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

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 $$k$$.