# The Generalised Jacobson-Morosov Theorem

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*Peter O’Sullivan*

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

# Table of Contents

## The Generalised Jacobson-Morosov Theorem

- Introduction 110 free
- Notation and Terminology 514 free
- Chapter 1. Affine Group Schemes over a Field of Characteristic Zero 716
- Chapter 2. Universal and Minimal Reductive Homomorphisms 3342
- Chapter 3. Groups with Action of a Proreductive Group 5968
- Chapter 4. Families of Minimal Reductive Homomorphisms 8392
- Bibliography 117126
- Index 119128 free