**Mathematical Surveys and Monographs**

Volume: 180;
2012;
380 pp;
Hardcover

MSC: Primary 20; 17;

Print ISBN: 978-0-8218-6920-8

Product Code: SURV/180

List Price: $96.00

Individual Member Price: $76.80

**Electronic ISBN: 978-0-8218-8510-9
Product Code: SURV/180.E**

List Price: $96.00

Individual Member Price: $76.80

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#### Supplemental Materials

# Unipotent and Nilpotent Classes in Simple Algebraic Groups and Lie Algebras

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*Martin W. Liebeck; Gary M. Seitz*

This book concerns the theory of unipotent elements in simple
algebraic groups over algebraically closed or finite fields, and
nilpotent elements in the corresponding simple Lie algebras. These
topics have been an important area of study for decades, with
applications to representation theory, character theory, the subgroup
structure of algebraic groups and finite groups, and the
classification of the finite simple groups.

The main focus is on obtaining full information on class
representatives and centralizers of unipotent and nilpotent
elements. Although there is a substantial literature on this topic,
this book is the first single source where such information is
presented completely in all characteristics. In addition, many of the
results are new—for example, those concerning centralizers of
nilpotent elements in small characteristics. Indeed, the whole
approach, while using some ideas from the literature, is novel, and
yields many new general and specific facts concerning the structure
and embeddings of centralizers.

#### Table of Contents

# Table of Contents

## Unipotent and Nilpotent Classes in Simple Algebraic Groups and Lie Algebras

- Cover Cover11 free
- Title page iii4 free
- Contents vii8 free
- Preface xi12 free
- Introduction 114 free
- Preliminaries 922 free
- Classical groups in good characteristic 3952
- Classical groups in bad characteristic: Statement of results 5972
- Nilpotent elements: The symplectic and orthogonal cases, 𝑝=2 6578
- Unipotent elements in symplectic and orthogonal groups, 𝑝=2 91104
- Finite classical groups 113126
- Tables of examples in low dimensions 119132
- Exceptional groups: Statement of results for nilpotent elements 129142
- Parabolic subgroups and labellings 133146
- Reductive subgroups 139152
- Annihilator spaces of nilpotent elements 153166
- Standard distinguished nilpotent elements 169182
- Exceptional distinguished nilpotent elements 203216
- Nilpotent classes and centralizers in 𝐸₈ 219232
- Nilpotent elements in the other exceptional types 263276
- Exceptional groups: Statement of results for unipotent elements 281294
- Corresponding unipotent and nilpotent elements 287300
- Distinguished unipotent elements 299312
- Non-distinguished unipotent classes 317330
- Proofs of theorems 1, 2 and corollaries 3–8 341354
- Tables of nilpotent and unipotent classes in the exceptional groups 351364
- Bibliography 373386
- Glossary of symbols 377390
- Index 379392 free
- Back Cover Back Cover1394

#### Readership

Research mathematicians interested in algebraic groups.