We apply the Atiyah–Singer index theorem and tensor products of elliptic complexes to cohomology transitive Lie algebroids. prove that Euler characteristic a representation algebroid $A$ over compact manifold $M$ vanishes unless $A=TM$, general Künneth formula. As applications, we give short proof vanishing result for principal bundle calculated using invariant differential forms, show certain ...

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...

<p style='text-indent:20px;'>We survey recent results on the local and global integrability of a Lie algebroid, as well infinitesimal multiplicative geometric structures it.</p>

We analyze the algebraic and geometric structures of deformations of Schwarz type topological field theories. Deformations of the Chern-Simons-BF theory and BF theories in n dimensions are analyzed. Two dimensionanl theory induces the Poisson structure and three dimensional theory induces the Courant algebroid structure on the target space as a sigma model. We generalize these structures to hig...

-semilattices with that in the category of modules over a given unital commutative quantale. The resulting structures are called quantale algebroids. We show that their constitute a monadic category and prove a representation theorem for them using the notion of nucleus adjusted for our needs. We also characterize the lattice of nuclei on a free quantale algebroid. At the end of the paper, we p...