We introduce the concept of Loday algebroids, a generalization of Courant algebroids. We define the naive cohomology and modular class of a Loday algebroid, and we show that the modular class of the double of a Lie bialgebroid vanishes. For Courant algebroids, we describe the relation between the naive and standard cohomologies and we conjecture that they are isomorphic when the Courant algebro...

In this paper some geometric structures on Finsler Lie algeboids are studied and h-basic distinguished connections introduced. Specially, Ichijy? connection that is a special investigated. The generalized Berwald algebroids presented, as particular case of Wagner-Ichijy? connection, studied. Moreover, the Wagner algebroid introduced equivalent conditions for space given. Finally, an optimal con...

Abstract We define solitons for the generalized Ricci flow on an exact Courant algebroid. then a family of flows left-invariant Dorfman brackets algebroid over simply connected nilpotent Lie group, generalizing bracket in way that might make this new useful study geometric such as flow. provide explicit examples both constructions Heisenberg group. also discuss solutions to

This paper extends previous results of the authors, concerning the behaviour of the equimultiple locus of algebroid surfaces under blowing–up, to arbitrary characteristic.

We introduce a new cohomology for Lie algebroids, and prove that it provides a differential graded Lie algebra which “controls” deformations of the structure bracket of the algebroid.

We prove some general results about the relation between the 1-cocycles of an arbitrary Lie algebroid A over M and the leaves of the Lie algebroid foliation on M associated with A. Using these results, we show that a E1(M)-Dirac structure L induces on every leaf F of its characteristic foliation a E1(F )-Dirac structure LF , which comes from a precontact structure or from a locally conformal pr...

We consider the category of anchored Banach vector bundles and we discuss the notion of semispray. Adding on the set of sections of an anchored Banach vector bundle a Lie bracket with some properties one gets the notion of Lie algebroid. We prove that the Lie algebroids form also a category. A Dirac structure on a Banach manifold M is defined as a subbundle of the big tangent bundle TM ⊕ T ∗M t...

The semi-classical data attached to stacks of algebroids in the sense of Kashiwara and Kontsevich are Maurer-Cartan elements on complex manifolds, which we call extended Poisson structures as they generalize holomorphic Poisson structures. A canonical Lie algebroid is associated to each Maurer-Cartan element. We study the geometry underlying these Maurer-Cartan elements in the light of Lie alge...

In this paper we study the differential systems on Leibniz algebroids. We introduce a class of almost metriplectic manifolds as a special case of Leibniz manifolds. Also, the notion of almost metriplectic algebroid is introduced. These types of algebroids are used in the presentation of associated differential systems. We give some interesting examples of differential systems on algebroids and ...