Conjugacy classes in semisimple algebraic groups / James E. Humphreys.

Format:

Book

Author/Creator:

Humphreys, James E., author.

Series:

Mathematical surveys and monographs ; no. 43.

Mathematical surveys and monographs ; volume 43

Language:

English

Subjects (All):

Linear algebraic groups.

Lie algebras.

Semisimple Lie groups.

Conjugacy classes.

Physical Description:

1 online resource (217 p.)

Place of Publication:

Providence, Rhode Island : American Mathematical Society, [1995]

Summary:

The book provides a useful exposition of results on the structure of semisimple algebraic groups over an arbitrary algebraically closed field. After the fundamental work of Borel and Chevalley in the 1950s and 1960s, further results were obtained over the next thirty years on conjugacy classes and centralizers of elements of such groups.

Contents:

""Contents""; ""Preface""; ""Frequently Used Symbols""; ""Chapter 0. Review of Semisimple Groups""; ""0.1. Jordan-Che valley Decomposition in Algebraic Group""; ""0.2. Semisimple and Reductive Groups""; ""0.3. Borel Subgroups""; ""0.4. Roots""; ""0.5. Root Groups""; ""0.6. Weyl Group""; ""0.7. Bruhat Decomposition""; ""0.8. Parabolic Subgroups""; ""0.9. Simple Algebraic Groups""; ""0.10. Classification""; ""0.11. Rational Representations and Weights""; ""0.12. Lie Algebras and the Adjoint Representation""; ""0.13. Ideals in g""; ""0.14. Morphisms""; ""0.15. A Key Geometric Argument""

""Chapter 1. Basic Facts about Classes and Centralizers""""1.1. Introduction""; ""1.2. Example: General and Special Linear Groups""; ""1.3. Centralizers of Nilpotent and Unipotent Matrices""; ""1.4. Orbits and Isotropy Groups""; ""1.5. Classes and Centralizers""; ""1.6. Minimum Dimension of a Centralizer""; ""1.7. Closures of Conjugacy Classes""; ""1.8. Morphisms and Subgroups""; ""1.9. Fields of Definition""; ""1.10. Global and Infinitesimal Centralizers""; ""1.11. Orbits in the Lie Algebra""; ""1.12. Connectedness Properties""; ""1.13. Centralizers of Unipotent Elements""

""1.14. Abelian Subgroups of Centralizers""""Chapter 2. Centralizers of Semisimple Elements""; ""2.1. Subgroups of Maximal Rank""; ""2.2. Centralizers of Semisimple Elements""; ""2.3. Regular Semisimple Elements""; ""2.4. Borel Subgroups and Regular Semisimple Elements""; ""2.5. The Set of Regular Semisimple Elements""; ""2.6. Overview of the Connectedness Theorem""; ""2.7. Roots of Unity in a Field""; ""2.8. Tori and Characters""; ""2.9. The Dual Group of a Reductive Group""; ""2.10. Semisimple Elements of Finite Order""; ""2.11. Connectedness Theorem""

""2.12. Characterization of Centralizers""""2.13. Proof of Carter's Criterion""; ""2.14. Example: Type C""; ""2.15. Deriziotis Criterion""; ""Chapter 3. The Adjoint Quotient""; ""3.1. Class Functions and Characters""; ""3.2. The Algebra of Class Functions""; ""3.3. Quotient of an Affine Variety by a Finite Group""; ""3.4. Semisimple Classes and Class Functions""; ""3.5. The Steinberg Map""; ""3.6. Rational Class Functions""; ""3.7. Similarity and Conjugacy""; ""3.8. Richardson's Finiteness Theorem""; ""3.9. Unipotent Classes in Good Characteristic""; ""3.10. Nilpotent Orbits""

""3.11. Lusztig's Finiteness Theorem""""Chapter 4. Regular Elements""; ""4.1. Unipotent Elements Lying in a Unique Borel Subgroup""; ""4.2. Dimension of the Unipotent Variety""; ""4.3. Existence of Regular Unipotent Elements""; ""4.4. Coxeter Elements of the Weyl Group""; ""4.5. Steinberg's Existence Proof""; ""4.6. Conjugacy of Regular Unipotent Elements""; ""4.7. Centralizer of a Regular Unipotent Element""; ""4.8. Jordan Decomposition of a Regular Element""; ""4.9. Borel Subgroups Containing a Regular Element""; ""4.10. Bijection Between Semisimple Classes and Regular Classes""

""4.11. Some Refinements""

Notes:

Description based upon print version of record.

Includes bibliographical references (pages 183-193) and index.

Description based on print version record.

ISBN:

1-4704-1274-8