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Gradings on Simple Lie Algebras Mathematical Surveys and Monographs Volume 189 Gradings on Simple Lie Algebras Alberto Elduque Mikhail Kochetov American Mathematical Society Atlantic Association for Research in the Mathematical Sciences http://dx.doi.org/10.1090/surv/189 Gradings on Simple Lie Algebras Mathematical Surveys and Monographs Volume 189 Gradings on Simple Lie Algebras Alberto Elduque Mikhail Kochetov American Mathematical Society Providence, RI Atlantic Association for Research in the Mathematical Sciences Halifax, Nova Scotia, Canada Editorial Committee of Mathematical Surveys and Monographs Ralph L. Cohen, Chair Robert Guralnick Benjamin Sudakov MichaelA.Singer Michael I. Weinstein Editorial Board of the Atlantic Association for Research in the Mathematical Sciences Jeannette Janssen, Director David Langstroth, Managing Editor Yuri Bahturin Theodore Kolokolnikov Robert Dawson Lin Wang 2010 Mathematics Subject Classification. Primary 17B70; Secondary 17B60, 16W50, 17A75, 17C50. For additional information and updates on this book, visit www.ams.org/bookpages/surv-189 Library of Congress Cataloging-in-Publication Data Elduque, Alberto. Gradings on simple Lie algebras / Alberto Elduque, Mikhail Kochetov. pages cm. — (Mathematical surveys and monographs ; volume 189) Includes bibliographical references and index. ISBN 978-0-8218-9846-8 (alk. paper) 1. Lie algebras 2. Rings (Algebra) 3. Jordan algebras. I. Kochetov, Mikhail, 1977– II. Title. QA252.3.E43 2013 512.482—dc23 2013007217 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Acquisitions Department, American Mathematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294 USA. Requests can also be made by e-mail to [email protected]. c 2013 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government. Printed in the United States of America. ∞ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at http://www.ams.org/ 10987654321 181716151413 To Pili, a mathematician, and to Eva, a mathematician to be. (A.E.) To the memory of my parents. (M.K.) Contents List of Figures ix Preface xi Conventions and Dependence among Chapters xiii Introduction 1 Chapter 1. Gradings on Algebras 9 1.1. General gradings and group gradings 9 1.2. The universal group of a grading 15 1.3. Fine gradings 16 1.4. Duality between gradings and actions 19 1.5. Exercises 25 Chapter 2. Associative Algebras 27 2.1. Graded simple algebras with minimality condition 28 2.2. Graded division algebras over algebraically closed fields 33 2.3. Classification of gradings on matrix algebras 38 2.4. Anti-automorphisms and involutions of graded matrix algebras 49 2.5. Exercises 60 Chapter 3. Classical Lie Algebras 63 3.1. Classical Lie algebras and their automorphism group schemes 64 3.2. ϕ-Gradings on matrix algebras 85 3.3. Type A 105 3.4. Type B 116 3.5. Type C 118 3.6. Type D 119 3.7. Exercises 121 Chapter 4. Composition Algebras and Type G2 123 4.1. Hurwitz algebras 123 4.2. Gradings on Cayley algebras 130 F F 4.3. Gradings on psl3( ), char = 3 137 4.4. Derivations of Cayley algebras and simple Lie algebras of type G2 140 4.5. Gradings on the simple Lie algebras of type G2 146 4.6. Symmetric composition algebras 149 4.7. Exercises 160 Chapter 5. Jordan Algebras and Type F4 163 5.1. The Albert algebra 164 vii viii CONTENTS 5.2. Construction of fine gradings on the Albert algebra 169 5.3. Weyl groups of fine gradings 178 5.4. Classification of gradings on the Albert algebra 184 5.5. Gradings on the simple Lie algebra of type F4 190 5.6. Gradings on simple special Jordan algebras 197 5.7. Exercises 206 Chapter 6. Other Simple Lie Algebras in Characteristic Zero 207 6.1. Fine gradings on the simple Lie algebra of type D4 207 6.2. Freudenthal’s Magic Square 224 6.3. Some nice gradings on the exceptional simple Lie algebras 239 6.4. Fine gradings on the simple Lie algebra of type E6 244 6.5. Fine gradings and gradings by root systems 259 6.6. Summary of known fine gradings for types E6, E7 and E8 265 6.7. Exercises 269 Chapter 7. Lie Algebras of Cartan Type in Prime Characteristic 271 7.1. Restricted Lie algebras 271 7.2. Construction of Cartan type Lie algebras 273 7.3. Automorphism group schemes 276 7.4. Gradings 287 7.5. Exercises 297 Appendix A. Affine Group Schemes 299 A.1. Affine group schemes and commutative Hopf algebras 299 A.2. Morphisms of group schemes 305 A.3. Linear representations 307 A.4. Affine algebraic groups 310 A.5. Infinitesimal theory 314 Appendix B. Irreducible Root Systems 321 Bibliography 323 Index of Notation 331 Index 333 List of Figures 2.1 Gradings, up to equivalence, on M2(F)whereF is an algebraically closed field, char F =2. 44 2.2 Gradings, up to equivalence, on M3(F)whereF is an algebraically closed field, char F =3. 46 4.1 Multiplication table of the split Cayley algebra 129 4.2 Gradings on the Cayley algebra over an algebraically closed field of characteristic different from 2 134 4.3 Gradings, up to equivalence, on the simple Lie algebra of type G2 over an algebraically closed field 149 4.4 Multiplication table of the split Okubo algebra 151 6.1 Dynkin diagram of D4 211 6.2 Fine gradings on the E-series. 268 ix Preface The aim of this book is to introduce the reader to the theory of gradings on Lie algebras, with a focus on the classification of gradings on simple finite-dimensional Lie algebras over algebraically closed fields. The classic example of such a grading is the Cartan decomposition with respect to a Cartan subalgebra in characteristic zero, which is a grading by a free abelian group. Since the 1960’s, there has been much work on gradings by other groups, starting with finite cyclic groups, and applications of such gradings to the theory of Lie algebras and their representations. We do not attempt to give a comprehensive survey of these results but rather to present a self-contained exposition of the classification of gradings on classical simple Lie algebras in characteristic different from 2 and on some non-classical simple Lie algebras in prime characteristic greater than 3. Other important algebras also enter the stage: matrix algebras, the octonions and the simple exceptional Jordan algebra. Most of the classification results presented here are recent and have not yet appeared in book form. This work started with the notes of two courses that the authors gave for the Atlantic Algebra Centre at Memorial University of Newfoundland: “Introduction to affine group schemes” (M. Kochetov, November–December 2008) and “Compo- sition algebras and their gradings” (A. Elduque, May 2009). Affine group schemes are an important tool for the study of gradings on finite-dimensional algebras in arbitrary characteristic, as we explain in Chapter 1. We give a brief exposition of the background on affine group schemes in Appendix A, with references to the literature on this subject. A reader who is interested exclusively in the case of characteristic zero will only need affine algebraic groups (in the “na¨ıve” sense) to follow this book. Apart from this, we assume that the reader is familiar with lin- ear algebra and with the basics on groups and algebras. The book is intended for specialists in Lie theory but may also serve as a textbook for graduate students (in conjunction with an introductory textbook on Lie algebras). In every chapter, at the beginning, we give a brief description of its main results and references to original works; at the end, we give a list of exercises on the covered material. This book would not have been written without the constant support, advice and encouragement of Yuri Bahturin, who himself greatly contributed to the study of gradings by arbitrary groups. It was his enthusiasm that convinced the authors to join efforts in the task of collecting, understanding, unifying and expanding the knowledge about gradings on simple Lie algebras. The second author would also like to use this opportunity to express his gratitude for all the help in his life and career given so generously by Professor Bahturin since becoming his thesis advisor a decade and a half ago. xi xii PREFACE The authors have benefited from discussions with many colleagues. Among them, our special thanks are due to Cristina Draper, who explained her results on gradings on exceptional simple Lie algebras long before they were publicly available. The first author acknowledges the support of the former Spanish Ministerio de Ciencia e Innovaci´on—Fondo Europeo de Desarrollo Regional (FEDER)1 and of the Diputaci´on General de Arag´on—Fondo Social Europeo (Grupo de Investigaci´on de Algebra).´ He would also like to thank Memorial University for hospitality during his visits to Newfoundland. The second author acknowledges the support of the Natural Sciences and Engi- neering Research Council (NSERC)2 of Canada and the hospitality of the University of Zaragoza during his visits to Spain. Both authors acknowledge the support of the Atlantic Association for Research in the Mathematical Sciences (AARMS) of Canada in the preparation of this book.
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