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# memo_has_moved_text(); The generalised Jacobson-Morosov theorem

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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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 $H \to K$ 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 $k$.