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Algebraic groups and differential Galois theory. (English) Zbl 1215.12001
Graduate Studies in Mathematics 122. Providence, RI: American Mathematical Society (AMS) (ISBN 978-0-8218-5318-4/hbk). xiv, 225 p. (2011).
The book under review is a substantially expanded version of the authors’ earlier lecture notes [Introduction to differential Galois theory. With an appendix by Juan J. Morales-Ruiz. Krakow: Politechnika Krakowska im. Tadeusza Kościuszki. 94 p. (2007; Zbl 1172.12002)]. Now as before, its objective is to provide a solid first introduction to the Galois theory of homogeneous linear differential equations, the so-called Picard-Vessiot theory, mainly based on the modern algebro-geometric framework of algebraic groups. However, in comparison with the former lecture notes, the current book has been conceived as a self-contained textbook addressed to graduate students in mathematics or physics, course instructors, and self-studying readers who are interested in the subject from the viewpoint of dynamical systems. Consequently, the prerequisites for this book are limited to undergraduate courses in modern abstract algebra, basic commutative algebra, and complex analysis, respectively, whereas detailed introductions to algebraic varieties and algebraic groups have now been included as integral constituent parts of the text.
As for the contents, the book consists of three major parts, each of which is divided into several chapters and sections, respectively.
Part 1 is titled “Algebraic Geometry” and comprises the first two chapters. While Chapter 1 develops the fundamentals of (abstract) affine and projective varieties, the topics treated in Chapter 2 are prevarieties and general varieties.
Part 2 is formed by the subsequent two chapters and comes with the heading “Algebraic Groups”. Chapter 3 provides an introduction to abstract algebraic groups and their basic properties, including affine algebraic groups, linear algebraic groups, homogeneous spaces, rational representations, and quotients of linear algebraic groups by closed normal subgroups. Many concrete examples illustrate the abstract material presented in this chapter, with elliptic curves serving as important types of nonlinear algebraic groups. Chapter 4 is then devoted to the relationship between, algebraic groups and Lie algebras, with particular emphasis on the Jordan decomposition in affine algebraic groups, the classical Lie-Kolchin Theorem on connected solvable subgroups of $$\text{GL}(n,\mathbb{C})$$, and the equivalence between the solvability of a connected linear algebraic group and the solvability of its associated Lie algebra. This chapter concludes with the classification of the subgroups of the special linear group $$\text{SL}(2,\mathbb{C})$$ in its classical geometric setting,mainly for the purpose of later use in the part on differential Galois theory.
Part 3 is designated “Differential Galois Theory” and turns to the gist of the present book. Occupying the second half of the entire text, this part contains the remaining four chapters. The material covered here appears to be closest to the contents of the authors’ above-mentioned earlier booklet (Zbl 1172.12002) from 2007 (loc. cit.), although various revisions, improvements, and enlargements have been made. Chapter 5 introduces differential rings, differential ring and field extensions, linear differential operators over a differential field and homogeneous linear differential equations over a differential field, with field of constants $$\mathbb{C}$$. Next the Picard-Vessiot extension of such a homogeneous linear differential equation is established and analyzed, thereby laying the foundations for elementary Picard-Vessiot theory as developed in the sequel. In Chapter 6, the authors define the differential Galois group of a homogeneous linear differential equation via its Picard-Vessiot extension and identify it as a linear algebraic group over $$\mathbb C$$ .Then they give a proof of the fundamental theorem of differential Galois theory, which establishes a bijective correspondence between intermediate differential fields of a Picard-Vessiot extension and closed subgroups of its differential Galois group. Furthermore, Liouville extensions and generalized Liouville extensions of differential fields are described, culminating in a characterization of linear differential equations which are solvable by quadratures as those having a differential Galois group with solvable identity component. The proofs of these central results are based on the various properties of algebraic groups and $$G$$-varieties as developed in Part 2 of the book.
Chapter 7 is devoted to the study of linear differential equations over the differential field $$\mathbb C(z)$$ of rational complex functions in one variable, including some classical results concerning Fuchsian differential equations and their monodromy groups. This chapter ends with a presentation of J. Kovačić’s powerful algorithm for computing Liouvillian solutions of linear differential equations of order 2 defined over $$\mathbb C(z)$$. This algorithm solves such a homogeneous linear differential equation of order 2 by quadratures, whenever it is solvable at all [cf. J. J. Kovačić, J. Symb. Comput. 2, 3–43 (1986; Zbl 0603.68035)]. Chapter 8, the final chapter of the book, briefly describes some further topics in differential Galois theory and related areas, mainly with a view toward current active research in these fields. This includes numerous suggestions for complementary and more advanced reading, with precise references as listed in the rich bibliography of the book.
Each chapter ends with a series of related exercises helping the reader assimilate the respective abstract material. Many of them supplement the material of the text, while others provide an insight into some topics beyond, the scope of this introductory textbook.
In addition, the whole exposition is interspersed with a great many instructive examples and elucidating remarks complementing the main text.
Altogether, this first introduction to algebraic groups and differential Galois theory stands out by its high degree of profundity, lucidity, systematic representation, mathematical rigour, and self-containedness. The authors have set a high value on both detailed and motivating explanations, including complete proofs of all the results discussed in the course of the text. No doubt, this book is an excellent first reading concerning modern differential Galois theory, and a useful source for teachers in the field as well. Also, the book may serve as a solid basis for studying the more recent standard texts on the subject such as “Lectures on differential Galois theory.” University Lecture Series. 7. Providence, RI: American Mathematical Society (AMS) (1994; Zbl 0855.12001) by A. R. Magid, “Galois theory of linear differential equations.” Grundlehren der Mathematischen Wissenschaften 328. Berlin: Springer (2003; Zbl 1036.12008) by I. van der Put and M. F. Singer, or the classic “Differential algebra and algebraic groups.” Pure and Applied Mathematics, 54. New York-London: Academic Press (1973; Zbl 0264.12102) by E. R. Kolchin, apart from the vast current research literature as suggested in Chapter 8.

##### MSC:
 12-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to field theory 12H05 Differential algebra 13B05 Galois theory and commutative ring extensions 14A05 Relevant commutative algebra 17B45 Lie algebras of linear algebraic groups 20G15 Linear algebraic groups over arbitrary fields 34M35 Singularities, monodromy and local behavior of solutions to ordinary differential equations in the complex domain, normal forms 12H20 Abstract differential equations 14L35 Classical groups (algebro-geometric aspects)
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