Actions and Invariants of Algebraic Groups

ISBN: 9781482239157 出版年：2017 页码：480 Ferrer Santos, Walter Ricardo Rittatore, Alvaro CRC Press

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