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. Sendrequestsfortranslationrightsandlicensedreprintstoreprint-permission@ams.org. (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