The Cohomology of Transitive Lie Algebroids

TL;DR

This paper investigates the cohomology of transitive Lie algebroids on manifolds, establishing a relationship between local and global cohomology groups and applying results to fiber bundles and Lie bialgebroids.

Contribution

It provides a new understanding of the kernel of the localization map in Lie algebroid cohomology and relates it to de Rham cohomology, with applications to fiber bundles and bialgebroids.

Findings

Kernel of localization map characterized by de Rham cohomology.

Relation between universal cover cohomology and original cohomology.

Applications to structure analysis of Lie bialgebroids.

Abstract

For a transitive Lie algebroid A on a connected manifold M and its a representation on a vector bundle F, we study the localization map Y^1: H^1(A,F)-> H^1(L_x,F_x), where L_x is the adjoint algebra at x in M. The main result in this paper is that: Ker Y^1_x=Ker(p^{1*})=H^1_{deR}(M,F_0). Here p^{1*} is the lift of H^1(\huaA,F) to its counterpart over the universal covering space of M and H^1_{deR}(M,F_0) is the F_0=H^0(L,F)-coefficient deRham cohomology. We apply these results to study the associated vector bundles to principal fiber bundles and the structure of transitive Lie bialgebroids.