Galois cohomology of real semisimple groups via Kac labelings

TL;DR

This paper computes the first Galois cohomology set of real semisimple groups using Kac labelings of affine Dynkin diagrams, providing a combinatorial approach to understanding their Galois cohomology.

Contribution

It introduces a method to determine H^1(R,G) for real semisimple groups via Kac labelings, extending previous techniques with a new combinatorial perspective.

Findings

Explicit computation of H^1(R,G) for various groups

Connection between Galois cohomology and affine Dynkin diagrams

Extension of Onishchik and Vinberg's method

Abstract

For a connected semisimple group G over the field of real numbers R, using a method of Onishchik and Vinberg, we compute the first Galois cohomology set H^1(R,G) in terms of Kac labelings of the affine Dynkin diagram of G.