Actions and Invariants of Algebraic Groups

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