ساختارهای اردن در جبرهای لی (بررسی‌ها و مونوگرافی‌های ریاضی)

This book explores applications of Jordan theory to the theory of Lie algebras. It begins with the general theory of nonassociative algebras and of Lie algebras and then focuses on properties of Jordan elements of special types. Then it proceeds to the core of the book, in which the author explains how properties of the Jordan algebra attached to a Jordan element of a Lie algebra can be used to reveal properties of the Lie algebra itself. One of the special features of this book is that it carefully explains Zelmanov's seminal results on infinite-dimensional Lie algebras from this point of view. The book is suitable for advanced graduate students and researchers who are interested in learning how Jordan algebras can be used as a powerful tool to understand Lie algebras, including infinite-dimensional Lie algebras. Although the book is on an advanced and rather specialized topic, it spends some time developing necessary introductory material, includes exercises for the reader, and is accessible to a student who has finished their basic graduate courses in algebra and has some familiarity with Lie algebras in an abstract algebraic setting -- Prové de l'editor Cover 1 Title page 4 Preface 12 Introduction 14 Chapter 1. Nonassociative Algebras 18 1.1. Definitions and notation 18 1.2. Multiplication algebra and centroid 23 1.3. Extended centroid and central closure 24 1.4. Nilpotency and local nilpotency 27 1.5. Martindale algebras of quotients 27 1.6. The split Cayley algebra 29 1.7. Exercises 30 Chapter 2. General Facts on Lie Algebras 32 2.1. Definitions and examples 32 2.2. Linear Lie algebras 38 2.3. Inner ideals of Lie algebras 43 2.4. Inheritance of primeness by ideals 46 2.5. Solvability and nilpotency 46 2.6. The locally nilpotent radical 48 2.7. A locally nilpotent radical for graded Lie algebras 53 2.8. The locally finite radical 58 2.9. Exercises 61 Chapter 3. Absolute Zero Divisors 64 3.1. Identities involving absolute zero divisors 64 3.2. A theorem on sandwich algebras 66 3.3. Absolute zero divisors generate a locally nilpotent ideal 73 3.4. Nondegenerate Lie algebras 74 3.5. Absolute zero divisors in the Lie algebra of a ring 77 3.6. Absolute zero divisors in Lie algebras of skew-symmetric elements 78 3.7. Exercises 80 Chapter 4. Jordan Elements 82 4.1. Identities involving Jordan elements 82 4.2. Jordan elements and abelian inner ideals 83 4.3. Jordan elements in nondegenerate Lie algebras 84 4.4. Minimal abelian inner ideals 87 4.5. On the existence of Jordan elements 87 4.6. Jordan elements in the Lie algebra of a ring 94 4.7. Jordan elements in Lie algebras of skew-symmetric elements 96 4.8. Exercises 98 Chapter 5. Von Neumann Regular Elements 100 5.1. Definition, examples, and first results 100 5.2. Jacobson–Morozov type results 101 5.3. Idempotents in Lie algebras 105 5.4. The socle of a nondegenerate Lie algebra 106 5.5. Principal filtrations 110 5.6. Exercises 112 Chapter 6. Extremal Elements 114 6.1. Definition and properties 114 6.2. Lie algebras generated by extremal elements 116 6.3. Jacobson–Morozov revisited 118 6.4. Simple Lie algebras with extremal elements 119 6.5. Exercises 123 Chapter 7. A Characterization of Strong Primeness 126 7.1. Orthogonality relations of adjoint operators 126 7.2. A characterization of strong primeness 130 Chapter 8. From Lie Algebras to Jordan Algebras 132 8.1. Linear Jordan algebras 132 8.2. The Jordan algebra attached to a Jordan element 143 8.3. Extremal elements and finitary Lie algebras 151 8.4. Clifford elements 153 8.5. The Kurosh problem for Lie algebras 160 8.6. Nil Lie algebras of finite width 162 8.7. Exercises 163 Chapter 9. The Kostrikin Radical 166 9.1. Definition y basic results 166 9.2. Lie algebras with enough Jordan elements 167 9.3. Lie algebras over a field of characteristic zero 170 9.4. Kostrikin radical versus Baer radical 173 9.5. Locally nondegenerate Lie algebras 174 9.6. Exercises 175 Chapter 10. Algebraic Lie Algebras and Local Finiteness 178 10.1. Strongly prime algebraic Lie PI-algebras 178 10.2. Algebraic Lie algebras of bounded degree 179 10.3. Exercises 183 Chapter 11. From Lie Algebras to Jordan Pairs 184 11.1. Linear Jordan pairs 184 11.2. From Jordan pairs to Lie algebras 193 11.3. Finite Z-gradings and Jordan pairs 197 11.4. Subquotient with respect to an abelian inner ideal 200 11.5. Lie notions by the Jordan approach 205 11.6. Exercises 210 Chapter 12. An Artinian Theory for Lie Algebras 214 12.1. Complemented inner ideals 214 12.2. Lifting idempotents 217 12.3. A construction of gradings of Lie algebras 220 12.4. Complemented Lie algebras 224 12.5. A unified approach to inner ideals 226 12.6. Exercises 228 Chapter 13. Inner Ideal Structure of Lie Algebras 230 13.1. Lie inner ideals of prime rings 230 13.2. Lie inner ideals of prime rings with involution 240 13.3. Point spaces 247 13.4. Inner ideals of rings with involution and minimal one-sided ideals 251 13.5. Inner ideals of the exceptional Lie algebras 258 13.6. Exercises 265 Chapter 14. Classical Infinite-Dimensional Lie Algebras 268 14.1. Simple Lie algebras with a finite Z-grading 268 14.2. Simple Lie algebras with minimal abelian inner ideals 269 14.3. Simple finitary Lie algebras revisited 271 14.4. Strongly prime Lie algebras with extremal elements 274 14.5. Locally finite Lie algebras with abelian inner ideals 276 14.6. Simple Jordan algebras generated by ad-nilpotent elements 279 14.7. Exercises 279 Chapter 15. Classical Banach–Lie algebras 282 15.1. Primitive Banach–Lie algebras and continuity of isomorphisms 282 15.2. Banach–Lie algebras with extremal elements 285 15.3. Compact elements in Banach–Lie algebras 295 15.4. Exercises 297 Bibliography 298 Index of Notations 306 Index 310 Back Cover 314