Levi decompositions of linear algebraic groups and nonabelian cohomology

George J. McNinch

Vol. 336 (2025), No. 1-2, 379–397

DOI: 10.2140/pjm.2025.336.379

Abstract

Let $k$ be a field, and let $G$ be a linear algebraic group over $k$ for which the unipotent radical $U$ of $G$ is defined and split over $k$. Consider a finite, separable field extension $\ell$ of $k$ and suppose that the group $G_{\ell }$ obtained by base-change has a Levi decomposition (over $\ell$). We continue here our study of the question previously investigated (Arch. Math. 100:1 (2013), 7–24): does $G$ have a Levi decomposition (over $k$)?

Using nonabelian cohomology we give some condition under which this question has an affirmative answer. On the other hand, we provide an(other) example of a group $G$ as above which has no Levi decomposition over $k$.

To the memory of Gary Seitz (1943–2023)

Keywords

linear algebraic groups, cohomology, Levi factors, reductive groups, unipotent groups

Mathematical Subject Classification

Primary: 20G15

Secondary: 20G07, 20G10

Milestones

Received: 1 July 2024

Revised: 25 August 2024

Accepted: 26 August 2024

Published: 26 May 2025