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 $H^1(A,K)\rightarrow H^1(A,G)$ 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 $G$.