李群与李代数(Ⅰ李理论基础李交换群影印版)(精)(国外数学名著系列)(Lie Groups and Lie Algebras Ⅰ Foundations of Lie Theory,Lie Transformation Groups)
点此进入淘宝搜索页搜索分類: 图书,科学与自然,数学,代数、数论、组合理论,
品牌: 奥尼契科
基本信息·出版社：科学出版社
·页码：235 页
·出版日期：2009年
·ISBN：7030235045/9787030235046
·条形码：9787030235046
·包装版本：1版
·装帧：精装
·开本：16
·正文语种：英语
·丛书名：国外数学名著系列
·外文书名：Lie Groups and Lie Algebras Ⅰ Foundations of Lie Theory,Lie Transformation Groups
产品信息有问题吗？请帮我们更新产品信息。
内容简介The book by Gorbatsevich, Onishchik and Vinberg is the first volume in a subseries of the Encyclopaedia devoted to the theory of Lie groups and Lie algebras.
The first part of the book deals with the foundations of the theory based on the classical global approach of Chevalley followed by an exposition of the alternative approach via the universal enveloping algebra and the Campbell-Hausdorff formula. It also contains a survey of certain generalizations of Lie groups.
The second more advanced part treats the topic of Lie transformation groups covering e.g. properties of orbits and stabilizers,homogeneous fibre bundles, Frobenius duality, groups of automorphisms of geometric structures, Lie algebras of vector fields and the existence of slices. The work of the last decades including the most recent research results is covered.
The book contains numerous examples and describes connections with topology, differential geometry, analysis and applications. It is written for graduate students and researchers in mathematics and theoretical physics.
目录
Introduction
Chapter 1. Basic Notions
1. Lie Groups, Subgroups and Homomorphisms
1.1 Definition of a Lie Group
1.2 Lie Subgroups
1.3 Homomorphisms of Lie Groups
1.4 Linear Representations of Lie Groups
1.5 Local Lie Groups
2. Actions of Lie Groups
2.1 Definition of an Action
2.2 Orbits and Stabilizers
2.3 Images and Kernels of Homomorphisms
2.4 Orbits of Compact Lie Groups
3. Coset Manifolds and Quotients of Lie Groups
3.1 Coset Manifolds
3.2 Lie Quotient Groups
3.3 The Transitive Action Theorem and the Epimorphism Theorem
3.4 The Pre-image of a Lie Group Under a Homomorphism
3.5 Semidirect Products of Lie Groups
4. Connectedness and Simply-connectedness of Lie Groups
4.1 Connected Components of a Lie Group
4.2 Investigation of Connectedness of the Classical Lie Groups
4.3 Covering Homomorphisms
4.4 The Universal Covering Lie Group
4.5 Investigation of Simply-connectedness of the Classical Lie Groups
Chapter 2. The Relation Between Lie Groups and Lie Algebras
1. The Lie Functor
1.1 The Tangent Algebra of a Lie Group
1.2 Vector Fields on a Lie Group
1.3 The Differential of a Homomorphism of Lie Groups
1.4 The Differential of an Action of a Lie Group
1.5 The Tangent Algebra of a Stabilizer
1.6 The Adjoint Representation
2. Integration of Homomorphisms of Lie Algebras
2.1 The Differential Equation of a Path in a Lie Group
2.2 The Uniqueness Theorem
2.3 Virtual Lie Subgroups
2.4 The Correspondence Between Lie Subgroups of a Lie Group and Subalgebras of Its Tangent Algebra
2.5 Deformations of Paths in Lie Groups
2.6 The Existence Theorem
2.7 Abelian Lie Groups
3. The Exponential Map
3.1 One-Parameter Subgroups
3.2 Definition and Basic Properties of the Exponential Map
3.3 The Differential of the Exponential Map
3.4 The Exponential Map in the Full Linear Group
3.5 Cartan's Theorem
3.6 The Subgroup of Fixed Points of an Automorphism of a Lie Group
4. Automorphisms and Derivations
4.1 The Group of Automorphisms
4.2 The Algebra of Derivations
4.3 The Tangent Algebra of a Semi-Direct Product of Lie Groups
5. The Commutator Subgroup and the Radical
5.1 The Commutator Subgroup
5.2 The Maltsev Closure
5.3 The Structure of Virtual Lie Subgroups
5.4 Mutual Commutator Subgroups
5.5 Solvable Lie Groups
5.6 The Radical
5.7 Nilpotent Lie Groups
Chapter 3. The Universal Enveloping Algebra
1. The Simplest Properties of Universal Enveloping Algebras
1.1 Definition and Construction
1.2 The Poincare-Birkhoff-Witt Theorem
1.3 Symmetrization
1.4 The Center of the Universal Enveloping Algebra
1.5 The Skew-Field of Fractions of the Universal Enveloping Algebra
2. Bialgebras Associated with Lie Algebras and Lie Groups
2.1 Bialgebras
2.2 Right Invariant Differential Operators on a Lie Group
2.3 Bialgebras Associated with a Lie Group
3. The Campbell-Hausdorff Formula
3.1 Free Lie Algebras
3.2 The Campbell-Hausdorff Series
3.3 Convergence of the Campbell-Hausdorff Series
Chapter 4. Generalizations of Lie Groups
1. Lie Groups over Complete Valued Fields
1.1 Valued Fields
1.2 Basic Definitions and Examples
1.3 Actions of Lie Groups
1.4 Standard Lie Groups over a Non-archimedean Field
1.5 Tangent Algebras of Lie Groups
2. Formal Groups
2.1 Definition and Simplest Properties
2.2 The Tangent Algebra of a Formal Group
2.3 The Bialgebra Associated with a Formal Group
3. Infinite-Dimensional Lie Groups
3.1 Banach Lie Groups
3.2 The Correspondence Between Banach Lie Groups and Banach Lie Algebras
3.3 Actions of Banach Lie Groups on Finite-Dimensional Manifolds
3.4 Lie-Frechet Groups
3.5 ILB- and ILH-Lie Groups
4. Lie Groups and Topological Groups
4.1 Continuous Homomorphisms of Lie Groups
4.2 Hilbert's 5-th Problem
5. Analytic Loops
5.1 Basic Definitions and Examples
5.2 The Tangent Algebra of an Analytic Loop
5.3 The Tangent Algebra of a Diassociative Loop
5.4 The Tangent Algebra of a Bol Loop
References
……[看更多目录]
序言要使我国的数学事业更好地发展起来，需要数学家淡泊名利并付出更艰苦地努力。另一方面，我们也要从客观上为数学家创造更有利的发展数学事业的外部环境，这主要是加强对数学事业的支持与投资力度，使数学家有较好的工作与生活条件，其中也包括改善与加强数学的出版工作。
从出版方面来讲，除了较好较快地出版我们自己的成果外，引进国外的先进出版物无疑也是十分重要与必不可少的。从数学来说，施普林格（springer）出版社至今仍然是世界上最具权威的出版社。科学出版社影印一批他们出版的好的新书，使我国广大数学家能以较低的价格购买，特别是在边远地区工作的数学家能普遍见到这些书，无疑是对推动我国数学的科研与教学十分有益的事。
这次科学出版社购买了版权，一次影印了23本施普林格出版社出版的数学书，就是一件好事，也是值得继续做下去的事情。大体上分一下，这23本书中，包括基础数学书5本，应用数学书6本与计算数学书12本，其中有些书也具有交叉性质。这些书都是很新的，2000年以后出版的占绝大部分，共计16本，其余的也是1990年以后出版的。这些书可以使读者较快地了解数学某方面的前沿，例如基础数学中的数论、代数与拓扑三本，都是由该领域大数学家编著的“数学百科全书”的分册。对从事这方面研究的数学家了解该领域的前沿与全貌很有帮助。按照学科的特点，基础数学类的书以“经典”为主，应用和计算数学类的书以“前沿”为主。这些书的作者多数是国际知名的大数学家，例如《拓扑学》一书的作者诺维科夫是俄罗斯科学院的院士，曾获“菲尔兹奖”和“沃尔夫数学奖”。这些大数学家的著作无疑将会对我国的科研人员起到非常好的指导作用。
当然，23本书只能涵盖数学的一部分，所以，这项工作还应该继续做下去。更进一步，有些读者面较广的好书还应该翻译成中文出版，使之有更大的读者群。
总之，我对科学出版社影印施普林格出版社的部分数学著作这一举措表示热烈的支持，并盼望这一工作取得更大的成绩。
文摘插图：
