| The localization of 1-cohomology of transitive Lie algebroids |
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| 引用本文： | CHEN Zhuo & LIU Zhangju Department of Mathematics,Capital Normal University,Beijing 100037,China, School of Mathematical Science,Peking University,Beijing 100871,China. The localization of 1-cohomology of transitive Lie algebroids[J]. 中国科学A辑(英文版), 2006, 49(2): 277-288. DOI: 10.1007/s11425-005-0174-2 |
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| 作者姓名： | CHEN Zhuo & LIU Zhangju Department of Mathematics Capital Normal University Beijing 100037 China School of Mathematical Science Peking University Beijing 100871 China |
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| 作者单位： | CHEN Zhuo & LIU Zhangju Department of Mathematics，Capital Normal University，Beijing 100037，China; School of Mathematical Science，Peking University，Beijing 100871，China |
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| 摘 要： | 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 rk: Hk(A,F) → Hk(Lx,Fx), called the localization map, where Lx is the adjoint algebra at x ∈ M. The main result in this paper is that if M is simply connected, or H (LX,FX) is trivial, then T 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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| 收稿时间： | 2005-05-31 |
| 修稿时间： | 2005-08-15 |
The localization of 1-cohomology of transitive Lie algebroids |
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| CHEN Zhuo,LIU Zhangju. The localization of 1-cohomology of transitive Lie algebroids[J]. Science in China(Mathematics), 2006, 49(2): 277-288. DOI: 10.1007/s11425-005-0174-2 |
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| Authors: | CHEN Zhuo LIU Zhangju |
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| Affiliation: | 1. Department of Mathematics, Capital Normal University, Beijing 100037, China
2. School of Mathematical Science, Peking University, Beijing 100871, China |
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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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| Keywords: | transitive Lie algebroid representation cohomology localization |
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