Lie algebroids and Lie pseudoalgebras arise from a wide variety of constructions in differential geometry; they have been introduced repeatedly into the geometry, physics and algebra literatures since the 1950s, under some 14 different terminologies. The first main part (Sections 2-5) of this survey describes the four principal classes of examples, emphazising that each arises by means of a gen...

The main aim of this paper is to give a Hopf algebroid approach the Picard-Vessiot theory linear differential matrix equations with coefficients in polynomial complex algebra. To end, we introduce general construction what call here \emph{the finite dual} co-commutative (right) and then apply first Weyl algebra viewed as universal enveloping Lie all vector fields on affine line. In way, for fix...

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...

A class of factorizable Poisson brackets is studied which includes almost all reasonable Poisson manifolds. In the simplest case these brackets can be associated with symplectic Lie algebroids (or, in another terminology, with triangular Lie bialgebroids associated to a nondegenerate r-matrix). The BRST theory is applied to describe the geometry underlying these brackets and to develop a covari...