Nonabelian cohomology of compact Lie groups

TL;DR

This paper proves the existence of A-invariant maximal compact subgroups in certain Lie groups and establishes a bijection between their nonabelian cohomology sets, generalizing classical results.

Contribution

It introduces a new generalization of Serre's classical result by establishing A-invariant maximal compact subgroups and a bijective cohomology map for Lie groups with automorphism actions.

Findings

Existence of A-invariant maximal compact subgroups in G.

Bijective correspondence between H^1(A,K) and H^1(A,G).

Generalization of Serre's classical result.

Abstract

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 $H^1(A,K)\to H^1(A,G)$ is bijective. This generalizes a classical result of Serre [6] and a recent result in [1].