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Actions and Invariants of Algebraic Groups

Walter Ferrer Santos and Alvaro Rittatore
Publisher: 
Chapman & Hall/CRC
Publication Date: 
2005
Number of Pages: 
454
Format: 
Hardcover
Series: 
Pure and Applied Mathematics 269
Price: 
89.96
ISBN: 
0-8247-5896-X
Category: 
Monograph
[Reviewed by
Gizem Karaali
, on
05/16/2006
]

This is a very solid text written as a self-contained introduction to the modern algebraic-geometric theory of invariants. In the Preface, the authors describe their work as an "introduction to geometric invariant theory a la Mumford — as presented in his seminal book Geometric Invariant Theory ." The latter first appeared as a research monograph in 1965, and brought two well-developed areas of mathematics together in a most elegant and powerful way: namely, algebraic geometry and invariant theory. The fundamental notions involved in geometric invariant theory are geometric quotients and (semi-)invariants of an algebraic variety X with respect to a (linearly reductive) algebraic group G acting on it. Mumford's 1965 monograph and its expanded versions published in 1982 and 1994 were influential in triggering frenzied research activity in this area, now affectionately called "GIT" by many, and have been read with great enthusiasm by all experts, including those differential geometers and topologists who approach similar problems via symplectic geometric methods. In one version or another, Mumford's tome became a classic and a must-read item for many advanced graduate students, too. Besides the excellent mathematics, it provides several leads to the literature for those who need them, and a very readable account of the research in the area up to the time it was written. However it is by no means an introductory text for a beginner.

Hence a book like the one under review is more than welcome for those mortals out there who have not yet been appropriately immersed in the language and framework of algebraic invariant theory. Ferrer Santos and Rittatore promise a "bridge between the basic theory of affine algebraic groups (that is inseparable from considerations related to the geometry of actions) and the more sophisticated theory mentioned above" and they deliver superbly. Their book is more than appropriate for a (necessarily ambitious) mathematician willing to put in the time and energy to learn this beautiful subject, possibly a graduate student who has completed a year-long graduate algebra sequence. With many exercises at the end of each chapter, varying from standard derivations to nontrivial examples and counterexamples for the concepts developed throughout, with several historical notes sprinkled along the way as well as in the introductory sections of each chapter, with very well placed references to results in the preceding chapters as well as those in the forthcoming ones, the book is a perfect text for an advanced course or self study.

The bulk of the text covers standard material found in various texts. It is a relief, however, to have all of it together in one package. The first few chapters provide a brief but sufficient overview of the requisite commutative algebra and algebraic geometry, and basic definitions and results about algebraic groups and their Lie algebras. The representation theory of algebraic groups is developed using the framework of Hopf algebras, and actions of algebraic groups and homogeneous spaces are studied in detail. By the end of the eighth chapter, the reader has already been introduced to the basic concepts of (semi-)invariants and geometric quotients and may move on smoothly on to Mumford. By that time, however, the authors' voices are familiar and the book is a pleasure to follow. Of course, a significant amount of effort and mental energy is still needed to follow and understand the material. The last five chapters of the book (respectively covering geometric reductivity, observable subgroups, affine homogeneous spaces, Hilbert's 14th problem, and more on quotient spaces) provide the reader a chance to learn about several major ideas of modern invariant theory.

The authors restrict themselves to the algebraically closed fields and algebraic varieties. For practical reasons, and also pedagogically, this is a sound decision. They often point out when these conditions may be weakened, but their choice makes the text much easier to follow and appreciate the theory. Overall this is an excellent text for anyone who is willing to put in the effort to learn this beautiful subject; it could even be a book to take with you to that deserted island since it is practically self-contained. Now if only there were solutions to (or at least a few hints for) some of the tougher exercises...

For a more mathematically detailed but very clear review see MathSciNet: MR2138858 (2006c:14067)


Gizem Karaali teaches at the University of California at Santa Barbara.

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

Dummy View - NOT TO BE DELETED