The localization of 1-cohomology of transitive Lie algebroids
type: Item
instance of: scholarly article
title: The localization of 1-cohomology of transitive Lie algebroids
zbMATH Open document ID: 1110.58018
author: Zhuo Chen
author: Zhangju Liu
published in: Science in China. Series A
publication date: 2006-12-06T00:00:00Z
review text: For a transitive Lie algebroid \(A\) on a connected manifold \(M\) and its representation on a vector bundle \(F\), a morphism of cohomology groups \(\gamma^k: H^k(A,F)\to H^k(L_x, F_x)\), the localization map, is defined. \(L_x\) is the adjoint algebra at \(x\in M\). The main result is the following: if \(M\) is simply connected, or \(H^0(L_x,F_x)\) is trivial, then \(\gamma^1\) is injective and the Lie algebroid 1-cohomology is totally determined by the 1-cohomology of its adjoint Lie algebra in the above two cases. See also \textit{A. Connes} [J. Math. Phys. 36, No. 11, 6194--6231 (1995; Zbl 0871.58008)] and \textit{A. Weinstein} [Lett. Math. Phys. 56, No. 1, 17--30 (2001; Zbl 1010.53060)].
reviewed by: Maido Rahula
Mathematics Subject Classification ID: 58H05
Mathematics Subject Classification ID: 17B60
Mathematics Subject Classification ID: 53C05
zbMATH DE Number: 5077574
zbMATH Keywords: transitive Lie algebroid
zbMATH Keywords: localization map
MaRDI profile type: Publication
full work available at URL: s11425-005-0174-2
OpenAlex ID: W1694317309
cites work: Q4733032
cites work: Poisson geometry
cites work: Open-string BRST cohomology for generalized complex branes
cites work: Poisson–Jacobi reduction of homogeneous tensors
cites work: Transverse measures, the modular class and a cohomology pairing for Lie algebroids
cites work: Lie Algebroids and Lie Pseudoalgebras
cites work: Q4425363
cites work: On transitive Lie bialgebroids and Poisson groupoids
cites work: Q5731354
DOI: 10.1007/S11425-005-0174-2
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description: scientific article; zbMATH DE number 5077574