Algebraic geometry 04 Linear algebraic groups, invariant theory 4

Two contributions on closely related subjects: the theory of linear algebraic groups and invariant theory, by well-known experts in the fields. The book will be very useful as a reference and research guide to graduate students and researchers in mathematics and theoretical physics. Contents......Page all_29622_to_00288.cpc0005.djvu Historical Comments......Page all_29622_to_00288.cpc0008.djvu 1.2. Morphisms......Page all_29622_to_00288.cpc0011.djvu 1.6. Non-Affine Varieties......Page all_29622_to_00288.cpc0012.djvu 2.1. The Definition of a Linear Algebraic Group......Page all_29622_to_00288.cpc0013.djvu 2.2. Some Basic Facts......Page all_29622_to_00288.cpc0017.djvu 2.3. G-Spaces......Page all_29622_to_00288.cpc0018.djvu 2.4. The Lie Algebra of an Algebraic Group......Page all_29622_to_00288.cpc0020.djvu 2.5. Quotients......Page all_29622_to_00288.cpc0022.djvu §3. Structural Properties of Linear Algebraic Groups......Page all_29622_to_00288.cpc0023.djvu 3.1. Jordan Decomposition and Related Results......Page all_29622_to_00288.cpc0024.djvu 3.2. Diagonalizable Groups and Tori......Page all_29622_to_00288.cpc0025.djvu 3.4. Connected Solvable Groups......Page all_29622_to_00288.cpc0027.djvu 3.5. Parabolic Subgroups and Borel Subgroups......Page all_29622_to_00288.cpc0029.djvu 4.1. Groups of Rank One......Page all_29622_to_00288.cpc0032.djvu 4.2. The Root Datum and the Root System......Page all_29622_to_00288.cpc0034.djvu 4.3. Basic Properties of Reductive Groups......Page all_29622_to_00288.cpc0037.djvu 4.4. Existence and Uniqueness Theorems for Reductive Groups......Page all_29622_to_00288.cpc0041.djvu 4.5. Classification of Quasi-simple Linear Algebraic Groups......Page all_29622_to_00288.cpc0043.djvu 4.6. Representation Theory......Page all_29622_to_00288.cpc0046.djvu 1.1. F-Structures on Affine Varieties......Page all_29622_to_00288.cpc0052.djvu 1.2. F-Structures on Arbitrary Varieties......Page all_29622_to_00288.cpc0053.djvu 1.3. Forms......Page all_29622_to_00288.cpc0054.djvu 1.4. Restriction of the Ground Field......Page all_29622_to_00288.cpc0055.djvu 2.1. Generalities About F-Groups......Page all_29622_to_00288.cpc0056.djvu 2.2. Quotients......Page all_29622_to_00288.cpc0058.djvu 2.3. Forms......Page all_29622_to_00288.cpc0059.djvu 2.4. Restriction of the Ground Field......Page all_29622_to_00288.cpc0060.djvu 3.1. F-Tori......Page all_29622_to_00288.cpc0061.djvu 3.2. F-Tori in F-Groups......Page all_29622_to_00288.cpc0063.djvu 3.3. Split Tori in F-Groups......Page all_29622_to_00288.cpc0064.djvu 4.1. Solvable Groups......Page all_29622_to_00288.cpc0065.djvu 4.2. Sections......Page all_29622_to_00288.cpc0066.djvu 4.3. Elementary Unipotent Groups......Page all_29622_to_00288.cpc0067.djvu 4.5. Basic Results About Solvable F-Groups......Page all_29622_to_00288.cpc0068.djvu 5.1. Split Reductive Groups......Page all_29622_to_00288.cpc0069.djvu 5.2. Parabolic Subgroups......Page all_29622_to_00288.cpc0070.djvu 5.3. The Small Root System......Page all_29622_to_00288.cpc0072.djvu 5.4. The Groups G(F)......Page all_29622_to_00288.cpc0076.djvu 5.5. The Spherical Tits Building of a Reductive F-Group......Page all_29622_to_00288.cpc0078.djvu 6.1. Isomorphism Theorem......Page all_29622_to_00288.cpc0079.djvu 6.2. Existence......Page all_29622_to_00288.cpc0081.djvu 6.3. Representation Theory of F-Groups......Page all_29622_to_00288.cpc0088.djvu 1.1. Algebraic Subalgebras......Page all_29622_to_00288.cpc0090.djvu 2.1. Locally Compact Fields......Page all_29622_to_00288.cpc0092.djvu 2.2. Real Lie Groups......Page all_29622_to_00288.cpc0095.djvu 3.1. Lang's Theorem and its Consequences......Page all_29622_to_00288.cpc0098.djvu 3.2. Finite Groups of Lie Type......Page all_29622_to_00288.cpc0101.djvu 3.3. Representations of Finite Groups of Lie Type......Page all_29622_to_00288.cpc0103.djvu 4.1. The Apartment and Affine Dynkin Diagram......Page all_29622_to_00288.cpc0105.djvu 4.2. The Affine Building......Page all_29622_to_00288.cpc0108.djvu 4.3. Tits System, Decompositions......Page all_29622_to_00288.cpc0111.djvu 4.4. Local Fields......Page all_29622_to_00288.cpc0112.djvu 5.1. Adele Groups......Page all_29622_to_00288.cpc0113.djvu 5.2. Reduction Theory......Page all_29622_to_00288.cpc0116.djvu 5.3. Finiteness Results......Page all_29622_to_00288.cpc0119.djvu References......Page all_29622_to_00288.cpc0122.djvu Contents......Page all_29622_to_00288.cpc0127.djvu Conventions and Notation......Page all_29622_to_00288.cpc0131.djvu 0.1. The Subject of Invariant Theory......Page all_29622_to_00288.cpc0133.djvu 0.2. Sources of Invariant Theory......Page all_29622_to_00288.cpc0135.djvu 0.3. Geometric Methods......Page all_29622_to_00288.cpc0136.djvu 0.4. Invariants of the Symmetric Group......Page all_29622_to_00288.cpc0137.djvu 0.6. Invariants of a Linear Operator......Page all_29622_to_00288.cpc0138.djvu 0.7. Unimodular Invariants of a Quadratic Form......Page all_29622_to_00288.cpc0139.djvu 0.9. Invariants of a System of Vectors......Page all_29622_to_00288.cpc0140.djvu 0.10. Applications to Projective Geometry......Page all_29622_to_00288.cpc0142.djvu 0.12. Invariants of Binary Forms......Page all_29622_to_00288.cpc0144.djvu 0.13. Invariants of Binary Polyhedral Groups......Page all_29622_to_00288.cpc0146.djvu 0.14. Invariants of a Ternary Cubic Form......Page all_29622_to_00288.cpc0149.djvu 1.1. Regular and Rational Actions......Page all_29622_to_00288.cpc0150.djvu 1.2. Embedding Theorems......Page all_29622_to_00288.cpc0152.djvu 1.3. Orbits......Page all_29622_to_00288.cpc0153.djvu 1.4. Stabilizers......Page all_29622_to_00288.cpc0155.djvu 1.5. Inheritance of Orbits......Page all_29622_to_00288.cpc0156.djvu 2.1. Introduction......Page all_29622_to_00288.cpc0157.djvu 2.2. The Graph of an Action......Page all_29622_to_00288.cpc0158.djvu 2.3. Separation of Orbits in General Position......Page all_29622_to_00288.cpc0159.djvu 2.4. Rational Quotient......Page all_29622_to_00288.cpc0160.djvu 2.5. Sections......Page all_29622_to_00288.cpc0161.djvu 2.7. Birational Classification of Actions......Page all_29622_to_00288.cpc0163.djvu 2.8. Relative Sections......Page all_29622_to_00288.cpc0164.djvu 2.9. The Rationality Problem......Page all_29622_to_00288.cpc0166.djvu 3.1. Introduction......Page all_29622_to_00288.cpc0168.djvu 3.2. Connection Between Integral and Rational Invariants......Page all_29622_to_00288.cpc0169.djvu 3.3. Basic Invariants......Page all_29622_to_00288.cpc0170.djvu 3.4. Hilbert's Theorem on Invariants......Page all_29622_to_00288.cpc0172.djvu 3.5. Constructive Invariant Theory......Page all_29622_to_00288.cpc0173.djvu 3.6. Hilbert's Fourteenth Problem......Page all_29622_to_00288.cpc0174.djvu 3.7. Grosshans Subgroups......Page all_29622_to_00288.cpc0175.djvu 3.8. Chevalley Sections......Page all_29622_to_00288.cpc0177.djvu 3.9. Properties of the Algebra of Invariants......Page all_29622_to_00288.cpc0180.djvu 3.10. Facts about Poincare Series......Page all_29622_to_00288.cpc0181.djvu 3.11. The Poincare Series of the Algebra of Invariants......Page all_29622_to_00288.cpc0183.djvu 3.12. Covariants......Page all_29622_to_00288.cpc0185.djvu 3.13. The Global Module of Covariants......Page all_29622_to_00288.cpc0187.djvu 3.14. The Algebra of Covariants......Page all_29622_to_00288.cpc0188.djvu 4.2. The Geometric Quotient......Page all_29622_to_00288.cpc0189.djvu 4.4. Construction of the Quotient for an Action of a Reductive Group on an Affine Variety......Page all_29622_to_00288.cpc0191.djvu 4.5. Igusa's Criterion......Page all_29622_to_00288.cpc0194.djvu 4.6. Construction of the Quotient for an Action of a Reductive Group on an Arbitrary Variety......Page all_29622_to_00288.cpc0195.djvu 4.7. Homogeneous Spaces......Page all_29622_to_00288.cpc0197.djvu 4.8. Homogeneous Fiber Spaces......Page all_29622_to_00288.cpc0198.djvu 5.1. Introduction......Page all_29622_to_00288.cpc0200.djvu 5.2. Asymptotic Cones......Page all_29622_to_00288.cpc0201.djvu 5.3. The Hilbert-Mumford Criterion......Page all_29622_to_00288.cpc0202.djvu 5.4. The Support Method......Page all_29622_to_00288.cpc0203.djvu 5.5. The Characteristic of a Nilpotent Element......Page all_29622_to_00288.cpc0205.djvu 5.6. Stratification and Resolution of Singularities of the Null-Cone......Page all_29622_to_00288.cpc0209.djvu 6.1. Slices: Statement of the Problem......Page all_29622_to_00288.cpc0211.djvu 6.2. Excellent Morphisms......Page all_29622_to_00288.cpc0212.djvu 6.3. Étale Slices......Page all_29622_to_00288.cpc0213.djvu 6.4. Stabilizers of Points in a Neighborhood of a Closed Orbit......Page all_29622_to_00288.cpc0215.djvu 6.6. Étale Slices and Analytic Slices......Page all_29622_to_00288.cpc0216.djvu 6.7. Structure of Fibers of the Quotient Morphism......Page all_29622_to_00288.cpc0217.djvu 6.8. The Theorem on Reaching the Boundary of an Orbit by Means of a One-Parameter Subgroup......Page all_29622_to_00288.cpc0219.djvu 6.9. Luna's Stratification......Page all_29622_to_00288.cpc0220.djvu 6.10. Sheets......Page all_29622_to_00288.cpc0223.djvu 6.11. Closedness of Orbits: Luna's Criterion......Page all_29622_to_00288.cpc0225.djvu 6.12. Closedness of Orbits: the Kempf-Ness Criterion......Page all_29622_to_00288.cpc0226.djvu 6.13. The Closed Orbit Contained in the Closure of a Given Orbit......Page all_29622_to_00288.cpc0228.djvu 6.14. The Moment Mapping......Page all_29622_to_00288.cpc0230.djvu 7.1. Introduction......Page all_29622_to_00288.cpc0232.djvu 7.2. Existence Theorems for S.G.P......Page all_29622_to_00288.cpc0233.djvu 7.3. S.G.P. for Linear Actions......Page all_29622_to_00288.cpc0236.djvu 7.4. Closed Orbits in General Position......Page all_29622_to_00288.cpc0239.djvu 7.5. S.G.P., Chevalley Sections, and Stability......Page all_29622_to_00288.cpc0240.djvu 8.1. Good Properties in Invariant Theory......Page all_29622_to_00288.cpc0242.djvu 8.2. Inheritance of Good Properties......Page all_29622_to_00288.cpc0244.djvu 8.3. Comparison of the Algebras of Invariants of Finite and Connected Reductive Linear Groups......Page all_29622_to_00288.cpc0245.djvu 8.4. The Case of a Two-Dimensional Quotient......Page all_29622_to_00288.cpc0247.djvu 8.5. Adjoint Groups of Graded Lie Algebras (0-Groups)......Page all_29622_to_00288.cpc0248.djvu 8.6. Polar Groups......Page all_29622_to_00288.cpc0250.djvu 8.7. Enumeration of Semisimple Linear Groups with Good Properties......Page all_29622_to_00288.cpc0251.djvu 8.8. Weierstrass Sections......Page all_29622_to_00288.cpc0252.djvu 9.1. Polarization......Page all_29622_to_00288.cpc0254.djvu 9.2. Reduction of the First Fundamental Theorem......Page all_29622_to_00288.cpc0255.djvu 9.3. Invariants of Systems of Vectors and Linear Forms......Page all_29622_to_00288.cpc0257.djvu 9.4. Relations Between Invariants of Systems of Vectors and Linear Forms......Page all_29622_to_00288.cpc0258.djvu 9.5. Invariants of Tensors......Page all_29622_to_00288.cpc0260.djvu Summary Table......Page all_29622_to_00288.cpc0263.djvu References......Page all_29622_to_00288.cpc0267.djvu The problems being solved by invariant theory are far-reaching generalizations and extensions of problems on the'reduction to canonical form'of various is almost the same thing, projective geometry. objects of linear algebra or, what Invariant theory has a ISO-year history, which has seen alternating periods of growth and stagnation, and changes in the formulation of problems, methods of solution, and fields of application. In the last two decades invariant theory has experienced a period of growth, stimulated by a previous development of the theory of algebraic groups and commutative algebra. It is now viewed as a branch of the theory of algebraic transformation groups (and under a broader interpretation can be identified with this theory). We will freely use the theory of algebraic groups, an exposition of which can be found, for example, in the first article of the present volume. We will also assume the reader is familiar with the basic concepts and simplest theorems of commutative algebra and algebraic geometry; when deeper results are needed, we will cite them in the text or provide suitable references.