“Levi decompositions of linear algebraic groups and non-abelian cohomology” (manuscript)

[2024-08-26] Accepted for publication in Pacific J. Math. Special issue in memory of Gary Seitz

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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 in (McNinch 2013): does \(G\) have a Levi decomposition (over \(k\))?

Using non-abelian 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 \(k\).