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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: http://dx.doi.org/10.1090/S0065-9266-10-00603-4
Published electronically: March 12, 2010
MathSciNet review: 2667427
Keywords: Algebraic group, reductive group, Jacobson–Morosov, representation, group action
MSC (2000): Primary 14L15; Secondary 20G15

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Table of Contents


Chapters

  • 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

Abstract


We consider homomorphisms from an affine group scheme over a field of characteristic zero to a proreductive group . Using a general categorical splitting theorem, André and Kahn proved that for every 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 is the additive group over . As well as universal homomorphisms, we consider more generally homomorphisms which are minimal, in the sense that factors through no proper proreductive subgroup of . For fixed , it is shown that the minimal with reductive are parametrised by a scheme locally of finite type over .

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