Jordan Structures in Lie Algebras

Description

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. Cover Title page Preface Introduction Chapter 1. Nonassociative Algebras 1.1. Definitions and notation 1.2. Multiplication algebra and centroid 1.3. Extended centroid and central closure 1.4. Nilpotency and local nilpotency 1.5. Martindale algebras of quotients 1.6. The split Cayley algebra 1.7. Exercises Chapter 2. General Facts on Lie Algebras 2.1. Definitions and examples 2.2. Linear Lie algebras 2.3. Inner ideals of Lie algebras 2.4. Inheritance of primeness by ideals 2.5. Solvability and nilpotency 2.6. The locally nilpotent radical 2.7. A locally nilpotent radical for graded Lie algebras 2.8. The locally finite radical 2.9. Exercises Chapter 3. Absolute Zero Divisors 3.1. Identities involving absolute zero divisors 3.2. A theorem on sandwich algebras 3.3. Absolute zero divisors generate a locally nilpotent ideal 3.4. Nondegenerate Lie algebras 3.5. Absolute zero divisors in the Lie algebra of a ring 3.6. Absolute zero divisors in Lie algebras of skew-symmetric elements 3.7. Exercises Chapter 4. Jordan Elements 4.1. Identities involving Jordan elements 4.2. Jordan elements and abelian inner ideals 4.3. Jordan elements in nondegenerate Lie algebras 4.4. Minimal abelian inner ideals 4.5. On the existence of Jordan elements 4.6. Jordan elements in the Lie algebra of a ring 4.7. Jordan elements in Lie algebras of skew-symmetric elements 4.8. Exercises Chapter 5. Von Neumann Regular Elements 5.1. Definition, examples, and first results 5.2. Jacobson–Morozov type results 5.3. Idempotents in Lie algebras 5.4. The socle of a nondegenerate Lie algebra 5.5. Principal filtrations 5.6. Exercises Chapter 6. Extremal Elements 6.1. Definition and properties 6.2. Lie algebras generated by extremal elements 6.3. Jacobson–Morozov revisited 6.4. Simple Lie algebras with extremal elements 6.5. Exercises Chapter 7. A Characterization of Strong Primeness 7.1. Orthogonality relations of adjoint operators 7.2. A characterization of strong primeness Chapter 8. From Lie Algebras to Jordan Algebras 8.1. Linear Jordan algebras 8.2. The Jordan algebra attached to a Jordan element 8.3. Extremal elements and finitary Lie algebras 8.4. Clifford elements 8.5. The Kurosh problem for Lie algebras 8.6. Nil Lie algebras of finite width 8.7. Exercises Chapter 9. The Kostrikin Radical 9.1. Definition y basic results 9.2. Lie algebras with enough Jordan elements 9.3. Lie algebras over a field of characteristic zero 9.4. Kostrikin radical versus Baer radical 9.5. Locally nondegenerate Lie algebras 9.6. Exercises Chapter 10. Algebraic Lie Algebras and Local Finiteness 10.1. Strongly prime algebraic Lie PI-algebras 10.2. Algebraic Lie algebras of bounded degree 10.3. Exercises Chapter 11. From Lie Algebras to Jordan Pairs 11.1. Linear Jordan pairs 11.2. From Jordan pairs to Lie algebras 11.3. Finite ℤ-gradings and Jordan pairs 11.4. Subquotient with respect to an abelian inner ideal 11.5. Lie notions by the Jordan approach 11.6. Exercises Chapter 12. An Artinian Theory for Lie Algebras 12.1. Complemented inner ideals 12.2. Lifting idempotents 12.3. A construction of gradings of Lie algebras 12.4. Complemented Lie algebras 12.5. A unified approach to inner ideals 12.6. Exercises Chapter 13. Inner Ideal Structure of Lie Algebras 13.1. Lie inner ideals of prime rings 13.2. Lie inner ideals of prime rings with involution 13.3. Point spaces 13.4. Inner ideals of rings with involution and minimal one-sided ideals 13.5. Inner ideals of the exceptional Lie algebras 13.6. Exercises Chapter 14. Classical Infinite-Dimensional Lie Algebras 14.1. Simple Lie algebras with a finite ℤ-grading 14.2. Simple Lie algebras with minimal abelian inner ideals 14.3. Simple finitary Lie algebras revisited 14.4. Strongly prime Lie algebras with extremal elements 14.5. Locally finite Lie algebras with abelian inner ideals 14.6. Simple Jordan algebras generated by ad-nilpotent elements 14.7. Exercises Chapter 15. Classical Banach–Lie algebras 15.1. Primitive Banach–Lie algebras and continuity of isomorphisms 15.2. Banach–Lie algebras with extremal elements 15.3. Compact elements in Banach–Lie algebras 15.4. Exercises Bibliography Index of Notations Index Back Cover