eBook ISBN:  9781470405304 
Product Code:  MEMO/198/924.E 
List Price:  $69.00 
MAA Member Price:  $62.10 
AMS Member Price:  $41.40 
eBook ISBN:  9781470405304 
Product Code:  MEMO/198/924.E 
List Price:  $69.00 
MAA Member Price:  $62.10 
AMS Member Price:  $41.40 

Book DetailsMemoirs of the American Mathematical SocietyVolume: 198; 2009; 84 ppMSC: Primary 17; 20; 22; 51
This work is devoted to the isomorphism problem for split KacMoody groups over arbitrary fields. This problem turns out to be a special case of a more general problem, which consists in determining homomorphisms of isotropic semisimple algebraic groups to KacMoody groups, whose image is bounded. Since KacMoody groups possess natural actions on twin buildings, and since their bounded subgroups can be characterized by fixed point properties for these actions, the latter is actually a rigidity problem for algebraic group actions on twin buildings. The author establishes some partial rigidity results, which we use to prove an isomorphism theorem for KacMoody groups over arbitrary fields of cardinality at least \(4\). In particular, he obtains a detailed description of automorphisms of KacMoody groups. This provides a complete understanding of the structure of the automorphism group of KacMoody groups over ground fields of characteristic \(0\).
The same arguments allow to treat unitary forms of complex KacMoody groups. In particular, the author shows that the Hausdorff topology that these groups carry is an invariant of the abstract group structure.
Finally, the author proves the nonexistence of cocentral homomorphisms of KacMoody groups of indefinite type over infinite fields with finitedimensional target. This provides a partial solution to the linearity problem for KacMoody groups.

Table of Contents

Chapters

Introduction

Acknowledgements

Chapter 1. The objects: KacMoody groups, root data and Tits buildings

Chapter 2. Basic tools from geometric group theory

Chapter 3. KacMoody groups and algebraic groups

Chapter 4. Isomorphisms of KacMoody groups: an overview

Chapter 5. Isomorphisms of KacMoody groups in characteristic zero

Chapter 6. Isomorphisms of KacMoody groups in positive characteristic

Chapter 7. Homomorphisms of KacMoody groups to algebraic groups

Chapter 8. Unitary forms of KacMoody groups


RequestsReview Copy – for publishers of book reviewsPermission – for use of book, eBook, or Journal contentAccessibility – to request an alternate format of an AMS title
 Book Details
 Table of Contents
 Requests
This work is devoted to the isomorphism problem for split KacMoody groups over arbitrary fields. This problem turns out to be a special case of a more general problem, which consists in determining homomorphisms of isotropic semisimple algebraic groups to KacMoody groups, whose image is bounded. Since KacMoody groups possess natural actions on twin buildings, and since their bounded subgroups can be characterized by fixed point properties for these actions, the latter is actually a rigidity problem for algebraic group actions on twin buildings. The author establishes some partial rigidity results, which we use to prove an isomorphism theorem for KacMoody groups over arbitrary fields of cardinality at least \(4\). In particular, he obtains a detailed description of automorphisms of KacMoody groups. This provides a complete understanding of the structure of the automorphism group of KacMoody groups over ground fields of characteristic \(0\).
The same arguments allow to treat unitary forms of complex KacMoody groups. In particular, the author shows that the Hausdorff topology that these groups carry is an invariant of the abstract group structure.
Finally, the author proves the nonexistence of cocentral homomorphisms of KacMoody groups of indefinite type over infinite fields with finitedimensional target. This provides a partial solution to the linearity problem for KacMoody groups.

Chapters

Introduction

Acknowledgements

Chapter 1. The objects: KacMoody groups, root data and Tits buildings

Chapter 2. Basic tools from geometric group theory

Chapter 3. KacMoody groups and algebraic groups

Chapter 4. Isomorphisms of KacMoody groups: an overview

Chapter 5. Isomorphisms of KacMoody groups in characteristic zero

Chapter 6. Isomorphisms of KacMoody groups in positive characteristic

Chapter 7. Homomorphisms of KacMoody groups to algebraic groups

Chapter 8. Unitary forms of KacMoody groups