Lie algebroids are by no means natural as an infinitesimal counterpart of groupoids. In this paper we propose a functorial construction called Nishimura algebroids for an infinitesimal counterpart of groupoids. Nishimura algebroids, intended for differential geometry, are of the same vein as Lawvere’s functorial notion of algebraic theory and Ehresmann’s functorial notion of theory called sketc...

in this paper, we present algebroids and crossed modules of algebroids. we also define pullbackcrossed module of algebroids.

Inspired by recent works of Zang Liu, Alan Weinstein and Ping Xu, we introduce the notions of CC algebroids and non asymmetric Courant algebroids and study these structures. It is shown that CC algebroids of rank greater than 3 are the same as Courant algebroids up to a constant factor, though the definition of CC algebroids is much simpler than that of Courant algebroids,requiring only 2 axiom...

in this paper, we presented algebroids and crossed modules of algebroids. we also presented pullback crossed module of algebroids and we then define pushout crossed modules of algebroids.

Lie n-algebroids and Lie infinity algebroids are usually thought of exclusively in supergeometric or algebraic terms. In this work, we apply the higher derived brackets construction to obtain a geometric description of Lie n-algebroids by means of brackets and anchors. Moreover, we provide a geometric description of morphisms of Lie n-algebroids over different bases, give an explicit formula fo...

We introduce the concept of para-Hopf algebroid and define their cyclic cohomology in the spirit of Connes-Moscovici cyclic cohomology for Hopf algebras. Para-Hopf algebroids are closely related to, but different from, Hopf algebroids. Their definition is motivated by attempting to define a cyclic cohomology theory for Hopf algebroids in general. We show that many of Hopf algebraic structures, ...

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...

In this note we give a definition of semiinfinite cohomology for Tate Lie algebras using the language of differential graded Lie algebroids with curvature (CDG Lie algebroids). 2000 Math. Subj. Class. 17-XX.

We examine Lie (super)algebroids equipped with a homological section, i.e., an odd section that ‘self-commutes’, we refer to such algebroids as inner Q-algebroids: these provide natural examples of suitably “superised” Q-algebroids in the sense Mehta. Such are generalisation Q-manifolds and superalgebras element. Amongst other results, show that, via derived bracket formalism, space sections Q-...

We investigate Atiyah algebroids, i.e. the infinitesimal objects of principal bundles, from the viewpoint of Lie algebraic approach to space. First we show that if the Lie algebras of smooth sections of two Atiyah algebroids are isomorphic, then the corresponding base manifolds are necessarily diffeomorphic. Further, we give two characterizations of the isomorphisms of the Lie algebras of secti...