We define and make initial study of Lie groupoids equipped with a compatible homogeneity (or graded bundle) structure, such objects we will refer to as weighted Lie groupoids. One can think of weighted Lie groupoids as graded manifolds in the category of Lie groupoids. This is a very rich geometrical theory with numerous natural examples. Note that VB-groupoids, extensively studied in the recen...

We study the family of vertex algebras associated with vertex algebroids, constructed by Gorbounov, Malikov, and Schechtman. As the main result, we classify all the (graded) simple modules for such vertex algebras and we show that the equivalence classes of graded simple modules one-to-one correspond to the equivalence classes of simple modules for the Lie algebroids associated with the vertex ...

Jacobi algebroids (i.e. a ‘Jacobi versions’ of Lie algebroids) are studied in the context of graded Jacobi brackets on graded commutative algebras. This unifies varios concepts of graded Lie structures in geometry and physics. A method of describing such structures by classical Lie algebroids via certain gauging (in the spirit of E.Witten’s gauging of exterior derivative) is developed. One cons...

We study the stability of singular points for smooth Poisson structures as well as general Lie algebroids. We give sufficient conditions for stability lying on the first (not necessarily linear) approximation of the given Poisson structure or Lie algebroid at a singular point. The main tools used here are the classical Lichnerowicz-Poisson cohomology and the deformation cohomology for Lie algeb...

<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{...

We show that quasi-Poisson structures can be identified with Dirac structures in suitable Courant algebroids. This provides a geometric way to construct Lie algebroids associated with quasi-Poisson spaces.

Weshow that a double Lie algebroid, togetherwith a chosen decomposition, is equivalent to a pair of 2-term representations up tohomotopy satisfying compatibility conditions which extend the notion of matched pair of Lie algebroids. We discuss in detail the double Lie algebroids arising from the tangent bundle of a Lie algebroid and the cotangent bundle of a Lie bialgebroid.

We express any Courant algebroid bracket by means of a metric connection, and construct a Courant algebroid structure on any orthogonal, Whitney sum E⊕C where E is a given Courant algebroid and C is a flat, pseudo-Euclidean vector bundle. Then, we establish the general expression of the bracket of a transitive Courant algebroid, that is, a Courant algebroid with a surjective anchor, and describ...

In this paper the properties of Lie algebroids with Poisson structures are investigated. We generalize some results of Fernandes [1] regarding linear contravariant connections on Poisson manifolds at the level of Lie algebroids. In the last part, the notions of complete and horizontal lifts on the prolongation of Lie algebroid are studied and their compatibility conditions are pointed out.

After a brief summary of the main properties of Poisson manifolds and Lie algebroids in general, we survey recent work on the modular classes of Poisson and twisted Poisson manifolds, of Lie algebroids with a Poisson or twisted Poisson structure, and of Poisson–Nijenhuis manifolds. A review of the spinor approach to the modular class concludes the paper.