Nonabelian cohomology with coefficients in Lie groups

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 $H^1(A,K)\to H^1(A,G)$ is bijective. In this case we also show that $H^1(A,G)$ is always finite. When $A=\ZZ$ and $G$ is compact, we show that for every maximal torus $T$ of the identity component $G_0^\ZZ$ of the group of invariants $G^\ZZ$, $H^1(\ZZ,T)\to H^1(\ZZ,G)$ is surjective if and only if the $\ZZ$-action on $G$ is 1-semisimple, which is also equivalent to that all fibers of $H^1(\ZZ,T)\to H^1(\ZZ,G)$ are finite. When $A=\Zn$, we show that $H^1(\Zn,T)\to H^1(\Zn,G)$ is always surjective, where $T$ is a maximal compact torus of the identity component $G_0^{\Zn}$ of $G^{\Zn}$. When $A$ is cyclic, we also interpret some properties of $H^1(A,G)$ in terms of twisted conjugate actions of $G$.