Jordan Structures in Lie Algebras

314 Pages·2019·2.717 MB·English

by  Antonio Fernandez Lopez (author)

Mathematical
Surveys
and
Monographs
Volume 240
Jordan Structures
in Lie Algebras
Antonio Fernández López
Jordan Structures
in Lie Algebras

Mathematical
Surveys
and
Monographs
Volume 240
Jordan Structures
in Lie Algebras
Antonio Fernández López
EDITORIAL COMMITTEE
Robert Guralnick, Chair Benjamin Sudakov
Natasa Sesum ConstantinTeleman
2010 Mathematics Subject Classification. Primary 17B05, 17C10;
Secondary 17B60, 17B65, 17C65.
For additional informationand updates on this book, visit
www.ams.org/bookpages/surv-240
Library of Congress Cataloging-in-Publication Data
Names: L´opez,AntonioFerna´ndez,1952-author.
Title: JordanstructuresinLiealgebras/AntonioFerna´ndezL´opez.
Description: Providence,RhodeIsland: AmericanMathematicalSociety,[2019]|Series: Mathe-
maticalsurveysandmonographs;volume240|Includesbibliographicalreferencesandindex.
Identifiers: LCCN2019010955|ISBN9781470450861(alk. paper)
Subjects: LCSH:Jordanalgebras. |Liealgebras.
Classification: LCCQA252.5.L6652019|DDC512/.482–dc23
LCrecordavailableathttps://lccn.loc.gov/2019010955
Copying and reprinting. Individual readersofthispublication,andnonprofit librariesacting
for them, are permitted to make fair use of the material, such as to copy select pages for use
in teaching or research. Permission is granted to quote brief passages from this publication in
reviews,providedthecustomaryacknowledgmentofthesourceisgiven.
Republication,systematiccopying,ormultiplereproductionofanymaterialinthispublication
ispermittedonlyunderlicensefromtheAmericanMathematicalSociety. Requestsforpermission
toreuseportionsofAMSpublicationcontentarehandledbytheCopyrightClearanceCenter. For
moreinformation,pleasevisitwww.ams.org/publications/pubpermissions.
Sendreq[e-mail].
(cid:2)c 2019bytheAmericanMathematicalSociety. Allrightsreserved.
TheAmericanMathematicalSocietyretainsallrights
exceptthosegrantedtotheUnitedStatesGovernment.
PrintedintheUnitedStatesofAmerica.
(cid:2)∞ Thepaperusedinthisbookisacid-freeandfallswithintheguidelines
establishedtoensurepermanenceanddurability.
VisittheAMShomepageathttps://www.ams.org/
10987654321 242322212019
To Conchi, my wife, who shared my dreams

Contents
Preface xi
Introduction 1
Chapter 1. Nonassociative Algebras 5
1.1. Definitions and notation 5
1.2. Multiplication algebra and centroid 10
1.3. Extended centroid and central closure 11
1.4. Nilpotency and local nilpotency 14
1.5. Martindale algebras of quotients 14
1.6. The split Cayley algebra 16
1.7. Exercises 17
Chapter 2. General Facts on Lie Algebras 19
2.1. Definitions and examples 19
2.2. Linear Lie algebras 25
2.3. Inner ideals of Lie algebras 30
2.4. Inheritance of primeness by ideals 33
2.5. Solvability and nilpotency 33
2.6. The locally nilpotent radical 35
2.7. A locally nilpotent radical for graded Lie algebras 40
2.8. The locally finite radical 45
2.9. Exercises 48
Chapter 3. Absolute Zero Divisors 51
3.1. Identities involving absolute zero divisors 51
3.2. A theorem on sandwich algebras 53
3.3. Absolute zero divisors generate a locally nilpotent ideal 60
3.4. Nondegenerate Lie algebras 61
3.5. Absolute zero divisors in the Lie algebra of a ring 64
3.6. Absolute zero divisors in Lie algebras of skew-symmetric elements 65
3.7. Exercises 67
Chapter 4. Jordan Elements 69
4.1. Identities involving Jordan elements 69
4.2. Jordan elements and abelian inner ideals 70
4.3. Jordan elements in nondegenerate Lie algebras 71
4.4. Minimal abelian inner ideals 74
4.5. On the existence of Jordan elements 74
4.6. Jordan elements in the Lie algebra of a ring 81
vii
viii CONTENTS
4.7. Jordan elements in Lie algebras of skew-symmetric elements 83
4.8. Exercises 85
Chapter 5. Von Neumann Regular Elements 87
5.1. Definition, examples, and first results 87
5.2. Jacobson–Morozov type results 88
5.3. Idempotents in Lie algebras 92
5.4. The socle of a nondegenerate Lie algebra 93
5.5. Principal filtrations 97
5.6. Exercises 99
Chapter 6. Extremal Elements 101
6.1. Definition and properties 101
6.2. Lie algebras generated by extremal elements 103
6.3. Jacobson–Morozov revisited 105
6.4. Simple Lie algebras with extremal elements 106
6.5. Exercises 110
Chapter 7. A Characterization of Strong Primeness 113
7.1. Orthogonality relations of adjoint operators 113
7.2. A characterization of strong primeness 117
Chapter 8. From Lie Algebras to Jordan Algebras 119
8.1. Linear Jordan algebras 119
8.2. The Jordan algebra attached to a Jordan element 130
8.3. Extremal elements and finitary Lie algebras 138
8.4. Clifford elements 140
8.5. The Kurosh problem for Lie algebras 147
8.6. Nil Lie algebras of finite width 149
8.7. Exercises 150
Chapter 9. The Kostrikin Radical 153
9.1. Definition y basic results 153
9.2. Lie algebras with enough Jordan elements 154
9.3. Lie algebras over a field of characteristic zero 157
9.4. Kostrikin radical versus Baer radical 160
9.5. Locally nondegenerate Lie algebras 161
9.6. Exercises 162
Chapter 10. Algebraic Lie Algebras and Local Finiteness 165
10.1. Strongly prime algebraic Lie PI-algebras 165
10.2. Algebraic Lie algebras of bounded degree 166
10.3. Exercises 170
Chapter 11. From Lie Algebras to Jordan Pairs 171
11.1. Linear Jordan pairs 171
11.2. From Jordan pairs to Lie algebras 180
11.3. Finite Z-gradings and Jordan pairs 184
11.4. Subquotient with respect to an abelian inner ideal 187
11.5. Lie notions by the Jordan approach 192
11.6. Exercises 197
CONTENTS ix
Chapter 12. An Artinian Theory for Lie Algebras 201
12.1. Complemented inner ideals 201
12.2. Lifting idempotents 204
12.3. A construction of gradings of Lie algebras 207
12.4. Complemented Lie algebras 211
12.5. A unified approach to inner ideals 213
12.6. Exercises 215
Chapter 13. Inner Ideal Structure of Lie Algebras 217
13.1. Lie inner ideals of prime rings 217
13.2. Lie inner ideals of prime rings with involution 227
13.3. Point spaces 234
13.4. Inner ideals of rings with involution and minimal one-sided ideals 238
13.5. Inner ideals of the exceptional Lie algebras 245
13.6. Exercises 252
Chapter 14. Classical Infinite-Dimensional Lie Algebras 255
14.1. Simple Lie algebras with a finite Z-grading 255
14.2. Simple Lie algebras with minimal abelian inner ideals 256
14.3. Simple finitary Lie algebras revisited 258
14.4. Strongly prime Lie algebras with extremal elements 261
14.5. Locally finite Lie algebras with abelian inner ideals 263
14.6. Simple Jordan algebras generated by ad-nilpotent elements 266
14.7. Exercises 266
Chapter 15. Classical Banach–Lie algebras 269
15.1. Primitive Banach–Lie algebras and continuity of isomorphisms 269
15.2. Banach–Lie algebras with extremal elements 272
15.3. Compact elements in Banach–Lie algebras 282
15.4. Exercises 284
Bibliography 285
Index of Notations 293
Index 297