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Nonabelian cohomology with coefficients in Lie groups
Authors:Jinpeng An  Zhengdong Wang
Institution:School of Mathematical Science, Peking University, Beijing, 100871, People's Republic of China ; School of Mathematical Science, Peking University, Beijing, 100871, People's Republic of China
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$.

Keywords:Nonabelian cohomology  Lie group  twisted conjugate action  
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