Algebraic Groups and Differential Galois Theory

✍ Scribed by Teresa Crespo, Zbigniew Hajto

• Publisher: American Mathematical Society
• Year: 2011
• Tongue: English
• Leaves: 241
• Series: Graduate Studies in Mathematics 122
• Category: Library

✦ Synopsis

Differential Galois theory has seen intense research activity during the last decades in several directions: elaboration of more general theories, computational aspects, model theoretic approaches, applications to classical and quantum mechanics as well as to other mathematical areas such as number theory.

This book intends to introduce the reader to this subject by presenting Picard-Vessiot theory, i.e. Galois theory of linear differential equations, in a self-contained way. The needed prerequisites from algebraic geometry and algebraic groups are contained in the first two parts of the book. The third part includes Picard-Vessiot extensions, the fundamental theorem of Picard-Vessiot theory, solvability by quadratures, Fuchsian equations, monodromy group and Kovacic's algorithm. Over one hundred exercises will help to assimilate the concepts and to introduce the reader to some topics beyond the scope of this book.

This book is suitable for a graduate course in differential Galois theory. The last chapter contains several suggestions for further reading encouraging the reader to enter more deeply into different topics of differential Galois theory or related fields.

Readership: Graduate students and research mathematicians interested in algebraic methods in differential equations, differential Galois theory, and dynamical systems.

✦ Table of Contents

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 algebraic group
3.2. Connected algebraic groups
3.3. Subgroups and morphisms
3.4. Linearization of afne 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
7.3.1. Determination of the possible cases.
7.3.2. The algorithm for case 1.
7.3.3. The algorithm for case 2.
7.3.4. The algorithm for case 3.
Exercises

Chapter 8 Suggestions for Further Reading

Bibliography

Index