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.
Author(s): Antonio Fernandez Lopez (author)
Series: Mathematical Surveys and Monographs
Publisher: American Mathematical Society
Year: 2019
Language: English
Pages: 299
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
