The focus of this book is the Galois theory of homogeneous linear differential equations: Picard-Vessiot theory. The authors point out in their preface that this theory “parallels the Galois theory of algebraic equations. The notions of splitting field, Galois group,and solvability by radicals have their counterparts in [respectively] the notions of Picard-Vessiot extension, differential Galois group, and solvability by quadratures.”
Furthermore, “the differential Galois group of a homogeneous linear differential equation has a structure of linear algebraic group,” so it is a variety: we have the Zariski topology to work with. This is very exciting indeed, and opens up vast prospects, but the authors wisely take it slowly: “we have decided to develop the theory of algebraic varieties and linear algebraic groups in the same way that books on classical Galois theory include some chapters on group, ring, and field theories.” Still, they are careful to “[include] complete proofs, both of the results on algebraic geometry and algebraic groups which are needed in Picard-Vessiot theory and of the results on Picard-Vessiot theory itself.” This is altogether proper in an AMS Graduate Studies in Mathematics text, of course, and affords the neophyte reader the additional benefit of learning algebraic geometry “in context” even as the main thrust is differential Galois theory.
Crespo and Hajto also take pains to develop some Lie theory, at the close of what is de facto a prelude to the subject that forms their stated objective. To wit, the book under review is split into three parts, a presentation of algebraic geometry through varieties; a discussion of algebraic groups followed by a discussion of their connection to Lie algebras; and then, as the climactic third movement, so to speak, a closing trio of chapters dealing with the meat and potatoes of the business at hand.
We get, first, in Chapter 5, a development of Picard-Vessiot extensions. Then, in Chapter 6, a treatment of the Galois correspondence and its attendant foliage. The correspondence states that there is an order-reversing lattice bijection between Zariski closed subgroups of the ambient differential Galois group of a Picard-Vessiot extension and the so-called differential fields intermediate between the base field and said extension. (“Watson! The parallel is exact!”) Finally, in Chapter 7, the authors go on to deal with linear differential equations over the field of rational functions in a single complex variable (transcendental) over C.
The latter material is remarkably deep and very elegant. Consider, for example the following gorgeous result (discussed on p.174): the differential Galois group attached to a Fuchsian differential equation is the Zariski closure of the DE’s monodromy group. Very pretty.
The authors go on to discuss at length Kovacic’s algorithm for computing “Liouvillian solutions” to a second order linear DE over C(z).
The last chapter of the book, Chapter 8, coming in at only four pages, is devoted to a scholarly and careful analysis of the literature in this field, replete with suggestions for future reading on the part of the reader.
As befits a graduate text on what is really a rather specialized (if eminently accessible) area, Algebraic Groups and Differential Galois Theory abounds with examples, and the authors have presented the student with plenty of exercises.
Thus, this well-crafted book certainly serves its intended purpose well: it is a very good self-contained introduction to Picard-Vessiot theory. Furthermore, for those who already have a strong enough background in algebraic geometry (of actually rather a modest flavor: schemes are nowhere to be found in the pages of this book) and are comfortable with Lie algebras at a relatively mild level, jumping in at around p. 121 is entirely safe (modulo the occasional flipping back, of course). So, Algebraic Groups and Differential Galois Theory succeeds in several ways: it serves the targeted graduate student as well as the more experienced mathematician new to Picard-Vessiot theory. It is a very nice book indeed.
Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.