Actions and Invariants of Algebraic Groups
Crc Press Inc (Verlag)
978-0-8247-5896-7 (ISBN)
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Actions and Invariants of Algebraic Groups presents a self-contained introduction to geometric invariant theory that links the basic theory of affine algebraic groups to Mumford's more sophisticated theory. The authors systematically exploit the viewpoint of Hopf algebra theory and the theory of comodules to simplify and compactify many of the relevant formulas and proofs.
The first two chapters introduce the subject and review the prerequisites in commutative algebra, algebraic geometry, and the theory of semisimple Lie algebras over fields of characteristic zero. The authors' early presentation of the concepts of actions and quotients helps to clarify the subsequent material, particularly in the study of homogeneous spaces. This study includes a detailed treatment of the quasi-affine and affine cases and the corresponding concepts of observable and exact subgroups.
Among the many other topics discussed are Hilbert's 14th problem, complete with examples and counterexamples, and Mumford's results on quotients by reductive groups. End-of-chapter exercises, which range from the routine to the rather difficult, build expertise in working with the fundamental concepts. The Appendix further enhances this work's completeness and accessibility with an exhaustive glossary of basic definitions, notation, and results.
ALGEBRAIC GEOMETRY
Introduction
Commutative Algebra
Algebraic subsets of the Affine Space
Algebraic Varieties
Deeper Results on Morphisms
Exercises
LIE ALGEBRAS
Introduction
Definitions and Basic Concepts
The Theorems of F. Engel and S. Lie
Semisimple Lie Algebras
Cohomology of Lie Algebras
The Theories of H. Weyl and F. Levi
p-Lie Algebras
Exercises
ALGEBRAIC GROUPS: BASIC DEFINITIONS
Introduction
Definitions and Basic Concepts
Subgroups and Homomorphisms
Actions of Affine Groups on Algebraic Varieties
Subgroups and Semidirect Products
Exercises
ALGEBRAIC GROUPS: LIE ALGEBRAS AND REPRESENTATIONS
Introduction
Hopf Algebras and Algebraic Groups
Rational G-Modules
Representations of SL(2)
Characters and Semi-Invariants
The Lie Algebra Associated to an Affine Algebraic Group
Explicit Computations
Exercises
ALGEBRAIC GROUPS: JORDAN DECOMPOSITION AND APPLICATIONS
Introduction
The Jordan Decomposition of a Single Operator
The Jordan Decompostiion of an Algebra Homomorphism and of a Derivation
Jordan Decomposition for Coalgebras
Jordan Decomposition for an Affine Algebraic Group
Unipotency and Semisimplicity
The Solvable and the Unipotent Radical
Structure of Solvable Groups
The Classical Groups
Exercises
ACTIONS OF ALGEBRAIC GROUPS
Introduction
Actions: Examples and First Properties
Basic Facts about te Geometry of the Orbits
Categorical and Geometric Quotients
The Subalgebras of Invariants
Induction and Restriction of Representations
Exercises
HOMOGENEOUS SPACES
Introduction
Embedding H-Modules inside G-Modules
Definition of Subgroups in Terms of Semi-Invariants
The Coset Space G/H as a Geometric Quotient
Quotients by Normal Subgroups
Applications and Examples
Exercises
ALGEBRAIC GROUPS AND LIE ALGEBRAS IN CHARACTERISTIC ZERO
Introduction
Correspondence Between Subgroups and Subalgebras
Algebraic Lie Algebras
Exercises
REDUCTIVITY
Introduction
Linear and Geometric Reductivity
Examples of Linearly and Geometrically Reductive Groups
Reductivity and the Structure of the Group
Reductive Groups are Linearly Reductive in Characteristic Zero
Exercises
OBSERVABLE SUBGROUPS OF AFFINE ALGEBRAIC GROUPS
Introduction
Basic Definitions
Induction and Observability
Split and Strong Observability
The Geometric Characterization of Observability
Exercises
AFFINE HOMOGENEOUS SPACES
Introduction
Geometric Reductivity and Observability
Exact Subgroups
From Quasi-Affine to Affine Homogeneous Spaces
Exactness, Reynolds Operators, Total Integrals
Affine Homogeneous Spaces and Exactness
Affine Homogeneous Spaces and Reductivity
Exactness and Integrals for Unipotent Groups
Exercises
HILBERT'S FOURTEENTH PROBLEM
Introduction
A Counterexample to Hilbert's 14th Problem
Reductive Groups and Finite Generation of Invariants
V. Popov's Converse to Nagata's Theorem
Partial Positive Answers to Hilbert's 14th Problem
Geometric characterization of Grosshans Pairs
Exercises
QUOTIENTS
Introduction
Actions by Reductive Groups: The Categorical Quotient
Actions by Reductive Groups: The Geometric Quotient
Canonical Forms of Matrices: A Geometric Perspective
Rosenlicht's Theorem
Further Results on Invariants of Finite Groups
Exercises
APPENDIX: Basic Definitions and Results
Bibliography
Author Index
Glossary of Notation
Index
| Erscheint lt. Verlag | 26.4.2005 |
|---|---|
| Reihe/Serie | Chapman & Hall/CRC Monographs and Research Notes in Mathematics |
| Mitarbeit |
Herausgeber (Serie): Earl Taft |
| Zusatzinfo | 4 Illustrations, black and white |
| Verlagsort | Bosa Roca |
| Sprache | englisch |
| Maße | 152 x 229 mm |
| Gewicht | 771 g |
| Themenwelt | Mathematik / Informatik ► Mathematik ► Algebra |
| ISBN-10 | 0-8247-5896-X / 082475896X |
| ISBN-13 | 978-0-8247-5896-7 / 9780824758967 |
| Zustand | Neuware |
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