CDATA[Contents Introduction 7 Chapter 1. General Theorems 8 1. Lie's and Engel's Theorems 8 1.1. Lie's Theorem 8 1.2. Generalization of Lie's Theorem 10 1.3. Engel's Theorem and Corollaries to It 11 1.4. An Analogue of Engel's Theorem in Group Theory 12 2. The Cartan Criterion 13 2.1. Invariant Bilinear Forms 13 2.2. Criteria of Solvability and Semisimplicity 13 2.3. Factorization into Simple Factors 14 3. Complete Reducibility of Representations and Triviality of the Cohomology of Semisimple Lie Algebras 15 3.1. Cartan's Criterion of Complete Reducibility 15 3.2. The Casimir Operator 15 3.3. Theorems on the Triviality of Cohomology 16 3.4. Complete Reducibility of Representations 16 3.5. Reductive Lie Algebras 17 4. Levi's Theorem 18 4.1. Decomposition 18 4.2. Existence of a Lie Group with a Given Tangent Algebra 19 4.3. Malcev's Theorem 20 4.4. Classification of Lie Algebras with a Given Radical 20 5. Linear Lie Groups 21 5.1. Basic Notions 21 5.2. Some Examples 22 5.3. Ado's Theorem 24 5.4. Criteria of Linearizability for Lie Groups. Linearizer 24 5.5. Sufficient Linearizability Conditions 25 5.6. Structure of Linear Lie Groups 27 6. Lie Groups and Algebraic Groups 27 6.1. Complex and Real Algebraic Groups 27 6.2. Algebraic Subgroups and Subalgebras 28 6.3. Semisimple and Reductive Algebraic Groups 29 6.4. Polar Decomposition 31 6.5. Chevalley Decomposition 32 7. Complexification and Real Forms 33 7.1. Complexification of Real Lie Algebras 33 7.2. Complexification and Real Forms of Lie Groups 35 7.3. Universal Complexification of a Lie Group 36 8. Splittings of Lie Groups and Lie Algebras 38 8.1. Definition of Split Lie Groups and Lie Algebras 38 8.2. Malcev Splittable Lie Groups and Lie Algebras 39 8.3. Theorem on the Existence and Uniqueness of Splittings 40 9. Cartan Subalgebras and Subgroups. Weights and Roots 41 9.1. Regular Representations of Nilpotent Lie Algebras 41 9.2. Weights and Roots with Respect to a Nilpotent Subalgebra 43 9.3. Cartan Subalgebras 43 9.4. Cartan Subalgebras and Root Decompositions of Semisimple Lie Algebras 45 9.5. Cartan Subgroups 46 Chapter 2. Solvable Lie Groups and Lie Algebras 48 1. Examples 48 2. Triangular Lie Groups and Lie Algebras 49 3. Topology of Solvable Lie Groups and Their Subgroups 50 3.1. Canonical Coordinates 50 3.2. Topology of Solvable Lie Groups 51 3.3. Algebraic Lie Groups 52 3.4. Topology of Subgroups of Solvable Lie Groups 53 4. Nilpotent Lie Groups and Lie Algebras 53 4.1. Definitions and Examples 53 4.2. Canonical Coordinates 56 4.3. Cohomology of Outer Automorphisms 56 5. Nilpotent Radicals in Lie Algebras and Lie Groups 58 5.1. Nilpotent Radical 58 5.2. Nilpotent Radical 58 5.3. Unipotent Radical 59 6. Some Classes of Solvable Lie Groups and Lie Algebras 59 6.1. Characteristically Nilpotent Lie Algebras 59 6.2. Filiform Lie Algebras 61 6.3. Nilpotent Lie Algebras of Class 2 62 6.4. Exponential Lie Groups and Lie Algebras 63 6.5. Lie Algebras and Lie Groups of Type (I) 65 7. Linearizability Criterion for Solvable Lie Groups 66 Chapter 3. Complex Semisimple Lie Groups and Lie Algebras 67 1. Root Systems 67 1.1. Abstract Root Systems 67 1.2. Root Systems of Reductive Groups 68 1.3. Root Decompositions and Root Systems for Classical Complex Lie Algebras 72 1.4. Weyl Chambers and Simple Roots 73 1.5. Borel Subgroups and Subalgebras 76 1.6. The Weyl Group 77 1.7. The Dynkin Diagram and the Cartan Matrix 79 1.8. Classification of Admissible Systems of Vectors and Root Systems 82 1.9. Root and Weight Lattices 83 1.10. Chevalley Basis 85 2. Classification of Complex Semisimple Lie Groups and Their Linear Representations 86 2.1. Uniqueness Theorem for Lie Algebras 86 2.2. Uniqueness Theorems for Linear Representations 88 2.3. Existence Theorems 90 2.4. Global Structure of Connected Semisimple Lie Groups 91 2.5. Classification of Connected Semisimple Lie Groups 92 2.6. Linear Representations of Connected Reductive Algebraic Groups 94 2.7. Dual Representations and Bilinear Invariants 96 2.8. The Kernel and the Image of a Locally Faithful Linear Representation 99 2.9. The Casimir Operator and Dynkin Index 100 2.10. Spinor Group and Spinor Representation 102 3. Automorphisms and Gradings 104 3.1. Description of Automorphisms 104 3.2. Quotient of the Group of Automorphisms 105 3.3. Homogeneous Automorphisms and Nilpotent Elements 106 3.4. Fixed Points of Semisimple Automorphisms 107 3.5. One-dimensional Tori of Automorphisms and Z-gradings 108 3.6. Canonical Form of an Inner Semisimple Automorphism 110 3.7. Inner Automorphisms of Finite Order and Z-gradings of Inner Type 112 3.8. Quotients Associated with a Component of the Group of Automorphisms 115 3.9. Generalized Root Decomposition 117 3.10. Canonical Form of an Outer Semisimple Automorphism 119 3.11. Outer Automorphisms of Finite Order and Z-gradings of Outer Type 121 3.12. Jordan Gradings of Classical Lie Algebras 123 3.13. Jordan Gradings of Exceptional Lie Algebras 127 Chapter 4. Real Semisimple Lie Groups and Lie Algebras 127 1. Classification of Real Semisimple Lie Algebras 127 1.1. Real Forms of Classical Lie Groups and Lie Algebras 128 1.2. Compact Real Form 131 1.3. Real Forms and Involutory Automorphisms 133 1.4. Involutory Automorphisms of Complex Simple Algebras 134 1.5. Classification of Real Simple Lie Algebras 135 2. Compact Lie Groups and Complex Reductive Groups 137 2.1. Some Properties of Linear Representations of Compact Lie Groups 137 2.2. Self-adjointness of Reductive Algebraic Groups 138 2.3. Algebraicity of a Compact Lie Group 139 2.4. Some Properties of Extensions of Compact Lie Groups 139 2.5. Correspondence Between Real Compact and Complex Reductive Lie Groups 141 2.6. Maximal Torus in Compact Lie Groups 142 3. Cartan Decomposition 143 3.1. Cartan Decomposition of a Semisimple Lie Algebra 143 3.2. Cartan Decomposition of a Semisimple Lie Group 145 3.3. Conjugacy of Maximal Compact Subgroups of Semisimple Lie Groups 147 3.4. Topological Structure of Lie Groups 148 3.5. Linearizer of a Connected Semisimple Lie Groups 149 3.6. Classification of Semisimple Lie Groups 151 4. Real Root Decomposition 153 4.1. Maximal Diagonizable Subalgebras 154 4.2. Real Root Systems 154 4.3. Satake Diagrams 156 4.4. Maximal Semisimple Subalgebras 157 4.5. Maximal Connected Triangular Subgroups 159 4.6. Cartan Subalgebras of a Real Semisimple Lie Algebra 162 5. Exponential Mapping for Semisimple Lie Groups 165 5.1. Image of the Exponential Mapping 165 5.2. Index of an Element of a Lie Group 164 5.3. Index of Simple Lie Groups 165 Chapter 5. Models of Exceptional Lie Algebras 167 1. Models Associated with the Cayley Algebra 167 1.1. The Cayley Algebra 167 1.2. The Algebra * 169 1.3. Exceptional Jordan Algebra 172 1.4. The Algebra * 170 1.5. The Algebra * 172 1.6. The Algebra * 175 1.7. Unified Construction of Exceptional Lie Algebras 177 2. Models Associated with Gradings of Exceptional Lie Algebras 178 Chapter 6. Subgroups and Subalgebras of Semisimple Lie Groups and Lie Algebras 182 1. Regular Subalgebras and Subgroups 182 1.1. Regular Subalgebras of Complex Reductive Lie Algebras 182 1.2. Description of Semisimple Regular Subalgebras 184 1.3. Parabolic Subalgebras and Subgroups 187 1.4. Subalgebras of Semisimple Lie Algebras and Flag Manifolds 188 1.5. Parabolic Subalgebras of Real Semisimple Lie Algebras 190 1.6. Non-isomorphic Maximal Subalgebras 192 2. The Semisimple Subalgebras and Nilpotent Elements 193 2.1. Three-dimensional Simple Subalgebras 193 2.2. Simple Lie Algebras of Classical Lie Algebras 195 2.3. Principal and Semiprincipal Three-dimensional Simple Subalgebras 197 2.4. Minimal Ambient Regular Subalgebras 199 2.5. Minimal Ambient Compact Regular Subalgebras 200 3. Semisimple Subalgebras and Subgroups 203 3.1. Semisimple Subalgebras of Complex Classical Groups 203 3.2. Maximal Connected Subgroups of Complex Classical Lie Algebras 205 3.3. Semisimple Subalgebras of Exceptional Complex Lie Algebras 206 3.4. Semisimple Subalgebras of Real Semisimple Lie Algebras 207 Chapter 7. On the Classification of Arbitrary Lie Groups and Lie Algebras of a Given Dimension 209 1. Classification of Lie Groups of Small Dimension 209 1.1. Lie Algebras of Dimension ≤3 209 1.2. Connected Lie Groups of Dimension ≤3 209 2. The Space of Lie Algebras, Deformations and Contractions 213 2.1. The Space of Lie Algebras 213 2.2. Orbital of the Lie Algebra of the Group GL on 214 2.3. Deformations of Lie Algebras 219 2.4. Contractions of Lie Algebras 220 2.5. Contractions of Lie Algebras 221 2.6. Rigid Lie Algebras 222 Tables 224 References 237 Author Index 245 Subject Index 246
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