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.