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The generalised Jacobson–Morosov theorem

About this Title

Peter O’Sullivan, School of Mathematics and Statistics F07, University of Sydney NSW 2006, Australia

Publication: Memoirs of the American Mathematical Society
Publication Year: 2010; Volume 207, Number 973
ISBNs: 978-0-8218-4895-1 (print); 978-1-4704-0587-8 (online)
DOI: https://doi.org/10.1090/S0065-9266-10-00603-4
Published electronically: March 12, 2010
Keywords: Algebraic group, reductive group, Jacobson–Morosov, representation, group action
MSC: Primary 20G15; Secondary 13A50, 14L30

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Table of Contents

Chapters

  • Introduction
  • Notation and Terminology
  • 1. Affine Group Schemes over a Field of Characteristic Zero
  • 2. Universal and Minimal Reductive Homomorphisms
  • 3. Groups with Action of a Proreductive Group
  • 4. Families of Minimal Reductive Homomorphisms

Abstract

We consider homomorphisms $H \to K$ from an affine group scheme $H$ over a field $k$ of characteristic zero to a proreductive group $K$. Using a general categorical splitting theorem, André and Kahn proved that for every $H$ there exists such a homomorphism which is universal up to conjugacy. We give a purely group-theoretic proof of this result. The classical Jacobson–Morosov theorem is the particular case where $H$ is the additive group over $k$. As well as universal homomorphisms, we consider more generally homomorphisms $H \to K$ which are minimal, in the sense that $H \to K$ factors through no proper proreductive subgroup of $K$. For fixed $H$, it is shown that the minimal $H \to K$ with $K$ reductive are parametrised by a scheme locally of finite type over $k$.

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