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The localization of 1-cohomology of transitive Lie algebroids

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Abstract

For a transitive Lie algebroid A on a connected manifold M and its representation on a vector bundle F, we define a morphism of cohomology groups ϒ?κ: H κ(A, F) → H κ(L χ, F χ), called the localization map, where L χ is the adjoint algebra at χ ∈ M. The main result in this paper is that if M is simply connected, or H0(L χ, F χ) is trivial, then ϒ1 is injective. This means that the Lie algebroid 1-cohomology is totally determined by the 1-cohomology of its adjoint Lie algebra in the above two cases.

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Authors and Affiliations

  1. Department of Mathematics, Capital Normal University, Beijing, 100037, China

    Chen Zhuo

  2. School of Mathematical Science, Peking University, Beijing, 100871, China

    Liu Zhangju

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  1. Chen Zhuo
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  2. Liu Zhangju
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Correspondence to Chen Zhuo.

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Chen, Z., Liu, Z. The localization of 1-cohomology of transitive Lie algebroids. SCI CHINA SER A 49, 277–288 (2006). https://doi.org/10.1007/s11425-005-0174-2

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  • DOI: https://doi.org/10.1007/s11425-005-0174-2

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