eBook ISBN:  9781470405878 
Product Code:  MEMO/207/973.E 
List Price:  $73.00 
MAA Member Price:  $65.70 
AMS Member Price:  $43.80 
eBook ISBN:  9781470405878 
Product Code:  MEMO/207/973.E 
List Price:  $73.00 
MAA Member Price:  $65.70 
AMS Member Price:  $43.80 

Book DetailsMemoirs of the American Mathematical SocietyVolume: 207; 2010; 120 ppMSC: Primary 20; Secondary 13; 14
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 grouptheoretic 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

Chapters

Introduction

Notation and Terminology

1. Affine Group Schemes over a Field of Characteristic Zero

2. Universal and Minimal Reductive Homomorphisms

3. Groups with Action of a Proreductive Group

4. Families of Minimal Reductive Homomorphisms


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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 grouptheoretic 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 \(k\).

Chapters

Introduction

Notation and Terminology

1. Affine Group Schemes over a Field of Characteristic Zero

2. Universal and Minimal Reductive Homomorphisms

3. Groups with Action of a Proreductive Group

4. Families of Minimal Reductive Homomorphisms