We define the Poisson quasi-Nijenhuis structures with background on Lie algebroids and we prove that to any generalized complex structure on a Courant algebroid which is the double of a Lie algebroid is associated such a structure. We prove that any Lie algebroid with a Poisson quasi-Nijenhuis structure with background constitutes, with its dual, a quasi-Lie bialgebroid. We also prove that any ...

We introduce and study the notion of representation up to homotopy of a Lie algebroid, paying special attention to examples. We use representations up to homotopy to define the adjoint representation of a Lie algebroid and show that the resulting cohomology controls the deformations of the structure. The Weil algebra of a Lie algebroid is defined and shown to coincide with Kalkman’s BRST model ...

A VB–algebroid is essentially defined as a Lie algebroid object in the category of vector bundles. There is a one-to-one correspondence between VB–algebroids and certain flat Lie algebroid superconnections, up to a natural notion of equivalence. In this setting, we are able to construct characteristic classes, which in special cases reproduce characteristic classes constructed by Crainic and Fe...

We discuss relations of linear Nambu-Poisson structures to Filippov algebras and define a Filippov algebroid – a generalization of a Lie algebroid. We also prove results describing multiplicative Nambu-Poisson structures on Lie groups. In particular, it is shown that simple Lie groups do not admit multiplicative Nambu-Poisson structures of order > 2.

We show that the Gerstenhaber algebra of the 1-jet Lie algebroid of a Jacobi manifold has a canonical exact generator, and discuss duality between its homology and the Lie algebroid cohomology. We also give a new example of a Lie bialgebroid on Poisson manifolds.

A notion of an algebroid-a generalization of a Lie algebroid structure on a vector bundle is introduced. We show that many objects of the differential calculus on a manifold M associated with the canonical Lie algebroid structure on TM can be obtained in the framework of a general algebroid. Also a compatibility condition which leads, in general, to a concept of a bialgebroid. 0 Introduction. T...

<abstract><p>In this paper, first we introduce the notion of a $ \mathsf{VB} $-Lie 2 $-algebroid, which can be viewed as categorification algebroid. The tangent prolongation Lie $-algebroid is naturally. We show that after choosing splitting, there one-to-one correspondence between $-algebroids and flat superconnections 2-algebroid on 3-term complex vector bundles. Then $-$ \mathsf{...

It is shown that a quantum groupoid (or a QUE algebroid, i.e., deformation of the universal enveloping algebra of a Lie algebroid) naturally gives rise to a Lie bialgebroid as a classical limit. The converse question, i.e., the quantization problem, is raised, and it is proved for all regular triangular Lie bialgebroids. For a Poisson manifold P , the existence of a star-product is shown to be ...

As a continuation of previous papers, we study the concept of a Lie algebroid structure on an affine bundle by means of the canonical immersion of the affine bundle into its bidual. We pay particular attention to the prolongation and various lifting procedures, and to the geometrical construction of Lagrangian-type dynamics on an affine Lie algebroid.