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代數群和微積分 版權信息
- ISBN：9787040510133
- 條形碼：9787040510133 ; 978-7-04-051013-3
- 裝幀：平裝-膠訂
- 冊數：暫無
- 重量：暫無
- 所屬分類：>
代數群和微積分 本書特色
微分Galois理論在*近的數十年中已經成為諸多方向上的研究熱點。本書是自封閉的，通過展示Picard-Vessiot理論，即線性偏微分方程的Galois理論，將讀者帶入主題。書中的*部分和第二部分給出了所需的代數幾何和代數群的先導知識，第三部分包括Picard-Vessiot擴張、Picard-Vessiot理論的基本定理、求積法的可解性、Fuchs方程、單值群和Kovacic算法。書中的100多道習題可以幫助讀者深入理解相關的概念并擴展了部分主題。本書可作為研究生的微分Galois理論課程的教學參考書。*后一章中包含的擴展閱讀的若干建議激勵讀者進入微分Galois理論或相關領域的更深入的不同主題。
代數群和微積分 內容簡介
微分Galois理論在*近的數十年中已經成為諸多方向上的研究熱點。本書是自封閉的，通過展示Picard-Vessiot理論，即線性偏微分方程的Galois理論，將讀者帶入主題。書中的**部分和第二部分給出了所需的代數幾何和代數群的先導知識，第三部分包括Picard-Vessiot擴張、Picard-Vessiot理論的基本定理、求積法的可解性、Fuchs方程、單值群和Kovacic算法。書中的100多道習題可以幫助讀者深入理解相關的概念并擴展了部分主題。 本書可作為研究生的微分Galois理論課程的教學參考書。*后一章中包含的擴展閱讀的若干建議激勵讀者進入微分Galois理論或相關領域的更深入的不同主題。
代數群和微積分 目錄
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 algorithmExercises
Chapter 8．Suggestions for Further Reading
Bibliography
Index
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 algorithmExercises
Chapter 8．Suggestions for Further Reading
Bibliography
Index
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