Journal of Lie Theory 19 (2009), No. 2, 231--236
Copyright Heldermann Verlag 2009
Nonabelian Cohomology of Compact Lie Groups
Jinpeng An
School of Mathematical Sciences, Beijing University, Beijing 100871, P. R. China
anjinpeng@gmail.com
Ming Liu
School of Mathematical Sciences, Beijing University, Beijing 100871, P. R. China
mingliulm@yahoo.com.cn
Zhengdong Wang
School of Mathematical Sciences, Beijing University, Beijing 100871, P. R. China
zdwang@pku.edu.cn
[Abstract-pdf]
Given a Lie group $G$ with finitely many components and a compact Lie group $A$ which acts on $G$ by automorphisms, we prove that there always exists an $A$-invariant maximal compact subgroup $K$ of $G$, and that for every such $K$, the natural map $H^1(A,K)\rightarrow H^1(A,G)$ is bijective. This generalizes a classical result of Serre and a recent result of the first and third named authors of the current paper.
Keywords: Nonabelian cohomology, compact Lie group, maximal compact subgroup.
MSC: 20J06, 22E15, 57S15
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