Jordan Structures in Lie Algebras

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
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