Computing Galois cohomology of a real linear algebraic group

TL;DR

This paper presents a computational method to determine the first Galois cohomology set of real linear algebraic groups, facilitating explicit calculations of 1-cocycles and their equivalences.

Contribution

It introduces an algorithmic approach, implemented on computer, for computing Galois cohomology sets of real algebraic groups, including explicit cocycle classification.

Findings

Successfully computes H^1(R,G) for various groups

Provides explicit representatives for cohomology classes

Enables practical calculations in algebraic group theory

Abstract

Let G be a linear algebraic group, not necessarily connected or reductive, over the field of real numbers R. We describe a method, implemented on computer, to find the first Galois cohomology set H^1(R,G). The output is a list of 1-cocycles in G. Moreover, we have an implemented algorithm that, given a 1-cocycle z in Z^1(R,G), finds the cocycle in the computed list to which z is equivalent, together with an element of G(C) realizing the equivalence.