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Author: Walter Ferrer Santos, Alvaro Rittatore


Publisher: CRC (2005)


Binding: Hardcover, 472 pages


pricer: $99.95


ISBN-10: 082475896X


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Actions and Invariants of Algebraic Groups presents a self-contained introduction to geometric invariant theory that links the basic theory of affine algebraic groups to Mumford's more sophisticated theory. The authors systematically exploit the viewpoint of Hopf algebra theory and the theory of comodules to simplify and compactify many of the relevant formulas and proofs. The first two chapters introduce the subject and review the prerequisites in commutative algebra, algebraic geometry, and the theory of semisimple Lie algebras over fields of characteristic zero. The authors' early presentation of the concepts of actions and quotients helps to clarify the subsequent material, particularly in the study of homogeneous spaces. This study includes a detailed treatment of the quasi-affine and affine cases and the corresponding concepts of observable and exact subgroups. Among the many other topics discussed are Hilbert's 14th problem, complete with examples and counterexamples, and Mumford's results on quotients by reductive groups. End-of-chapter exercises, which range from the routine to the rather difficult, build expertise in working with the fundamental concepts. The Appendix further enhances this work's completeness and accessibility with an exhaustive glossary of basic definitions, notation, and results. ---------------------Features--------------------- ¡¤ Provides a solid, self-contained introduction to the field of geometric invariant theory ¡¤ Builds the foundation needed to approach the specialized literature in geometric invariant theory ¡¤ Pays particular attention to the study of quotient varieties ¡¤ Presents a careful study of Hilbert's fourteenth problem along with Mumford's results on quotients by reductive groups ¡¤ Includes a long list of examples and end-of-chapter exercises ---------------------Contents--------------------- 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




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