Computing Galois cohomology of a real linear algebraic group

TL;DR

The paper provides a comprehensive, computer-implemented framework to compute the first Galois cohomology set ${\rm H}^1({\mathds R},{\bf G})$ for real linear algebraic groups ${\bf G}$, including non-connected and non-reductive cases. It combines Sarkarian-style reductions (Levi decompositions, Sansuc’s lemma) with explicit torus and Weyl-group techniques to reduce to tractable subproblems, such as computing ${\rm H}^1$ for tori and connected reductive groups, and then lifts to the full group via dévissage. The authors implement the methods in GAP (with SageMath interfaces) and demonstrate practical performance on groups like ${\rm SO}(p,q)$, providing explicit representatives of cocycle classes and explicit realizations of equivalences. This enables explicit orbit classifications, forms over $\mathds R$, and related applications in geometry and physics, while contributing a suite of algorithms for nonabelian and higher cohomology in real algebraic groups.

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.

Theorems & Definitions (41)

• Remark 3
• Remark 1.2
• Lemma 1.3
• proof
• Definition 2.1
• Example 2.5
• Definition 2.7
• Lemma 2.10: well-known
• proof
• Proposition 3.2