Advanced Courses in Mathematics CRM Barcelona Centre de Recerea Matemätiea Managing Editor: Manuel Castellet Vesselin Drensky Edward Formanek Polynomial Identity Rings Springer Basel AG Authors' addresses: Vesselin Drensky Edward Formanek Institute of Mathematics and Informatics Department of Mathematics Bulgarian Academy of Sciences Pennsylvania State University 1113 Sofia University Park, PA 16802 Bulgaria USA [email protected] [email protected] 2000 Mathematical Subject Classification 16R, 15A24, 15A72 A CIP catalogue record for this book is available from the Library of Congress, Washington D.C., USA Bibliografische Information Der Deutschen Bibliothek Die Deutsche Bibliothek verzeichnet diese Publikation in der Deutschen Nationalbibliografie; detaillierte bibliografische Daten sind im Internet über <http://dnb.ddb.de> abrufbar. ISBN 978-3-7643-7126-5 ISBN 978-3-0348-7934-7 (eBook) DOI 10.1007/978-3-0348-7934-7 This work is subject to copyright. 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TCF00 ISBN 978-3-7643-7126-5 987654321 www.birkhauser-science.com Contents Foreword vii A Combinatorial Aspects in PI-Rings Vesselin Drensky 1 Introduction 3 1 Basic Properties of PI-algebras 1.1 PI-algebras - Definitions and Examples 5 1.2 T-ideals and Varieties of Algebras .... 8 1.3 Generic Matrices . . . . . . . . . . . . . 12 1.4 Polynomial Identities of the Grassmann Algebra 15 2 Quantitative Approach to PI-algebras 2.1 Co dimensions and Hilbert Series 19 2.2 Background on Representation Theory of Groups 25 2.3 Group Representations and PI-algebras ..... 32 3 The Amitsur-Levitzki Theorem 3.1 The Amitsur-Levitzki Theorem and Related Results 37 3.2 Polynomial Identities of Low Degree - a Survey 42 4 Central Polynomials for Matrices 4.1 The Approach of Razmyslov . 49 4.2 More Central Polynomials 56 5 Invariant Theory of Matrices 5.1 A Background on Invariant Theory . 59 5.2 The Generic Trace Algebra 61 5.3 Invariants of 2 x 2 Matrices 67 6 The Nagata-Higman Theorem 6.1 The Nagata-Higman Theorem and its Applications. 75 6.2 Bounds for the Class of Nilpotency ......... . 79 7 The Shirshov Theorem for Finitely Generated PI-algebras 7.1 The Shirshov Theorem and the Kurosh Problem 87 7.2 Growth of PI-algebras ................ . 97 Contents Vi 8 Growth of Co dimensions of PI-algebras 8.1 The Theorem of Regev for the Tensor Product of PI-algebras 103 8.2 The Theory of Kerner and Growth of Codimensions ..... 110 Bibliography 119 B Polynomial Identity Rings Edward Formanek 131 Introduction 133 1 Polynomial Identities 137 2 The Amitsur-Levitzki Theorem 143 3 Central Polynomials 147 4 Kaplansky's Theorem 151 5 Theorems of Amitsur and Levitzki on Radicals 155 6 Posner's Theorem 159 7 Every PI-ring Satisfies a Power of the Standard Identity 161 8 Azumaya Algebras 163 9 Artin's Theorem 169 10 Chain Conditions 173 11 Hilbert and Jacobson PI-Rings 177 12 The Ring of Generic Matrices 179 13 The Generic Division Ring of Two 2 x 2 Generic Matrices 183 14 The Center of the Generic Division Ring 185 15 Is the Center of the Generic Division Ring a Rational Function Field? 189 Bibliography 193 Index 197 Foreword Algebras satisfying polynomial identities, called PI-algebras, have been one of the major research fields in the last 20 years, since they constitute a reasonably big class containing the finite dimensional algebras and the commutative algebras and enjoy many of their main properties. These notes correspond to the lectures delivered by the authors in the Ad vanced Course on Polynomial Identity Rings, held at the Centre de Recerca Matematica in Bellaterra (Barcelona), July 2003. One strand of lectures concen trated on the combinatorial side of polynomial identity rings (PI-rings), while a second explained the structural side of PI-rings. Thus these notes, which are re vised and smoothed version of the lecture notes delivered to the participants of the Advanced Course, are organised in two parts corresponding to these two lecture series. The first part, due to V. Drensky, is devoted to introducing the reader to the combinatorial aspects of PI-theory. The purpose of this chapters is to present sev eral results which form the foundation of the combinatorial theory of PI-algebras, including a survey on some recent results on related topics. The second part, due to E. Formanek, is an introduction to the structural aspects of polynomial identity rings, including the Amitsur-Levitzki theorem, cen tral polynomials, the theorems of Kaplansky, Posner and Artin, and the ring of generic matrices. Besides our indebtedness to the Centre de Recerca Matematica, thanks are due to Ferran Cedo, the course co-ordinator, for making it possible, and to the CRM secretaries, Consol Roca, Maria Julia and Neus Portet, for their assistance. Special thanks go to all the participants of the course for their interest in the event and for their very helpful contributions during the afternoon sessions. E. Formanek V. Drensky Part A Combinatorial Aspects in PI-Rings Vesselin Drensky Introduction The class of PI-algebras, i.e., algebras satisfying polynomial identities, is a reason ably big class containing the finite dimensional and the commutative algebras and enjoying many of their properties. Traditionally the theory of PI-algebras has two aspects, structural and combinatorial, with considerable overlap between them. The purpose of these lecture notes is to present several results which form the foundation of the combinatorial theory of PI-agebras: The Amitsur-Levitzki theorem; the construction of central polynomials for matrices; the polynomial identities of matrices and their relation with invariant theory; the Nagata-Higman theorem on the nil potency of nil algebras of bounded index; the Shirshov theorem for finitely generated PI-algebras; the Regev theorem for the tensor product of PI-algebras. In many places in the text a survey on some recent results on related topics is given, such as the negative solution of the Specht problem in positive charac teristic, applications of computers to the polynomial identities of matrices, central polynomials of low degree for matrices of any size, growth of PI-algebras and their co dimension sequences. Also, a couple of results which usually are not included in courses on PI-algebras are presented, such as the lower bound of Kuzmin for the class of nilpotency in the Nagata-Higman theorem. I hope that the included open problems, comments and references to the current situation in the discussed field will be useful for the orientation of the reader. I tried to make the exposition accessible for young mathematicians with standard background on linear algebra and elements of ring and group theory. Nevertheless I believe that professional mathematicians working on PI-theory also will find the text useful. There are several books which can serve for further reading on some of the topics included in these lecture notes. Among them are the general books on PI-algebras by Procesi [Pr2]' Jacobson p] and Rowen [Rwl]. The books by For manek [F6] , Kerner [Ke5] and Razmyslov [Ra5] are devoted to more specific topics. Recently, I wrote the book [DrlO] which deals with combinatorial theory of free algebras and PI-algebras. I tried to minimize its intersection with the present text. V. Drensky et al., Polynomial Identity Rings © Birkhäuser Verlag 2004 4 Introduction Acknowledgements I am very grateful to Ed Formanek for the useful discussions and suggestions about the topics of the course. I am also very thankful to many colleagues and friends. I want especially to mention my advisers Georgy Genov from the University of Sofia and Yuri Bahturin from the Moscow State University, as well as S.A. Amitsur, L.A. Bokut, O.M. Di Vincenzo, M. Domokos, M.B. Gavrilov, A. Giambruno, A.R. Kerner, P. Koshlukov, V.N. Latyshev, A.P. Popov, Yu.P. Razmyslov, A. Regev, I.P. Shestakov, A.L. Shmelkin, LB. Volichenko, and E.I. Zelmanov. Finally, I am deeply indebted to my family and especially to my wife for the family atmosphere, the support and the understanding as long as I was working on this text. This project was also partially supported by Grant MM-l106/2001 of the Bulgarian Foundation for Scientific Research.