of the Nijenhuis torsion of an endomorphism of a Lie algebroid to that of a relation, and we prove that the torsion of the relation defined by a bi-Hamiltonian structure vanishes. Following Gelfand and Dorfman, we…

In this paper we present some approaches to classification of almost complex structures and to construction of local or formal pseudoholomorphic mapping from one almost complex manifold to another. The corresponding criteria are given in…

Using tools from Dirac geometry and through an explicit construction, we show that every Poisson homogeneous space of any Poisson Lie group admits an integration to a symplectic groupoid. Our theorem follows from a more general result which…