Modular Lie Algebras and their Representations (Chapman & Hall/CRC Pure and Applied Mathematics Book 116)

This Book Presents An Introduction To The Structure And Representation Theory Of Modular Lie Algebras Over Fields Of Positive Characteristic. It Introduces The Beginner To The Theory Of Modular Lie Algebras And Is Meant To Be A Reference Text For Researchers. Cover......Page 1 Half Title......Page 2 Title Page......Page 8 Copyright Page......Page 9 Preface......Page 10 Contents......Page 14 1.1 Basic Definitions of Lie Algebras and Representations......Page 18 1.2 Some Basic Module Operations......Page 28 1.3 Nilpotent Lie Algebras; Engel's Theorem......Page 31 1.4 Primary Decomposition; Cartan Subalgebras......Page 39 1.5 Solvable Lie Algebras; Lie's Theorem......Page 45 1.6 Examples of Low Dimension......Page 50 1.7 The Solvable Radical and Cartan's Criterion for Solvability......Page 55 1.8 The Universal Enveloping Algebra......Page 65 1.9 Filtrations......Page 69 References......Page 77 2.1 p-Mappings and Restricted Lie Algebras......Page 78 2.2 Existence of p-Mappings; Restrictable Lie Algebras......Page 87 2.3 Some Properties of the p-Mapping......Page 94 2.4 Tori and Cartan Decomposition......Page 102 2.5 Restricted Enveloping Algebras and Universal p-Envelopes......Page 107 References......Page 115 Chapter 3 Filtered and Graded Lie Algebras......Page 116 3.1 Filtered Lie Algebras......Page 117 3.2 Graded Lie Algebras......Page 124 3.3 The Interrelation Between Filtrations and Gradations; Criteria for Simplicity......Page 128 3.4 Subsidiary Results on Filtrations and Gradations......Page 135 3.5 Realizations as Derivation Algebras......Page 141 References......Page 157 4.1 The Construction Processes......Page 158 4.2 The Generali zed Jacobson-Witt Algebra W(n;m̳)......Page 163 4.3 The Special Algebra S(n;m̳)......Page 170 4.4 The Hamiltonian Algebra H(2r;m̳)......Page 179 4.5 The Contact Algebra K(2r + l; m̳)......Page 186 4.6 Associative Forms of Graded Cartan-Type Lie Algebras......Page 195 4.7 Generators of Restricted Cartan-Type Lie Algebras......Page 203 4.8 The Derivation Algebras of the Restricted Cartan-Type Lie Algebras......Page 208 References......Page 217 5.1 The Center of the Universal Enveloping Algebra......Page 218 5.2 Irreducible Representations......Page 222 5.3 Reduced Enveloping Algebras......Page 229 5.4 Frobenius Algebras......Page 232 5.5 Lie Algebras with Completely Reducible Representations......Page 236 5.6 Induced Representations......Page 241 5.7 A Criterion for Irreducibility......Page 246 5.8 Representations of Solvable Lie Algebras......Page 254 5.9 Examples......Page 261 References......Page 265 6.1 The Prime Spectrum of a Ring......Page 266 6.2 Noetherian Rings......Page 270 6.3 Properties of the Mapping Spec(A) ➔ Spec(R)......Page 274 6.4 The Going-Down Property......Page 279 6.5 The Prime Spectrum of U(L)......Page 283 6.6 The Maximal Dimension of Irreducible Modules......Page 289 6.7 Examples......Page 292 References......Page 295 Notation......Page 296 Bibliography......Page 298 Index......Page 316