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arXiv · 0712.4228

The Cohomology of Transitive Lie Algebroids

Z. Chen·Z. -J. Liu
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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.

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Z. Chen, Z. -J. Liu. 2007-12-27. The Cohomology of Transitive Lie Algebroids. https://doi.org/10.1142/9789812779649_0006

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