Actions and Invariants of Algebraic Groups (Chapman & Hall/CRC Monographs and Research Notes in Mathematics)

Description

Actions and Invariants of Algebraic Groups, Second Edition presents a self-contained introduction to geometric invariant theory starting from the basic theory of affine algebraic groups and proceeding towards more sophisticated dimensions." Building on the first edition, this book provides an introduction to the theory by equipping the reader with the tools needed to read advanced research in the field. Beginning with commutative algebra, algebraic geometry and the theory of Lie algebras, the book develops the necessary background of affine algebraic groups over an algebraically closed field, and then moves toward the algebraic and geometric aspects of modern invariant theory and quotients. Cover Half Title Title Page Copyright Page Table of Contents Preface to the Second Edition Preface to the First Edition Enumeration of items and cross-references 1: Algebraic geometry: basic definitions and results 1 Introduction 2 Commutative algebra 2.1 Ring and field extensions 2.2 Hilbert's Nullstellensatz 2.3 Separability 2.4 Faithfully flat ring extensions 2.5 Regular local rings 3 Algebraic subsets of the affine space 3.1 Basic definitions 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 Affine algebraic varieties 4.4 Algebraic varieties 5 Exercises 2: Algebraic varieties 1 Introduction 2 Morphisms of algebraic varieties 3 Complete varieties 4 Singular points and normal varieties 5 The Proj variety associated to a graded algebra 6 Deeper results on morphisms 7 Algebraic varieties and k-schemes 8 Exercises 3: Lie algebras 1 Introduction 2 Definitions 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 4: Algebraic groups: basic definitions 1 Introduction 2 Definitions and basic concepts 3 Subgroups and homomorphisms 4 Actions of affine groups on algebraic varieties 5 Subgroups and semidirect products 6 Exercises 5: Algebraic groups: Lie algebras and representations 1 Introduction 2 Hopf algebras and algebraic groups 3 Rational G-modules 4 The category of rational G-modules 5 Representations of SL2 6 Characters and semi-invariants 7 The Lie algebra associated to an affine algebraic group 8 Explicit computations 9 Exercises 6: Algebraic groups: Jordan decomposition and applications 1 Introduction 2 The Jordan decomposition of a single operator 3 The Jordan decomposition of an algebra homomorphism and of a derivation 4 The Jordan decomposition for coalgebras 5 The Jordan decomposition for an affine 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 7: Actions of algebraic groups 1 Introduction 2 Actions: examples and first properties 3 Basic facts about the geometry of the orbits 4 Categorical and geometric quotients 5 Affinized quotients 6 The subalgebra of invariants 7 Induction and restriction of representations 8 Exercises 8: Homogeneous spaces 1 Introduction 2 Embedding H-modules inside G-modules 3 Definition 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 7 Exercises 9: Algebraic groups and Lie algebras in characteristic zero 1 Introduction 2 Correspondence between subgroups and subalgebras 3 Algebraic Lie algebras 4 Exercises 10: Reductivity 1 Introduction 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 6 Exercises 11: Observable subgroups of affine algebraic groups 1 Introduction 2 Basic definitions 3 Induction and observability 4 Split and strong observability 5 The geometric characterization of observability 6 Exercises 12: Affine homogeneous spaces 1 Introduction 2 Geometric reductivity and observability 3 Exact subgroups 4 From quasi-affine to affine homogeneous spaces 5 Exactness, Reynolds operators, total integrals 6 Affine homogeneous spaces and exactness 7 Affine homogeneous spaces and reductivity 8 Exactness and integrals for unipotent groups 9 Exercises 13: Hilbert's 14th problem 1 Introduction 2 A counterexample to Hilbert's 14th problem 3 Reductive groups and finite 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 7 Exercises 14: Quotient varieties: basic results 1 Introduction 2 Actions by reductive groups: the semigeometric quotient 3 Actions by reductive groups: the geometric quotient 4 Canonical forms of matrices: a geometric perspective 5 Rosenlicht's theorem 6 Induced actions and homogeneous fiber bundles 7 Revisiting affinized quotients 8 Further results on invariants of finite groups 8.1 Invariants of graded algebras 8.2 Polynomial subalgebras of polynomial algebras 8.3 The case of a group generated by reflections 8.4 The degree of the fundamental invariants for a finite group 9 Exercises 15: Observable actions of affine algebraic groups 1 Introduction 2 Basic definitions 3 Observable actions and unipotency 4 The geometry of observable actions 5 The algebraic viewpoint on observable actions 6 Observable actions of reductive groups 7 Exercises 16: Quotient varieties: an introduction to geometric invariant theory 1 Introduction 2 One parameter subgroups and actions of Gm 2.1 One parameter subgroups 2.2 Actions of Gm on affine varieties 3 Reductive groups acting on affine algebraic varieties 3.1 Stable points — affine case 3.2 Semistable points — affine case 3.3 Hilbert–Mumford criterion in the affine case 4 Actions of reductive groups on projective varieties 4.1 Linear actions on the projective space 4.2 Actions on projective varieties 5 Exercises Appendix: basic definitions and results 1 Introduction 2 Notations 2.1 Category theory 2.2 General topology 2.3 Linear algebra 2.4 Group theory 3 Rings and modules 4 Representations Bibliography Glossary of Notations Author Index Index