652 Lie Algebras and Related Topics Workshop on Lie Algebras in Honor of Helmut Strade’s 70th Birthday May 22–24, 2013 Università degli Studi di Milano-Bicocca, Milano, Italy Marina Avitabile Jörg Feldvoss Thomas Weigel Editors AmericanMathematicalSociety Lie Algebras and Related Topics Workshop on Lie Algebras in Honor of Helmut Strade’s 70th Birthday May 22–24, 2013 Università degli Studi di Milano-Bicocca, Milano, Italy Marina Avitabile Jörg Feldvoss Thomas Weigel Editors e d a r t S t u m el H of y s e t r u o C 652 Lie Algebras and Related Topics Workshop on Lie Algebras in Honor of Helmut Strade’s 70th Birthday May 22–24, 2013 Università degli Studi di Milano-Bicocca, Milano, Italy Marina Avitabile Jörg Feldvoss Thomas Weigel Editors AmericanMathematicalSociety Providence,RhodeIsland EDITORIAL COMMITTEE Dennis DeTurck, Managing Editor Michael Loss Kailash Misra Martin J. Strauss 2010 Mathematics Subject Classification. Primary 17B50, 17B20,17B10, 17B70, 17B37, 17B60, 16S34, 16W10, 17A36, 16S37. Library of Congress Cataloging-in-Publication Data Lie algebras and related topics : workshop on Lie algebras in honor of Helmut Strade’s 70th birthday : May 22–24, 2013, Universit`a degli Studi di Milano-Bicocca, Milano, Italy / Marina Avitabile,J¨orgFeldvoss,ThomasWeigel,editors. pagescm. –(Contemporarymathematics;volume652) Includesbibliographicalreferences. ISBN978-1-4704-1023-0(alk. paper) 1.Liealgebras—Congresses. I.Strade,Helmut,1942– II.Avitabile,Marina,1971– III.Feldvoss,Jo¨rg,1960– IV.Weigel,Thomas(ThomasS.) QA252.3.L544 2015 512(cid:2).482–dc23 2015013482 DOI:http://dx.doi.org/10.1090/conm/652 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 select pages for useinteachingorresearch. 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TheAmericanMathematicalSocietyretainsallrights exceptthosegrantedtotheUnitedStatesGovernment. PrintedintheUnitedStatesofAmerica. (cid:2)∞ Thepaperusedinthisbookisacid-freeandfallswithintheguidelines establishedtoensurepermanenceanddurability. VisittheAMShomepageathttp://www.ams.org/ 10987654321 201918171615 Contents Preface vii List of Participants ix List of Talks xi Grading Switching for Modular Non-Associative Algebras Marina Avitabile and Sandro Mattarei 1 Gradings by Groups on Cartan Type Lie Algebras Yuri Bahturin, Mikhail Kochetov, and Jason McGraw 15 Simple Locally Finite Lie Algebras of Diagonal Type Alexander Baranov 47 Okubo Algebras: Automorphisms, Derivations and Idempotents Alberto Elduque 61 Constructing Semisimple Subalgebras of Real Semisimple Lie Algebras Paolo Faccin and Willem A. de Graaf 75 Tilting Modules over the Path Algebra of Type A, Polytopes, and Catalan Numbers Lutz Hille 91 Lie Identities on Skew Elements in Group Algebras Gregory T. Lee, Sudarshan K. Sehgal, and Ernesto Spinelli 103 Regular Derivations of Truncated Polynomial Rings Alexander Premet 123 Lie Properties of Restricted Enveloping Algebras Salvatore Siciliano and Hamid Usefi 141 Generic Semisimplicity of Reduced Enveloping Algebras Serge Skryabin 153 Modular Quantizations of Lie Algebras of Cartan Type H via Drinfel’d Twists Zhaojia Tong, Naihong Hu, and Xiuling Wang 173 Some Problems in the Representation Theory of Simple Modular Lie Algebras Georgia Benkart and Jo¨rg Feldvoss 207 Conjugacy of Nilpotent Elements in Characteristic p Willem A. De Graaf 229 v vi CONTENTS Problems on Lie Properties of Skew and Symmetric Elements of Group Rings Gregory T. Lee, Sudarshan K. Sehgal, and Ernesto Spinelli 231 Some Problems on Lie Properties of Restricted Enveloping Algebras Salvatore Siciliano and Hamid Usefi 235 Open Questions on Modular Lie Algebras Serge Skryabin 237 Koszul Lie Algebras Thomas Weigel 241 Preface The study of Lie groups goes back to the end of the nineteenth century when Sophus Lie investigated symmetry groups of differential equations. The latter can be studied locally by means of the Lie algebra structure on the tangent space in the identity element of the group. Later, an interest in algebraic groups emerged from studying problems in number theory and algebraic geometry. According to thealgebraic natureofthese objects, itwasquitenatural(andoftenalsouseful) to consideralgebraicgroupsdefinedoverfieldsofarbitrarycharacteristic,orevenmore generally, over arbitrary commutative rings. It turned out quite early that many of the classical results over the real or complex numbers and the correspondence between the group and its associated Lie algebra, which is essential in many of the proofs, are no longer valid over fields of non-zero characteristic. So modular Lie algebras (that is, Lie algebras over fields of prime characteristic) were studied in itsownrightandverysoonseveralapplications(forexample,topurelyinseparable field extensions or to the restricted Burnside problem) appeared. This volume originates from a Workshop on Lie Algebras which took place at the Universita` degli Studi di Milano-Bicocca from May 22nd to May 24th, 2013. The workshop featured invited talks by 13 mathematicians from around the world and was attended by 39 participants from 12 countries, including several graduate students. The aim of the workshop was to bring together some of the leading experts in the field of Lie (super)algebras (simple, locally finite, modular, or real), their gradings, the ideal structure of their universal enveloping algebras, as well as some computational aspects, and several related topics such as representation theory, W-algebras, algebraic groups, quantum groups, Weyl algebras, and the Lie structure of group algebras. Established researchers were joined by younger colleagues with the goal of promoting the exchange of ideas and establishing new collaborations. Lie algebras are at the core of several areas of mathematics, for example, Lie groups, algebraic groups, quantum groups, representation theory, homogeneous spaces,integrablesystems,andalgebraictopology. Thetalksgivenattheworkshop covereredabroadspectrumofthesetopics. Thefirstpartoftheseproceedingscom- binesresearchpaperswithsurveypapers, allwrittenbytheinvitedspeakers(alone or with co-authors). The second part consists of several collections of problems on modular Lie algebras, their representations, and the conjugacy of their nilpotent elements as well as the Koszulity of (restricted) Lie algebras and Lie properties of group algebras or restricted universal enveloping algebras. Theclassificationoffinite-dimensionalsimpleLiealgebrasoveranalgebraically closed field of prime characteristic p>3 has been one of the milestones in Lie the- ory. The workshop was dedicated to Helmut Strade, one of the main architects of vii viii PREFACE this extraordinary mathematical achievement. This volume will represent a snap- shot of the state of the art of the post-classification period. Many applications concerning the structure of an arbitrary finite-dimensional modular Lie algebra require a detailed knowledge of the structure of the simple Lie algebras and their representations. Intheproofoftheclassificationmanyofthepropertiesofmodular simple Lie algebras were established. Since many details of the classification have onlybeenpublishedrecentlyinthethreebookswrittenbyHelmutStrade,itisnow possible to attack many open problems in the structure and representation theory of Lie algebras in prime characteristic. The same phenomenon has been observed after the classification of finite simple groups was completed. This volume exhibits some of the progress in this vibrant area of mathematics. We especially hope that the open problems in the second part will inspire future research in Lie algebras and also in some related topics. The organizers wish to thank the Dipartimento di Matematica e Applicazioni at the Universit`a degli Studi di Milano-Bicocca for the hospitality, the logistical support and—last but not least—the financial funding. We are also very grateful to the editorial staff of Contemporary Mathematics, and in particular to Chris- tine Thivierge for her patience and help in editing these proceedings. Our special thanks go to the anonymous referees for their insight and the high standards em- ployed in writing their reports. Finally, we would like to take this opportunity to express our appreciation to the participants for making the workshop such a suc- cess, to the speakers for their very interesting talks, and especially to the authors for contributing to this volume. Marina Avitabile, J¨org Feldvoss, and Thomas Weigel List of Participants Marina Avitabile Alberto Elduque Universita` degli Studi di Universidad de Zaragoza, Spain Milano-Bicocca, Italy Paolo Faccin Yuri Bahturin Universita` di Trento, Italy Memorial University of Newfoundland, J¨org Feldvoss Canada University of South Alabama, USA Alexander Baranov Norberto Gavioli University of Leicester, UK Universita` de L’Aquila, Italy Raimundo de Arau´jo Bastos Sebastian Herpel Universidade de Bras´ılia, Brasil Technische Universita¨t Kaiserslautern, Germany Georgia Benkart University of Wisconsin-Madison, USA Lutz Hille Westfa¨lische Wilhelms-Universita¨t Martino Borello Mu¨nster, Germany Universita` degli Studi di Milano-Bicocca, Italy Minguan Hu Universita` degli Studi di Sofiane Bouarroudj Milano-Bicocca, Italy United Arab Emirates University, Naihong Hu United Arab Emirates East China Normal University, PR Giovanna Carnovale China Universita` di Padova, Italy Nataliya Ivanova Ilaria Castellano National Academy of Sciences of Universita` degli Studi di Bari Aldo Ukraine at Kiev, Ukraine Moro, Italy Sandro Mattarei Universita` di Trento, Italy Francesca Dalla Volta Universita` degli Studi di Christos A. Pallikaros Milano-Bicocca, Italy University of Cyprus, Cyprus Willem de Graaf Federico W. Pasini Universita` di Trento, Italy Universita` degli Studi di Milano-Bicocca, Italy Lino Di Martino Universita` degli Studi di Alexander Premet Milano-Bicocca, Italy University of Manchester, UK ix