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  李群与李代数I：...
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《李群和李代数》共3本，由该领域国际权威的专家撰写，涵盖了该领域所有重要的方向和理论，内容非常全面，有很高的参考价值。

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• Contents
  Introduction 4
  Chapter 1. Basic Notions 6
  §1. Lie Groups, Subgroups and Homomorphisms 6
  1.1 Definition of a Lie Group 6
  1.2 Lie Subgroups 7
  1.3 Homomorphisms of Lie Groups 9
  1.4 Linear Representations of Lie Groups 9
  1.5 Local Lie Groups 11
  §2. Actions of Lie Groups 12
  2.1 Definition of an Action 12
  2.2 Orbits and Stabilizers 12
  2.3 Images and Kernels of Homomorphisms 14
  2.4 Orbits of Compact Lie Groups 14
  §3. Coset Manifolds and Quotients of Lie Groups 15
  3.1 Coset Manifolds 15
  3.2 Lie Quotient Groups 17
  3.3 The Transitive Action Theorem and the Epimorphism Theorem 18
  3.4 The Pre-image of a Lie Group Under a Homomorphism 18
  3.5 Semidirect Products of Lie Groups 19
  §4. Connectedness and Simply-connectedness of Lie Groups 21
  4.1 Connected Components of a Lie Group 21
  4.2 Investigation of Connectedness of the Classical Lie Groups 22
  4.3 Covering Homomorphisms 24
  4.4 The Universal Covering Lie Group 26
  4.5 Investigation of Simply-connectedness of the Classical Lie Groups 27
  Chapter 2. The Relation Between Lie Groups and Lie Algebras 29
  §1. The Lie Functor 29
  1.1 The Tangent Algebra of a Lie Group 29
  1.2 Vector Fields on a Lie Group 31
  1.3 The Differential of a Homomorphism of Lie Groups 32
  1.4 The Differential of an Action of a Lie Group 34
  1.5 The Tangent Algebra of a Stabilizer 35
  1.6 The Adjoint Representation 35
  §2. Integration of Homomorphisms of Lie Algebras 37
  2.1 The Differential Equation of a Path in a Lie Group 37
  2.2 The Uniqueness Theorem 38
  2.3 Virtual Lie Subgroups 38
  2.4 The Correspondence Between Lie Subgroups of a Lie Group and Subalgebras of Its Tangent Algebra 39
  2.5 Deformations of Paths in Lie Groups 40
  2.6 The Existence Theorem 41
  2.7 Abelian Lie Groups 43
  §3. The Exponential Map 44
  3.1 One-Parameter Subgroups 44
  3.2 Definition and Basic Properties of the Exponential Map 44
  3.3 The Differential of the Exponential Map 46
  3.4 The Exponential Map in the Full Linear Group 47
  3.5 Cartan’s Theorem 47
  3.6 The Subgroup of Fixed Points of an Automorphism of a Lie Group 48
  §4. Automorphisms and Derivations 48
  4.1 The Group of Automorphisms 48
  4.2 The Algebra of Derivations 50
  4.3 The Tangent Algebra of a Semi-Direct Product of Lie Groups 51
  §5. The Commutator Subgroup and the Radical 52
  5.1 The Commutator Subgroup 52
  5.2 The Malcev Closure 53
  5.3 The Structure of Virtual Lie Subgroups 54
  5.4 Mutual Commutator Subgroups 55
  5.5 Solvable Lie Groups 56
  5.6 The Radical 57
  5.7 Nilpotent Lie Groups 58
  Chapter 3. The Universal Enveloping Algebra 59
  §1. The Simplest Properties of Universal Enveloping Algebras 59
  1.1 Definition and Construction 60
  1.2 The Poincaré-Birkhoff-Witt Theorem 61
  1.3 Symmetrization 63
  1.4 The Center of the Universal Enveloping Algebra 64
  1.5 The Skew-Field of Fractions of the Universal Enveloping Algebra 64
  §2. Bialgebras Associated with Lie Algebras and Lie Groups 66
  2.1 Bialgebras 66
  2.2 Right Invariant Differential Operators on a Lie Group 67
  2.3 Bialgebras Associated with a Lie Group 68
  §3. The Campbell-Hausdorff Formula 70
  3.1 Free Lie Algebras 70
  3.2 The Campbell-Hausdorff Series 71
  3.3 Convergence of the Campbell-Hausdorff Series 73
  Chapter 4. Generalizations of Lie Groups 74
  §1. Lie Groups over Complete Valued Fields 74
  1.1 Basic Definitions 74
  1.2 Valued Fields and Examples 75
  1.3 Actions of Lie Groups 75
  1.4 Standard Lie Groups over a Non-archimedean Field 76
  1.5 Tangent Algebras of Lie Groups 76
  §2. Formal Groups 78
  2.1 Definition and Simplest Properties 78
  2.2 The Tangent Algebra of a Formal Group 79
  2.3 The Bialgebra Associated with a Formal Group 80
  §3. Infinite-Dimensional Lie Groups 80
  3.1 Banach Lie Groups 81
  3.2 The Correspondence Between Banach Lie Groups and Banach Lie Algebras 82
  3.3 Actions of Banach Lie Groups on Finite-Dimensional Manifolds 83
  3.4 Lie-Frechet Groups 84
  3.5 ILB- and ILH-Lie Groups 85
  §4. Lie Groups and Topological Groups 86
  4.1 Continuous Homomorphisms of Lie Groups 87
  4.2 Hilbert’s 5-th Problem 87
  §5. Analytic Loops 88
  5.1 Basic Definitions and Examples 88
  5.2 The Tangent Algebra of an Analytic Loop 89
  5.3 The Tangent Algebra of a Diassociative Loop 90
  5.4 The Tangent Algebra of a Bol Loop 91
  References 92