This book contains two contributions on closely related subjects: the theory of linear algebraic groups and invariant theory. The first part is written by T. A. Springer, a well-known expert in the first mentioned field. He presents a comprehensive survey, which contains numerous sketched proofs and he discusses the particular features of algebraic groups over special fields (finite, local, and global). The authors of part two-E. B. Vinberg and V. L. Popov-are among the most active researchers in invariant theory. The last 20 years have been a period of vigorous development in this field due to the influence of modern methods from algebraic geometry. The book will be very useful as a reference and research guide to graduate students and researchers in mathematics and theoretical physics.
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目录
I. Linear Algebraic Groups Introduction Historical Comments Chapter 1 Linear Algebraic Groups over an Algebraically Closed Field §1. Recollections from Algebraic Geometry 1.1. Affine Varieties 1.2. Morphisms 1.3. Some Topological Properties 1.4. Tangent Spaces 1.5. Properties of Morphisms 1.6. Non-Affine Varieties §2. Linear Algebraic Groups，Basic Definitions and Properties 2.1. The Definition of a Linear Algebraic Group 2.2. Some Basic Facts 2.3. G-Spaces 2.4. The Lie Algebra of an Algebraic Group 2.5. Quotients §3. Structural Properties of Linear Algebraic Groups 3.1. Jordan Decomposition and Related Results 3.2. Diagonalizable Groups and Tori 3.3. One-Dimensional Connected Groups 3.4. Connected Solvable Groups 3.5. Parabolic Subgroups and Borel Subgroups 3.6. Radicals，Semi-simple and Reductive Groups §4. Reductive Groups 4.1. Groups of Rank One 4.2. The Root Datum and the Root System 4.3. Basic Properties of Reductive Groups 4.4. Existence and Uniqueness Theorems for Reductive Groups 4.5. Classification of Quasi-simple Linear Algebraic Groups 4.6. Representation Theory Chapter 2 Linear Algebraic Groups over Arbitrary Ground Fields §1. Recollections from Algebraic Geometry 1.1 F-Structures on Afine Varieties 1.2. F-Structures on Arbitrary Varieties 1.3. Forms 1.4. Restriction of the Ground Field §2. F-Groups，Basic Properties 2.1. Generalities About F-Groups 2.2. Quotients 2.3. Forms 2.4. Restriction of the Ground Field §3. Tori 3.1. F-Tori 3.2. F-Tori in F-Groups 3.3. Split Tori in F-Groups §4. Solvable Groups 4.1. Solvable Groups 4.2. Sections 4.3. Elementary Unipotent Groups 4.4. Properties of Split Solvable Groups 4.5. Basic Results About Solvable F-Groups §5. Reductive Groups 5.1. Split Reductive Groups 5.2. Parabolic Subgroups 5.3. The Small Root System 5.4. The Groups G(F) 5.5. The Spherical Tits Building of a Reductive F-Group §6. Classification of Reductive F-Groups 6.1. Isomorphism Theorem 6.2. Existence 6.3. Representation Theory of F-Groups Chapter 3 Special Fields §1. Lie Algebras of Algebraic Groups in Characteristic Zero 1.1. Algebraic Subalgebras §2. Algebraic Groups and Lie Groups 2.1. Locally Compact Fields 2.2. Real Lie Groups §3. Linear Algebraic Groups over Finite Fields 3.1. Lang's Theorem and its Consequences. 3.2. Finite Groups of Lie Type 3.3. Representations of Finite Groups of Lie Type §4. Linear Algebraic Groups over Fields with a Valuation 4.1. The Apartment and Affine Dynkin Diagram 4.2. The Affine Building 4.3. Tits System，Decompositions 4.4. Local Fields §5. Global Fields 5.1. Adele Groups 5.2. Reduction Theory 5.3. Finiteness Results 5.4. Galois Cohomology References II. Invariant Theory Preface Conventions and Notation Introduction 0.1. The Subject of Invariant Theory 0.2. Sources of Invariant Theory 0.3. Geometric Methods 0.4. Invariants of the Symmetric Group 0.5. Orthogonal Invariants of a Vector 0.6. Invariants of a Linear Operator 0.7. Unimodular Invariants of a Quadratic Form 0.8. Orthogonal Invariants of a Quadratic Form 0.9. Invariants of a System of Vectors 0.10. Applications to Projective Geometry 0.11. Unordered Sets of Points of the Projective Line and Hyperelliptic Curves 0.12. Invariants of Binary Forms 0.13. Invariants of Binary Polyhedral Groups 0.14. Invariants of a Ternary Cubic Form §1. Actions of Algebraic Groups 1.1. Regular and Rational Actions 1.2. Embedding Theorems 1.3. Orbits 1.4. Stabilizers 1.5. Inheritance of Orbits §2. Rational Invariants 2.1. Introduction 2.2. The Graph of an Action 2.3. Separation of Orbits in General Position 2.4. Rational Quotient 2.5. Sections 2.6. Special Groups 2.7. Birational Classification of Actions 2.8. Relative Sections 2.9. The Rationality Problem §3. Integral Invariants and Covariants 3.1. Introduction 3.2. Connection Between Integral and Rational Invariants 3.3. Basic Invariants 3.4. Hilbert's Theorem on Invariants 3.5. Constructive Invariant Theory 3.6. Hilbert's Fourteenth Problem 3.7. Grosshans Subgroups 3.8. Chevalley Sections 3.9. Properties of the Algebra of Invariants 3.10. Facts about Poincare Series 3.11. The Poincare Series of the Algebra of Invariants 3.12. Covariants 3.13. The Global Module of Covariants 3.14. The Algebra of Covariants §4. Quotients 4.1. Introduction 4.2. The Geometric Quotient 4.3. The Categorical Quotient 4.4. Construction of the Quotient for an Action of a Reductive Group on an Aftine Variety 4.5. Igusa's Criterion 4.6. Construction of the Quotient for an Action of a Reductive Group on an Arbitrary Variety 4.7. Homogeneous Spaces 4.8. Homogeneous Fiber Spaces §5. The Null-Cone 5.1. Introduction 5.2. Asymptotic Cones 5.3. The Hilbert-Mumford Criterion 5.4. The Support Method 5.5. The Characteristic of a Nilpotent Element 5.6. Stratification and Resolution of Singularities of the Null-Cone §6. The Fine Geometry of an Actio 6.1. Slices：Statement of the Problem 6.2. Éxcellent Morphisms 6.3. Gtale Slices 6.4. Stabilizers of Points in a Neighborhood of a Closed Orbit 6.5. Slice at a Nonsingular Point 6.6. Étale Slices and Analytic Slices 6.7. Structure of Fibers of the Quotient Morphism 6.8. The Theorem on Reaching the Boundary of an Orbit by Means of a One-Parameter Subgroup 6.9. Luna's Stratification 6.10. Sheets 6.11. Closedness of Orbits：Luna's Criterion 6.12. Closedness of Orbits：the Kempf-Ness Criterion 6.13. The Closed Orbit Contained in the Closure of a Given Orbit 6.14. The Moment Mapping §7. Stabilizers in General Position 7.1. Introduction 7.2. Existence Theorems for S.G.P 7.3. S.G.P. for Linear Actions 7.4. Closed Orbits in General Position 7.5. S.G.P., Chevalley Sections, and Stability §8. Reductive Linear Groups with a Free Algebra of Invariants 8.1. Good Properties in Invariant Theory 8.2. Inheritance of Good Properties 8.3. Comparison of the Algebras of Invariants of Finite and Connected Reductive Linear Groups 8.4. The Case of a Two-Dimensional Quotient 8.5. Adjoint Groups of Graded Lie Algebras (6-Groups) 8.6. Polar Groups 8.7. Enumeration of Semisimple Linear Groups with Good Properties 8.8. Weierstrass Sections §9. Classical Invariant Theory 9.1. Polarization 9.2. Reduction of the First Fundamental Theorem 9.3. Invariants of Systems of Vectors and Linear Forms 9.4. Relations Between Invariants of Systems of Vectors and Linear Forms 9.5. Invariants of Tensors Summary Table References
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