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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)
Published electronically: March 12, 2010
MathSciNet review: 2667427
Keywords:Algebraic group, reductive group, Jacobson–Morosov, representation, group action
MSC: Primary 14L15; Secondary 20G15

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Table of Contents


  • Introduction
  • Notation and terminology
  • Chapter 1. Affine group schemes over a field of characteristic zero
  • Chapter 2. Universal and minimal reductive homomorphisms
  • Chapter 3. Groups with action of a proreductive group
  • Chapter 4. Families of minimal reductive homomorphisms


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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