Linearity for actions on vector groups.
type: Item
instance of: scholarly article
title: Linearity for actions on vector groups.
zbMATH Open document ID: 1316.20052
author: George McNinch
published in: Journal of Algebra
publication date: 2014-09-08T00:00:00Z
reviewed by: Michael Bate
Mathematics Subject Classification ID: 20G15
Mathematics Subject Classification ID: 17B45
Mathematics Subject Classification ID: 20G05
Mathematics Subject Classification ID: 20G10
zbMATH DE Number: 6340907
zbMATH Keywords: algebraic groups
zbMATH Keywords: representations
zbMATH Keywords: actions on vector groups
zbMATH Keywords: equivariant isomorphisms
zbMATH Keywords: cohomology
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2013.08.030
OpenAlex ID: W2077199490
cites work: A geometric approach to complete reducibility.
cites work: Q3999894
cites work: Q3584740
cites work: Q3922829
cites work: Q5602064
cites work: Q5484519
cites work: Finite-Dimensional Division Algebras over Fields
cites work: Q4440466
cites work: Levi decompositions of a linear algebraic group.
cites work: Nilpotent orbits over ground fields of good characteristic
cites work: Q4414310
cites work: Jordan algebras and algebraic groups.
cites work: On unipotent algebraic \(G\)-groups and 1-cohomology.
cites work: Représentations linéaires irréductibles d'un groupe réductif sur un corps quelconque.
DOI: 10.1016/J.JALGEBRA.2013.08.030
review text: Let \(k\) be a field. The \textit{vector groups} in the title of this paper are linear algebraic groups over \(k\) which are isomorphic to a product of copies of the additive group \(\mathbb G_a\). Suppose \(U\) is such a vector group and \(G\) is a linear algebraic group over \(k\) acting on \(U\) by group automorphisms. Then the action of \(G\) on \(U\) gives rise to a linear action on the Lie algebra \(\mathrm{Lie}(U)\). The aim of this paper is to provide conditions on \(G\) which ensure that there is a \(G\)-equivariant isomorphism \(U\simeq\mathrm{Lie}(U)\); in such a situation we call the action of \(G\) on \(U\) \textit{linear}.NEWLINENEWLINE The main result of the paper is the following: assume that the geometric unipotent radical of \(G\) (that is, the unipotent radical of \(G\) of the algebraic closure of \(k\)) arises by base change from a subgroup \(R\) of \(G\), and that \(G\) acts by automorphisms on a vector group \(U\). If \(\mathrm{Lie}(U)\) is a simple module for the identity component \(G^0\), then the action of \(G\) on \(U\) is linear.NEWLINENEWLINE The assumption on the unipotent radical allows the author to avoid possible issues which can arise if the unipotent radical ``cannot be seen'' over \(k\). Under the stronger condition that the subgroup \(R\) in the theorem is also split over \(k\), the author is able to apply his result to the action of \(G\) on the group \(R\) itself to deduce the existence of a Levi factorisation of \(G\) over \(k\), and the conjugacy of any two Levi factors under the action of \(R(k)\), building on his earlier work [Transform. Groups 15, No. 4, 937-964 (2010; Zbl 1221.20034)].
description: scientific article; zbMATH DE number 6340907