ACTIONS AND INVARIANTS OF ALGEBRAIC GROUPS
CONTENTS CHAPTER 1 ALGEBRAIC GEOMETRY 1 INTRODUCTION 2 COMMUTATIVE ALGEBRA 2 1 RING AND ?ELD EXTENSIONS 2 2 HILBERT’S NULLSTELLENSATZ 2 3 SEPARABILITY 2 4 FAITHFULLY ?AT RING EXTENSIONS 2 5 REGULAR LOCAL RINGS 3 ALGEBRAIC SUBSETS OF THE A?NE SPACE 3 1 BASIC DE?NITIONS 3 2 THE ZARISKI TOPOLOGY 3 3 POLYNOMIAL MAPS MORPHISMS 4 ALGEBRAIC VARIETIES 4 1 SHEAVES ON TOPOLOGICAL SPACES 4 2 THE MAXIMAL SPECTRUM 4 3 A?NE ALGEBRAIC VARIETIES 4 4 ALGEBRAIC VARIETIES 4 5 MORPHISMS OF ALGEBRAIC VARIETIES 4 6 COMPLETE VARIETIES 4 7 SINGULAR POINTS AND NORMAL VARIETIES 5 DEEPER RESULTS ON MORPHISMS 6 EXERCISES CHAPTER 2 LIE ALGEBRAS 2 DE?NITIONS AND BASIC CONCEPTS 3 THE THEOREMS OF F ENGEL AND S LIE 4 SEMISIMPLE LIE ALGEBRAS 5 COHOMOLOGY OF LIE ALGEBRAS 6 THE THEOREMS OF H WEYL AND F LEVI 7 P–LIE ALGEBRAS 8 EXERCISES CHAPTER 3 ALGEBRAIC GROUPS: BASIC DE?NITIONS 3 SUBGROUPS AND HOMOMORPHISMS 4 ACTIONS OF A?NE GROUPS ON ALGEBRAIC VARIETIES 5 SUBGROUPS AND SEMIDIRECT PRODUCTS CHAPTER 4 ALGEBRAIC GROUPS: LIE ALGEBRAS AND REPRESENTATIONS 2 HOPF ALGEBRAS AND ALGEBRAIC GROUPS 3 RATIONAL G–MODULES 5 CHARACTERS AND SEMI–INVARIANTS 6 THE LIE ALGEBRA ASSOCIATED TO AN A?NE ALGEBRAIC GROUP 7 EXPLICIT COMPUTATIONS CHAPTER 5 ALGEBRAIC GROUPS: JORDAN DECOMPOSITION AND APPLICATIONS 2 THE JORDAN DECOMPOSITION OF A SINGLE OPERATOR 3 THE JORDAN DECOMPOSITION OF AN ALGEBRA HOMOMORPHISM AND OF A DERIVATION 4 JORDAN DECOMPOSITION FOR COALGEBRAS 5 JORDAN DECOMPOSITION FOR AN A?NE ALGEBRAIC GROUP 6 UNIPOTENCY AND SEMISIMPLICITY 7 THE SOLVABLE AND THE UNIPOTENT RADICAL 8 STRUCTURE OF SOLVABLE GROUPS 9 THE CLASSICAL GROUPS 9 1 THE GENERAL LINEAR GROUP GLN 9 2 THE SPECIAL LINEAR GROUP SLN (CASE A) 9 3 THE PROJECTIVE GENERAL LINEAR GROUP PGLN(K) (CASE A) 9 4 THE SPECIAL ORTHOGONAL GROUP SON (CASES B,D) 9 5 THE SYMPLECTIC GROUP SPN, N = 2M (CASE C) 10 EXERCISES CHAPTER 6 ACTIONS OF ALGEBRAIC GROUPS 2 ACTIONS: EXAMPLES AND ?RST PROPERTIES 3 BASIC FACTS ABOUT THE GEOMETRY OF THE ORBITS 4 CATEGORICAL AND GEOMETRIC QUOTIENTS 5 THE SUBALGEBRA OF INVARIANTS 6 INDUCTION AND RESTRICTION OF REPRESENTATIONS 7 EXERCISES CHAPTER 7 HOMOGENEOUS SPACES 2 EMBEDDING H–MODULES INSIDE G–MODULES 3 DE?NITION OF SUBGROUPS IN TERMS OF SEMI–INVARIANTS 4 THE COSET SPACE G/H AS A GEOMETRIC QUOTIENT 5 QUOTIENTS BY NORMAL SUBGROUPS 6 APPLICATIONS AND EXAMPLES CHAPTER 8 ALGEBRAIC GROUPS AND LIE ALGEBRAS IN CHARACTERISTIC ZERO 2 CORRESPONDENCE BETWEEN SUBGROUPS AND SUBALGEBRAS 3 ALGEBRAIC LIE ALGEBRAS 4 EXERCISES CHAPTER 9 REDUCTIVITY 2 LINEAR AND GEOMETRIC REDUCTIVITY 3 EXAMPLES OF LINEARLY AND GEOMETRICALLY REDUCTIVE GROUPS 4 REDUCTIVITY AND THE STRUCTURE OF THE GROUP 5 REDUCTIVE GROUPS ARE LINEARLY REDUCTIVE IN CHARACTERISTIC ZERO CHAPTER 10 OBSERVABLE SUBGROUPS OF A?NE ALGEBRAIC GROUPS 2 BASIC DE?NITIONS 3 INDUCTION AND OBSERVABILITY 4 SPLIT AND STRONG OBSERVABILITY 5 THE GEOMETRIC CHARACTERIZATION OF OBSERVABILITY CHAPTER 11 A?NE HOMOGENEOUS SPACES 2 GEOMETRIC REDUCTIVITY AND OBSERVABILITY 3 EXACT SUBGROUPS 4 FROM QUASI–A?NE TO A?NE HOMOGENEOUS SPACES 5 EXACTNESS, REYNOLDS OPERATORS, TOTAL INTEGRALS 6 A?NE HOMOGENEOUS SPACES AND EXACTNESS 7 A?NE HOMOGENEOUS SPACES AND REDUCTIVITY 8 EXACTNESS AND INTEGRALS FOR UNIPOTENT GROUPS 9 EXERCISES CHAPTER 12 HILBERT’S 14TH PROBLEM 2 A COUNTEREXAMPLE TO HILBERT’S 14TH PROBLEM 3 REDUCTIVE GROUPS AND ?NITE GENERATION OF INVARIANTS 4 V POPOV’S CONVERSE TO NAGATA’S THEOREM 5 PARTIAL POSITIVE ANSWERS TO HILBERT’S 14TH PROBLEM 6 GEOMETRIC CHARACTERIZATION OF GROSSHANS PAIRS CHAPTER 13 QUOTIENTS 2 ACTIONS BY REDUCTIVE GROUPS: THE CATEGORICAL QUOTIENT 3 ACTIONS BY REDUCTIVE GROUPS: THE GEOMETRIC QUOTIENT 4 CANONICAL FORMS OF MATRICES: A GEOMETRIC PERSPECTIVE 5 ROSENLICHT’S THEOREM 6 FURTHER RESULTS ON INVARIANTS OF ?NITE GROUPS 6 1 INVARIANTS OF GRADED ALGEBRAS 6 2 POLYNOMIAL SUBALGEBRAS OF POLYNOMIAL ALGEBRAS 6 3 THE CASE OF A GROUP GENERATED BY RE?ECTIONS 6 4 THE DEGREE OF THE FUNDAMENTAL INVARIANTS FOR A ?NITE GROUP