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Nonabelian cohomology with coefficients in Lie groups


Authors: Jinpeng An and Zhengdong Wang
Journal: Trans. Amer. Math. Soc. 360 (2008), 3019-3040
MSC (2000): Primary 20J06, 22E15, 57S15, 57S20
DOI: https://doi.org/10.1090/S0002-9947-08-04278-5
Published electronically: January 25, 2008
MathSciNet review: 2379785
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Abstract: In this paper we prove some properties of the nonabelian cohomology $ H^1(A,G)$ of a group $ A$ with coefficients in a connected Lie group $ G$. When $ A$ is finite, we show that for every $ A$-submodule $ K$ of $ G$ which is a maximal compact subgroup of $ G$, the canonical map $ H^1(A,K)\rightarrow H^1(A,G)$ is bijective. In this case we also show that $ H^1(A,G)$ is always finite. When $ A=\mathbb{Z}$ and $ G$ is compact, we show that for every maximal torus $ T$ of the identity component $ G_0^\mathbb{Z}$ of the group of invariants $ G^\mathbb{Z}$, $ H^1(\mathbb{Z},T)\rightarrow H^1(\mathbb{Z},G)$ is surjective if and only if the $ \mathbb{Z}$-action on $ G$ is $ 1$-semisimple, which is also equivalent to the fact that all fibers of $ H^1(\mathbb{Z},T)\rightarrow H^1(\mathbb{Z},G)$ are finite. When $ A=\mathbb{Z}/n\mathbb{Z}$, we show that $ H^1(\mathbb{Z}/n\mathbb{Z},T) \rightarrow H^1(\mathbb{Z}/n\mathbb{Z},G)$ is always surjective, where $ T$ is a maximal compact torus of the identity component $ G_0^{\mathbb{Z}/n\mathbb{Z}}$ of $ G^{\mathbb{Z}/n\mathbb{Z}}$. When $ A$ is cyclic, we also interpret some properties of $ H^1(A,G)$ in terms of twisted conjugate actions of $ G$.


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Additional Information

Jinpeng An
Affiliation: School of Mathematical Science, Peking University, Beijing, 100871, People’s Republic of China
Address at time of publication: Department of Pure Mathematics, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1
Email: anjinpeng@gmail.com

Zhengdong Wang
Affiliation: School of Mathematical Science, Peking University, Beijing, 100871, People’s Republic of China
Email: zdwang@pku.edu.cn

DOI: https://doi.org/10.1090/S0002-9947-08-04278-5
Keywords: Nonabelian cohomology, Lie group, twisted conjugate action.
Received by editor(s): September 17, 2005
Received by editor(s) in revised form: March 14, 2006
Published electronically: January 25, 2008
Additional Notes: This work was supported by the 973 Project Foundation of China (#TG1999075102).
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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