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The Generalised Jacobson-Morosov Theorem
Peter O'Sullivan, University of Sydney, NSW, Australia

Memoirs of the American Mathematical Society
2010; 120 pp; softcover
Volume: 207
ISBN-10: 0-8218-4895-X
ISBN-13: 978-0-8218-4895-1
List Price: US$73
Individual Members: US$43.80
Institutional Members: US$58.40
Order Code: MEMO/207/973
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The author considers 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. The author gives 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, the author considers 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 \(k\).

Table of Contents

  • Introduction
  • Notation and terminology
  • Affine Group schemes over a field of characteristic zero
  • Universal and minimal reductive homomorphisms
  • Groups with action of a proreductive group
  • Families of minimal reductive homomorphisms
  • Bibliography
  • Index
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