微分Galois理论在最近的数十年中已经成为诸多方向上的研究热点。克雷斯波、哈杰托著的《代数群和微分Galois理论（英文版）（精）》是自封闭的，通过展示Picard-Vessiot理论，即线性偏微分方程的Galois理论，将读者带入主题。书中的第一部分和第二部分给出了所需的代数几何和代数群的先导知识，第三部分包括Picard-Vessiot扩张、Picard-Vessiot理论的基本定理、求积法的可解性、Fuchs方程、单值群和Kovacic算法。书中的100多道习题可以帮助读者深入理解相关的概念并扩展了部分主题。
本书可作为研究生的微分Galois理论课程的教学参考书。最后一章中包含的扩展阅读的若干建议激励读者进入微分Galois理论或相关领域的更深入的不同主题。

Preface
Introduction
Part 1. Algebraic Geometry
Chapter 1. Affine and Projective Varieties
1.1. Affine varieties
1.2. Abstract affine varieties
1.3. Projective varieties
Exercises
Chapter 2. Algebraic Varieties
2.1. Prevarieties
2.2. Varieties
Exercises
Part 2. Algebraic Groups
Chapter 3. Basic Notions
3.1. The notion of Mgebraic group
3.2. Connected algebraic groups
3.3. Subgroups and morphisms
3.4. Linearization of affine algebraic groups
3.5. Homogeneous spaces
3.6. Characters and semi-invariants
3.7. Quotients
Exercises
Chapter 4. Lie Algebras and Algebraic Groups
4.1. Lie algebras
4.2. The Lie algebra of a linear algebraic group
4.3. Decomposition of algebraic groups
4.4. Solvable algebraic groups
4.5. Correspondence between algebraic groups and Lie algebras
4.6. Subgroups of SL(2, C)
Exercises
Part 3. Differential Galois Theory
Chapter 5. Picard-Vessiot Extensions
5.1. Derivations
5.2. Differential rings
5.3. Differential extensions
5.4. The ring of differential operators
5.5. Homogeneous linear differential equations
5.6. The Picard-Vessiot extension
Exercises
Chapter 6. The Galois Correspondence
6.1. Differential Galois group
6.2. The differential Galois group as a linear algebraic group
6.3. The fundamental theorem of differential Galois theory
6.4. Liouville extensions
6.5. Generalized Liouville extensions
Exercises
Chapter 7. Differential Equations over C(z)
7.1. Fuchsian differential equations
7.2. Monodromy group
7.3. Kovacic's algorithm Exercises
Chapter 8. Suggestions for Further Reading
Bibliography
Index