Groups, Rings and Group Rings

Author: Antonio Giambruno | Cesar Polcino Milies | Sudarshan K. Sehgal

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Groups, Rings and Group Rings

© 2006 by Taylor & Francis Group, LLC

PURE AND APPLIED MATHEMATICS A Program of Monographs, Textbooks, and Lecture Notes

EXECUTIVE EDITORS Earl J. Taft Rutgers University Piscataway, New Jersey

Zuhair Nashed University of Central Florida Orlando, Florida

EDITORIAL BOARD M. S. Baouendi University of California, San Diego Jane Cronin Rutgers University Jack K. Hale Georgia Institute of Technology S. Kobayashi University of California, Berkeley Marvin Marcus University of California, Santa Barbara W. S. Massey Yale University Anil Nerode Cornell University

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Freddy van Oystaeyen University of Antwerp, Belgium Donald Passman University of Wisconsin, Madison Fred S. Roberts Rutgers University David L. Russell Virginia Polytechnic Institute and State University Walter Schempp Universität Siegen Mark Teply University of Wisconsin, Milwaukee

LECTURE NOTES IN PURE AND APPLIED MATHEMATICS Recent Titles J. Cagnol et al., Shape Optimization and Optimal Design J. Herzog and G. Restuccia, Geometric and Combinatorial Aspects of Commutative Algebra G. Chen et al., Control of Nonlinear Distributed Parameter Systems F. Ali Mehmeti et al., Partial Differential Equations on Multistructures D. D. Anderson and I. J. Papick, Ideal Theoretic Methods in Commutative Algebra Á. Granja et al., Ring Theory and Algebraic Geometry A. K. Katsaras et al., p-adic Functional Analysis R. Salvi, The Navier-Stokes Equations F. U. Coelho and H. A. Merklen, Representations of Algebras S. Aizicovici and N. H. Pavel, Differential Equations and Control Theory G. Lyubeznik, Local Cohomology and Its Applications G. Da Prato and L. Tubaro, Stochastic Partial Differential Equations and Applications W. A. Carnielli et al., Paraconsistency A. Benkirane and A. Touzani, Partial Differential Equations A. Illanes et al., Continuum Theory M. Fontana et al., Commutative Ring Theory and Applications D. Mond and M. J. Saia, Real and Complex Singularities V. Ancona and J. Vaillant, Hyperbolic Differential Operators and Related Problems G. R. Goldstein et al., Evolution Equations A. Giambruno et al., Polynomial Identities and Combinatorial Methods A. Facchini et al., Rings, Modules, Algebras, and Abelian Groups J. Bergen et al., Hopf Algebras A. C. Krinik and R. J. Swift, Stochastic Processes and Functional Analysis: A Volume of Recent Advances in Honor of M. M. Rao S. Caenepeel and F. van Oystaeyen, Hopf Algebras in Noncommutative Geometry and Physics J. Cagnol and J.-P. Zolésio, Control and Boundary Analysis S. T. Chapman, Arithmetical Properties of Commutative Rings and Monoids O. Imanuvilov, et al., Control Theory of Partial Differential Equations Corrado De Concini, et al., Noncommutative Algebra and Geometry A. Corso, et al., Commutative Algebra: Geometric, Homological, Combinatorial and Computational Aspects G. Da Prato and L. Tubaro, Stochastic Partial Differential Equations and Applications – VII L. Sabinin, et al., Non-Associative Algebra and Its Application K. M. Furati, et al., Mathematical Models and Methods for Real World Systems A. Giambruno, et al., Groups, Rings and Group Rings © 2006 by Taylor & Francis Group, LLC

Groups, Rings and Group Rings

Edited by

Antonio Giambruno Universita` di Palermo Palermo, Italy

César Polcino Milies Universidade de São Paulo São Paulo, Brazil

S.K. Sehgal University of Alberta Edmonton, Alberta, Canada

Boca Raton London New York

© 2006 by Taylor & Francis Group, LLC

C5815_Discl.fm Page 1 Tuesday, October 4, 2005 10:39 AM

Published in 2006 by Chapman & Hall/CRC Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2006 by Taylor & Francis Group, LLC Chapman & Hall/CRC is an imprint of Taylor & Francis Group No claim to original U.S. Government works Printed in the United States of America on acid-free paper 10 9 8 7 6 5 4 3 2 1 International Standard Book Number-10: 1-58488-581-5 (Softcover) International Standard Book Number-13: 978-1-58488-581-8 (Softcover) Library of Congress Card Number 2005053799 This book contains information obtained from authentic and highly regarded sources. Reprinted material is quoted with permission, and sources are indicated. A wide variety of references are listed. Reasonable efforts have been made to publish reliable data and information, but the author and the publisher cannot assume responsibility for the validity of all materials or for the consequences of their use. No part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, please access www.copyright.com (http://www.copyright.com/) or contact the Copyright Clearance Center, Inc. (CCC) 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. CCC is a not-for-profit organization that provides licenses and registration for a variety of users. For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. Library of Congress Cataloging-in-Publication Data Groups, rings and group rings / [edited by] Antonio Giambruno, César Polcino Milies, S.K. Sehgal. p. cm. -- (Lecture notes in pure and applied mathematics ; 247) Includes bibliographical references and index. ISBN 1-58488-581-5 1. Group theory--Congresses. 2. Rings (Algebra)--Congresses. 3. Group rings--Congresses. I. Giambruno, A. II. Milies, César Polcino. III. Sehgal, Sudarshan K., 1936- IV. Lecture notes in pure and applied mathematics ; v 247. QA174.G768 2006 512'.2--dc22

2005053799

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Preface

This volume contains the proceedings of the Conference on “Groups, Rings, and Group Rings” held in Ubatuba, Brazil, from July 26 to 31, 2004, in honor of the sixtieth birthday of Prof. C´esar Polcino Milies. This was an occasion for bringing together experts from different areas of algebra ranging from group theory to associative and nonassociative algebras to coding theory. The meeting was attended by more than 80 participants and experts from several different countries including Argentina, Belgium, Brazil, Chile, Canada, Germany, Italy, Israel, Russia, Spain, Switzerland, Ukraine, Thailand and USA. The conference featured invited main lectures by E. Aljadeff, N. Andruskiewitsch, Y. Bahturin, G. Cliff, A. del R´ıo, R. Exel, M. Ferrero, V. Futorny, A. Giambruno, W. Herfort, W. Kimmerle, V. Kirichenko, B. K¨ uhlshammer, S. Lopez-Permouth, S. Sehgal, I. Shestakov, M. Shirvani, L. Shneerson, S. Sidki, and E. Zelmanov and several other talks of shorter length. The papers of most of the principal speakers and some other contributions are included in the present volume. The editors wish to express their appreciation to the agencies and the institutions that contributed financial support: Capes, CNPq, FAPESP, FINEP, Grupo Votorantim, Pr´o-Reitorias de Pesquisa e de P´os-Gradua¸c˜ao, and the Instituto de Matem´atica e Estat´ıstica of the Universidade de S˜ao Paulo.

Antonio Giambruno Sudarshan K. Sehgal

© 2006 by Taylor & Francis Group, LLC

List of Participants

Eli Aljadeff Technion, Israel

´ Angel del R´ıo Universidad de Murcia, Spain

Nicol´ as Andruskiewitsch Universidad Nacional de C´ordoba, Argentina

Michael Dokuchaev Universidade de S˜ao Paulo, Brasil

Alfredo J. Aragona Vallejo Universidade de S˜ao Paulo, Brasil

Ann Dooms Vrije Universiteit Brussel, Belgium

Yuri A. Bahturin Memorial University of Newfoundland, Canada

Flaviana Dutra Universidade Federal de Minas Gerais, Brasil

Luiz G.X. de Barros Senac, Brasil

Ruy Exel Universidade Federal de Santa Catarina, Brasil

Laurent Bartholdi ´ Ecole Polytechnique F´ed´erale, Switzerland

Bernardo Felzenszwalb Universidade Federal do Rio de Janeiro, Brasil

Osnel Broche Cristo Universidade de S˜ao Paulo, Brasil

Raul A. Ferraz Universidade de S˜ao Paulo, Brasil

Sheila Campos Chagas Universidade de Bras´ılia, Brasil

Vitor O. Ferreira Universidade de S˜ao Paulo, Brasil

Gladys Chalom Universidade de S˜ao Paulo, Brasil

Miguel Ferrero Universidade Federal do Rio Grande do Sul, Brasil

Gerald Cliff University of Alberta, Canada

Leila Maria Vasconcellos Figueiredo Universidade de S˜ao Paulo, Brasil

Mariana Cornelissen Universidade Federal de Minas Gerais, Brasil

Vyacheslav Futorny Universidade de S˜ao Paulo, Brasil

Wagner Cortes Universidade Estadual de Maring´a, Brasil

Antonio Giambruno Universit` a di Palermo, Italy

© 2006 by Taylor & Francis Group, LLC

Andr´e Gimenez Bueno Universidade de S˜ao Paulo, Brasil

Burkhard K¨ ulshammer Friedrich-Schiller-Universit¨ at, Germany

Adilson Gon¸calves Universidade Federal do Rio de Janeiro, Brasil

Alicia Labra Universidad de Chile, Chile

Jairo Z. Gon¸calves Universidade de S˜ao Paulo, Brasil

Jo˜ ao Lazzarin Universidade Federal do Rio Grande do Sul, Brasil

Carlos Grossi Universidade Estadual de Campinas, Brasil

Yuanlin Li Brock University, Canada

Henrique Guzzo, Jr. Universidade de S˜ao Paulo, Brasil

Alexander Lichtman University of Wisconsin, USA

Wolfgang Herfort Vienna University of Technology, Austria

Thierry Petit Lob˜ ao Universidade Federal da Bahia, Brasil

Edson Iwaki Universidade de S˜ao Paulo, Brasil

Jonas Lopes Universidade de S˜ao Paulo, Brasil

Sandra Jorge Universidade Federal de Minas Gerais, Brasil

Pedro Lopes Universidade T´ecnica de Lisboa, Portugal

Stanley O. Juriaans Universidade de S˜ao Paulo, Brasil

Sergio R. L´ opez-Permouth Ohio University, USA

Iryna Kashuba Universidade de S˜ao Paulo, Brasil

Val´eria Ostete Jannis Luchetta Universidade de S˜ao Paulo, Brasil

Marina Khibina Arnaldo Mandel Institute of Engineering of Thermophy- Universidade de S˜ao Paulo, sics of National Academy, Ukraine Brasil Wolfgang Kimmerle Universit¨ at Stuttgart, Germany Vladimir Kirichenko Kyiv Taras Shevchenko Ukraine

Eduardo Marcos Universidade de S˜ao Paulo, Brasil Lucia S.I. Murakami University, Universidade de S˜ao Paulo, Brasil

Plamen Koshlukov Universidade Estadual de Campinas, Brasil

© 2006 by Taylor & Francis Group, LLC

Boris V. Novikov Kharkov National University, Ukraine

Serge Ovsienko Kiev Tars Shevchenko University, Ukraine

Lev Shneerson The City University of New York, USA

Henrique Panzarelli Universidade de S˜ao Paulo, Brasil

Said N. Sidki Universidade de Bras´ılia, Brasil

Luiz Antonio Peresi Universidade de S˜ao Paulo, Brasil

Maria Sobral Singer Universidade de S˜ao Paulo, Brasil

C´esar Polcino Milies Universidade de S˜ao Paulo, Brasil

Antonio Calixto Souza Filho Universidade de S˜ao Paulo, Brasil

Nora´ı R. Rocco Universidade de Bras´ılia, Brasil

Maria Trushina Universidade de S˜ao Paulo, Brasil

Virg´ınia Rodrigues Universidade Federal do Rio Grande do Sul, Brasil

Ana Cristina Vieira Universidade Federal de Minas Gerais, Brasil

Javier S´ anchez Serd` a Universitat Aut`onoma de Barcelona, Spain

Benjamin John Wilson Universidade de S˜ao Paulo, Brasil

Nguyen Van Sanh Mahidol University, Thailand

Pavel Zalesskii Universidade de Bras´ılia, Brasil

Sudarshan K. Sehgal University of Alberta, Canada

Andrei Zavarnitsine Universidade de S˜ao Paulo, Brasil

Ivan Shestakov Universidade de S˜ao Paulo, Brasil

Efim Zelmanov University of California, USA

Mazi Shirvani University of Alberta, Canada

Natalia Zhukavets Universidade de S˜ao Paulo, Brasil

© 2006 by Taylor & Francis Group, LLC

Contributors

Eli Aljadeff Department of Mathematics Technion Haifa, Israel ´s Andruskiewitsch Nicola Facultad de Matem´atica, Astronom´ıa y F´ısica Universidad Nacional de C´ordoba C´ ordoba, Argentina Yuri A. Bahturin Department of Mathematics and Statistics Memorial University of Newfoundland St. John’s, Newfoundland, Canada

Gerald Cliff Department of Mathematical and Statistical Sciences University of Alberta Edmonton, Alberta, Canada ´ Angel del R´ıo Departamento de Matem´aticas Universidad de Murcia Murcia, Spain Onofrio M. Di Vincenzo Dipartimento di Matematica Universit` a degli Studi di Bari Bari, Italy

Hai Q. Dinh Department of Mathematical Sciences Gunther Bergauer Institute for Analysis and Scientific Kent State University Warren, Ohio Computation University of Technology Michael Dokuchaev Vienna, Austria Instituto de Matem´atica e Estat´ıstica Universidade de S˜ao Paulo Osnel Broche Cristo S˜ ao Paulo, Brasil Instituto de Matem´atica e Estat´ıstica Universidade de S˜ao Paulo Ann Dooms S˜ ao Paulo, Brasil Department of Mathematics Vrije Universiteit Brussel Gladys Chalom Brussels, Belgium Instituto de Matem´atica e Estat´ıstica Universidade de S˜ao Paulo Raul A. Ferraz S˜ ao Paulo, Brasil Instituto de Matem´atica e Estat´ıstica Universidade de S˜ao Paulo Orin Chein S˜ ao Paulo, Brasil Temple University Philadelphia, PA

© 2006 by Taylor & Francis Group, LLC

Vitor O. Ferreira Instituto de Matem´atica e Estat´ıstica Universidade de S˜ao Paulo S˜ ao Paulo, Brasil Walter R. Ferrer Santos Centro de Matem´atica Facultad de Ciencias Universidad de la Rep´ ublica Montevideo, Uruguay Miguel Ferrero Instituto de Matem´atica Universidade Federal do Rio Grande do Sul Porto Alegre, Brasil Vyacheslav Futorny Instituto de Matem´atica e Estat´ıstica Universidade de S˜ao Paulo S˜ ao Paulo, Brasil Antonio Giambruno Dipartimento di Matematica e Applicazioni Universit` a di Palermo Palermo, Italy ´ Gimenez Bueno Andre Instituto de Matem´atica e Estat´ıstica Universidade de S˜ao Paulo S˜ ao Paulo, Brasil Carmen Rosa Giraldo Vergara Faculdade de Filosofia, Ciˆencias e Letras Funda¸c˜ ao Educacional Monsenhor Messias Sete Lagoas, Brasil Jairo Z. Gonc ¸ alves Instituto de Matem´atica e Estat´ıstica Universidade de S˜ao Paulo S˜ ao Paulo, Brasil

© 2006 by Taylor & Francis Group, LLC

Edgar G. Goodaire Memorial University of Newfoundland St. John’s, Newfoundland, Canada Marcus Greferath Department of Mathematics University College Dublin Dublin, Ireland Nadiya M. Gubareni Institute of Econometrics and Computer Science Technical University of Cz¸estochowa Cz¸estochowa, Poland Henrique Guzzo, Jr. Instituto de Matem´atica e Estat´ıstica Universidade de S˜ao Paulo S˜ ao Paulo, Brasil Darrell Haile Department of Mathematics Indiana University Bloomington, Indiana Wolfgang Herfort Institute for Analysis and Scientific Computation University of Technology Vienna, Austria ¨ fert Christian Ho Institut f¨ ur Geometrie und Topologie Fachbereich Mathematik Universit¨ at Stuttgart Stuttgart, Germany Eric Jespers Department of Mathematics Vrije Universiteit Brussel Brussels, Belgium Stanley O. Juriaans Instituto de Matem´atica e Estat´ıstica Universidade de S˜ao Paulo S˜ ao Paulo, Brasil

Wolfgang Kimmerle Institut f¨ ur Geometrie und Topologie Fachbereich Mathematik Universit¨ at Stuttgart Stuttgart, Germany Vladimir V. Kirichenko Department of Mechanics and Mathematics Kyiv Taras Shevchenko University Kiev, Ukraine Plamen Koshlukov Instituto de Matem´atica e Estat´ıstica Universidade Estadual de Campinas Campinas, Brasil ¨ lshammer Burkhard Ku Mathematisches Institut Friedrich-Schiller-Universit¨ at Jena, Germany Daniela La Mattina Dipartimento di Matematica e Applicazioni Universit` a di Palermo Palermo, Italy Guilherme Leal Instituto de Matem´atica Universidade Federal do Rio de Janeiro Rio de Janeiro, Brasil Yuanlin Li Department of Mathematics Brock University St. Catharines, Ontario, Canada ´ pez-Permouth Sergio R. Lo Department of Mathematics Ohio University Athens, Ohio

© 2006 by Taylor & Francis Group, LLC

´ria O. Luchetta Vale Instituto de Matem´atica e Estat´ıstica Universidade de S˜ao Paulo S˜ ao Paulo, Brasil Zbigniew Marciniak Institute of Mathematics Warsaw University Warsaw, Poland David McNeilly University of Alberta Department of Mathematical and Statistical Sciences Edmonton, Alberta, Canada ´ctor Merklen He Instituto de Matem´atica e Estat´ıstica Universidade de S˜ao Paulo S˜ ao Paulo, Brasil Paola Misso Dipartimento di Matematica e Applicazioni Universit` a di Palermo Palermo, Italy Lucia S.I. Murakami Instituto de Matem´atica e Estat´ıstica Universidade de S˜ao Paulo S˜ ao Paulo, Brasil Michael Natapov Department of Mathematics Technion Haifa, Israel Boris V. Novikov Kharkov National University Kharkov, Ukraine Konstantin Okunev Novosibirsk State University Novosibirsk, Russia

Michael M. Parmenter Department of Mathematics and Statistics Memorial University of Newfoundland St. John’s, Newfoundland, Canada

Sudarshan K. Sehgal Department of Mathematical and Statistical Sciences University of Alberta Edmonton, Alberta, Canada

Arturo Pianzola Department of Mathematical and Statistical Sciences University of Alberta Edmonton, Alberta, Canada

Ivan Shestakov Instituto de Matem´atica e Estat´ıstica Universidade de S˜ao Paulo S˜ ao Paulo, Brasil and Sobolev Institute of Mathematics Novosibirsk, Russia

Antonio Pita Departamento de Matem´aticas Universidad de Murcia Murcia, Spain ´sar Polcino Milies Ce Instituto de Matem´atica e Estat´ıstica Universidade de S˜ao Paulo S˜ ao Paulo, Brasil Manuel Ruiz Mar´ın Departamento de M´etodos Cuantitativos e Inform´aticos Universidad Polit´ecnica de Cartagena Cartagena, Spain

© 2006 by Taylor & Francis Group, LLC

Mazi Shirvani Department of Mathematical and Statistical Sciences University of Alberta Edmonton, Alberta, Canada Angela Valenti Dipartimento di Metodi e Modelli Matematici Universit` a di Palermo Palermo, Italy Ana Cristina Vieira Departamento de Matem´atica Instituto de Ciˆencias Exatas Universidade Federal de Minas Gerais Belo Horizonte, Brasil

Contents

1. On fine gradings on central simple algebras Eli Aljadeff, Darrell Haile, and Michael Natapov

1

2. On observable module categories Nicol´ as Andruskiewitsch and Walter R. Ferrer Santos

11

3. Group gradings on integral group rings Yuri A. Bahturin and Michael M. Parmenter

25

4. Profinite graphs – comparing notions Gunther Bergauer and Wolfgang Herfort

33

5. Lie identities in symmetric elements in group rings: A survey Osnel Broche Cristo and Manuel Ruiz Mar´ın

43

6. Irreducible morphisms in subcategories Gladys Chalom and H´ector Merklen

57

7. Bol loops with a unique nonidentity commutator/associator Orin Chein and Edgar G. Goodaire

73

8. Weil representations of symplectic groups Gerald Cliff and David McNeilly

81

9. Gradings and graded identities for the upper triangular matrices over an infinite field Onofrio M. Di Vincenzo, Plamen Koshlukov, and Angela Valenti 10. Structure of some classes of repeated-root constacyclic codes over integers modulo 2m Hai Q. Dinh

91

105

11. Units in noncommutative orders Ann Dooms and Eric Jespers

119

12. Idempotents in group algebras and coding theory Raul A. Ferraz, Val´eria O. Luchetta, and C´esar Polcino Milies

137

13. Finitely generated constants of free algebras Vitor O. Ferreira and Lucia S.I. Murakami

149

14. Partial actions of groups on semiprime rings Miguel Ferrero

155

15. Representations of affine Lie superalgebras Vyacheslav Futorny

163

16. On algebras and superalgebras with linear codimension growth Antonio Giambruno, Daniela La Mattina, and Paola Misso

173

17. On spectra of group rings of finite abelian groups Andr´e Gimenez Bueno and Michael Dokuchaev

183

© 2006 by Taylor & Francis Group, LLC

18. Wedderburn decomposition of small rational group algebras Carmen Rosa Giraldo Vergara

191

19. Some questions on skewfields Jairo Z. Gon¸calves

201

20. On the role of rings and modules in algebraic coding theory Marcus Greferath and Sergio R. L´ opez-Permouth

205

21. Semiperfect rings with T -nilpotent prime radical Nadiya M. Gubareni and Vladimir V. Kirichenko

217

22. The structure of the baric algebras Henrique Guzzo, Jr.

233

23. On torsion units of integral group rings of groups of small order Christian H¨ ofert and Wolfgang Kimmerle

243

24. On a conjecture of Zassenhaus for metacyclic groups Stanley O. Juriaans and Sudarshan K. Sehgal

253

25. Nilpotent blocks revisited Burkhard K¨ ulshammer

263

26. Decomposition of central units of integral group rings Yuanlin Li and Michael M. Parmenter

275

27. Generic units in ZC Zbigniew Marciniak and Sudarshan K. Sehgal

279

28. On quasi-Frobenius semigroup algebras Boris V. Novikov

287

29. Twisted loop algebras and Galois cohomology Arturo Pianzola

293

− 30. Presentation of the group of units of ZD16 ´ Antonio Pita and Angel del R´ıo

305

31. Engel theorem for Jordan superalgebras Ivan Shestakov and Konstantin Okunev

315

32. A characterization of centre-by-finite subgroups of division algebras Mazi Shirvani 33. Isomorphic rational group algebras Ana Cristina Vieira and Guilherme Leal

© 2006 by Taylor & Francis Group, LLC

321 331

On fine gradings on central simple algebras Eli Aljadeff Department of Mathematics, Technion, Haifa 32000, Israel [email protected] Darrell Haile Department of Mathematics, Indiana University, Bloomington, IN 47405 [email protected] Michael Natapov Department of Mathematics, Technion, Haifa 32000, Israel [email protected]

1. Introduction Let G be a group and R an associative ring. A G−grading on R is a decomposition of R as an additive group into the direct sum of subgroups R = ⊕g∈G Rg such that Rg Rh ⊆ Rgh for any g, h ∈ G. The subgroup Rg is called the homogeneous component of degree g and elements of Rg are called homogeneous of degree g. The subgroup Re , where e is the identity element of G, is called the identity component. The support of R is defined by Supp R = {g ∈ G : Rg 6= 0}. If H = Supp R is a subgroup of G, we say that the group H supports the G−grading on R. If R is an algebra over a field F , a G−grading on R is called fine if dimF (Rg ) ≤ 1 for all g ∈ G. We refer the reader to [5] for more details on G–gradings on algebra over a field F. Our first object in this paper is to observe that the notion of a fine grading on a central simple algebra is basically the same as the notion of a projective basis. Recall that a finite dimensional F –algebra R is said to have a projective basis if it has a basis {a1 , a2 , . . . , an } of invertible elements, such that for every pair i, j there is an m such that ai aj = λij am for some λij ∈ F × . It is often convenient to view such an algebra R as a twisted group algebra. Let Γ be the subgroup of R× generated by a1 , a2 , . . . , an and F × , and let G = Γ/F × , then R is isomorphic to the twisted group algebra F α G where the cohomology class α ∈ H 2 (G, F × ) is defined by the central extension 1 −→ F × −→ Γ −→ G −→ 1 1

© 2006 by Taylor & Francis Group, LLC

2

E. Aljadeff, D. Haile, and M. Natapov

(Recall that for each finite group G and two-cocycle f in Z 2 (G, F × ) one can form the twisted group algebra F f G = ⊕σ∈G F xσ where xσ xτ = f (σ, τ )xστ for all σ, τ ∈ G. The resulting F –algebra depends (up to F –isomorphism) only on the class α = [f ] in H 2 (G, F × ), and so we denote the algebra by F α G.) The precise statement that relates projective bases to fine gradings for F –central simple algebras is as follows: Theorem 1. Let F be a field. Let R be a finite dimensional central simple algebra over a field F graded by a group G such that dimF (Rg ) ≤ 1 for all g ∈ G. Then H = Supp R is a subgroup of G and R ∼ = F α H for some 2 × α ∈ H (H, F ). The converse of Theorem 1 is clear. If R is an F –algebra isomorphic to a twisted group algebra F α H, α ∈ H 2 (H, F × ), then H supports a fine grading on R over F . Theorem 1 is proved in Section 2. We will call a group H, such that F α H is F –central simple for some F and α ∈ H 2 (H, F × ), a group of central type (nonclassically). The class α, such that F α H is F –central simple, is said to be nondegenerate. Recall that classically a finite group Γ is of central type if it has anpirreducible complex representation of the maximal possible degree, namely [Γ : Z(Γ)] (where Z(Γ) denotes the center of Γ). It is easy to see that H is of central type nonclassically if and only if there exists a classical central type group Γ such that H ∼ = Γ/Z(Γ). Isaacs and Howlett proved, using the classification of finite simple groups, that such Γ, and hence H, is solvable ([6, Theorem 7.3]). In this paper, we refer to “central type” only in the nonclassical sense. Thus by Theorem 1 we obtain that the groups H that support fine gradings of central simple algebras are precisely the groups of central type. Theorem 1 allows us to give a precise description of the groups H that support fine gradings of F –central division algebras. Let Λp be the following list of p–groups: Q (1) H is abelian of symmetric type, that is H ∼ = (Zpni × Zpni ) n n (2) H ∼ = H1 × H2 where H1 = Zpn o Zpn = h π, σ | σ p = π p = s

1 and σπσ −1 = π p +1 i where 1 ≤ s < n and 1 6= s if p = 2 and H2 is an abelian group of symmetric type of exponent ≤ ps (3) H ∼ = H1 × H2 where ¯ ¿ À ¯ π 2n+1 = σ 2n = τ 2 = 1, στ = τ σ, ¯ H1 = Z2n+1 o (Z2n × Z2 ) = π, σ, τ ¯ σπσ −1 = π 3 , τ πτ −1 = π −1 and H2 is an abelian group of symmetric type of exponent ≤ 2 Combining Theorem 1 with [1, Theorem 1] and [2, Corollary 3] we have

Theorem 2. Let H be a finite group that is the support of a fine grading on an F –central division algebra D. Then

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On fine gradings on central simple algebras

3

(1) H is a nilpotent group. (2) If H ∼ = P1 × . . . × Pr is the decomposition of H into the product of its Sylow pi –subgroups then Pi belongs to Λpi . In particular, the commutator subgroup H 0 of H is cyclic. Remark: In a future paper [8] Natapov shows that any group in Λp supports a fine grading on an F –central division algebra for some F . Moreover, if H∼ = P1 ×. . .×Pr , Pi are pi –groups in Λpi (pi are different primes), then there exists a field F (independent of i) such that Pi supports a fine grading on an F –central division algebra Di , 1 ≤ i ≤ r. Then, it is clear that H supports a fine grading on an F –central division algebra, namely D = D1 ⊗F . . . ⊗ Dr . Theorem 2 and the above remark give a complete classification of the groups that support fine gradings on finite dimensional division algebras over their center. The main bulk of this paper is devoted to showing that if a nilpotent group H supports a fine grading on Mr (D) then the index of Mr (D) (= √ ind(D) = dimF D) puts a strong constraint on H. More precisely: Theorem 3. Let H be a nilpotent group that supports a fine grading on an F –central simple algebra R = Mr (D), D a division algebra. Then there exists a subquotient S of H of order ≥ ind(D)2 whose commutator subgroup S 0 is cyclic. Theorem 3 is proved in Section 3. 2. Fine gradings and projective bases In this Section we prove Theorem 1. We follow the proof of [5, Theorem 5]. Claim 1: The identity component Re = F · 1. Proof. PLet aσ be a homogeneous element of degree σ, then Rσ = F aσ . Write 1 = σ∈H λσ aσ where λσ are scalars in F . We claim that λσ = 0 for all σ 6= e. SupposePthere is τ 6= e, τ ∈ H, such that λτ 6= 0. Then writing ag = ag · 1 = σ∈H λσ ag aσ for all g ∈ G we get that λτ ag aτ = 0, or, equivalently, ag aτ = 0 for all g ∈ G. It follows Raτ = 0, a contradiction. Hence, 1 = λe ae . In other words 1 ∈ Re and Claim 1 follows. ¤ Claim 2: Let g ∈ H and 0 6= ag ∈ Rg . Then ag is invertible. Proof. Consider the two-sided ideal Rag R in R. It is a nonzero ideal, hence Rag R = R. Write the unity element 1 as an element of Rag R: 1=

X i

Ã

X

! λiσ aσ

σ∈H

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Ã ag

X τ ∈H

! µiτ aτ

=

X i,σ,τ

λiσ µiτ aσ ag aτ

4

E. Aljadeff, D. Haile, and M. Natapov

where λiσ and µiτ are in F . Since 1 is a homogeneous element of degree e, there exist σ0 and τ0 in G such that σ0 gτ0 = e and aσ0 ag aτ0 is a nonzero and hence invertible element of Re = F . This proves the claim. ¤ It follows that Rg Rh 6= 0 for any g, h ∈ H and therefore Rgh 6= 0 or, equivalently, gh ∈ Supp R = H. That is H is a finite subset of G closed under the multiplication. Therefore H is a subgroup of G. Since dim(Rg ) = 1 for all g ∈ H it follows that ord(H) = dim(R). Now pick homogeneous elements ai ∈ Rgi for every gi ∈ H. Then the set {ai } is a basis of R and ai aj = λij am for some λij ∈ F × . Thus R is isomorphic to F α H where α = [f ] ∈ H 2 (H, F × ) is defined by f (gi , gj ) = λij . 3. Fine gradings on central simple algebras supported by nilpotent subgroups In this Section we prove Theorem 3. Observe first that if H is a nilpotent group that supports a fine grading on an F –central simple algebra R then R ∼ = F α H is isomorphic to F α1 P1 ⊗ α α F 2 P2 ⊗ · · · ⊗ F m Pm , where P1 , P2 , . . . , Pm are the Sylow pi –subgroups of H and αi is the restriction α to Pi . Furthermore, if Pi has a subquotient Si of order ≥ ind(F αi Pi ) with cyclic commutator then S = S1 × · · · × Sm is a subquotient of H such that ord(S) ≥ ind(F α H) and the commutator subgroup S 0 is cyclic. So, this reduces the analysis to p–groups. Let G be a finite p–group which supports the grading on an F –central simple algebra R. Write R = F α G. By the proof of [4, Theorem 1.2, Claim II] F contains a nontrivial p–th root of unity. In particular char(F ) 6= p. Let Γ be the subgroup of (F α G)× given by the central extension π

α : 1 −→ F × −→ Γ −→ G −→ 1 Clearly F α G is spanned as an F -vector space (and even as an abelian group) by Γ. We often write F α G = F (Γ). For every σ ∈ G we choose an element uσ in Γ such that π(uσ ) = σ. We start with the following reduction lemma. Lemma 4. Let G be a p–group and F α G an T F –central simple algebra. Suppose that G0 is not trivial. Let U ⊆ Z(G) G0 be a subgroup of order p. If F α U is not simple then a. F α U ∼ = F α U e1 ⊕...⊕F α U ep ∼ = F e1 ⊕...⊕F ep where ei ’s are mutually orthogonal idempotents. b. There is a subgroup T , U ⊂ T ⊂ G of index p such that 2 × 2 × resG T (α) ∈ Im(inf : H (T /U, F ) → H (T, F ))

c. F α G ∼ = Mp (F β H) where H ∼ = T /U and β ∈ H 2 (H, F × ) T /U infT (β) = resG T (α).

© 2006 by Taylor & Francis Group, LLC

with

On fine gradings on central simple algebras

5

Proof. Let σ be a generator of U√and let upσ = a ∈ F × . Since F contains a nontrivial p–th root of unity F ( p a)/F is a √ Kummer extension. Clearly, if a is not a p–th power in F × then F α U ∼ = F ( p a) which is simple. Therefore b where U b = hb−1 uσ i. We obtain a = bp , b ∈ F × , and hence F α U = F U b∼ b e1 ⊕ ... ⊕ F U b ep = F e1 ⊕ ... ⊕ F ep where e1 = 1/p(1 + b−1 uσ + ... + FU = FU −1 p−1 (b uσ ) ). To see (b) and (c) note that since U is normal in G, the group G acts on F α U by conjugation. Moreover, since F α G is F –central the action permutes transitively the simple components of F α U , [3, Lemma 1.2]. Let T ⊂ G be b. the stabilizer of F α U e1 . Clearly, T is of index p in G and centralizes U −1 α ∼ b b Let T = π (T ) ⊂ Γ. By [3, Theorem 1.4], F G = Mp (F e1 (T e1 )) where F e1 (Tbe1 ) is the algebra spanned by Tbe1 over F e1 ∼ = F . We show that T /U β G ∼ b F e1 (T e1 ) = F (T /U ) where infT (β) = resT (α). Splicing together the short exact sequences b −→ Tb −→ Tbe1 −→ 1 1 −→ U and res(α) : 1 −→ F × −→ Tb −→ T −→ 1, we obtain a commutative diagram with exact rows and columns:

b −−−−→ 1 −−−−→ U   y

1   y

1   y

F×   y

F×   y

Tb −−−−→ Tbe1 −−−−→ 1     y y

1 −−−−→ U −−−−→ T −−−−→ T /U −−−−→ 1     y y 1 1 If we let β be the cohomology class which corresponds to the right column T /U then we get infT (β) = resG T (α). Finally, identifying F and F e1 we get a surjective map of F –algebras F β (T /U ) −→ F e1 (Tbe1 ). The proof is completed by a dimension argument.

¤

We continue with a factorization lemma. It generalizes a similar statement for division algebras (see [1]). The proof is identical and hence is omitted.

© 2006 by Taylor & Francis Group, LLC

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E. Aljadeff, D. Haile, and M. Natapov

Lemma 5. Let P be a normal subgroup of G and assume F α P is F –central simple. Then F α G ∼ = F α P ⊗ F β G/P for some β ∈ H 2 (G/P, F × ). Next we consider the restriction of the cohomology class α ∈ H 2 (G, F × ) to the commutator subgroup G0 . Lemma 6. Let F α G be F –central simple. Then resG G0 (α) has a representative 2–cocycle whose values are p–power roots of unity. Proof. Consider the group Γ ⊆ (F α G)× , which is generated by representatives of G and F × in F α G. Observe that Γ is center by finite, so by a theorem of Schur [10, Chapter 2, Theorem 9.8] its commutator subgroup Γ0 is finite. It is easy to see that F × Γ0 /F × = G0 . It follows that F (Γ0 ), the subalgebra generated by Γ0 , is isomorphic to the twisted group algebra F α G0 . Since Γ0 is finite there exists a representative f ∈ α that takes finite values on G0 , that is, for every σ, τ in G0 , f (σ, τ ) ∈ µ ⊂ F × , where µ denotes the group of roots of unity in k. More than that, by the [10, Lemma of Chapter 2, §9], since G is a p-group, then values of f are p-power roots of unity. ¤ We say that a cohomology class is of finite type if it can be represented by a cocycle which takes finite values in F × . Lemma 7. Let E = Zp × Zp be an elementary abelian group of order p2 . Let F α E be a twisted group algebra over F . Let σ, τ be generators of E and uσ , uτ be representatives of σ, τ in F α E. Then a. The algebra F α E is F –central simple if and only if uσ and uτ do not commute. b. Suppose uσ and uτ do not commute. If α ∈ H 2 (E, F × ) is of finite type then 1. If p is odd then F α E ∼ = Mp (F ) 2. If p = 2 then F α E ∼ = M2 (F ) or F α E ∼ = H where H is the nonsplit quaternion algebra (−1, −1)F . c. Suppose uσ and uτ commute. If α ∈ H 2 (E, F × ) is of finite type then F α E is not a field. Proof. Consider first the case where uσ and uτ commute. Then the algebra F α E is commutative, in particular it is not F –central. Furthermore, if α is of finite type then F α E contains a finite noncyclic group generated by uσ , uτ and hence F α E is not a field. Assume now that uσ and uτ do not commute. Since σ and τ are commuting elements of order p in E, the commutator (uσ1 , uσ2 ) = ξp - a p–primitive root of unity. Write upσ1 = x ∈ F × and upσ2 = y ∈ F × , then, clearly, F α E is isomorphic to the symbol algebra (x, y)p over F , in particular it is F –central simple. Now, if α ∈ H 2 (E, F × ) is of finite type, then F α E has the form (ζ1 , ζ2 )p where ζ1 , ζ2 are p–power roots of unity. It is well known that such a symbol

© 2006 by Taylor & Francis Group, LLC

On fine gradings on central simple algebras

7

is a split algebra except the quaternion algebra (−1, −1)F , which may be nonsplit in the case where F does not contain a 4th root of unity. ¤ The following proposition is the key for the proof of the Theorem. Proposition 8. Let G be a p–group and F α G an F –central simple algebra. If the commutator G0 is not cyclic then there exist a subquotient H of G and a class β ∈ H 2 (H, F × ) such that F α G ∼ = Mp (F β H) or F α G ∼ = H ⊗ F β H. T 0 Proof. Consider the nontrivial subgroup N = Z(G) G of G. Let Np denote the p–elementary abelian subgroup of N . By Lemma 4 we may assume that the algebra F α G is reduced, that is for every subgroup U of N of order p, the corresponding subalgebra F α U is a field. We proceed as in the proof of [4, Theorem 1.2]. Case (1): Np is not cyclic. Let P = hσ1 , σ2 i be any subgroup of order p2 in Np . Every cyclic subgroup of P generates a cyclotomic field extension. In particular, for i = 1, 2, upσi = ζi where ζi is a p–power root of unity. We may assume that ζ1 = ζ2t for some integer t. We claim that uσ1 and uσ2 do not commute. Otherwise, we get (uσ1 u−t )p = 1, that is, the restriction of α T 0 σ2 −t to the subgroup hσ1 σ2 i ⊆ Z(G) G is trivial, a contradiction. Hence, by Lemma 7a, F α P is simple and by Lemma 5 we have F αG ∼ = F αP ⊗ F β H where H is isomorphic to G/P . Since the restriction of α on P is of finite type, by Lemma 7b we obtain that – If p is an odd prime or p = 2 and the quaternion algebra (−1, −1)F is split then F α G ∼ = Mp (F β H). – If p = 2 and H = (−1, −1)F is not split then F α G ∼ = H ⊗ F β H. Case (2): Np is cyclic. A close inspection of the proof of [4, Theorem 1.2] shows that there is a subgroup P in G0 , which is normal in G and isomorphic to Zp × Zp . Let σ1 , σ2 be generators of P . Since P is not central, we cannot proceed as in Case (1). We have two possibilities. Case (2.1): F α P is a simple algebra. We claim that uσ1 and uσ2 do not commute, for otherwise F α P is a commutative simple algebra that is, by Lemma 7c, not a field, a contradiction. Recall that the restriction of α to G0 ⊇ P is of finite type. Therefore, as in the Case (1), applying Lemmas 5 and 7b we get either F α G ∼ = Mp (F β H) or F α G ∼ = H ⊗ F β H with H ∼ = G/P . α Case T (2.2): F P is not simple. Let us show that this case is impossible. Since P Z(G) is not trivial, we may assume that σ1 ∈ Z(G). Then hσ1 i is the unique subgroup of P of order p, which is normal in G. Since F α G is reduced, F α hσ1 i is a field. Note that there exists a subgroup K of P such that resG K (α) = 1, for if not, arguing as in Case (1), we have that F α P is isomorphic to a symbol algebra, contradicting our assumption. Of course, we may assume that K is generated by σ2 , and hence there exists a representing 2–cocycle c : G×G →

© 2006 by Taylor & Francis Group, LLC

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E. Aljadeff, D. Haile, and M. Natapov

F × of α such that c(σ2i , σ2j ) = 1 for all i, j. Furthermore, since (uσ1 , uσ2 ) = 1 (Lemma 7a) we may assume also c(σ2i , x) = c(x, σ2i ) = 1 for all x ∈ P , and all i. Consider the decomposition (1)

F α P = F α P e1 ⊕ ... ⊕ F α P ep

P where ei = 1/p( k ξik ukσ2 ) are central mutually orthogonal idempotents (here ξi are different p-roots of unity and ξ1 = 1). Note that F α P e1 is isomorphic to the field F α hσ1 i. The group G acts on F α P by conjugation and hence on the decomposition (1). Since the algebra F α G is F –central simple, the group G permutes transitively the simple components of F α P . Let T be the stabilizer of the component F α P e1 , or equivalently the centralizer of e1 . Note that T is of index p in G. On the other hand, since P is normal in G and hσ2 i is not normal, the centralizer of hσ2 i in G is of index p and in fact is equal to T . This implies that all elements uτ , τ ∈ CG (σ2 ) commute with uσ2 , contradicting a well-known characterization of a nondegenerate 2–cocycle on a central type group, see e.g. [7, Problems 11.5, 11.8]. ¤ We now turn to the proof of Theorem 3. Proof of Theorem 3. As mentioned above it is sufficient to consider the case where H is a p–group. By repeated application of the proposition the result follows easily in cases where p is odd or p = 2 and the quaternion algebra (−1, −1)F is split. Of course, if after s(< n) applications of the proposition we obtain a subquotient Hs such that Hs0 is cyclic, there is nothing to prove. Otherwise F α H ∼ = Mpn (D) is isomorphic to Mpn (F βn Hn ) which says that F βn Hn ∼ = D. By [2, Corollary 3] we have that Hn ∈ Λp and in particular Hn0 is cyclic. We therefore assume that p = 2 and H = (−1, −1)F is not split. In √ particular −1 ∈ / F × . We claim that it is sufficient to show that if H ⊗ β ∼ n F Hn = M2 (D) then Hn has a cyclic commutator subgroup (as above we may assume that after s < n applications of the proposition the group Hs has a noncyclic commutator): Since H ⊗F H ∼ = M4 (F ) it follows that if the quaternion algebra H is factored out an even number of times we obtain F αH ∼ = M2n (F βn Hn ) ∼ = M2n (D) and hence Hn ∈ Λ2 . The remaining case yields H ⊗ M2n−1 (F βn Hn ) ∼ = M2n (D) and hence H ⊗ F βn Hn ∼ = M2 (D). 0 Assume Hn is not cyclic. If F βn Hn is reduced, it follows from the proof of Proposition 8 that a quaternion algebra can be factored out from F βn Hn . Clearly, this is impossible by an index argument. We conclude that F βn Hn is not reduced. T Let U ⊆ Z(Hn ) Hn0 be a subgroup of order 2 such that F βn U is not simple. By Lemma 4 there is a subgroup T ⊃ U of index 2 in Hn such that F βn Hn ∼ = M2 (F β T /U ). It follows that H ⊗ F βn Hn ∼ = M2 (H ⊗ F β T /U ) ∼ =

© 2006 by Taylor & Francis Group, LLC

On fine gradings on central simple algebras

9

M2 (D). In particular H ⊗ F β T /U is a division algebra over F . Now, since Hn0 is not cyclic, the subgroup Hn0 /U ⊂ T /U is not trivial. We claim that β restricted to Hn0 /U is of finite type. To see this let µ = {±1} ⊂ F × and consider the commutative diagram with exact columns: H 2 (Hn0 /U, µ)  i y∗

−−−−→

inf

H 2 (Hn0 , µ)  i y∗

inf

H 2 (Hn0 , F × )  p∗ y

β ∈ H 2 (Hn0 /U, F × ) −−−−→  p∗ y inf

H 2 (Hn0 /U, F × /µ) −−−−→ H 2 (Hn0 , F × /µ) We know that inf (β) ∈ H 2 (Hn0 , F × ) is of finite type and hence it is in the image of i∗ . This implies p∗ inf (β) ∗ (β)) = 0. But since √ = inf (p 1 × × × H (U, F /µ) = Hom(U, F /µ) = 0 ( −1 ∈ / F ) the map inf in the bottom line is injective [9, Chapter VII, §6, Proposition 5]. This implies p∗ (β) = 0 and hence β restricted to Hn0 /U has finite values and the claim is proved. Now, β is not trivial on Hn0 /U since F β T /U is a √ division algebra. β It follows that F T /U contains a field isomorphic to F ( −1) and hence H ⊗ F β T /U cannot be a division algebra. This completes the proof of the Theorem. ¤ Acknowledgments The first author is grateful to Plamen Koshlukov for useful discussions held during the meeting in Ubatuba, Brazil. References [1] E. Aljadeff and D. Haile, Division algebras with a projective basis, Israel J. Math. 121 (2001) 173–198. [2] E. Aljadeff, D. Haile, and M. Natapov, Projective bases of division algebras and groups of central type, to appear in Israel J. Math. [3] E. Aljadeff and J. Sonn, Projective Schur algebras have abelian splitting fields, J. Algebra 175 (1995) 179–187. [4] E. Aljadeff and J. Sonn, Projective Schur algebras of nilpotent type are Brauer equivalent to radical algebras, J. Algebra 220 (1999) 401–414. [5] Yu. Bahturin, S. Sehgal, and M. Zaicev, Group gradings on associative algebras, J. Algebra 241 (2001) 677–698. [6] R. Howlett and I. Isaacs, On groups of central type, Mathematische Zeitschrift 179 (1982) 555–569. [7] I. Isaacs, Character Theory of Finite Groups, Dover Publications, New York, 1994. [8] M. Natapov, Projective bases of division algebras and groups of central type II, preprint. [9] J.-P. Serre, Local Fields, Springer-Verlag, New York, 1979. [10] M. Suzuki, Group Theory I, Springer–Verlag, New York, 1982.

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On observable module categories Nicol´ as Andruskiewitsch Facultad de Matem´atica, Astronom´ıa y F´ısica, Universidad Nacional de C´ordoba, CIEM – CONICET, (5000) Ciudad Universitaria, C´ordoba, Argentina [email protected] Walter R. Ferrer Santos Centro de Matem´atica, Facultad de Ciencias, Universidad de la Rep´ ublica, Igu´a 4225 esq. Mataojo, CP 11400, Montevideo, Uruguay [email protected] ¡Feliz cumplea˜ nos C´esar!

Introduction Module categories should play the same fundamental role in the theory of tensor categories, as representations do in the theory of groups. They were introduced in [Be] and studied in several papers on category theory; they appeared also in mathematical physics; see for example [FK]. Recently a systematic study of module categories over fusion, or more generally finite, tensor categories was undertaken in [O1, O2, ENO, EO]. In particular, indecomposable module categories over the category of representations of a finite group were classified in [O1] (characteristic 0) and [EO] (arbitrary characteristic). In the papers [ENO, EO], the authors consider rigid tensor categories with appropriate finiteness conditions. One of the motivations for the discussions in the present paper is the study of module categories over the tensor category G M of rational modules over an algebraic group G. Furthermore, we are interested in the induction functor from the category H M of representations of a closed subgroup H of G to G M. For this, one needs to consider any rational module, not only the rigid (= finite-dimensional) ones. We are naturally led to the notions of ind-rigid and geometric tensor categories, see definition 1.2. The purpose of the present paper is to begin the study of a class of module categories over a tensor category C that we call observable module categories. These module categories are simple in a suitable sense (that we introduce 1991 Mathematics Subject Classification. 16W30. 11

© 2006 by Taylor & Francis Group, LLC

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N. Andruskiewitsch and W.R. Ferrer Santos

in this paper). If C = G M then the archetypical example is the module category H M where H is an observable subgroup of G. We extend some well-known results on observable subgroups to the setting of quotients of Hopf algebras. The notion of observable subgroups has the following geometric characterization: a closed subgroup H of an algebraic group G is observable iff the homogeneous space G/H is quasi-affine. This suggests that the study of observable module categories could have a “non-commutative algebra” flavor: they should correspond to “non-commutative quasi-affine varieties.” Throughout, k denotes an arbitrary algebraically closed field, and all vector spaces, algebras, varieties, etc. are considered over k. 1. Preliminaries In this section we recall some basic definitions of the theory of tensor categories and of the theory of affine algebraic groups. We also introduce the main definitions we shall work with. 1.1. Serre subcategories. Let A be an abelian category and let A0 be a full abelian subcategory of A. Recall that • A0 is a Serre subcategory of A if subobjects, quotients, and finite direct sums of objects in A0 are again in A0 . • A0 is a closed subcategory of A if any object in A that is a colimit of objects in A0 is itself in A0 . • A0 generates A if any object in A is colimit of objects in A0 . An example of the use of these abstract notions in a very general version of duality theory is the following result. See also references in [NT]. Lemma 1.1. [NT, Lemma 3.2]. Let A and B be two Grothendieck categories. Let A0 and B0 be two generating Serre subcategories of A and B, respectively. If F : A0 → B0 , G : B0 → A0 is an equivalence (resp. duality) then there exists a unique extension F : A → B, G : B → A which is an equivalence (resp. a pair of contravariant right adjoint functors). ¤ 1.2. Tensor categories. In this paper, by tensor category we understand an abelian k-linear tensor category such that the unit object 1 is simple as in [BK], except that we do not assume rigidity. This is because we are primarily interested in the category of rational modules over an algebraic group, which is not rigid. Given an object X in a tensor category C, a right dual of X is an object X ∗ provided with maps ev : X ∗ ⊗ X → 1,

© 2006 by Taylor & Francis Group, LLC

coev : 1 → X ⊗ X ∗ ,

On observable module categories

13

called “evaluation” and “coevaluation,” such that the compositions coev ⊗ id

id ⊗ ev

X −−−−−−→ X ⊗ X ∗ ⊗ X −−−−−→ X id ⊗ coev

ev ⊗ id

X ∗ −−−−−−→ X ∗ ⊗ X ⊗ X ∗ −−−−−→ X ∗

(1.1) (1.2)

are the identities of X and of X ∗ , respectively. Note that if X 6= 0 has a right dual then coev : 1 → X ⊗ X ∗ is monic. Indeed, if coev is not monic then it is 0, as 1 is simple. But coev ⊗ id is monic, thus X = 0. Analogously, ev : X ∗ ⊗ X → 1 is epi. Similarly, a left dual of Y ∈ C is an object ∗ Y provided with maps ev : Y ⊗ ∗ Y → 1,

coev : 1 → ∗ Y ⊗ Y,

still called “evaluation and coevaluation,” such that the compositions ∗

coev ⊗ id

id ⊗ ev

Y −−−−−−→ ∗ Y ⊗ Y ⊗ ∗ Y −−−−−→ ∗ Y id ⊗ coev

ev ⊗ id

Y −−−−−−→ Y ⊗ ∗ Y ⊗ Y −−−−−→ Y

(1.3) (1.4)

are the identities of ∗ Y and of Y , respectively. An object is rigid if it has a right and left dual. If X ∈ C is rigid and Y, Z ∈ C then C(Y, X⊗Z) ' C(X ∗ ⊗Y, Z), ∗

C(Y, Z⊗X ) ' C(Y ⊗X, Z).

(1.5) (1.6)

See [BK, Lemma 2.1.6]. A tensor category is rigid if every object is rigid. Our main interest here is in a weaker notion of “ind-rigid tensor categories.” Let C be a tensor category and let Crig be the full subcategory of rigid objects. Then Crig is a rigid monoidal subcategory of C, but not necessarily a Serre subcategory. A subobject or a quotient object of a rigid object need not be rigid. For example, let R be a commutative k-algebra (resp. any associative k-algebra with unit) and C = R M the category of R-modules (resp. C = R MR the category of R-bimodules) with ⊗ = ⊗R . Then an object M is rigid if and only if it is finitely generated and projective as R-module (resp. both as a left and right module); but quotients and submodules of finitely generated projective modules in general do not inherit these properties. On the other hand, if X and Y are rigid objects then X ⊕ Y is rigid. Definition 1.2. An object in a tensor category is ind-rigid if it is the colimit of a family of rigid objects. A tensor category is ind-rigid if (a) any object is ind-rigid, (b) any subobject or quotient object of a rigid object is rigid. An ind-rigid tensor category is geometric if in addition

© 2006 by Taylor & Francis Group, LLC

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N. Andruskiewitsch and W.R. Ferrer Santos

(c) the tensor product functor is exact in both variables, and (d) preserves colimits (on both sides). Remark 1.3. If C is an ind-rigid tensor category then any object of C is the colimit of a family of rigid sub-objects - take the epi-monic decomposition of the arrows of an arbitrary colimit of rigid objects. Remark 1.4. If C is an ind-rigid tensor category then Crig is a Serre subcategory generating C. Thus, if C is a Grothendieck category then the duality (contravariant) functor ∗ : Crig → Crig extends uniquely to ∗ : C → C. Remark 1.5. In an ind-rigid tensor category, the simple objects are necessarily rigid. Indeed, let S be a simple object and express it as a colimit of rigid objects {dj : j ∈ J}. Then at least one of the arrows dj → S of the associated cone is nonzero - otherwise S = 0 - hence S = Im dj is rigid. Thus, not every tensor category is ind-rigid. For instance, the category of representations of a Hopf algebra H is not ind-rigid if H has a simple infinite-dimensional module, e.g., H = U (sl(2, k)). Remark 1.6. If C is an ind-rigid tensor category and (d’) the tensor product functor preserves colimits on one side, then (d) holds. Indeed, assume that it preserves colimits on the left side. Given x ∈ C, write it as a colimit x = colimj xj of rigid objects xj . If y = colimi yi in C then ¡ ¢ x⊗y = (colimj xj )⊗(colimi yi ) ' colimj xj ⊗(colimi yi ) ¡ ¢ ¡ ¢ ' colimj colimi xj ⊗yi ' colimi colimj xj ⊗yi ¢ ¡ ¢ ¡ ' colimi (colimj xj )⊗yi ' colimi x⊗yi , the first isomorphism since xj ⊗

is exact because xj is rigid.

Remark 1.7. Condition (c) fails in general, e.g., in R M or R MR . It is true in rigid tensor categories, see [BK, Prop. 2.1.8]. Actually, the proof of [BK, Prop. 2.1.8] shows that X ⊗ and ⊗ X are exact in an arbitrary tensor category if X is rigid. We do not know if (c) and (d) follow from (a) and (b), i.e., if any ind-rigid tensor category is geometric. A natural example of geometric tensor category is the category of comodules over a Hopf algebra with bijective antipode. 1.3. Tensor functors and module categories. Recall that a tensor functor between tensor categories is an additive functor that preserves the tensor and the unit “up to a natural isomorphism.” Technically, a tensor functor from a tensor category C to a tensor category D is a triple (F, ζ, φ) where F : C → D is an additive functor, φ : 1 → F (1) is an isomorphism in D, and ζX,Y : F (X)⊗F (Y ) → F (X⊗Y ) is a natural

© 2006 by Taylor & Francis Group, LLC

On observable module categories

15

transformation such that F (aX,Y,Z )ζX⊗Y,Y (ζX,Y ⊗ idF (Z) ) = ζX,Y ⊗Z (idF (X) ⊗ζY,Z )aF (X),F (Y ),F (Z) , (1.7) F (lX )ζ1,X (φ⊗ idF (X) ) = lF (X) ,

(1.8)

F (rX )ζX,1 (idF (X) ⊗φ) = rF (X) ,

(1.9)

for all X, Y, Z ∈ C. If F : C → C 0 is a tensor functor and X is a rigid object of C then F (X) is a rigid object of C 0 and F preserves duals. Thus F induces a monoidal 0 . functor Frig : Crig → Crig A module category over a tensor category C is an abelian k-linear category M provided with an exact bifunctor ⊗ : C × M → M (the action) and natural isomorphisms mX,Y,M : (X ⊗ Y ) ⊗ M → X ⊗ (Y ⊗ M ),

lM : 1 ⊗ M → M,

X, Y ∈ C, M ∈ M, such that the following diagrams commute: ((X ⊗ Y ) ⊗ Z) ⊗ M

UUUU UUmUX⊗Y,Z,M UUUU UUUU U*

j

aX,Y,Z ⊗idjjjjj

jj jjjj jt jjj

(X ⊗ (Y ⊗ Z)) ⊗ M

(X ⊗ Y ) ⊗ (Z ⊗ M )

mX,Y ⊗Z,M

mX,Y,Z⊗M

²

id⊗mY,Z,M

X ⊗ ((Y ⊗ Z) ⊗ M )

² / X ⊗ (Y ⊗ (Z ⊗ M ))

(1.10) and (X ⊗ 1) ⊗ M

mX,1,Y

OOO OOrOX ⊗id OOO OO'

/ X ⊗ (1 ⊗ M ) o id⊗lM ooo oo o o ow oo

(1.11)

X ⊗M

for all X, Y, Z ∈ C and M ∈ M. In the definition, “exact bifunctor” means “it is exact in both variables.” By analogy with the axioms (c) and (d) in the definition of geometric tensor category, we shall also assume that the action satisfies the following axiom: • the action functor ⊗ preserves colimits (on both sides). Let M, M0 be two module categories over a tensor category C. A module functor from M to M0 is a triple (F, b, u) where F : M → M0 is an additive functor, bX,Y : F (X⊗Y ) → F (X)⊗F (Y ) is a natural isomorphism and u : F (1) → 1 is an isomorphism, satisfying some natural compatibilities, see [O2]. By a submodule category of a module M we understand a Serre subcategory stable under the action.

© 2006 by Taylor & Francis Group, LLC

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N. Andruskiewitsch and W.R. Ferrer Santos

Remark 1.8. As explained in [EO], it is worth restricting attention to the class of exact module categories. Recall that M is an exact tensor category over a finite tensor category C if P ⊗X ∈ M is projective for any P ∈ C projective an any X ∈ M. Module functors from an exact module category are always exact. Now, the inclusion functor from a full abelian subcategory of an abelian category is exact precisely when the domain is a Serre subcategory. This justifies our definition of submodule category. The trivial module category is M = 0. A module category is indecomposable if it is not isomorphic to a direct sum of two nontrivial submodule categories [O1]. Definition 1.9. A nontrivial module category over a tensor category C is simple if any proper submodule category is trivial. 1.4. Algebraic groups. In this paper, we consider affine algebraic groups defined over k by “subgroup” we mean a closed subgroup. If G is an affine algebraic group, we denote as k[G] the Hopf algebra of polynomial functions on G. Also we denote as G M, the category of all rational left G-modules (for the basic properties of this category see [FR]). In particular it is well known that G M is an abelian ind-rigid tensor category, where G Mrig is the full subcategory of finite-dimensional rational G-modules. Clearly k[G] ∈ G M when it is equipped with the left translation: (x, f ) 7→ x · f , for x ∈ G and f ∈ k[G]. If G is an affine algebraic group and H a closed subgroup, then the coset space G/H is naturally an algebraic G–variety. In the above situation the restriction functor ResG H : G M → H M is a tensor functor. Then we can consider H M as module category over G M, that is X ⊗ N = X|H ⊗ N , for a rational H-module N and a rational G-module X. The restriction functor has an adjoint, that we denote as IndG H : H M → G M, called the induction functor. Explicitly, H IndG H (M ) = (k[G]⊗M ),

M ∈ H M,

where the superscript H means that we are taking H-invariants. Note that IndG H has usually infinite dimension; this is why we can not work directly with rigid tensor categories. The counit of this adjunction EM : IndG H (M ) → M for M ∈ H M, is called the evaluation map. 1.5. Observable subgroups. Let G be an algebraic group and H a closed subgroup. Recall that a rational character η : H → k is extendible, if there is a nonzero polynomial f ∈ k[G] such that for all x ∈ H, x · f = η(x)f . The following properties are equivalent: (a) G/H is a quasi–affine variety. (b) All rational characters χ : H → k are extendible.

© 2006 by Taylor & Francis Group, LLC

On observable module categories

17

(c) For all H–modules M ∈ H M, the evaluation map EM is surjective. See [FR, Ch. 10] for the relevant definitions and proofs. If any of (a), (b), (c) holds then one says that H is observable in G. Here are some examples of subgroups that are observable (or not): • If the only rational character χ : H → k is the trivial one, then H is observable in any algebraic group containing it. Thus, semisimple and unipotent algebraic subgroups are always observable. • Finite subgroups are observable. • Normal subgroups are observable. • Finite-index subgroups are observable. • A parabolic subgroup of a semisimple algebraic group is not observable. 2. Observable tensor categories 2.1. Definition and basic properties. Let F : C → D be a tensor functor between arbitrary tensor categories C and D. Consider D as module category over C via F ; that is, c ⊗ d := F (c) ⊗ d,

c ∈ C, d ∈ D.

To illustrate the concepts in action, we prove the following result. Lemma 2.1. Assume that F has a right adjoint I : D → C. Then the full subcategory D0 of D with objects D0 = {d ∈ D : I(d) = 0} is stable under action by objects of Crig . If I preserves colimits then D0 is a full abelian subcategory with the same kernels and cokernels as in D. Proof. Let d ∈ D0 . If c ∈ Crig , x ∈ C, then C(x, I(c⊗d)) ' C(x, I(F (c)⊗d)) ' D(F (x), F (c)⊗d) ' D(F (c)∗ ⊗F (x), d) ' D(F (c∗ ⊗x), d) ' C(c∗ ⊗x, I(d)) = 0, where we have used (1.5). Thus c⊗d ∈ D0 for any c ∈ Crig as claimed. Since I is a right adjoint, it preserves limits [M, Ch. V, §5, Th. 1]. If I preserves colimits then D0 is closed under colimits. Thus D0 is closed under kernels and cokernels and is a full abelian subcategory of D. ¤ Here is one of the main definitions in the present paper. Definition 2.2. Let F : C → D be a tensor functor between arbitrary tensor categories C and D and assume that F has a right adjoint I : D → C. We say that the category D is observable over C if the counit for the adjunction εd : F (I(d)) → d is an epimorphism for all d ∈ D.

© 2006 by Taylor & Francis Group, LLC

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N. Andruskiewitsch and W.R. Ferrer Santos

Here recall that εd corresponds to idI(d) via D(F (I(d)), d) ' C(I(d), I(d)). The motivation for the above definition comes from the theory of algebraic groups, see Section 1.4. Indeed, a subgroup H is observable in an algebraic group G if and only if H M is observable in G M. Theorem 2.3. In the notations above, if D is ind-rigid and observable over C, then it is simple as a module category over C. Proof. Let N ⊂ D be a nonzero submodule category. Take 0 6= m ∈ N and write it as a colimit in D of rigid objects {dj : j ∈ J}, dj ∈ Drig . One of the arrows of the cone fj : dj → m is nonzero, since otherwise m = 0. Using the standard epi, monic factorization of fj we obtain a pair of morphisms in D and an object 0 6= n ∈ D, h : dj → n, g : n → m with fj = gh, h epi and g monic. As N is closed under subobjects, it follows that n ∈ N and as n is a quotient of dj , it is rigid. In other words, N contains at least one rigid object n. In this situation the morphism εn∗ ⊗id : I(n∗ )⊗n = F (I(n∗ ))⊗n → n∗ ⊗n is an epimorphism by the observability hypothesis. Since N is a submodule category, I(n∗ ) ⊗ n ∈ N and we conclude that n∗ ⊗ n ∈ N , because N is closed under quotients. Using the monomorphism given by the coevaluation map coev : 1 → n∗ ⊗ n, we conclude that 1 ∈ N . Consider now an arbitrary object d ∈ D; in this situation the morphism εd ⊗ id : I(d) ⊗ 1 = F (I(d)) ⊗ 1 → d ⊗ 1 ' d is epi and we conclude that d ∈ N . Thus N = D. ¤ 2.2. Observable quotients of Hopf algebras. Now let A be a Hopf algebra with bijective antipode and let MA be the tensor category of right A-comodules. It is well known that MA satisfies the properties required to be geometric. A Let π : A → B be a Hopf algebra quotient and let Res = ResB A : M → B M be the corresponding “restriction” functor; that is, if M is a right comodule over A via ρ : M → M ⊗ A then Res M = M|B is the vector space M considered as a right comodule over B via (id ⊗π)ρ. Clearly, Res is a tensor functor. If g ∈ G(B) is a group-like and M is a right B-comodule then M g denotes the isotypic component of type g, that is M g = {m ∈ M : ρ(m) = m ⊗ g}. If g = 1, we denote M B = M 1 the subcomodule of invariant elements. Let M ∈ MB . If we consider A ∈ MA via the comultiplication, then A|B ⊗M ∈ MB with coaction ρ1 ; explicitly, ρ1 (a⊗m) = a(1) ⊗m(0) ⊗π(a(2) )m(1) ,

a ∈ A, m ∈ M.

There is another right coaction of A on A⊗M by ρ2 (a⊗m) = a(2) ⊗m⊗S(a(1) ),

© 2006 by Taylor & Francis Group, LLC

a ∈ A, m ∈ M.

On observable module categories

19

It is easy to see that these two coactions commute. Then the space of invariants (A|B ⊗M )B with respect to ρ1 is a right comodule over A via ρ2 . In this situation we can define a map EM : (A|B ⊗M )B → M - the evaluation map - by EM

¡X

¢ X fi ⊗mi = ε(fi )mi ,

i

i

X

fi ⊗mi ∈ A⊗M.

i

Then EM is a map of B-comodules, natural on M . The following result is well known; see [F]. We sketch a proof for completeness. The proof does not differ from the analogous proof for algebraic groups, see [FR, Th. 6.6.11]. B → MA , Proposition 2.4. The “induction” functor Ind = IndA B : M given by B IndA M ∈ MB , B (M ) = (A|B ⊗M ) ,

is a right adjoint to ResB A . The counit for the adjunction is the map EM . Proof. (Sketch). Let N ∈ MA . Given ϕ ∈ HomB (N, M ), define ϕ e ∈ Hom(N, A|B ⊗M ) by X ϕ(n) e = S −1 (n(1) )⊗n(0) , n ∈ N. Straightforward calculations show that: (a) The image of ϕ e is contained in (A|B ⊗M )B . (b) ϕ e is a morphism of A-comodules and EM ϕ e = ϕ. ϕ e

N JJ / (A|B ⊗M )B JJ JJ J EM ϕ JJJ J$ ² M (c) A morphism of A-comodules N → (A|B ⊗M )B such that EM ϕ e=ϕ coincides necessarily with ϕ. e This proves the Proposition. ¤ We are then in the situation of definition 2.2; it is natural to introduce the following notion. Definition 2.5. A Hopf algebra quotient π : A → B is observable over A (or B is observable over A) if the evaluation map EM : (A|B ⊗M )B → M is an epimorphism for all M ∈ MB . As for algebraic groups, there are alternative characterizations of observability. The proof of the following result mimics the proof of the analogous result for algebraic groups, see [FR, Ch. 10].

© 2006 by Taylor & Francis Group, LLC

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N. Andruskiewitsch and W.R. Ferrer Santos

Theorem 2.6. Let π : A → B be a Hopf algebra quotient. The following are equivalent: (a) B is observable over A. (b) For every finite-dimensional M ∈ MB there exists a finite-dimensional N ∈ MA and an epimorphism of B-comodules ϕ : N|B → M . (c) For every finite-dimensional M ∈ MB there exists a finite-dimensional N ∈ MA and a monomorphism of B-comodules M → N|B . Furthermore, if any of these conditions hold then (d) For every g ∈ G(B), there exists f ∈ (A|B )g such that π(f ) = g. Proof. (a) =⇒ (b): Let s : M → (A|B ⊗M )B → M be a linear section of the evaluation map EM . The A-comodule N generated by the image s(M ) is finite-dimensional and ϕ = π|N : N|B → M is an epimorphism of B-comodules. (b) =⇒ (a): Assume first that M is finite-dimensional. Let ϕ : N|B → M be an epimorphism of B-comodules and let ϕ e ∈ Hom(N, (A|B ⊗M )B ) be the morphism of A-comodules such that EM ϕ e = ϕ. Since ϕ is surjective, EM is surjective. If M has arbitrary dimension, then it is the union of its finite-dimensional subcomodules and the claim follows from the naturality of the evaluation map EM . (b) ⇐⇒ (c): by duality. (b) =⇒ (d): Let M = kg and let N ∈ MA finite-dimensional provided with an epimorphism of B-comodules ϕ : N|B → kg. If n ∈ N satisfies ϕ(n) = g and α ∈ N ∗ satisfies α(n) = 1 then the matrix coefficient f = φα,n ∈ A satisfies π(f ) = g. ¤ Remark 2.7. As we have seen, the implication (d) =⇒ (b) is also valid for algebraic groups. It is likely that this implication is not always true. The proof in [FR] makes use of the modules of antisymmetric tensors of a given rational module, not a comodule in the case of a general Hopf algebra. As for algebraic groups, cosemisimple Hopf algebra quotients are observable. Theorem 2.8. Let π : A → B be a Hopf algebra quotient. If B is cosemisimple then B is observable over A. Proof. Note that the following diagram commutes: π⊗ id / (B⊗M )B OOO OOO eM O E EM OOOO O' ²

(A|B ⊗M )B

M

© 2006 by Taylor & Francis Group, LLC

On observable module categories

21

¡ ¢ eM : (B⊗M )B → M is again given by E eM P fi ⊗mi = P ε(fi )mi , where E i i P e f ⊗m ∈ B⊗M . Now it is clear that E is an isomorphism, since i M i i B B (B⊗M ) = IndB (M ) is adjoint to the identity functor. Thus, we have to prove that π⊗ id : (A|B ⊗M )B → (B⊗M )B is surjective. But A|B ⊗M ' ker(π⊗ id)⊕(B⊗M ) since B is cosemisimple, and the needed surjectivity follows taking the invariant submodules at both sides of this isomorphism. ¤ 3. Linearized Sheaves Let G be an affine algebraic group. Let X be a G–variety and let G MX be the abelian category of G–linearized sheaves on X, see [CPS, p. 453]. If F is a G–linearized and if x ∈ X, we denote by Fx the stalk of the sheaf at the point x. Then for every pair x ∈ X, g ∈ G there exists a linear isomorphism ϕg,x : Fx → Fg·x , regular on the pair (g, x), in such a way that if h ∈ G, the following diagram commutes: ϕg,x

/ Fg·x FF FF ϕh,g·x ϕhg,x FF " ²

Fx F F

F(hg)·x

In particular, notice that the structure sheaf OX ∈ G MX . Then G MX is a module category over G M. Indeed, if M ∈ G M and F ∈ G MX , we define M ⊗ F ∈ G MX as the presheaf associated to the sheaf (M ⊗ F)(U ) = M ⊗ F(U ), U open in X. Hence, for all x ∈ X, (M ⊗ F)x = M ⊗ Fx and ψg,x : M ⊗ Fx → M ⊗ Fg·x is ψg,x = g · ⊗ ϕg,x . If T : Y → X is a morphism of algebraic varieties and G is a sheaf on Y , then we denote as T∗ (G) the “direct image” sheaf on X defined as T∗ (G)(U ) = G(T −1 U ), for U ⊂ X open; see [H, p. 65]. Assume that X and Y are G–varieties and that T intertwines the G–actions. If the sheaf G is G–linearized, then T∗ (G) is also G–linearized. Moreover, in this situation, we have a module functor T∗ : G MY → G MX . Lemma 3.1. Let X be a G–variety and let Y be a closed G–stable subvariety of X. Then we can identify G MY with a submodule category of G MX . Proof. If Y is a closed subvariety of an arbitrary variety X and ι is the inclusion, then ι∗ (G)x = Gx if x ∈ Y and zero otherwise, for any sheaf G on Y [H, Ex. 1.19, p. 68]. This implies that we can identify G MY with the full abelian subcategory of G MX consisting of those G–linearized sheaves F on X such that FP for any P ∈ / Y . It is clear that subobjects and quotients of objects in G MY are again in G MY . Thus, G MY can be identified with a submodule category of G MX . ¤

© 2006 by Taylor & Francis Group, LLC

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N. Andruskiewitsch and W.R. Ferrer Santos

Remark 3.2. Actually, G MY can be identified with a closed submodule category of G MX , cf. subsection 1.1, because of [H, Ex. 1.12, p. 67]. We wonder whether it is important to distinguish closed submodule categories. Lemma 3.3. Let X be a G–variety. If the the category G MX is a simple module category, then the action of G on X is transitive. Assume further that the orbit map G → X is separable (this is automatic if k has characteristic 0). Then X ' G/H for a closed subgroup H of G with the left regular action. Proof. Let Y be a closed orbit for the action of G on X. The submodule category G MY is not zero since OY ∈ G MY . Thus, G MY = G MX . Hence, the sheaf OX is in G MY , and this means that OX,x = 0 for all x ∈ X \ Y , therefore X = Y . The second statement is a well-known fact in the theory of algebraic groups. ¤ Let H be a closed subgroup of G and let π : G → G/H be the canonical projection. Theorem 3.4. (a) The functor L : H M → G MG/H given by ¡ ¢ LN (U ) = H OG (π −1 (U )) ⊗ N for a rational H-module N and U open subset of G/H, is an equivalence of categories, whose inverse is given by taking the fiber at the identity: F 7→ FeH ,

F ∈ G MG/H .

(b) L is a module functor. If K is a subgroup of H then the following diagram commutes: L

HM

−−−−→

G MG/H

KM

−−−−→

L

G MG/K ,

  Resy

 π y ∗

where π : G/H → G/K is the canonical projection. Proof. (a) is [CPS, Th. 2.7]. The proof of (b) is straightforward.

¤

Theorem 3.5. Assume that k has characteristic 0. If X is a G–variety then the following are equivalent: (a) G MX is isomorphic to an observable module category. (b) X ' G/H for a closed subgroup H observable in G. Proof. By Theorem 2.3, if such an isomorphism exists then G MX should be simple. Then Lemma 3.3 says that X ' G/H, and Theorem 3.4, that G MX ' H M. The claim now follows from the characterization of observable subgroups, see subsection 1.4. ¤

© 2006 by Taylor & Francis Group, LLC

On observable module categories

23

The following problems arise naturally: Is the converse to Lemma 3.3 true? That is, what are the closed subgroups H such that H M is simple? More generally, is any simple module category over G M twist-equivalent (in some appropriate sense) to H M with H closed subgroup of G? Acknowledgments Results in this paper were obtained during visits of W.F.S. to the Universidad de C´ordoba, November 2003, and of N.A. to the Universidad de la Rep´ ublica, November 2004. These visits had partial funding, respectively, by the Fundaci´ on Antorchas and PEDECIBA. The work of N.A. was partially supported by CONICET, Agencia C´ordoba Ciencia, ANPCyT-FONCyT, TWAS (Trieste) and Secyt (UNC). N.A. thanks Sonia Natale and Blas Torrecillas for interesting conversations. The work of W.F.S. was partially supported by CSIC-Udelar and Dinacyt-MEC, Uruguay. W.F.S. thanks Ignacio L´ opez for many exchanges concerning category theory. References [BK] [Be] [CPS] [ENO] [EO] [F] [FR] [FK] [H] [M] [NT] [O1] [O2]

B. Bakalov and A. Kirillov, Jr., Lectures on tensor categories and modular functors, University Lecture Series 21, Amer. Math. Soc., Providence, RI, 2001. J. B´enabou, Introduction to bicategories, Reports of the Midwest Category Seminar (1967), Lecture notes in Math. 47, pp. 1–77, Springer, Berlin. E. Cline, B. Parshall, and L. Scott, A Mackey imprimitivity theory for algebraic groups, Math. Z. 182, pp. 447–471 (1983). P. Etingof, D. Nikshych, and V. Ostrik, On fusion categories, Annals Math., to appear, math.QA/0203060. P. Etingof and V. Ostrik, Finite tensor categories, Mosc. Math. J. 4 (2004), no. 3, 627–654, 782–783. math.QA/0301027. W. Ferrer Santos, Cohomology of comodules, Pacific J. Math. 109 (1983), 179– 213. W. Ferrer Santos and A. Rittatore, Actions and invariants of algebraic groups, Series: Pure and Applied Mathematics, 268, Dekker-CRC Press, Florida (2005). J. Fr¨ ohlich and T. Kerler, Quantum groups, quantum categories and quantum field theory, Lecture Notes in Math. 1542, Springer-Verlag, Berlin (1993). R. Hartshorne, Algebraic Geometry, 6th. corrected printing, Springer Verlag, (1993). S. Mac Lane, Categories for the Working Mathematician, 2nd. edition, Springer Verlag (1998). C. Nastasescu and B. Torrecillas, Morita duality for Grothendieck categories with applicatinos to coalgebras, Commun. Alg., to appear. V. Ostrik, Module categories over the Drinfeld double of a finite Group, Int. Math. Res. Not. 27, 1507–1520 (2003). V. Ostrik, Module categories, weak Hopf algebras and modular invariants, Transform. Groups, 2 8, 177–206 (2003).

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Group gradings on integral group rings Yuri A. Bahturin Department of Mathematics and Statistics, Memorial University of Newfoundland, St. John’s, NL, A1C 5S7, Canada [email protected] Michael M. Parmenter Department of Mathematics and Statistics, Memorial University of Newfoundland, St. John’s, NL, A1C 5S7, Canada [email protected] To Professor C´esar on his 60th birthday. Abstract. In this paper we study group gradings on an integral group ring Z G. We show that if G abelian and the grading group is any finite abelian 2-group then any grading is “trivial,” that is, all elements of G are homogeneous. In the case where G is nonabelian, we deal only with grading by the cyclic group C2 of order 2. We indicate certain “natural” gradings and show that if G is finite nilpotent, any C2 -grading of Z G is one of them.

1. Introduction Let A be an associative algebra over a commutative ring LR with 1. We say that A is graded by a finite abelian group H if A = h∈H Ah where each Ah is an R-submodule of A and Ah1 Ah2 ⊂ Ah1 h2 for all h1 , h2 ∈ H. Our main concern in this paper will be the determination of all possible gradings by abelian groups on integral group rings, that is, when A = ZG and H is abelian. This paper continues our research on gradings in groups rings starter in [BP]. 2. Abelian groups The main significance of this section is to show that if G is a finite abelian group and ZG is graded by C2 , the cyclic group of order 2, then all elements of G are homogeneous. It follows that the only nontrivial C2 -gradings of ZG in this case are of the form ZG = ZH ⊕ (ZH)g where H is a subgroup of index 2 in G and g ∈ / H. 25

© 2006 by Taylor & Francis Group, LLC

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Yu. A. Bahturin and M.M. Parmenter

The result we actually prove (Theorem 2.3) is quite a bit more general. We start by recalling two results. The first lemma is well known but we include it for completeness. Lemma 2.1. Assume that an R-algebra A (R a commutative ring with 1) is graded by a finite abelian group H. Then (a) If 1 is the multiplicative identity of A then 1 is in Ae (where e is the identity of H). P (b) If z ∈ Z(A), the center of A, and z = zg where zg is in Ag for all g in H, then zg is in Z(A) for all g in H. P Proof. (a) Assume that 1 = ag where ag is in PAg for each g. Let y ∈ Ah be any homogeneous element of A. Then y = ag y and ag y is in Agh for each g. It follows that y = ae y, and similarly y = yae . We conclude that ae is a multiplicative identity for A, as desired. P (b)P Let y ∈ Ah be any homogeneous element of A. Then yzg = yz = zy = zg y. Since H is abelian and yzg is in Ahg = Agh , and zg y is in Agh , we have yzg = zg y for all g ∈ H. Hence zg is in Z(A) as desired. ¤ We will need the following important result about central units in integral group rings. Theorem 2.2. [MS, Corollary 7.3.3] If α is a central unit of finite order in ZG, then α belongs to ±Z(G). ¤ Theorem 2.3. Assume ZG is graded by a finite abelian 2-group K. If g ∈ Z(G) is of finite order then g is homogeneous. Proof. We proceed by induction on |K|. First assume |K| = 2, that is K = {e, x}, and ZG = Ae ⊕ Ax . Say g = αe + αx where αe ∈ Ae and αx ∈ Ax . Since g is central in ZG, according to Lemma 2.1, αe and αx are both central in ZG. Note that 1 = g n = (αe + αx )n . In the binomial expansion of (αe + αx )n terms are alternately in Ae and Ax . Since 1 ∈ Ae , it follows that (αe − αx )n = 1 also. Since αe − αx is a central unit of finite order in ZG, αe − αx must belong to ± G so that g − 2αx ∈ ± G. This can only happen if either αx = 0 or αx = g and in either case we have that g is homogeneous, as desired. Next assume |K| > 2 and the result is true if ZG is graded L by any finite abelian 2-group K1 with 2 ≤ |K1 | < |K|. We have ZG = k∈K Ak . Assume to the contrary that P g ∈ G is not homogeneous so in the homogeneous decomposition g = k∈K αk of g there exist αx 6= 0 and αy 6= 0 for some x 6= y in K. Note that if L is any subgroup of K we can grade ZG by K/L in the usual way. It follows by the inductive hypothesis that xy −1 ∈ L for all nonidentity subgroups L of K. This implies that K must be cyclic and xy −1 = z where z is the unique element of order 2 in K. It also implies (since this works for any such x, y) that g = αy + αyz . But now we can conclude using exactly the same reasoning as seen before that αy − αyz is

© 2006 by Taylor & Francis Group, LLC

Group gradings on integral group rings

27

also a central unit of finite order in ZG and hence belongs to ±G. This gives the desired contradiction, so g is homogeneous, as desired. ¤ Remarks. (1) Theorem 2.3 says that if G is a finite abelian group and ZG is graded by a finite abelian 2-group K then all elements of G are homogeneous. In addition, if G is a Hamiltonian 2-group then all units of finite order in ZG are trivial and the proof of Theorem 2.3 shows that the same result holds. (2) The proof of Theorem 2.3 can also be extended to the case where ZG is graded by Cp , the cyclic group of order an odd prime p, as long as we know that central units of finite order in the group ring Z[ε](G) are trivial (where ε is a primitive pth root of 1). Berman’s proof of the result uses the fact that |r| ≥ 1 for all nonzero r in Z[ξ], and this √still holds in Z[ξ](G) when p = 3. Indeed, a typical element in Z[− 21 + 23 i] √ p looks like a = z0 + z1 (− 21 + 23 i) and |a| = z02 + z12 − z0 z1 , proving √

our claim. If we set ξ = − 21 + 23 i then the proof of Theorem 2.3 in the case when ZG is graded by C3 will look like the following. Say ZG is graded by {e, x, x2 }. Then also Z[ξ]G is graded by {e, x, x2 }. Let g = α0 + α1 + α2 with αi ∈ Axi . Then α0 , α1 , α2 are central units and g n = (α0 + α1 + α2 )n = 1 for some natural number n. If we expand the previous relation then it splits into three components X X X αi αj αk = 0. αi αj αk = 0, αi αj αk = 1, i+j+k=0

i+j+k=1

i+j+k=2

If we replace αi by αi ξ j then these relations will hold and so α0 + α1 ξ + α2 ξ 2 and α0 + α1 ξ 2 + α2 ξ are also central units of finite order. So α0 + α1 ξ + α2 ξ 2 = ξ i g1 and α0 + α1 ξ 2 + α2 ξ = ξ j g2 for some g1 , g2 ∈ G and some integers i, j. Hence (1 − ξ)(α1 + (1 + ξ)α2 ) = g − ξ i g1 . Since 1 − ξ is a nonunit in Z[ξ] this forces g = g1 . Similarly g = g2 . Now we have the system  α0 + α1 + α2 = g  α0 + α1 ξ + α2 ξ 2 = ξ i g  α0 + α1 ξ 2 + α2 ξ = ξ j g. If α0 6= 0 then ξ i = ξ j = 1 and we easily derive α1 = α2 = 0 hence g = α0 ∈ Ae . If α0 = 0 but α1 6= 0 then ξ i = ξ, ξ j = ξ 2 and α2 = 0 hence g = α1 ∈ Ax . Otherwise g = α2 ∈ Ax2 . (3) Continuing on from remark 2, induction can also be applied showing that Theorem 2.3 holds if ZG is graded by a finite abelian group of order 2α 3β .

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(4) Again continuing on from remark 2, Berman’s proof fails for any p ≥ 5 (if p ≥ 7 then | − 1 + ε| < 1 and if p = 5 then |1 + ε2 | < 1). However we do not know whether Theorem 2.3 holds in this case. We do know that if G is such that ZG has no nonzero nilpotent elements, for example if G is abelian, and ZG is graded by Cp = hxip and Axj 6= 0 for some 0 < j < p then Axi 6= 0 for all i. Indeed, if α 6= 0 is in Axj then αs ∈ Axjs for all s. But αs 6= 0 because ZG has no nonzero nilpotent elements. So if ZG has no nonzero nilpotent elements and |G| < p then any grading of ZG by Cp is trivial. 3. General Results In this section we will investigate C2 -gradings of ZG for arbitrary finite groups G. If G has a subgroup H of index 2 and g ∈ G \ H then an obvious C2 -grading is given by ZG = ZH ⊕ (ZH)g. Additionally, ZG = γ(ZH) ⊕ γ((ZH)g) is a C2 -grading for any automorphism γ of ZG. Question. Do the above represent all possible C2 -gradings of ZG? The answer to this question is still not clear. Shortly we will give two results, Theorem 3.2 and (especially) Corollary 3.3, which describe what appears to be a more general method of constructing C2 -gradings. In fact, Theorem 3.4 shows that this method characterizes all C2 -gradings of ZG for a large class of finite groups G. A number of specific examples to this construction are presented later in Section 4, but we have been unable to verify that gradings arising from this method could not also be obtained by a suitable choice of H and γ in the example mentioned above. Our first result will be applied to integral group rings but is stated in a much more general setting. Proposition 3.1. Let R be a ring with 1 such that (R, +) is 2-torsion free. Then there is a 1 − 1 correspondence between gradings of R by C2 and automorphisms θ of R such that θ2 = I (the identity automorphism) and θ(r) − r ∈ 2R for all r ∈ R. Proof. First assume R = R0 ⊕R1 is a C2 -grading. Let θ be the automorphism of R defined by θ(x + y) = x − y where x ∈ R0 , y ∈ R1 . Then θ satisfies the given conditions. Conversely assume θ is any automorphism of R satisfying the given conditions. Define R0 = {r ∈ R | θ(r) = r} and R1 = {r ∈ R | θ(r) = −r}. Clearly, R02 ⊂ R0 , R12 ⊂ R0 R0 R1 ⊂ R1 , and R1 R0 ⊂ R1 . The second assumption tells us that if r ∈ R then r + θ(r) = 2r0 and r − θ(r) = 2r1 for some r0 , r1 ∈ R. By the first assumption, we know that r + θ(r) is in R0 , so θ(2r0 ) = 2r0 . Similarly, r − θ(r) is in R1 so θ(2r1 ) = −2r1 . Since (R, +) has no 2-torsion, r0 ∈ R0 and r1 ∈ R1 . Also, 2r = 2r0 + 2r1 , which implies (as above)

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that r = r0 + r1 . Since R0 ∩ R1 = 0 (again using no 2-torsion), we have R = R0 ⊕ R1 is a C2 -grading. Finally, observe that the lack of 2-torsion in (R, +) tells us that the relations just described are inverses of each other. ¤ In addition to the statement of Proposition 3.1, the precise nature of the 1 − 1 correspondence will be used later on. Examples. (1) Let H be a subgroup of index 2 in a group G. Define ιH to be the automorphism of ZG induced by ιH (g) = g if g ∈ H, ιH (g) = −g if g ∈ / H. Then ιH satisfies the required conditions, and the corresponding grading is ZG = ZH ⊕ (ZH)g where g ∈ / H. (2) More generally, if H is a subgroup of index 2 in G and γ is any automorphism of ZG then θ = γ ◦ ιH ◦ γ −1 satisfies the required conditions. The corresponding grading is ZG = γ(ZH) ⊕ γ((ZH)g), noted earlier. We now proceed to show that if R = ZG then the automorphisms of the type described in Proposition 3.1 can be obtained from certain types of units in the rational group algebra QG. Specific examples of such construction are given in Section 4. For a large class of finite groups, including finite nilpotent groups, Theorems 3.2 and 3.4 together give a characterization of all C2 -gradings of ZG. In the following results ιH has the same meaning as in the previous example. In addition, when H = G ιH is the identity automorphism. Theorem 3.2. Let H be a subgroup of index ≤ 2 in G and β an automorphism of G such that β(H) = H (we will also use β to represent its extension to an automorphism of ZG). Assume that a unit u in QG satisfies the following two conditions: (i) For any g ∈ G, (u(β ◦ ιH (u)))β 2 (g)(u(β ◦ ιH (u)))−1 = g; (ii) Inn u, the inner automorphism of QG induced by u, is an automorphism of ZG when restricted to ZG. Moreover, uβ(g)u−1 − g ∈ 2ZG for all g ∈ G. Then, restricted to ZG, the automorphism (Inn u) ◦ β ◦ ιH satisfies the conditions of Proposition 3.1. Proof. Let θ = (Inn u) ◦ β ◦ ιH . For any s ∈ ZG, θ2 (s) = θ(u((β ◦ ιH )(s))u−1 ) = u((β ◦ ιH )(u((β ◦ ιH )(s))u−1 ))u−1 = u((β ◦ ιH )(u))((β ◦ ιH ◦ β ◦ ιH )(s))((β ◦ ιH )(u−1 ))u−1 = (u((β ◦ ιH )(u)))(β 2 (s))(u((β ◦ ιH )(u)))−1

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since β(H) = H. Property (i) now tells us that θ2 (s) = s, so θ2 = I. Also, if g ∈ G then θ(g) equals uβ(g)u−1 or −uβ(g)u−1 . In either case property (ii) tells us that θ(g) − g ∈ 2ZG. ¤ Later we will give examples of units satisfying the conditions of Theorem 3.2. In all such examples β will be the identity automorphism and u will be in ZG, so we restate Theorem 3.2 in this simpler setting. Corollary 3.3. Let H be a subgroup of index ≤ 2 in G. Assume that a unit u in ZG satisfies the following two conditions. (i) uιH (u) is central in ZG; (ii) ugu−1 − g ∈ 2ZG for all g ∈ G. Then, if Inn u denotes the inner automorphism of ZG, the automorphism θ = (Inn u) ◦ ιH satisfies the conditions of Proposition 3.1. We now show that, for a large class of groups, the conditions in Theorem 3.2 characterize C2 -gradings of ZG. We recall first one of the Zassenhaus Conjectures (see [S, S2, p. 205]). AUT. If θ is an augmentation preserving automorphism of ZG, then there exist β ∈ Aut G and w ∈ QG such that θ(g) = wβ(g)w−1 for all g ∈ G. Roggenkamp and Scott showed that AUT holds for finite nilpotent groups, and also give an example of a finite metabelian group where AUT does not hold (see S2, Section 37, for more information). Theorem 3.4. Let G be a group such that AUT holds in ZG and assume θ is an automorphism of ZG satisfying the conditions of Proposition 3.1. Then θ = (Inn u) ◦ β ◦ ιH for some subgroup H of index ≤ 2 in G, some automorphism β of G (extended to ZG) satisfying β(H) = H and some unit u of QG satisfying conditions (i) and (ii) of Theorem 3.2. Proof. Let ZG = R0 ⊕ R1 be the C2 -grading associated with θ, θ(x0 + x1 ) = x0 − x1 where xi ∈ Ri for each i. If ε is the augmentation map in ZG, define H = {g ∈ G | ε(θ(g)) = 1}. It is easily checked that H is a subgroup of index ≤ 2 in G. Moreover, θ ◦ ιH is an augmentation preserving automorphism of ZG, so the fact that AUT holds tells us that θ ◦ ιH = Innu ◦ β where β is an automorphism of ZG induced by an automorphism of G, u is a unit of QG and Innu is as described in the first sentence of condition (ii) of Theorem 3.2. We then get θ = Innu ◦ β ◦ ιH . As in the proof of Theorem 3.2 we see that θ2 = I implies (u((β ◦ ιH )(u)))((β ◦ ιH ◦ β ◦ ιH )(g))(u((β ◦ ιH )(u)))−1 = g. for all g ∈ G. This means that the augmentation of the lefthand side is 1 for all g and this implies β(H) = H. It is easy to see that this means condition (i) is satisfied. Also θ(g)−g ∈ 2ZG for all g implies condition (ii) is satisfied, and the proof is complete. ¤

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As noted earlier, finite nilpotent groups are known to satisfy AUT, so Theorem 3.4 applies to this case. However, we can get some additional information. Proposition 3.5. Let G be finite nilpotent and assume ZG = R0 ⊕ R1 , graded by C2 . If R1 ⊂ ∆(G) then R1 = 0. Proof. We proceed by induction on |G|. If G is abelian then Theorem 2.3 tells us all group elements are homogeneous. Since R1 ⊂ ∆(G) we must have G ⊂ R0 and so R0 = ZG and R1 = 0. Assume the result holds for all nilpotent groups of order less than |G| and choose z ∈ Z(G) of prime order p. Theorem 2.3 tells us z is homogeneous, and since R1 ⊂ ∆(G) this means z ∈ R0 . Thus (1 − z)ZG is a graded ideal of ZG, and so the grading on ZG induces a C2 -grading on ZG/(1 − ¯0 ⊕ R ¯ 1 . By the induction hypothesis R ¯ 1 = 0, so z)ZG ∼ = Z(G/hzi) = R R1 ⊂ (1 − z)ZG. Let r1 ∈ R1 , so r1 = s(1 − z) for some s ∈ ZG. Now s = s0 + s1 , si ∈ Ri for each i. Since 1 − z is in R0 this implies r1 = s1 (1 − z). Since s1 ∈ R1 , s1 = t(1 − z) and, again, if t = t0 + t1 , then s1 = t1 (1 − z) giving r1 = t1 (1 − z)2 . Continuing we get r1 is in (1 − z)n ZG for all n. Since (1 − z)p is in p ZG, r1 = 0. ¤ Returning to the proof of Theorem 3.4, when G is finite nilpotent Proposition 3.5 tells us that if θ is not the identity automorphism then θ cannot be augmentation preserving (since R1 6= 0). It follows that when G is nilpotent the subgroup H obtained in this theorem cannot be equal to G, a fact that is not at all clear without Proposition 3.5 (an may not be true for general finite G).

4. Examples We now present some examples of units which satisfy the conditions in Corollary 3.3. (a) If u is a central unit in ZG, both conditions of Corollary 3.3 hold. But then θ = ιH so we have nothing new. (b) Let G = ha, b | a8 = 1, b2 = 1, ba = a5 bi. Take H = ha2 , bi. If u = (1 + a + a2 )2 − a ˆ, then uιH (u) = ((1 + a + a2 )2 − a ˆ)((1 − a + a2 )2 − ιH (ˆ a)) = (1 + a2 + a4 )2 − 2(1 + a2 + a4 + a6 )

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which is central. To check condition (ii) we need only consider uau−1 − a and ubu−1 − b. The first equals 0 and ubu−1 − b = [(1 + a + a2 )2 − a ˆ]b[(1 + a3 + a6 )2 − a ˆ] − b = [(1 + a + a2 )2 − a ˆ][(1 + a7 + a6 )2 − a ˆ]b − b = [(1 + a + a2 )2 − a ˆ]2 a4 b − b = (−8 − 6a + 6a3 + 9a4 + 6a5 − 6a7 )a4 b − b which is in 2ZG. (c) Let G = ha, b | a8 = 1, b2 = 1, ba = a3 bi and take H = ha2 , bi. If u = (1 + a + a2 )4 − 10ˆ a, then it is not hard to see that conditions (i) and (ii) are satisfied. (d) Let G = ha, b | a16 = 1, b2 = 1, ba = a7 bi and take H = ha2 , bi. If u = (1 + a + a2 )8 − 410ˆ a, again conditions (i) and (ii) are satisfied. Cautioning Example. Finally here is an example when a unit satisfies conditions (i) and (ii) of Corollary 3.3 but does NOT give a new grading. Let G be the dihedral group of order 8, that is, G = ha, b | a4 = 1, b2 = 1, ba = a3 bi. Let H = ha2 , bi. If u = 1 + (1 − b)a(1 + b) then it is easily checked that u satisfies conditions (i) and (ii). But the resulting grading is the same as 1 ZG = v(ZH)v −1 ⊕ v((ZH)a)v −1 where v = 1 + (1 − b)a(1 + b) in QG. 2 Acknowledgments This research was supported in part by NSERC. References [BP] [MS] [S] [S2]

Yu. A. Bahturin and M.M. Parmenter, Generalized commutativity in group algebras, Canad. Math. Bull. 46 (2003), 14-25. C. Polcino Milies and S.K. Sehgal, An Introduction to Group Rings, Algebras and Applications, Kluwer, 2002. S.K. Sehgal, Topics in Group Rings, Marcel Dekker, 1978. S.K. Sehgal, Units in Integral Group Rings, Longmans, 1999.

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Profinite graphs – comparing notions Gunther Bergauer and Wolfgang Herfort Institute for Analysis and Scientific Computation, University of Technology, Vienna, Austria [email protected] [email protected] Dedicated to C. Polcino Milies, S. Sehgal, and S. Sidki. Abstract. Defining a profinite graph to be a projective limit of an inverse system of finite graphs can lead to different objects depending upon which of the (usually) equivalent concepts of a finite graph is employed.

1. Categories of (profinite) graphs Graphs can be oriented - each edge has an origin and a terminal vertex or nonoriented. In the first situation we shall talk about digraphs, otherwise about graphs. Starting with precise definitions, let us recall Serre’s definition of a graph [5], which we would like to term a Serre digraph. Definition 1.1. A digraph consists of a set Γ together with two functions o, t : Γ → Γ satisfying ot = t and to = o. The vertices V (Γ) of the digraph is the set o(Γ) ∪ t(Γ) and the set of (directed) edges E(Γ) is Γ \ V (Γ). A quadruple (Γ, o, t, ι) where (Γ, o, t) is a digraph and ι an involution acting on Γ with set of fixed points V (Γ) is a Serre digraph. Morphisms in this category are morphisms of digraphs φ : Γ → ∆ with φι = ιφ. Serre uses bar ¯ for ι. A morphism is strict if φ(E(Γ)) ⊆ E(φ(Γ)). Any concept of a profinite graph, known to the authors, arises by • replacing the sets Γ, V (Γ) by boolean spaces (i.e., projective limits of finite sets; see Definition 1.2 below) and letting o, t (and, after having given Definition 1.5, δ) be continuous with analogous properties as in the respective finite situation. 33

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• characterizing any such profinite graph as a projective limit of an inverse system of finite graphs and morphisms in the respective category. Definition 1.2. A boolean or profinite space is a compact totally disconnected Hausdorff space. Alternatively it is the projective limit of an inverse system of finite sets [4]. A category of boolean spaces has objects, boolean spaces, and morphisms continuous functions, termed maps. For a finite group G a boolean space X is a G-space, if G acts continuously upon X. Morphisms in the category of G-spaces are maps respecting the action. The above recipe leads to profinite graphs as defined in [4]. Definition 1.3. A profinite digraph is a boolean space Γ which is a digraph with o and t maps. A morphism φ : Γ → ∆ is a morphism of digraphs which is a map. Alternatively, Γ is the projective limit of an inverse system of finite digraphs. When E(Γ) is closed the graph is boolean in the sense of [2], where it was shown that every such graph is the projective limit of an inverse system of finite digraphs with strict morphisms. Then morphisms are strict maps. A profinite Serre digraph is a Serre digraph which at the same time is a hιi-space. Such a graph is the projective limit of finite Serre digraphs. A different concept of graph is found, e.g., in [6], although no definition for morphisms is given. We ¡find ¢ it convenient to denote for any set X the set of X all n-element subsets by n and at most n-element subsets by F≤n (X). For the profinite situation the Vietoris topology on the set of all closed subsets of a boolean space appears to be suitable. Definition 1.4. For a topological space X, the Vietoris-topology on the set C(X) of all nonempty closed subsets of X has a basis of open sets W (O1 , . . . , Ok ) := {A ∈ C(X) | ∀j = 1, . . . , k : A ∩ Oj 6= ∅ ∧ A ⊆

k [

Oj }

j=1

with k ∈ NI and Oj running through all clopen subsets of X. S Since, for clopen sets O1 , . . . , Ok , U1 , . . . , Ul , setting O := ki=1 Oi and S U := lj=1 Uj , as a consequence of the definition one has W (O1 ∩ U, . . . , Ok ∩ U, O ∩ U1 , . . . , O ∩ Ul ) = W (O1 , . . . , Ok ) ∩ W (U1 , . . . , Ul ), these sets indeed form a basis for the topology. We claim that every W (O1 , . . . , Ok ) is also closed. Indeed, again as an immediate consequence

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S of the definition, setting O := ki=1 Oi , and letting dash denote the complement of a subset of X in X, one finds 0

C(X) \ W (O1 , . . . , Ok ) = W (O , O1 , . . . , Ok ) ∪

k [

W (Oi0 ),

i=1

to be a finite union of open sets, i.e., it is closed as well. When X is finite and discrete, then C(X) consists of all nonempty subsets and it is not hard to see that the Vietoris-topology coincides with the discrete topology on C(X). When (Xα , φαβ ) is an inverse system of finite sets, then there is an inverse system (C(Xα ), φ∗αβ ), where φ∗αβ sends each set A ∈ C(X) to {φαβ (a) | a ∈ A}. Then, when X = lim Xα , it turns out ←−α that C(X) = lim C(Xα ) and the Vietoris-topology coincides with the one, ←−α when C(X) is considered as a profinite space. The definitions show that for any boolean space X the sets F≤n (X) for all n ∈ NI are closed subsets of C(X). Definition 1.5. A graph (Γ, δ) consists of the following data: (1) a set Γ and a subset V (Γ) (the vertices); (2) a function δ : Γ → F≤2 (V (Γ)) satisfying δ(v) = {v} for all v ∈ V (Γ). Then E(Γ) := Γ \ V (Γ) is the set of its edges. When v ∈ δ(e) we say that v and e are adjacent. A function φ : Γ → Γ0 is a morphism if δ 0 φ = φ∗ δ. Here φ∗ denotes the function from F≤2 (Γ) → F≤2 (Γ0 ) induced by φ, i.e., φ∗ ({a, b}) := {φ(a), φ(b)} for any a, b ∈ V (Γ). A Serre graph is a triple (Γ, δ, ι) with (Γ, δ) a graph and ι an involution acting on Γ with fixed point set V (Γ). A profinite graph is a graph (Γ, δ) with Γ a boolean space, V (Γ) a closed subset of Γ, F≤2 (Γ) equipped with the Vietoris topology (Definition 1.4) and δ a map. Morphisms in the category of profinite graphs are morphisms of the underlying graphs which at the same time are maps. A profinite Serre graph is a Serre graph (Γ, δ, ι) with (Γ, δ) a profinite graph and Γ at the same time a hιi-space. An edge e with δ(e) containing a single element only is a loop. Let us remark: (1) There is the notion of ¡simple graph [3]. It is stipulated that δ is ¢ V injective with values in 2 . When δ is not injective, the graph has multiple edges and is sometimes called a multigraph. (2) It is common to use strict morphisms in graph theory [3], i.e., edges go to edges and vertices to vertices - no mixing. During the present article we allow collapsing edges for better coherence with Definition 1.1.

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(3) In finite graph theory these concepts are equivalent at least for objects in the respective categories in the following sense: given a digraph, one easily constructs a Serre digraph by introducing an inverse edge e¯ for every e. Conversely for a Serre digraph select from each pair of edges exactly one. Every digraph can be produced from a Serre digraph in this manner. Next, given a digraph Γ, define δ : E → F≤2 (V ) by setting δ(e) := {o(e), t(e)}. Similarly as before one can see that each digraph gives rise to a unique graph and conversely, every graph arises in this way. Our main concern is showing that in the profinite situation, (1) from each digraph (Γ, o, t) one can produce a profinite Serre digraph by “doubling” the space of edges with Γ naturally embedded, but there exists a profinite Serre digraph not containing a naturally embedded profinite digraph. (2) from every profinite digraph one can construct a profinite graph by “forgetting” orientation, but not every profinite graph arises in this way. (3) every profinite Serre graph gives rise to a profinite graph and conversely (Lemma 2.6). (4) every Serre digraph gives rise to a profinite graph by “identifying” each edge e with e¯ but not every profinite graph arises in this way. Taking (3) into account, our comments can be displayed in a diagram: digraph

Lemma 2.4 Example 3.6 -

@

Lemma 2.3 @ Corollary 3.5

@ R @

Serre digraph

¡ ¡Lemmata 2.5 & 2.6 Example 3.4 ¡ ª ¡

graph

When the underlying boolean space(s) are second countable, then, similarly as in the discrete situation, the concepts are equivalent, i.e., the arrows in the diagram can be “reverted” (see Lemmata 2.3 and 2.4). 2. Relations between the concepts Definition 2.1. Given a profinite graph (Γ, δ), then a profinite digraph (Γ, o, t) is a coherent digraph presentation provided δ(m) = {o(m), t(m)}, whenever m ∈ Γ. Lemma 2.2. Let G ∼ = C2 and X a G-space. When X is second countable or C2 acts freely there is a continuous section from X/G to X. Proof. This follows from Lemmata 5.6.5 and 5.6.7 in [4].

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˜ δ) be defined by Lemma 2.3. Given a digraph (Γ, o, t), let the graph (Γ, ˜ ˜ setting E(Γ) := E(Γ), V (Γ) := V (Γ) and δ(m) := {o(m), t(m)}. Then ˜ δ). (Γ, o, t) is a coherent digraph presentation of (Γ, ˜ Every second countable profinite graph (Γ, δ) has a coherent digraph presentation. Proof. The continuity of δ is a consequence of F≤2 (V (Γ)) being homeomorphic to Γ × Γ/hαi, where α permutes the coordinates. ˜ δ) be a second countable profinite graph. For the second statement, let (Γ, ˜ ˜ Then Y := Γ × Γ is a C2 -space where the involution permutes coordinates ˜ By Lemma 2.2 there exists a continwith Y /C2 homeomorphic to F≤2 (Γ). uous section σ : Y /C2 → Y . For every edge define o(e) := σ(δ(e)C2 /C2 ) and let t(e) be the edge in δ(e) \ {o(e)} if the latter is nonempty and o(e) else. ¤ Lemma 2.4. When (Γ, o, t) are the data of a profinite digraph then data (Γ0 , δ 0 , ι) of a profinite Serre digraph can be constructed as follows: Set Γ0 := ΓqV (Γ) Γopp and let ι denote the evident involution sending each edge e ∈ E(Γ) to eopp ∈ Γopp and each vertex v ∈ V (Γ) to v = v opp ∈ V (Γopp ). Here (Γopp , δ opp ) is the digraph with (oopp , topp ) := (t, o). Every second countable profinite Serre graph arises in this manner. Proof. Observe the existence of a homeomorphism ψ : Γ → Γopp sending v ∈ V (Γ) to v and e ∈ E(Γ) to eopp ∈ E(Γopp ). Then ψ (and its inverse) give rise to an involutory homeomorphism ι of ΓqΓopp which factors via the identification of the identical copies of V (Γ) in the fixed set V (Γ)qV (Γopp ) under the involution inside ΓqΓopp . Finally set δ 0 (e) is equal to δ 0 (eopp ) := {o(e), t(e)}. The proof of the last statement is similar to the one in Lemma 2.3. ¤ In the situation of the previous Lemma we say that (Γ, o, t) is an oriented shadow of (Γ0 , δ 0 , ι). Lemma 2.5. Given a profinite Serre digraph (Γ, o, t, ι) a profinite Serre graph (Γ0 , δ 0 , ι) can be constructed by setting Γ0 := Γ/hιi and δ 0 (¯ e) := {o(e), t(e)}. Here bar denotes passing to the hιi-orbit. Proof. We omit the easy proof.

¤

Lemma 2.6. For a profinite Serre graph (Γ, δ, ι) let Γ0 be the quotient of Γ under the action of the involution and define δ 0 by setting δ 0 (m) := {m, m}. ¯ 0 0 Then (Γ , δ ) are the data of a profinite graph. Conversely, given a profinite graph (Γ, δ) let h be an isomorphism of Γ onto a copy Γ0 of Γ. Form the profinite graph ΓqΓ0 and amalgamate by identifying v with h(v) for all v in V (Γ). For all e ∈ E(Γ) define δ 0 (e) := δ(e) =: δ 0 (h(e)). Then (Γ0 , δ 0 , ι) is a profinite Serre graph whose quotient according to the first part of the Lemma is (Γ, δ).

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Proof. The proof is straightforward.

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¤

3. Examples We shall employ the following result about the nonexistence of continuous sections. Lemma 3.1. There is an example of C2 -space X with no continuous section from the quotient space X/C2 to X and C2 fixing a single point of X. Moreover, X × X is a C2 -space with action permuting “coordinates” and no continuous section from (X × X)/C2 to X × X. Proof. An example is given on p. 194 in [4]. Let X with G = hαi ∼ = C2 be an example. Then, sending x ∈ X to (x, xα ) provides an embedding of the hαi-space X into X × X such that we can extend the action of α by permuting coordinates in X × X and we find it convenient to identify X with its image in X × X. Suppose there exists a continuous section σ : (X × X)/hαi → X × X. Let σ0 denote the restriction of σ to X/hαi. Then, whenever (x, y) = σ0 (x0 G/G) for some x0 ∈ X one verifies {x, y} = {x0 , xα0 } ⊆ X. Therefore σ0 ψ : X/hαi → X would be a continuous section, a contradiction. ¤ In light of Lemma 2.4, since in a profinite Serre digraph all edges occur in pairs, one might expect, by factoring hιi to arrive at a digraph representing its oriented shadow. Our next example shows that this can fail to happen. Example 3.2. Let X and α be as in Lemma 3.1. Define a profinite Serre graph (Γ, ι, δ) as follows: Γ is the disjoint union of its space of edges E and its space of vertices V both homeomorphic to X. Let ι act upon E like α and trivially upon V . For every edge x ∈ X set δ(x) := {x, xα }, where x and xα are vertices in V . We claim that there is no profinite digraph (Γ0 , o, t) such that Γ0 has the same vertices as Γ, edges X/hαi and δ(x) = {o(π(x)), t(π(x))} for every x ∈ E(Γ), where π : E → E/hαi denotes the canonical projection. Indeed (Γ0 , o, t) cannot exist, otherwise oπ induces a map from X to X which represents a continuous section from X/hαi to X, a contradiction. To give an example of a profinite graph not arising as a result of the procedure described in Lemma 2.3 the next result will be needed. Lemma 3.3. Let a G ∼ = C2 = hιi and X a G-space. Construct from it a profinite graph (Γ, δ) by setting Γ := X qX G X/G (here X G is the set of G-fixed points and we identify this set inside X with its image in X/G), V (Γ) := X and defining δ by setting δ(x) := {x} when x ∈ X and by δ(xG/G) := {x, xι } for all xG/G ∈ X/G. Suppose Γ possesses a coherent presentation as a profinite digraph. Then there exists a continuous section σ : X/G → X.

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Proof. Observe the existence of a homeomorphism ψ : im(δ) → X/G. Then, due to coherence, there is a commutative diagram o

Γ

ι

- X-

-X

π

δ

π

? ? ? ? ? ? 1X/G ψ - X/G- X/G im(δ) Since π, when restricted to o(Γ) is a homeomorphism, it is a continuous section σ as claimed. ¤

Example 3.4. Let α be an involution and (X, hαi) an hαi-space with a single fixed point x0 ∈ X, and no continuous section σ : X/hαi −→ X. Define a profinite graph (Γ, δ) by setting E(Γ) := X/hαi, V (Γ) := X, for all x ∈ E(Γ) defining δ(x) := {x, xα }, to be interpreted as an element in X/hαi, and, for all x˙ ∈ V (Γ) setting δ(x) ˙ := {x}. ˙ As a boolean space it is the disjoint union of its sets of edges and vertices respectively (hence a profinite graph in the sense of Gildenhuys and Ribes [2]). We will show that ˜ δS ) with Γ/hιi ˜ there is no profinite Serre digraph (Γ, = Γ. ˜ identifies with V (Γ) = X. Hence δS , Suppose it does. Then clearly V (Γ) ˜ maps onto the subset D := {(x, xα ) | x ∈ X} of when restricted to E(Γ) X ×X. Let hµi denote the canonical action, permuting coordinates in X ×X. ˜ coincides with Note D to be an invariant subset. Observe next that E(Γ) ˜ onto Γ = Γ/hιi, ˜ the preimage of E(Γ) under the canonical projection from Γ ˜ is closed, i.e., compact. Since ι acts freely upon E(Γ), ˜ deduce hence E(Γ) ˜ from Lemma 2.2 the existence of a continuous section λ : E(Γ) → E(Γ). There is a commutative diagram of profinite actions ˜ E(Γ)

δS

?

∼ =

µ

ι

˜ E(Γ)

-D

δS

? -D

-X

α

∼ =

? -X

˜ D/hµi, which in turn induces homeomorphisms between the spaces E(Γ), and X/hαi, giving rise to an embedding of X/hαi, thereby using λ. Finally by means of the isomorphisms indicated in the diagram and δS embed the latter closed set into X to become a continuous section for X/hαi in X, a contradiction. Corollary 3.5. There is no profinite digraph such that “forgetting” along the lines of Lemma 3.3 yields the profinite graph described in Example 3.4. Proof. Suppose it exists. Then, by virtue of Lemma 2.4, it must possess a profinite Serre digraph, contradicting the findings in Example 3.4. ¤

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Example 3.6. We give an example of a profinite Serre digraph (Γ, o, t, ι) with no profinite digraph giving rise to Γ. Let X be the boolean space from Lemma 3.1 and let α be the involution. Set E = E(Γ) := X, V = V (Γ) := X/hαi, let o and t coincide with the canonical projection from E = X onto V = X/hαi and ι := α on E and the identity on V . Suppose there is a profinite digraph (Γ0 , o0 , t0 ) giving rise to Γ as described in Lemma 2.4. Then one may identify the vertices of Γ with those of Γ0 . Due to our construction, E would be the disjoint union of two copies of E(Γ0 ) to be permuted by ι. Each of them gives rise to a continuos section from X/hαi to X, a contradiction. 4. Compact realizations of profinite graphs Let I := [0, 1] denote the closed unit interval. A continuum X which is the finite union of arcs intersecting pairwise in at most two points is sometimes called a graph. One can give a combinatorial description of X by using the cut set P , the set E of edges being the connected components of X \ P and a function δ : E → F≤2 (P ) for describing adjacencies, i.e., by a graph in the sense of Definition 1.5. In this section we consider the following generalization: Definition 4.1. A topological space X is a topological graph, provided it contains a closed totally disconnected subset P such that X\P has connected components homeomorphic to the open unit interval (0, 1). P is a set of cut points. Let (X, P ) be a compact topological graph. Then P is a boolean space. One would like to store its data into a profinite graph and, conversely, recover from the profinite graph the compact topological graph (X, P ). To this end let c(X) denote the set of the closures of all connected components (all are homeomorphic to either I or the circle S 1 ). Define a graph Γ by letting V (Γ) be the set of all singleton sets {p} with p ∈ P and E(Γ) := c(X). Let δ(e) := ∂e be the topological boundary of e in X. It is not hard to see how to recover (X, P ) as a set and we omit a formal presentation. For recovering the topology one needs “topological” information on c(X). Example 4.2. Let X := [0, 1] and P := {0} ∪ { n1 | n ∈ NI}. A natural way of encoding (X, P ) by means of a graph Γ is by defining vertices NI ∪ {∞} (n 7→ n1 and ∞ 7→ 0), edges NI and δ(n) := {n, n + 1}. Equip Γ with a topology by letting each n ∈ NI be open (as a vertex and as an edge) as well as the sets {k ∈ NI | k ≥ n} ∪ {∞} of vertices and {k ∈ NI | k ≥ n} ∪ {∞} as a set of edges together with the vertex at infinity. A set theoretic reconstruction of (X, P ) yields the interval [0, ∞) with P := ZZ ∩ [0, ∞) together with an isolated point ∞. The “topological” information given to us is the understanding, that ∞ is the one point compactification, thus one arrives indeed at a homeomorphic copy of [0, 1] with the homeomorphism sending P to (ZZ ∩ [0, ∞)) ∪ {∞}.

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Remark 4.3. Since any profinite graph Γ is compact, we shall from now on restrict ourselves to compact topological graphs (X, P ) with the set c(X) ∪ {{p} | p ∈ P } encoding Γ, a closed subset of C(X). The following space (X, P ) does not meet this condition: Example 4.4. Consider a family {Kn }∞ n=1 of circles with Kn passing 1 through the three points {(0, −1), ( n , 0), (0, 1)}. Let Y be the union of all the circles Kn . Define X := Y ∩ {(x, y) ∈ IR2 | x ≥ 0} ∪ {0} × [−1, 1] and let P := {(0, −1), (0, 0), (0, 1)}. Then c(X) = {Kn ∩ X | n ∈ NI} ∪ {{0} × [−1, 0], {0} × [0, 1]}. However the sequence of arcs {Kn ∩ X}∞ n=1 converges w.r.t. the Vietoris topology to the set {0} × [−1, 1], which does not belong to c(X). When a profinite graph Γ possesses a profinite digraph representation, it is easy to construct from it a compact topological graph (X, P ) such that ˜ := Γ × I and identify {v} × I with (v, 0) its Γ is our profinite graph: Form Γ for every v ∈ V (Γ). Next identify each (o(e), 0) with (e, 0) and (t(e), 1) with (e, 1) for every e ∈ E(Γ). In particular, any second countable version of a profinite graph can be interpreted as the data structure of a compact topological graph. The authors would like to thank the referee for valuable remarks. Acknowledgments Dr. Bergauer is supported by FWF Project Nr. P17477-N13. Dr. Herfort is grateful for the great hospitality and support during the Conference. References [1] Topology: A geometric account of general topology, homotopy types and the fundamental groupoid. Rev., updated and expanded ed. Ellis Horwood Series in Mathematics and its Applications. Chichester (UK), New York (1988). [2] D. Gildenhuys and L. Ribes, Profinite groups and Boolean graphs, Pure and Applied Algebra 12, (1978), 21-47. [3] W. Imrich and S. Klavzar, Product graphs. Structure and recognition. With a foreword by Peter Winkler., John Wiley & Sons, Ltd. 2000. [4] L. Ribes and P.A. Zalesskii, Profinite Groups, Springer 2000. [5] J.-P. Serre, Trees, Springer 1980. [6] W.T. Tutte, Graph theory, Encyclopedia of Mathematics and its Applications, 21. Cambridge University Press, Cambridge, 2001.

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Lie identities in symmetric elements in group rings: A survey Osnel Broche Cristo Instituto de Matem´atica e Estat´ıstica, Universidade de S˜ao Paulo, Caixa Postal 66281, S˜ao Paulo, SP, 05315-970, Brasil [email protected] Manuel Ruiz Mar´ın Departamento de M´etodos Cuantitativos e Inform´aticos, Universidad Polit´ecnica de Cartagena, Paseo Alfonso XIII, 50 30.203 Cartagena, Spain [email protected] To Professor C´esar Polcino on the occasion of his 60th birthday. Abstract. In this survey we gather some results concerning polynomial identities (resp. group identities) in the set of symmetric elements (resp. symmetric units), and when these identities are transferred to the whole group ring (resp. unit group). We consider several kinds of involutions, not only the classical involution.

1. Introduction Let R be a ring with unity and ∗ a ring involution on R. The set of ∗-symmetric elements, or just symmetric when the involution ∗ is clear from the context, is R∗ = {r ∈ R| r∗ = r}. We are going to denote by U(R) the group of units of R and by U∗ (R) = {u ∈ U(R)| u∗ = u} the set of symmetric units. In general the set R∗ (respectively U∗ (R)) is not a ring (resp. a subgroup). In fact R∗ (resp. U∗ (R)) is a ring (resp. subgroup) if and only if the symmetric elements (resp. units) commute. Several papers have dealt with questions of how various algebraic properties of the set R∗ affect the structure of the whole ring. Similar question may be posed by making assumptions about the symmetric units or subgroup they generate. Definition 1.1. Let A be an R-algebra and S ⊂ A be a subset. We say that S satisfies a polynomial identity (PI for short) if there exists a nonzero 43

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polynomial f (z1 , . . . , zn ) in the polynomial ring Rhz1 , . . . , zn i in the noncommuting variables z1 , . . . , zn with f (s1 , . . . , sn ) = 0 for all si ∈ S. The set of units S ∩U(A) of the set S is said to satisfy a group identity (GI for short) if there exists a nontrivial word w(x1 , . . . , xn ) in the free group generated by x1 , . . . , xn such that w(u1 , . . . , un ) = 1 for all u1 , . . . , un ∈ S ∩ U(A). Since results about symmetric units in rings with involution seem to be difficult to obtain, one way to begin the study of these units is to try to mimic known results for the symmetric elements. A fundamental result of this kind is the following theorem of Amitsur [1]. Theorem 1.2. [1] If R is an algebra with involution whose symmetric elements S satisfy a polynomial identity, then R itself satisfies a polynomial identity. Most of the results concerning identities of the symmetric elements and symmetric units, and how these identities are transferred to the whole ring or its group of units respectively, have been stated in the context of group rings. Let R be a commutative ring with unity and let G be a group. We denote by RG the group ring of G over R. This ring is a ring with involution. Any involution on the group G can be extended R-linearly to an involution on RG. Let ∗ be an R-linear involution in RG. We define the set of symmetric elements in the group G by G∗ = {g ∈ G| g ∗ = g}. Notice that an element α of RG is ∗-symmetric if and only if  ∗ X X X αg g ∗ = αg g = α, α∗ =  αg g  = g∈G

g∈G

g∈G

i.e., if and only if αg = αg∗ , for all g ∈ G. Thus, (RG)∗ is generated, as an R−module, by the set S = {g + g ∗ | g ∈ G with g ∗ 6∈ G∗ } ∪ G∗ . Therefore (RG)∗ satisfy a multilinear polynomial identity if and only if S satisfy the same identity. A complete classification of group algebras satisfying a polynomial identity was given by Isaacs and Pasmman [13] for characteristic zero and Passman in [26] for characteristic p. Taking into account Theorem 1.2 we resume these results in the following theorem. Theorem 1.3. Let R be a field of characteristic p and G be a group. Let ∗ be an involution of the group algebra RG. Then the following are equivalents. (i) (RG)∗ satisfies a PI. (ii) RG satisfies a PI.

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(iii) G has a p−abelian subgroup of finite index. Recall that a group G is said to be p-abelian if the derived subgroup G0 is a finite p-group. We say that G is 0-abelian if it is abelian. Some specific PI’s as Lie nilpotence, Lie n-Engel, and commutativity have also been studied. We will call these identities Lie-identities. Recall that a set S of an R-algebra A is Lie nilpotent if, for some integer n ≥ 2, [x1 , x2 , . . . , xn ] = 0, for any x1 , . . . , xn ∈ S, where [x, y] = xy − yx denote the Lie bracket and inductively [x1 , . . . , xn ] = [[x1 , . . . , xn−1 ], xn ]. Also, recall that S is Lie n-Engel if [x, y, y, . . . , y ] = 0, for all x, y ∈ S. | {z } n

As we will see in the following sections, that the set of symmetric elements satisfies a Lie-identity does not imply that this Lie-identity is always satisfied by the whole algebra. In this survey we are going to collect results concerning Lie-identities in the set of symmetric elements and symmetric units of a group ring and when these identities are transferred to the whole ring or the whole group of units respectively. The rest of the paper is structured as follows. In Section 2 we collect the results concerning commutativity, Lie nilpotence, and Lie n-Engel for the set of symmetric elements in group rings RG when the involution ∗ is the classical involution on the group G, that is g ∗ = g −1 . In Section 3 we deal with the commutativity of the symmetric elements when the involution ∗ is any group involution which is extended by linearity to the whole group ring. In Section 4 we present the results concerning the commutativity of the symmetric elements when the involution is not a group involution extended by linearity. The involutions treated in this section are the oriented involutions introduced by Novikov [23] in the context of K-theory and its extension by considering an arbitrary group involution. Finally in Section 5 we collect the results concerning GI property, nilpotence, and commutativity of the set of symmetric units. The first two properties have been studied only for the classical involution, while the commutativity has been studied for any group involution extended R-linearly to the whole group ring and also for oriented involutions. 2. Symmetric elements under the classical involution Let R be a ring with unity and G be a group. Throughout this section we are going to denote by ∗ the classical involution, which is defined by setting g∗ = P g −1 , for all g ∈ G, and extending by linearity to RG; i.e., P ∗ ( αg g) = αg g −1 . Notice that the set of symmetric elements in the group G are the elements of order 2, that is G∗ = {g ∈ G| g 2 = 1}. In this section we gather some results concerning Lie identities of symmetric elements in group rings, when they are satisfied, and when these properties are transferred to the whole group ring RG. We start with the

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more simple identity, the commutativity of (RG)∗ , or equivalently when (RG)∗ , the set of symmetric elements, form a ring. The following lemma states that if the polynomial identities like commutativity or Lie n-Engel are satisfied, then the symmetric group elements are central. Lemma 2.1. [7, 15] Let R be a commutative ring with unity and G be a group. (i) If (RG)∗ is commutative then G∗ ⊆ Z(G). (ii) If char(R) is odd prime and (RG)∗ is Lie n−Engel then G∗ ⊆ Z(G). Thus, (RG)∗ is commutative if and only if S \ G∗ is commutative. Furthermore, if char(R) is odd prime then (RG)∗ is Lie nilpotent (resp. Lie n−Engel) if and only if S \ G∗ is Lie nilpotent (resp. Lie n−Engel). To study these Lie-identities in S \ G∗ the following Lemma is very useful. Lemma 2.2. [7, 15] Let R be a commutative ring with unity and G be a group. (i) If (RG)∗ is commutative then for any g, h ∈ G one of the following condition holds: (a) gh is equal to either hg, hg −1 or h−1 g; (b) char(R) = 2, gh = h−1 g −1 and the orders of g, h are both equal to 4. (ii) If char(R) is odd prime, (RG)∗ is Lie n−Engel and g, h ∈ G such that [g + g −1 , h + h−1 ] = 0 then gh is equal to either hg, hg −1 or h−1 g. Notice that Lemma 2.2 justifies the distinction between the characteristic of the ring R equal to 2 and different from 2. The following results state when the symmetric elements (RG)∗ commute, if the characteristic of the ring R is different from 2. Theorem 2.3. [7] Let G be a nonabelian group and let R be a commutative ring of characteristic different from 2. Then, (RG)∗ is a commutative ring if and only if G is a Hamiltonian 2-group. Recall that a group G is a Hamiltonian 2-group if G ∼ = Q8 × E where Q8 is the quaternion group of order 8 and E is an elementary abelian 2group. Thus, as a consequence of Theorem 2.3 one deduces that if (RG)∗ is a commutative ring then G is a torsion group. In the case in which the characteristic of the ring R is 2 we have the following result: Theorem 2.4. [7] Let G be a nonabelian group and let R be a commutative ring of characteristic 2. Then, (RG)∗ is a commutative ring if and only if G is the direct product of an elementary abelian 2-group and a group H, for which one of the following conditions holds.

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(i) H has an abelian subgroup A of index 2 and an element b of order 4, such that conjugation by b inverts each element of A; (ii) H is the direct product of the quaternion group of order 8 and the cyclic group of order 4, or the direct product of two quaternion groups of order 8; (iii) H is the central product of the group hx, y | x4 = y 4 = 1, x2 = (y, x)i with the quaternion group of order 8, where the nontrivial common element to the two central factors is x2 y 2 ; (iv) H is isomorphic to one of the groups H32 or H245 , where H32 = hx, y, u | x4 = y 4 = 1, x2 = (y, x), y 2 = u2 = (u, x), x2 y 2 = (u, y)i and H245 = hx, y, u, v | x4 = y 4 = (v, u) = 1, x2 = v 2 = (y, x) = (v, y), y 2 = u2 = (u, x), x2 y 2 = (u, y) = (v, x)i. Notice that when (RG)∗ is a ring and the exponent of the group G is different from 4 then G = H × E where E is an elementary abelian 2-group and H is a group with the property (i) of Theorem 2.4. If the exponent of G is 4, then H can be any of the groups listed in Theorem 2.4. This list of groups was given by V. Bovdi, Kov´ acs and Sehgal, in [4], answering the question of when the set of symmetric units of a modular group ring RG is a multiplicative group, assuming that R is a commutative ring of prime characteristic p and G is a locally finite p-group. In the more general context of Theorem 2.4 when the exponent of G is 4 the proof is reduced to verify the conditions of the result in [4] and to prove that the commutativity of the symmetric units implies the commutativity of the symmetric elements. Another property in (RG)∗ which has been studied is the Lie nilpotence. From now until the end of this section we assume that R is a field of characteristic p 6= 2. The characterization of the Lie nilpotence of (RG)∗ was given by Giambruno, Sehgal and Lee in two different papers. In the first one [10], Giambruno and Sehgal considered the case in which the group G has no 2elements. In the second one [15], Lee considered the case in which the group G contains 2-elements. These two results can be resumed in the following theorem. Theorem 2.5. [10, 15] Let R be a field of characteristic p 6= 2 and let G be a group. Then (RG)∗ is Lie nilpotent if and only if one of the following conditions holds: (i) Q8 6⊆ G and G is nilpotent and p-abelian. (ii) p > 2 and G ∼ = K × P where K is a Hamiltonian 2-group and P is a finite p-group. (iii) p = 0 and G is a Hamiltonian 2-group.

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Recall that the Lie nilpotence of RG was characterized by Passi, Passman and Sehgal in [24] (see also [27]) in terms of the group G and the characteristic of the ring R. A consequence of this characterization is the following result. Corollary 2.6. Let R be a field of odd prime characteristic and let G be a group not containing a copy of Q8 . Then (RG)∗ is Lie nilpotent if and only if RG is Lie nilpotent. In [10], Giambruno and Sehgal also proved the following result in a general context, showing the strong relationship between Lie nilpotence and commutativity of the symmetric elements. Theorem 2.7. [10] Let R be a semiprime ring with involution ∗ such that 2R = R. If R∗ is Lie nilpotent then R∗ is a commutative ring. Finally we consider a particular case of Lie nilpotence, the Lie n-Engel property. This property was characterized by Lee in [16]. The result is the following. Theorem 2.8. [16] Let R be a field of characteristic p 6= 2 and let G be a group. Then (RG)∗ is Lie n-Engel if and only if one of the following conditions holds: (1) Q8 6⊆ G and either (i) p > 2 and G is nilpotent and contains a normal p-abelian subgroup A with G/A a finite p-group or (ii) p = 0 and G is abelian. (2) p > 2 and G ∼ = K × P where K is a Hamiltonian 2-group and P is a nilpotent p-group of bounded exponent containing a normal subgroup A of finite index such that A0 is also finite; (3) p = 0 and G is a Hamiltonian 2-group. Recall that the Lie n-Engel property in group rings RG was characterized by Sehgal in [27]. A consequence of this characterization is the following corollary. Corollary 2.9. Let R be a field of characteristic prime odd and let G be a group not containing a copy of Q8 . Then (RG)∗ is Lie n−Engel if and only if RG is Lie m−Engel. 3. Symmetric elements under an arbitrary group involution Any group involution ϕ : G → G can be extended R-linearly to a ring involution ϕ : RG → RG. In this section we only deal with the question of when (RG)ϕ is a commutative ring for an arbitrary group involution ϕ. Properties like the Lie nilpotence or Lie n-Engel of (RG)ϕ are open questions except in the semiprime case (see Theorem 2.7).

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When ϕ is the classical involution the study of the commutativity of the symmetric elements was divided in two cases depending on the characteristic of the ring R, char(R) = 2 or char(R) 6= 2. The following theorem gives a characterization of when the symmetric elements commute in terms of the group G and the involution ϕ when the characteristic of the ring R is different from 2. Theorem 3.1. [14] Let ϕ be an involution on a nonabelian group G and let R be a commutative ring with char(R) 6= 2. The following are equivalent: (i) (RG)ϕ is commutative. (ii) The group G has the LC property, a unique nontrivial commutator s and the involution ϕ : G → G is given by ½ g if g ∈ Z(G) ϕ(g) = sg if g 6∈ Z(G) (iii) G/Z(G) ∼ = C2 × C2 and ϕ(g) = h−1 gh or ϕ(g) = g for all g, h ∈ G. In this case, (RG)ϕ = Z(RG). Groups with lack of commutativity property (“LC” for short) have been described in [12]. A group G has the LC property if for any pair of elements g, h ∈ G, it is the case that gh = hg if and only if g ∈ Z(G) or h ∈ Z(G) or gh ∈ Z(G). When the characteristic of R is 2 we have the following result. Theorem 3.2. [14] Let R be a commutative ring with char(R) = 2 and let G be a nonabelian group with involution ϕ. Then (RG)ϕ is commutative if and only if one of the following conditions holds: (i) G contains an abelian subgroup A of index 2 and b ∈ G \ Gϕ with b2 ∈ Gϕ such that ϕ(a) = b−1 ab for all a ∈ A. (ii) G contains a central subgroup Z such that G/Z is an elementary abelian 2-group and the involution ϕ : G → G is given by ϕ(g) = cg g where cg ∈ Z and the following properties are satisfied: (1) c2g = 1, (2) cg = 1 if and only if g ∈ Z, (3) cgh = cg ch (g, h) and if (g, h) 6= 1 we have that cgh = cg , ch or (g, h). Under the hypothesis of Theorem 3.2, if the group G has an element g with g 2 6∈ Gϕ then the group is described by condition (i). In this case Gϕ = Z(G) ⊆ A and there exists an element a0 ∈ Z(G) of order two so that the involution ϕ is given by ½ a0 g if g 6∈ A ϕ(g) = b−1 gb if g ∈ A On the other hand if for all g ∈ G, g 2 ∈ Gϕ condition (ii) of Theorem 3.2 holds.

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4. Symmetric elements under oriented involution One can wonder what happens with the commutativity of the symmetric elements if the involution on RG does not come from the group G. In this section we are going to consider an R-linear involution ψ on RG which does not come from the group G. Let σ : G → {±1} be a group homomorphism. Such a map is called an orientation of the group G. Novikov [23], introduced in the context of K-theory, an involution of RG, defined by  (4.1)



X

σ αg g  =

g∈G

X

αg σ(g)g −1

g∈G

which was subsequently studied by various authors (see [2], [6], [18] and [19]). Clearly, this involution does not come from a group involution. The above involution can also be constructed in a more general context. Given both an orientation σ : G → {±1} and a group involution ϕ : G → G, an oriented involution ψ of RG is defined by   X X ψ αg g  = σ(g)αg ϕ(g). g∈G

g∈G

Notice that if σ is trivial then ψ is the kind of involutions studied in the previous section. Now we assume that σ is nontrivial and hence the characteristic of the ring R must be different from 2. Let N be the kernel of σ. Then N is a subgroup of index 2 in G. It is obvious that the involution ψ coincides on the subring RN with the ring involution ϕ. Thus, if (RG)ψ is commutative then (RN )ϕ is commutative and, by Theorem 3.1, we know the struture of N and the action of ϕ on N . If (RG)ψ is a commutative ring, [G : N ] = 2 and the structure of the group N and the action of ϕ on N are both known. Despite this it is not an easy task to describe G and the action of ϕ on G. The following theorem gives us this characterization. Theorem 4.1. [8] Let R be a commutative ring with unity and let G be a nonabelian group with involution ϕ and nontrivial orientation homomorphism σ with kernel N . Then (RG)ψ is a commutative ring if and only if one of the following conditions holds: (i) N is an abelian group and (G \ N ) ⊂ Gϕ ; (ii) G and N have the LC property, and there exists a unique nontrivial commutator s such that the involution ϕ is given by ½ g if g ∈ N ∩ Z(G) or g ∈ (G \ N ) \ Z(G). ϕ(g) = sg if otherwise.

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(iii) char(R) = 4, G has the LC property, and there exists a unique nontrivial commutator s such that the involution ϕ is given by ½ g if g ∈ Z(G). ϕ(g) = sg if g 6∈ Z(G). In the case of when ψ is the involution of Novikov (ϕ is the classical involution) we have the following characterization. Theorem 4.2. Let R be a commutative ring with unity and let G be a nonabelian group with nontrivial orientation homomorphism σ with kernel N . Then (RG)σ is a commutative ring if and only if one of the following conditions holds: (i) N is an abelian group and (G \ N )2 = 1; (ii) N ∼ = hx, y | x4 = 1, y 2 = x2 , y −1 xy = x−1 i × E and G∼ = hx, y, g | x4 = 1, y 2 = x2 = g 2 , xy = x−1 , xg = x, y g = yi × E, where E is an elementary abelian 2-group; (iii) char(R) = 4 and G is a Hamiltonian 2-group. 5. Symmetric units In this section we gather results concerning Lie-identities and nilpotence of the set of symmetric units, U∗ (RG), and when these properties are transferred to the whole unit group U(RG). On the relation between group and polynomial identities Hartley made the following conjecture Hartley’s conjecture (1980). If the units group U(RG) of the group algebra RG of a torsion group G over a field R satisfies a group identity, then RG satisfies a polynomial identity. An affirmative answer was proved first by Giambruno, Jespers, Sehgal, and Valenti for infinite fields R in [9, 11] and next for finite fields by Liu in [20]. Passman [25] and Liu and Passman in [21] characterized when U(RG) satisfies a group identity for R a field of characteristic 0 and G a torsion group. We can ask if the identities at the unit level are also controlled by symmetric elements. We have the analogue of Hartley’s conjecture: Theorem 5.1. [11] Let R be an infinite field of characteristic 6= 2, let G be a torsion group, and let ∗ be the classical involution. If U∗ (RG) satisfies a group identity then RG satisfies a polynomial identity. In fact, one can say more. In the following theorem Giambruno, Sehgal, and Valenti characterized when the set of symmetric units, U∗ (RG) satisfies a group identity when R is a infinite field and G is a nonabelian torsion group.

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Theorem 5.2. [11] Let R be a infinite field and G a nonabelian torsion group. If char(R) = 0, U∗ (RG) satisfies GI if and only if G is a Hamiltonian 2-group. If char(R) = p > 2, then U∗ (RG) satisfies GI if and only if RG satisfies P I and G has a p-abelian subgroup of finite index and one of the following conditions holds: (i) Q8 6⊆ G and G0 is of bounded exponent pk , (ii) Q8 ⊆ G and the p-elements of G form a subgroup P and G/P is a Hamiltonian 2-group, or (iii) Q8 ⊆ G and G is of bounded exponent 4ps . The nilpotence of U∗ (RG) was characterized in terms of the group G by Lee in [17], dependent on whether or not Q8 is contained in G. Theorem 5.3. Let R be a field of characteristic p 6= 2 and G a torsion group. (i) If Q8 6⊆ G, then U∗ (RG) is nilpotent if and only if U(RG) is nilpotent. (ii) If Q8 ⊆ G then U∗ (RG) is nilpotent if and only if either a) p > 2 and G ∼ = Q8 × E × P where E is an elementary abelian 2-group and P is a finite p-group or b) p = 0 and G ∼ = Q8 × E where E is an elementary abelian 2group. As a consequence of this result and Theorem 2.5 we have the following corollary. Corollary 5.4. Let R be a field of characteristic p 6= 2 and G a torsion group. Then U∗ (RG) is nilpotent if and only if (RG)∗ is Lie nilpotent. Now we are left with the most simple property, the commutativity. When one wants to characterize the commutativity in the set of symmetric units one needs to construct these units. This is a very difficult task. Under some additional condition on the group G and the ring R, and when ∗ is the classical involution on G, we have the following result which gives us, in a very simple manner, how to construct symmetric units. Lemma 5.5. [7] Let G be a torsion group, R a commutative ring with unity of prime characteristic p 6= 2 and ∗ the classical involution on G. Assume that U∗ (RG) is commutative. Then: (i) If g is a 2’-element of G then g + g −1 ∈ U∗ (RG). (ii) If p 6= 3 and g is a 2-element of G then 1 + g + g −1 ∈ U∗ (RG). (iii) If p 6= 5 and g is a 2-element of G then 1 + g + g −1 + g 2 + g −2 ∈ U∗ (RG). Recall [3] that a ring R is called G-favourable if for any g ∈ G, of finite |g| order |g|, there is a nonzero αg ∈ R such that 1−αg is invertible in R. Notice

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that every infinite field is G-favourable. When the ring R is G-favourable, Bovdi in [3] showed that (g−αg )(g −1 −αg ) is a symmetric unit in the context of the classical involution. This idea was generalized by Jespers and Ruiz in [14] for arbitrary group involutions and by Broche and Polcino in [8] for oriented involutions. This generalization is resumed in the following lemma. Lemma 5.6. Let G be a torsion group with involution ϕ and nontrivial orientation homomorphism σ with kernel N . Let R be a G-favourable integral |g| domain. Let g ∈ G and αg ∈ R such that (1 − αg ) ∈ U(R). Define ½ (g − αg )(g ϕ − αg ) if g ∈ N u= (g + αg )(αg − g ϕ ) if g 6∈ N Then u ∈ Uψ (RG). The construction of these units is an essential step in the proof of the necessary condition of the following theorem. Theorem 5.7. [8, 14] Let G be a torsion group, R be a G-favourable integral domain and ψ an oriented involution. Then, Uψ (RG) is commutative if and only if (RG)ψ is commutative (or, equivalently, Uψ (RG) is a subgroup of U(RG) if and only if (RG)ψ is a subring of RG). Notice that if the orientation σ is trivial then ψ is the group involution ϕ extended R-linearly to the whole group ring RG. Now we study the commutativity of the set of symmetric units when ∗ is the classical involution. For this involution Bovdi and Parmenter [5] proved the following result for integral group rings. Theorem 5.8. [5] If U∗ (ZG) is a subgroup in U(ZG), then the set t(G) of elements of G of finite order is a subgroup in G, every subgroup of t(G) is normal in G and t(G) is either abelian or a Hamiltonian 2-group. Conversely, suppose that the group G satisfies the above conditions and G/t(G) is a right ordered group. Then U∗ (ZG) is a subgroup in U(ZG). Notice that if G is not a torsion group then additional conditions are needed to obtain the converse in Theorem 5.8. This also happens in other contexts and the reason is that very little is known about the units in the torsion-free part. Indeed the following appears as Problem 10 in [27]: If G is torsion free then all units U(RG) are trivial. The benefit of working with integral group rings is that quite a number of strong results on the unit groups U(ZG) are known. In particular, if u is a nontrivial bicylic unit then Marciniak and Sehgal [22] have shown that the group hu, u∗ i is free. Clearly uu∗ , u∗ u ∈ U∗ (ZG) but they do not commute. This latter fact is an essential step in the proof of the previous theorem. That the group hu, u∗ i is a nonabelian free group for u a bicyclic unit is not always true. For instance when ∗ is the involution introduced by Novikov (4.1) Li in [19] characterized when hu, u∗ i is a nonabelian free group.

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The following theorem characterized when the set of symmetric units form a multiplicative group when ∗ is the classical involution and the characteristic of the ring R is different from 2. When the characteristic of R is 2 still remains an open problem. Theorem 5.9. [7] Let G be a nonabelian torsion group, let R be a commutative ring of characteristic 0 or p 6= 2 and ∗ the classical involution on G. Then, U∗ (RG) form a multiplicative group if and only if G is a Hamiltonian 2-group. Notice that in the case in which char(R) = 0, Theorem 5.9 follows from Theorem 5.8. As a consequence of the previous result and Theorem 2.3 we have the following corollary. Corollary 5.10. Let G be a torsion group, let R be a commutative ring with unity of prime characteristic p 6= 2 and ∗ be the classical involution on G. Then U∗ (RG) is an abelian group if and only if (RG)∗ is a commutative ring. Notice that the previous result is not true when the group is torsion free. Indeed if G is a torsion free nilpotent group and R an infinite field, then the group algebra RG has only trivial units. Since G has no elements of order 2, the symmetric units commute (1 is the only symmetric unit), but the symmetric elements (RG)∗ do not necessarily commute. Acknowledgments Dr. Broche Cristo has been partially supported by FAPESP and CAPES of Brazil and Dr. Ruiz Mar´ın by Fundaci´ on S´eneca of Murcia. References [1] S.A. Amitsur, Rings with involution, Israel J. Math. 6 (1998), 99–106. [2] A.A. Bovdi, Unitarity of the multiplicative group of an integral group ring, Mat. USSR Sbornik, 47, 2, (1984), 377–389. [3] V. Bovdi, On symmetric units in group algebras, Comm. Algebra 29, 12 (2001), 5411–5422. [4] V. Bovdi, L.G. Kov´ acs, and S.K. Sehgal, Symmetric units in modular group algebras, Comm. Algebra, 24, 3 (1996), 803–808. [5] V. Bovdi and M.M. Parmenter, Symmetric units in integral group rings, Publ. Math. Debrecen, 50, 3-4 (1997), 369–372. [6] A.A. Bovdi and S.K. Sehgal, Unitary subgroup of integral group rings, Manuscripta Math., 76, (1992), 213–222. [7] O. Broche, Commutativity of symmetric elements in group rings, preprint, (2003). [8] O. Broche and C. Polcino, Symmetric elements under oriented involutions in group rings, preprint, (2004). [9] A. Giambruno, E. Jespers, and A. Valenti, Group identities on units of rings, Arch. Math. (Basel), 63(4) (1994), 291–296.

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[10] A. Giambruno and S.K. Sehgal, Lie nilpotence of group rings, Comm. Algebra 21 (1993), 4253–4261. [11] A. Giambruno, S.K. Sehgal, and A. Valenti, Symmetric units and group identities, Manuscripta Math., 96 (1998), 443–461. [12] E.G. Goodaire, E. Jespers, and C. Polcino Milies, Alternative Loop Rings, Elsevier, 1996. [13] I.M. Isaacs and D.S. Passman, Groups with representations of bounded degree, Canad. J. Math. 16 (1964), 299–309. [14] E. Jespers and M. Ruiz, On symmetric elements and symmetric units in group rings, preprint, (2004). [15] G.T. Lee, Group rings whose symmetric elements are Lie nilpotent, Proc. Amer. Math. Soc. 127 (11) (1999), 3153–3159. [16] G.T. Lee, The Lie n-Engel property in group rings , Comm. Alg., 28 (2000), 867–881. [17] G.T. Lee, Nilpotent symmetric units in group rings, Comm. Algebra, 31, 2 (2003), 581–608. [18] Y. Li, Units of Z(G × C2 ), Quaestiones Mathematicae, 21, (1998), 201–218. [19] Y. Li, Some results on bicyclic units in an integral group ring, Comm. in Algebra, 29, 3, 2001, 1339–1345. [20] C-H. Liu, Group algebras with units satisfying a group identity, Proc. Amer. Math. Soc. 127 (2) (1999), 337–341. [21] C-H. Liu and D. S. Passman Group algebras with units satisfying a group identity II, Proc. Amer. Math. Soc. 127 (2) (1999), 327–336. [22] Z.S. Marciniak and S.K. Sehgal, Constructing free subgroups of integral group ring units, Proc. Amer. Math. Soc. 125 (4) (1997), 1005–1009. [23] S.P. Novikov, Algebraic construction and properties of hermitian analogues of Ktheory over rings with involution from the viewpoint of Hamiltonian formalism. Applications to differential topology and the theory of characteristitic classes II, Izv. Akad. Nauk SSSR Ser. Mat., 34, 3, (1970), 475–500; English transl. in Math. USSR Izv., 4, 3, (1970). [24] I.B.S. Passi, D.S. Passman, and S.K. Sehgal, Lie solvable group rings, Cand. J. Math., 25 (1973), 748–757. [25] D.S. Passman, Group algebras whose units satisfy a group identity II, Proc. Amer. Math. Soc. 125 (2) (1997), 657–662. [26] D.S. Passman, Group rings satisfying a polynomial identity, J. Algebra 20 (1972) 221–225. [27] S.K. Sehgal, Topics in Group Rings, Marcel Dekker, New York, 1978.

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Irreducible morphisms in subcategories Gladys Chalom and H´ ector Merklen Instituto de Matem´atica e Estat´ıstica, Universidade de S˜ao Paulo, Caixa Postal 66281, S˜ao Paulo, SP, 05315-970, Brasil [email protected] [email protected]

1. Introduction The main objective of this note is to characterize the irreducible morphisms of a class C. As it is well known, these objects played a crucial role for the development of the modern theory of representations of algebras which was initially motivated by the monumental works of M. Auslander and I. Reiten. They proved the existence of almost split sequences (which now we prefer to call Auslander-Reiten sequences) for C = mod−Λ, thus solving completely, for this class, the problems of existence of almost split and irreducible morphisms arriving to or starting at an indecomposable. Several years afterwards, M. Auslander and S. Smalø solved the main problem for general classes C (see [2]). For the representation theory of Artin algebras it was clearly enough to study the categories of indecomposable modules, and so, to characterize irreducibles ending up or starting from an indecomposable module. On the other hand, reckoning with the fact that, in the last decades, the study of derived categories has become an important tool in several areas and, especially, in representation theory, the notion of irreducible morphisms for a class is certainly of major importance.

Next, we state the basic facts we are going to use. In section 2, it is shown that there is an intimate relation between almost split maps and irreducibles, and we characterize minimal, right almost split maps in terms of their indecomposable components (see theorem 1). In section 3, we characterize C−right approximations and show that their existence depends essentially on the finiteness of the so-called maximal approximating families (see theorem 3). Hence, in sections 2 and 3, the main problem is essentially solved in the case of existence of almost split maps which is guaranteed by the existence of 57

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approximations, so that it can be completely solved, in practice, for classes of finite type. In the last section some examples are given. 1.1. Preliminaries and notations. We will use the following standard notions and notations. Notations 1. – R (resp. k) is a commutative Artin ring (resp. a field); – Λ is an R−Artin algebra; – unless otherwise specified, module means finitely generated, left Λ−module; – unless otherwise specified, morphism means Λ−homomorphism; – Mod−Λ is the category of all Λ−modules; – mod−Λ is the category of all finitely generated Λ−modules; – a Krull-Schmidt category A, is an additive category where any object can be uniquely (up to isomorphism) decomposed in a direct sum of indecomposable objects; – a class, C, in a Krull-Schmidt category A, is a full subcategory which is closed under direct sums and direct summands (it is a KrullSchmidt subcategory of A); – indA is the category of indecomposable objects of A; – if C is a class in A, indC = indA ∩ C is the full subcategory of indecomposables of C; – if S is a set of indecomposables in a Krull-Schmidt category A the class C, generated by it (i.e., the minimal class containing it) is the class add(S) of all objects whose indecomposable components lie in S, i.e., the direct sums of objects belonging to S. (Sometimes we will find it convenient to consider the class addS whose objects are all direct sums (may be of infinite families) of indecomposables in S). Then C is determined by S (and conversely), which is its set of indecomposables. This is a very convenient way of characterizing classes: by giving the set of their indecomposables; – if C is any class and Γ a set of indecomposable objects, the notation CΓ will be used for the class generated by the indecomposables of C which belong to Γ; – if C is a class in a Krull-Schmidt category A, an ideal of C, I, consists of a family of subgroups I(x, y) ⊂ A(x, y) (x, y ∈ C ), such that A(x, y) · I(y, z) ⊂ I(x, z), I(x, y) · A(y, z) ⊂ I(x, z), ∀x, y, z ∈ A; – an ideal of A can also be looked at as a subcategory, without identities, satisfying conditions above whose objects are all the objects of A;

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– in particular, rad(C), is the ideal of C generated by the nonisomorphisms between indecomposable objects (see notations 2); – P is the class of projective modules; – 1N denotes the identity map of N ; – a morphism f : M → N is split monic (resp. split epic) if and only if there exists a morphism f˜ : N → M such that f˜f = 1M (resp. f f˜ = 1N ); – a nonzero, nonsplit C-morphism f : M → N is C−irreducible if for any C−factorization f = hg, either g is split monic or h is split epic (it is easy to see that, if f is irreducible and M decomposes: M = M 0 ⊕ M 00 , then the restrictions of f to M 0 , M 00 are nonzero; if it is irreducible and N decomposes, then, normally, one assumes that the corresponding components of f are nonzero); – any homomorphism f : ⊕i Mi → ⊕j Nj , especially when all modules are indecomposable, will be given by the matrix (fji )ij ,fij : Mi → Nj . Remark 1. Usually in what follows, if there is no mention to the class C or to the category A we assume it is just mod−Λ. We begin our study by remembering some well-known facts which we assemble in the following two propositions. Proposition 1. (1) For any indecomposable module, C, that is not simple projective (resp. simple injective) there is a minimal right (resp. left) almost split map gC : E → C (resp. fC : C → E) (which is, clearly, unique up to isomorphism). (2) For all nonprojective indecomposable C, this map gC is epic (resp. for all noninjective indecomposable C, fC is monic) and it defines in an obvious way the Auslander-Reiten sequence ending (resp. beginning) at C. (3) If C is an indecomposable, projective, nonsimple, gC is the inclusion of rad(C) into C. (By abuse of language, we say that, if C is simple projective, gC = 0 is the minimal, right, almost split map ending up at C.). (4) If C is an indecomposable injective, nonsimple, fC is the canonical epimorphism of C onto C/socC. (By abuse of language, we say that, if C is simple injective, gC = 0 is the minimal, left, almost split map starting up at C). (5) If C is not simple projective (resp. injective) the irreducible morphisms arriving at (resp. starting from) C are the components of gC (resp. fC ). Proposition 2. Let f : M → N be a homomorphism with one of the modules M, N indecomposable. Then,

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(1) f is split if and only if it does not belong to the radical. (2) f is irreducible if and only if f ∈ rad(mod − Λ) \ rad2 (mod − Λ) (3) If f is irreducible and g : M → N is congruent to f modulo rad2 (mod − Λ), then g is irreducible too. Remark 2. Items 1 and 3 of proposition 2, clearly are true for any class C, for it is easy to see that a morphism in the ideal rad2 (the square of the radical) can be written as the product of morphisms in the radical, so that it is not irreducible. On the other hand, item 2 is not true in general if both objects are decomposable (see proposition 3). However, item 3 allows for a convenient way of representing the irreducible morphisms. As a matter of fact, given two objects M, N , at least one of rad(M,N ) them indecomposable, the space Irr(M, N ) := rad represents all the 2 (M,N ) irreducible morphisms from M to N . It is generally called the space of the irreducible morphisms. This “space” is clearly a bimodule over the rings of endomorphisms of N and M and (up to isomorphisms) the irreducible morphisms correspond to the orbits over the automorphism groups. In particular, if Λ is a finite dimensional k-algebra (for k an algebraically closed field) the tops of the local rings of endomorphisms of indecomposables are isomorphic to k itself, so that, modulo the radical, Hom(M, N ) and rad(M, N ) are finite dimensional k-vector spaces. In this case, the irreducible maps are conveniently represented by bases of Irr(M, N ), all of them being isomorphic among themselves. Of course, in Irr(M, N ) it is always possible to choose (representative) maps whose components belonging to the square of the radical are equal to zero. This is what we will do in general. Proposition 3. Let C be a class, let M, N ∈ C and let f : M → N . Then, (1) If f is irreducible (in C), there is a decomposition f

M1 ⊕ M 0 → M1 ⊕ N 0 µ ¶ 1 0 f∼ = 0 f0 where f 0 : M 0 → N 0 is an irreducible morphism which is in radC. (Here, it is possible that M1 = 0.) (2) Conversely, if there is a decomposition µ ¶ 1 0 f= 0 f0 with f 0 : M 0 → N 0 an irreducible morphism, which is in rad (modΛ) then, f is irreducible. Proof.

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∼ M1 ⊕M 0 and N ∼ (1) Consider a decomposition M = = N1 ⊕N 0 such that M1 ∼ = N1 and no indecomposable summand of M 0 is isomorphic to any indecomposable summand of N 0 . The proof of the direct statement can be done by induction (on the f1

number of indecomposables summands of M1 ). Let P1 → P2 be an arbitrary indecomposable component of f . If all these components are nonisomorphisms, the statement is true (with M1 = 0). If this is not the case, the induction step is performed as follows. Choose f1 to be an isomorphism. Using convenient automorphisms, we can put P1 = P2 = M1 andf1 = 1M1 = 1. Then the decomposition for f is of the form µ

1 ∗ ∗ f2

¶

and, using again suitable automorphisms, we can take it to the form µ

1 0 0 f0

¶ .

The proof will be complete if we show that f 0 is irreducible. Let us begin with a factorization f 0 = h0 g 0 ,with h0 non split-epic, and let us consider, then, the following. µ

¶ µ ¶µ ¶ µ ¶ 1 0 1 0 1 0 1 0 = = . 0 f0 0 h0 0 g0 0 h0 g 0 µ ¶ µ ¶ 1 0 1 0 Since cannot be split-epic, must be split-monic, 0 h0 0 g0 so that g 0 is split-monic. µ ¶ 1 0 (2) Let us be given now f = : M = M1 ⊕ M 0 → N = 0 f0 M1 ⊕ N 0 , a homomorphism in C, with f 0 irreducible in C, and let us g h consider any factorization in C: f = hg, M1 ⊕ M 0 → H → M1 ⊕ N 0 . Hence, writing H = M1 ⊕ H1 (where H1 = ker h), g = (g1 , g 0 ) and h = (h1 , h0 ), we have h1 g1 = 1,

h1 g 0 = 0,

h0 g1 = 0,

h0 g 0 = f 0 ,

and, therefore, we obtain µ h=

1 ∗ 0 h01

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¶

µ and

g=

1 0 ∗ g10

¶ .

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Since either h01 is split epic or g10 split monic (because h01 g10 = f 0 ), it is easy to conclude that either h is split epic or g is split monic. This completes the proof. ¤ Proposition 4. (1) Let f : ⊕m i=1 Ci → E, f = (f1 , · · · , fm ) be a homomorphism in rad(C). Then, f is C−irreducible if and only if all components fi are. t (2) Let f : E → ⊕m i=1 Ci , f = (f1 , · · · , fm ) be a homomorphism in rad(C). • If f is irreducible in C, then each fi is irreducible in C. • If each fi is irreducible in C and if the family (Ci )1=1,··· ,m is additively disjoint (i.e., the Ci ’s do not have indecomposable summands in common), then f is irreducible in C. Proof. Since item 1 will become straightforward, we give details only for the proof of item 2. It is enough to work in the simple case when f has the form Ã

!

c d E −→ C ⊕ D. Considering products of the forms µ ¶ c 0 0 (c , d ) = c0 c + d0 d d µ ¶ µ 0 ¶ c cc cd0 (c0 , d0 ) = d dc0 dd0 and observing that c, d belong to the radical, we deduce that our map cannot be split. µ ¶ c (1) To show, for instance, that if is irreducible, then c is irred ducible too, we consider a factorization c = hC gC . It leads to µ ¶µ ¶ µ ¶ hC 0 gC c = , 0 1 d d from where, since f is irreducible, we get that either gC is split monic or hC is split epic. (2) For the reverse statement, let us assume that c and d are irreducible (in the radical), thatµ C and ¶ D are additively disjoint and µthat we ¶ c hC have a factorization = hg, with h : H → C ⊕ D, h = d hD and with g not split monic. Since we have hC g = c and hD g = d, hC and hD are split epic. Hence, C and D (up to isomorphism) are

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direct summands of H and, being additively disjoint, C ⊕ D is a direct summand of H. From this, it follows that h is split epic. ¤ We can write the following: n Corollary 1. Let f : ⊕m i=1 Ci → ⊕j=1 Dj , f = (fi,j ) be a homomorphism in rad(C) where the family (Dj )j=1,··· ,n is additively disjoint. Then f is irreducible in C if and only if each fi,j is irreducible in C.

Example 1. If the Ci ’s are not additively disjoint, there are morphisms which are not irreducible with irreducible components. To show this, it suffices to consider the factorization Ã ! 1 1 f E −→ C −→ C ⊕ C withf irreducible. We recall the definition of minimal morphisms. Definition 1. A non-zero morphism x : X → C (resp. x : C → X) in a class of morphisms ending up (resp. beginning at) C is called right minimal (resp. left minimal) if xϕ = x (resp. ϕx = x) implies that ϕ is an automorphism of X. In our context, instead of working with the usual definition of almost split maps, we introduce the following: Definition 2. A non-zero morphism x : X → C (resp. x : C → X) in the radical of a class C is called C−right almost split (resp. C−left almost split) if every morphism y : Y → C (resp. y : C → Y ), also in the radical of C, factors through x. Then, we get the following consequence. Proposition 5. Let C be a class of modules and let us consider a module in C, C = ⊕Ci (a decomposition into indecomposable summands). We have: (gi )

⊕gi

If Ei → Ci (∀i) is minimal right almost split in C, then E := ⊕Ei → ⊕Ci := C is also minimal right almost split in C (and conversely).

Hence, the existence of right almost split maps in C behaves nicely with respect to direct sums or direct-sum decompositions. Proof. The proof is straightforward for the direct statement. The converse results from the unicity of the minimal almost split map up to isomorphism: g If E → C is minimal right almost split, there is an isomorphism ϕ : E → ⊕Ei such that g = ⊕gi · ϕ. ¤ Remark 3. It is easy to see that minimal almost split maps are irreducible.

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2. Existence of right almost split maps In this section, unless otherwise specified, all modules and morphisms are assumed to belong to a fixed class C of mod−Λ and all concepts like irreducible, almost split, etc. are relative to this class. We think that the following proposition is well known. Basically, it asserts that, for any given object, the existence of certain right approximation from C guarantees the existence of a right almost split map for C. We recall the definitions of approximations. Definition 3. Given a class, C, and a module, M , an approximation of, or left approximation of or covariant approximation of M to C (if it exists) is any morphism x : M → X (with X ∈ C) which induces an epimorphism of the covariant functor Hom(X, ·)|C onto Hom(M, ·)|C . (In other words, a morphism x such that, any morphism f : M → C ∈ C factors through x: there exists some f˜, such that f = f˜x (clearly, if M is a direct sum and each summand has an approximation to a class then the direct sum of these gives a corresponding approximation of M )); x

M NN NNN NNN NNN f NN&

/X ~

C

~~

~ ~ ˜ ∃f

Definition 4. Given a class, C, and a module, M , an approximation, or right approximation or contravariant approximation from C to M (if it exists) is a morphism x : X → M (with X ∈ C) which induces an epimorphism of the contravariant functor Hom(·, X)|C onto Hom(·, M )|C . (In other words, a morphism x such that, any morphism f : C ∈ C → M factors through x: there exists some f˜, such that f = xf˜ (clearly, if M is a direct sum and each summand has an approximation from a class then the direct sum of these gives a corresponding approximation to M )). g

Proposition 6. Let C be in ind C, a class in mod−Λ, and let E → C be a minimal right almost split map in mod−Λ. x If there is a right approximation X → E from C to E, then gx is a right almost split map in C. Obviously, for a convenient choice of the indecomposable components of X, calling x0 the restriction of x to the components of X for which gx is different from 0, g0 = gx0 is minimal right almost split. On the other hand, if there is a right almost split map in C, g0 : X0 → C, and if there is an approximation from C to ker g := D = DTr(C), then there is an approximation from C to E. y

Proof. Given a nonzero map in the radical of C: Y → C, we have a y 0 such that y = gy 0 and, by the definition of x, we have also an y1 such that

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y 0 = xy1 , whence y = gxy1 . On the other hand, if g0 : X → C is C−right almost split, then there is a map x : X → E such that g0 = gx and it follows that (x, f x0 ) : X ⊕ X 0 → E (where x0 : X 0 → D is an approximation from C and where f : D → E is the map kernel of g) is a right approximation from C to E. In fact, for any map y : Y → E (with Y ∈ C), since there exists y0 : Y → X such that gy = g0 y0 , we deduce that g(y − xy0 ) = 0. This means that y − xy0 is of the form f x0 y1 . ¤ Corollary 2. Let C ∈ ind C • If there is a C−minimal right almost split map g : D → C, a map y : Y → C is irreducible if and only if y = gy 0 with y 0 split mono. x • If there is a minimal right approximation X0 →0 E as in the proposiy tion 6, then, a map in C:Y → C is irreducible if and only if y = g0 y˜ with y˜ split-monic. Proof. If y = g0 y˜, y being irreducible, y˜ is split-monic. Conversely, if this is so, the result follows from proposition 1. ¤ Notations 2. In what follows we use the notation RA to denote the radical of A. In case A = mod − Λ we simply represent this radical by R. The traditional, Auslander’s, notation rΛ or, simply, r is reserved for the Jacobson radical of the ring Λ. Definition 5. Let A be an additive, Krull-Schmidt category. A nonempty family of maps is said to be independent if none of them belongs to the ideal generated by the others. The following theorem gives, in a more precise way, the relation between minimal right almost split maps and irreducible morphisms. Theorem 1. Let C be a class in mod−Λ and let C ∈ ind(C) . f

(1) If there exists a C−right almost split map M → C, then the number (up to isomorphism) of independent C−irreducibles X → C is finite, and all nonsplit C−maps y : D → C factor through an irreducible: y = f y 0 (f a C−irreducible). (2) If any nonsplit C−map y : D → C factors through a C−irreducible and if the number (up to isomorphism) of independent C−irreducibles X → C is finite, then any direct sum of a complete set of representatives of these defines a C−minimal, right almost split map. The second item is also true in general (that is, for any Krull-Schmidt category A). Proof. Let us assume, first, that there are (up to isomorphism) an infinite fi

number of independent irreducible maps Xi → C. Since we must have factorizations fi = f fi0 , by corollary 2 each fi is a split mono, as they are

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independent, we have that the Xi are nonisomorphic, and we get a contra(f1 ,··· ,ft )

diction with the fact that C is finitely generated. Next, let ⊕i Xi −→ C where (fi )i=1,··· ,t represents (up to isomorphism) all the independent irreducibles. By hypothesis, any C−map in R factors through f . ¤ Definition 6. As usual, for an Artin algebra Λ, ΓΛ denotes its AR-quiver, that is the quiver whose vertices correspond to a choice of indecomposable modules and where the arrows correspond to certain irreducible maps. If there is no possibility of misunderstanding, we denote generically by Γ any connected component of this quiver. We refer to them as components of ΓΛ . A path, σ, in the AR-quiver is a finite sequence of arrows (the trivial path, i.e., the empty sequence, to the indecomposable M is the identity 1M ). By abuse of language, if α is an arrow, we will denote also α any irreducible homomorphism associated to it. If σ is a path, we will also use the notation σ for the composite of the irreducible morphisms associated to its arrows. Sometimes, it is referred to as a path morphism. Consider a map f : M → E, and a component Γ of ΓΛ , we will denote by fΓ the restriction of f to the direct sum of the Mi ’s belonging to Γ and call it the Γ−component of f . In the case of right almost split maps with all components finite (that is, that they do not belong to the infinite radical) we can be more specific. We see, in this case, that we have a generalization of what happens when C = mod − Λ. Theorem 2. Let C be a class in modΛ. (1) If there is a C−right almost split map, M → C, with finite components (i.e., a right almost split map “totally” in the component of C), then, up to isomorphism, there is only a finite number of irreducibles, which can be chosen to be the path-maps corresponding to some paths of M to C. The converse is also true. More precisely: (2) Let us assume that there is a finite number of (nontrivial) path-maps: σi : Xi → C (i = 1, · · · , r) such that all nonzero maps to C belong (σi )i

to the ideal generated in RC by the σi ’s, then ⊕i Xi → C is C−right almost split. Proof. The hypotheses imply that all irreducibles are linear combinations of path-maps. Since the summands with coefficients in the radical of R belong to RC , we can assume that all path-maps have as coefficients units of R (or of k). The first result follows. The second is immediate. ¤ Remark 4. It follows from the above facts that, when right almost split maps exist, the study and finding of irreducible maps is highly reduced to the case when the codomain-module, C, is indecomposable. Unfortunately,

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as far as we know at present, the only general case for which this is really so happens when C is of finite type (see proposition 7). Therefore, a main issue here, which deserves a separate work, is to obtain a general characterization of irreducibles (sitting in the radical) for an arbitrary class C of modules. Taking into account the examples in the last section, this general situation seems to be fairly complex. 3. The existence of right approximations The following result is fairly old, probably it goes back to works of Auslander. Fortunately, it solves our problems for the class P, which is of finite type. Proposition 7. Let Λ be an Artin algebra and C a class of modules. If C is of finite type, every module, E, has a right approximation from C. Let us recall that a class C is of finite type if indC is a finite set. Proof. It is clearly enough to prove the statement for the case when there is a unique indecomposable, C, in C. If fi : C → E gives an R− (resp. (f1 ,··· )

k−) generating set for HomΛ (C, E), the map ⊕i −→ E defines obviously a right approximation. ¤ Remark 5. As a consequence, if C is of finite type, every module C ∈ C is the codomain of a C minimal right almost split map, whose indecomposable components give all the C irreducibles with target C which are in the radical of C. Definition 7. C−approximating families. Let C be a class and E an indecomposable object. A family of maps (fi )i∈I , fi : Ci → E, is a C−approximating family to E if all Ci ’s belong to C and if it is maximal independent. For example, if E ∈ C, 1E is a (unitary) maximal approximating family. Theorem 3. Given a class C and an indecomposable E, in order that there is a C−right approximation to E it is necessary and sufficient that there exists a finite, maximal approximating family from C to E. x

Proof. Let X = ⊕ri=1 Xi → E be a right approximation from C to E, with x = (xi )i=1,··· ,r . Then, of course (eventually getting rid of some superfluous maps), we have that (xi )i is an approximating family. And it is clear that it is finite. Conversely, it is also immediate that such a family defines, in the obvious way, a right approximation. ¤ In the case of right approximation maps with all components finite (that is, that they do not belong to the infinite radical) we can be more specific (cf. theorem 2).

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Theorem 4. Let C be a class in mod−Λ and E an indecomposable module. x If there exists X → E, a right approximation from C to E with all its components finite, then there is also such an approximation given by paths-maps. More precisely: if X = ⊕ti=1,··· ,t Xi is a decomposition into indecomposables, then the family of all path-maps from the Xi ’s to E defines a right approximation from C to E. Proof. If x = (x1 , · · · , xt ), by assumption, each xi is a linear combination of path-morphisms. ¤ 3.1. Approximation for exact sequences and ARS sequences. Definition 8. Let C be a class in an additive Krull-Schmidt category. (1) We say that the sequence f

0 → X → Y

g

→ Z → 0

(∗)

is an exact sequence in C if it is an exact sequence, where any object belongs to C. (2) Given an exact sequence: f

0 → X → Y

g

→ Z → 0

(∗)

we say it has a (right) approximation from C if there exists an exact sequence in C and a homomorphism: 0 → 0 →

fr

gr

f

g

Xr → Yr → Zr → 0 ↓ xr ↓ yr ↓ zr X

→

Y

→

Z

(∗)h

→ 0

such that each down arrow is a (right) approximation from C. (3) Given an exact sequence: f

0 → X → Y

g

→ Z → 0

(∗∗)

we say it has a (left) approximation to C if there exists an exact sequence in C and a homomorphism: 0 → 0 →

f

X → Y → Z → 0 ↓ xl ↓ yl ↓ zl Xl

fl

→

Yl

gl

→

Zl

(∗∗)h

→ 0

such that each down arrow is a (left) approximation to C. It is interesting (and easy) to show that certain classes of modules always have approximations for “borderly” exact sequences. Proposition 8. Let C be a class in mod−Λ.

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(1) If C is closed for kernels and if Y has right approximations from C, any exact sequence f

0 → X → Y

g

→ Z → 0

(∗)

with Z ∈ C, has right approximations from C. (2) If C is closed for cokernels and if Y has left approximations to C, any exact sequence f

0 → X → Y

g

→ Z → 0

(∗∗)

with X ∈ C, has left approximations to C. Proof. We only prove the first statement because the proof of the second is analogous. In order to construct the C−exact sequence and the desired yr homomorphism, we begin (on the right) with Zr = Z, Yr → Y a right fr

approximation to Y and gr = gyr . Let us denote Xr → Yr the kernel of gr which, by hypothesis, is in C and let xr be the passage of yr to the kernels. We show that the sequence just constructed is a right approximation from C. Given D ∈ C and d : D → X, f d factors as f d = yr d0 . Since it is anihilated by g: gf d = (gyr )d0 = gr d0 = 0, d0 factors through fr : d0 = fr d” and we deduce that f d = yr d0 = yr fr d” = f xr d”. Since f is monic, this implies that d = xr d”, as we wanted to show. Now, the following diagram shows the desired approximation from C to (∗). 0 → 0 →

fr

gr

f

g

Xr → Y r → Z r → 0 ↓ xr ↓ yr ↓1 X

→

Y

→

Z

.

→ 0 ¤

Proposition 9. Let us assume that C is a class in mod−Λ which is functorially finite and closed under extensions. (1) Given an ARS sequence which ends at an indecomposable of C, its approximation from C is the corresponding C−ARS sequence, slightly modified. (2) Given an ARS sequence which begins at an indecomposable of C, its approximation to C is the corresponding C−ARS sequence slightly modified. Proof. This is essentially a corollary of proposition 8. Let us see the item 1. In the diagram obtained in that proposition: 0 → 0 →

fr

gr

f

g

Xr → Y r → Z r → 0 ↓ xr ↓ yr ↓1 X

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→

Y

→

Z

→ 0

,

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if the lower sequence is an ARS, we have that gr is a C−right almost split map, not necessarily minimal. Hence, by proposition 6, the C−ARS sequence “fits” in the middle of the diagram, as follows. 0 → 0 → 0 →

Xr ↓

fr

→

Yr ↓

f0

gr

→

Zr ↓1

→

0

g0

,

r r Xr0 → Yr0 → Z →0 ↓ x0r ↓ yr0 ↓1

X

f

→

Y

g

→

Z

→

0

where there is a direct summand of Xr0 , X”r , such that Yr = Yr0 ⊕ X”r and Xr = Xr0 ⊕ X”r . ¤ 4. Examples (1) If Λ is a finite representation type algebra, any class C has right approximations to any object (see remark 5). Therefore, for any indecomposable object C there is a minimal right almost split homomorphism ending at it and all relatively irreducible morphisms D → C are finite in number (up to isomorphisms). (2) Also, if C is a class of finite type the same result applies to any object C ∈ indC. In the particular case of P, the projectives, the minimal right almost split maps (for mod−Λ) are the inclusions of the radicals, and the minimal right approximations are the minimal projective covers of them. Hence, if P, Q are projective modules, Q indecomposable, the irreducible maps P → Q in P are of the form P → E → Q, E = rad(Q), the left map being the minimal projective cover and the right map being the inclusion. Hence, in most cases, an irreducible of P is not irreducible in mod−Λ. → (3) Let Λ be the Kronecker algebra k(· ·) and let C be generated by → one indecomposable, MT , in each tube T and the simple injective, I. Then, there is an infinite number of homomorphisms with target I and all of them are C−irreducible. On the other hand, there are no C−irreducibles with target MT , for any T . → (4) Let Λ be the Kronecker algebra k(· ·), with I the simple injec→ tive, and P its projective cover. Choose ten (or any other number) indecomposable nonisomorphic, simple regular modules Mi , in the stable tubular family. Let C be generated by the finite set given by the projective P , the injective I, and the chosen regular modules Mi . Then, for all i the morphisms P → Mi and Mi → I are irreducible in C and by theorem 1 the map ⊕Mi → I is the right almost split

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morphism that arrives in I, and similarly P → ⊕Mi is the left almost split morphism starting in P . But observe that there is not a relative Auslander-Reiten sequence in C. On the other hand, there are short exact sequences P → Mi ⊕Mj → I with all the components C−irreducible. (5) Let Λ be an Artin algebra which has as a component a regular tube T . Let S be a regular simple in T and I the injective envelope of S. Let C be generated by T and I. Then, there is an infinite number of maps of C with target I, but there is not one C−irreducible homomorphism with target I. Acknowledgments The authors express their sincere gratitude to Prof. Eduardo Marcos, whose careful reading and helpful remarks, had greatly improved this work. References [1] M. Auslander and S. Smalø, Preprojective modules in subcategories, J. Algebra 66, (1980) 61–122. [2] M. Auslander and S. Smalø, Almost split sequences in subcategories, J. Algebra 69 (1981), 426–454. [3] M. Auslander, I. Reiten, and S. Smalø, Representation Theory of Artin Algebras, Cambridge Univ. Press, 1995. [4] V. Bekkert and H. Merklen, Indecomposables in derived categories of gentle algebras, Alg. and Rep. Th. 6 (2003), 285–302. [5] V. Bekkert, E. Marcos, and H. Merklen, Indecomposables in derived categories of skewed-gentle algebras, Comm. Alg. 31 6 (2003), 2615–2654.

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Bol loops with a unique nonidentity commutator/associator Orin Chein Temple University, Philadelphia, PA 19122, USA [email protected] Edgar G. Goodaire Memorial University of Newfoundland, St. John’s, Newfoundland, A1C 5S7, Canada [email protected] To our friend C´esar in celebration of his 60 th birthday. Abstract. We study Bol loops with a unique nonidentity commutator/associator and use our findings to give a new construction of Bol loops. We also show that a Bol loop with a unique nonidentity commutator/associator that is minimally non-Moufang is, in fact, minimally nonassociative.

1. Introduction A loop is a pair (L, ·) where L is a set and (a, b) 7→ a · b is a binary operation on L relative to which there is a two-sided identity element and with the property that for each a ∈ L, the left and right translation maps R(a) and L(a), defined by xR(a) = xa,

xL(a) = ax,

are bijections. For a, b, c ∈ L, the commutator (a, b) and associator (a, b, c) are, respectively, the unique elements satisfying ab = ba(a, b),

ab · c = [a · bc](a, b, c).

2000 Mathematics Subject Classification. Primary 20N05. Key words and phrases. Bol loop, Moufang loop, nilpotent, commutator, associator. 73

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The left, middle, and right nuclei of L are defined, respectively, by Nλ (L) = {a ∈ L | (a, x, y) = 1 for all x, y ∈ L}, Nµ (L) = {a ∈ L | (x, a, y) = 1 for all x, y ∈ L}, Nρ (L) = {a ∈ L | (x, y, a) = 1 for all x, y ∈ L}. The nucleus of L is N (L) = Nλ (L) ∩ Nµ (L) ∩ Nρ (L). The centrum of L is C(L) = {a ∈ L | (a, x) = 1 for all x ∈ L} and the centre of L is Z(L) = N (L) ∩ C(L). A unique nonidentity commutator/associator in a loop L is an element s 6= 1 that is the only commutator and the only associator in L (other than 1). It is easy to see that such an element must lie in the centre of L. For example, if as 6= sa for some a ∈ L, then as = sa · s and cancelation gives a = sa, so s = 1, a contradiction. Thus s ∈ C(L). Similarly, one can show that s is in the nucleus and hence in the centre. Moreover, choosing a and b such that ab 6= ba, we have ab = (ba)s and ba = (ab)s, so s2 = 1. It follows (and we use this implicitly and frequently in the sequel) that if a loop L has a unique nonidentity commutator/associator, then, for any a, b, c ∈ L, ba = ab(a, b) and a · bc = [ab · c](a, b, c), the square brackets here actually unnecessary by centrality of the associator. A loop is Moufang if it satisfies the identity (xy ·z)y = x(y ·zy) and (right) Bol if it satisfies (xy·z)y = x(yz ·y).1 The (seemingly small) difference in the order of multiplication on the right side of the Moufang and Bol identities makes a world of difference to the loops. For example, Moufang loops are diassociative, meaning that the subloop generated by any pair of elements is a group. In fact, a basic result in the theory of Moufang loops states that if three elements of a Moufang loop associate in some order, then they generate a group (and hence associate in all orders). In particular, y · zy = yz · y for any y, z in a Moufang loop, so a Moufang loop is right Bol. Given a commutative, associative ring R with 1, one forms the loop ring RL in precisely the way the group ring is defined. It is rare for a loop ring to satisfy any interesting identity other than associativity [Pai55], so it came as a surprise when, in the early 1980s, a class of alternative, but not associative, loop rings was discovered [Goo83]. Such loops satisfy the right and left alternative laws (yx)x = yx2 ,

x(xy) = x2 y

but are not associative. An early observation about the underlying (necessarily Moufang) loops was that they had a unique nonidentity commutator/associator. More than ten years later, it was shown that Bol loops with 1A loop is left Bol if it satisfies the reflection of the right Bol identity, namely, y(z·yx) =

(y · zy)x. Throughout this paper, whether or not we say so explicitly, we assume that Bol loops are right Bol.

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a unique nonidentity commutator/associator have loop rings that satisfy the right, but not the left, alternative law [GR96, GR95]. Again, there was good reason to believe such loops could not exist [Kun98]. The comments just made suggest an investigation of Bol loops with a unique nonidentity commutator/associator. We do that here and use our findings in two different ways. We show that if a Bol loop with a unique nonidentity commutator/associator is minimally non-Moufang, then it is minimally nonassociative. Then we present a new way to construct Bol loops. 2. Some identities Bol loops satisfy some weak associativity conditions, but not others. For example, any (right) Bol loop satisfies the right alternative law (set z = 1 in the right Bol identity) and this, together with a straightforward induction argument can be used to show that Bol loops are power associative, meaning that powers of a single element are well defined. In fact, a Bol loop L is right power alternative: (xy n )y m = xy n+m for any integers n and m and any x, y ∈ L. On the other hand, a Bol loop which is not Moufang cannot even satisfy the left alternative law. We refer the reader to the text by Pflugfelder [Pfl90] where she will find more than enough background on Bol (and Moufang) loops for this paper. For the rest of this section, B denotes a (right) Bol loop with a unique nonidentity commutator/associator which we always label s. Lemma 2.1. For any x, y, z ∈ B, the associators (x, y, z) and (x, z, y) are equal. Proof. It is known that for any x, y, z, w ∈ B, one of the following three conditions holds [GR96]: D(x, y, z, w) :

(xy · z)w = x(yz · w) and (xw · z)y = x(wz · y)

E(x, y, z, w) :

(xy · z)w = x(wz · y) and (xw · z)y = x(yz · w)

F (x, y, z, w) :

(xy · z)w = (xw · z)y and x(yz · w) = x(wz · y).

Setting w = 1, we observe that for any x, y, z ∈ B, D(x, y, z, 1) :

xy · z = x · yz and xz · y = x · zy, or

E(x, y, z, 1) :

xy · z = x · zy and xz · y = x · yz, or

F (x, y, z, 1) :

xy · z = xz · y and x · yz = x · zy.

Note that D(x, y, z, 1) implies (x, y, z) = (x, z, y) = 1, E(x, y, z, 1) implies xz · y = x · yz = xy · z(x, y, z) = x · zy(x, y, z), so (x, z, y) = (x, y, z), and F (x, y, z, 1) implies xz · y = xy · z = x · yz(x, y, z) = x · zy(x, y, z) and so, again, (x, z, y) = (x, y, z). ¤ Lemma 2.2. For all x, y, z ∈ B, (x, yz, y) = (x, y, z).

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Proof. Using the Bol identity, we have (xy·z)y = x(yz·y) = (x·yz)y(x, yz, y). Cancelling y gives xy · z = (x · yz)(x, yz, y), hence the result. ¤ Lemma 2.3. Squares of elements of B are in the right and middle nuclei; that is, (x, y, z 2 ) = 1 = (x, z 2 , y) for all x, y, z ∈ B. Proof. In a (right) Bol loop, the middle and right nuclei coincide. Unable to find a reference for this well-known fact, we include a short argument here. The right Bol identity says (xz · y)z = x(zy · z) so, if z ∈ Nρ , the right nucleus, (xz · y)z = (x · zy)z. Cancelling z gives xz · y = x · zy, so z ∈ Nµ , the middle nucleus. On the other hand, if z ∈ Nµ and we take x, y ∈ B, then y = zy1 for some y1 , so xy · z = (x · zy1 )z = (xz · y1 )z = x(zy1 · z) (right Bol) = x · yz. Thus z ∈ Nρ too. In the present situation then, it suffices to show that (x, y, z 2 ) = 1 for all x, y, z ∈ B. To see why, observe that x(yz 2 ) = x(yz · z) = x(zy · z)(y, z) = (xz · y)z(y, z) = (x · zy)z(y, z)(x, z, y) = (x · yz)z(x, z, y) = (xy · z)z(x, z, y)(x, y, z). Using Lemma 2.1 and the fact the associators have square 1, the last expression here is (xy · z)z = (xy)z 2 . ¤ Lemma 2.4. Squares of elements of B are in the centrum. Proof. Let x, y ∈ B. Then xy 2 = xy · y = yx · y(x, y) = y · xy(x, y)(y, x, y) = y · yx(y, x, y) = y 2 x(y, x, y)(y, y, x) = y 2 x, using Lemma 2.1 again at the last step. ¤ Lemma 2.5. Squares of elements of B are central. Proof. In view of Lemmas 2.3 and 2.4, it suffices to show that squares are in the left nucleus of B. For this, and using Lemma 2.4 immediately, z 2 · xy = xy · z 2 = (xy · z)z = (x · yz)z(x, y, z) = (x · zy)z(x, y, z)(z, y) = (xz · y)z(x, y, z)(z, y)(x, z, y) = (xz·y)z(z, y), by Lemma 2.1. Now (xz·y)z(z, y) = (xz)(yz)(z, y)(xz, y, z) = (xz)(zy)(xz, y, z) = (xz · z)y(xz, y, z)(xz, z, y) = (xz · z)y, with another application of Lemma 2.1. This last term is xz 2 · y = z 2 x · y, giving the desired result. ¤ Lemma 2.6. For all x, y, z ∈ B, we have (z, xy) = (z, x)(z, y)(z, x, y). Proof. Let x, y, z ∈ B. Then xy ·z = (x·yz)(x, y, z) = (x·zy)(x, y, z)(y, z) = (xz · y)(x, y, z)(y, z)(x, z, y) = (xz · y)(y, z), by Lemma 2.1. Continuing, (xz · y)(y, z) = (zx · y)(y, z)(x, z) = (z · xy)(y, z)(x, z)(z, x, y) = (xy · z)(y, z)(x, z)(z, x, y)(z, xy), giving the result (because all associators and commutators are 1 or s, s is central and s2 = 1). ¤

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Setting z = x in Lemma 2.6 and noting that (a, b) = (b, a) for any a, b ∈ B gives Corollary 2.7. (xy, x) = (x, y)(x, x, y) for all x, y ∈ B. Lemma 2.8. For any four elements x, y, z, w ∈ B, we have (xy, z, w)(x, yz, w)(x, y, zw)(x, y, z)(y, z, w) = 1. Proof. For any x, y, z, w ∈ B, (xy · z)w = (x · yz)w(x, y, z) = x(yz · w)(x, y, z)(x, yz, w) = x(y · zw)(x, y, z)(x, yz, w)(y, z, w) = (xy · zw)(x, y, z)(x, yz, w)(y, z, w)(x, y, zw) = (xy · z)w(x, y, z)(x, yz, w)(y, z, w)(x, y, zw)(xy, z, w). ¤ Corollary 2.9. (xy, y, w) = (x, y, w)(y, y, w) (x, x, y)(y, y, x) for all x, y, w ∈ B.

and

(xy, y, x)

=

Proof. Setting z = y in Lemma 2.8, using the right Bol identity and remembering that squares are in the nucleus, we have (xy, y, w)(x, y, yw)(y, y, w) = 1. Each of the three associators here is central of order 2, so (xy, y, w) = (x, y, yw)(y, y, w). Now Lemmas 2.1 and 2.2 give (x, y, yw) = (x, yw, y) = (x, y, w), so we have the first identity of the lemma. The second follows upon setting w = x in the first and remembering that (x, y, x) = (x, x, y). ¤ 3. Minimally non-Moufang Bol loops In this section and the next, we use implicitly the facts that for any central element z, and any p, q, r in a loop, (zp, q, r) = (p, zq, r) = (p, q, zr) = (p, q, r) and (zp, q) = (p, zq) = (p, q). Over 100 years ago, G. A. Miller and and H. C. Moreno characterized nonabelian groups with the property that all subgroups are Abelian [MM03]. (See also [Sco64, §6.5].) Motivated by this work, the authors have, in recent years, begun a study of minimal nonassociativity in loops [CG01, CG03, CG02]. We call a loop minimally nonassociative (MNA) if it is not associative, but every proper subloop is associative. Similarly, it is natural to think about Bol loops that are not Moufang, but in which every proper subloop is Moufang. We call such a subloop minimally non-Moufang (MNM). It is not hard to see that an MNM Bol loop B must be generated by two elements. For this, note first that because B is not Moufang, there exist a, b ∈ B with ba · b 6= b · ab. Since the subloop ha, bi generated by a and b cannot be Moufang, it cannot be proper.

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Theorem 3.1. Let B be a finite Bol loop with a unique nonidentity commutator/associator. If B is minimally non-Moufang, then it is minimally nonassociative. Proof. As noted, there exist elements a, b ∈ B with (b, a, b) 6= 1 and B = ha, bi. Since all commutators and associators of B are central, B/Z (Z = Z(B)) is an Abelian group. Since this group can be generated by two elements but not by one (else it is easy to see that B would be a group), we have B/Z = hZai×hZbi. Since squares in B are central, hZai ∼ = hZbi ∼ = C2 , the cyclic group of order 2, so (3.1)

B = Z ∪ Za ∪ Zb ∪ Zab

and every element of B can be represented in the form za² bδ , with δ, ² ∈ {0, 1} and z ∈ Z. We must show that every proper subloop of B is associative. So let K be a proper subloop and let x, y, u be elements of K. If any of these elements is central, clearly (x, y, u) = 1. If two of these elements come from the same coset of B - see (3.1) - say x = z1 t, y = z2 t, u = z3 w, then centrality of the zi gives (u, x, y) = (w, t, t), which is 1 by the right alternative law in B. Since K is Moufang and u, x, y associate in one order, they associate in all orders. Suppose x, y, and u are representatives of the cosets Za, Zb, Zab, in some order. For instance, suppose x = z1 a, y = z2 b, and u = z3 ab. Then, (y, u, x) = (b, ab, a) = (y, xy, x) = 1, by diassociativity. Again, it follows that y, u, and x associate in all orders. All this shows that K is associative, the fact we wanted to establish. ¤ 4. Two-generator Bol loops Suppose B is a Bol loop that is not Moufang and that has a unique nonidentity commutator/associator s. Suppose furthermore that B can be generated by two elements a and b. As shown in the proof of Theorem 3.1, every element of B can be written in the form za² bδ , ², δ ∈ {0, 1}, z ∈ Z = Z(B). It is easy to see that a commutative Bol loop is Moufang, thus at least two of the elements a, b, ab do not commute. If these are a and ab, say, then centrality of a2 gives Za2 b = Zb so B = Z ∪ Za ∪ Zb1 ∪ Zab1 , where b1 = ab. Without loss of generality, then, we may assume that a and b do not commute. Thus (a, b) = s. Suppose (b, b, a) = 1. Then (b, a, b) = 1 by Lemma 2.1 and (b, ab, b) = (b, a, b) = 1 by Lemma 2.2. Similarly, if (a, a, b) = 1, then (a, b, a) = 1. Furthermore, since ba = abc with c central, we have (a, ab, b) = (a, ba, b), so (a, ab, b) = (a, ba, b) = (a, b, a) by Lemma 2.2. Suppose both (b, b, a) = 1 and (a, a, b) = 1. We claim that (x, y, u) = 1 for all x, y, u ∈ B (which is not true). For this, it is sufficient to assume that x, y, u ∈ {a, b, ab}. Of the 27 cases, associativity follows in 13 because of the right alternative law or the assumptions together with Lemma 2.1.

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Many other cases follow from Lemma 2.2, and, as we have mentioned, the observation that for any central element z, and any p, q, r in a loop, (zp, q, r) = (p, zq, r) = (p, q, zr) = (p, q, r). To wit, (a, a, ab) = (a, ab, a) = (a, a, b) = 1 (a, b, ab) = (a, ab, b) = (a, sba, b) = (a, ba, b) = (a, b, a) = (a, a, b) = 1 (b, a, ab) = (b, ab, a) = (a, b, a) = (b, b, a) = 1 (b, b, ab) = (b, ab, b) = (b, sba, b) = (b, ba, b) = (b, b, a) = 1 Those of the 27 cases involving associators with x = ab can be handled using Lemma 2.8. For instance, (ab, a, b)(a, b · ab, a)(a, b, ab · a)(a, b, ab)(b, ab, a) = 1. We have already noted that (a, b, ab) = (b, ab, a) = 1. Also, (a, b · ab, a) = (a, ab · b, a) = (a, ab2 , a) = (a, a, a) (since b2 is central), so (a, b · ab, a) = 1. Similarly, (a, b, ab · a) = 1, so (ab, a, b) = (ab, b, a) = 1. From this, and Lemmas 2.1 and 2.2, we get (ab, a, ab) = (ab, ab, a) = (ab, a, b) = 1, hence also (ab, ab, a) = 1. Continuing, (ab, b, ab) = (ab, ab, b) = (ab, sba, b) = (ab, ba, b) (since s is central), so (ab, b, ab) = (ab, b, b) = 1 and (ab, ab, b) = 1 too. All this shows, indeed that if (b, b, a) = (a, a, b) = 1, then B is associative, which is not true. Thus assume that (a, a, b) = s (the case (b, b, a) = s is completely similar) and let (b, b, a) = t. The following table shows the products of pairs of a, b, ab.

(4.1)

a

b

ab

a

a2

ab

a2 bs

b

abs

b2

ab2 st

ab a2 bs ab2 a2 b2 t For instance, a(ab) = a2 b(a, a, b) = a2 bs, and b(ab) = (ba)b(b, a, b) = (ab)b(b, b, a)s = (ab2 )st. Also (ab)(ab) = a(b · ab)(a, b, ab) = a(stab2 )(a, ab, b). Since s and t are central and (a, ab, b) = (a, sba, b) = (a, ba, b) = (a, b, a) = (a, a, b) = s, we obtain (ab)(ab) = a(stab2 )(a, ab, b) = sta2 b2 s = a2 b2 t. It is now clear how to compute the product of elements from any two of the cosets Z, Za, Zb, Zab and to express such a product as an element from one of the these cosets. Thus multiplication in B is determined by

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the Table (4.1). Furthermore, the table suggests a construction of Bol loops that appears to be “new.” Let m > 1 and n > 1 be even integers and let B = {ap bq sα tβ | α, β ∈ {0, 1}, p, q ∈ Z} together with the relations am = bn = 1, s2 = t2 = 1, and a2 , b2 , s, t central. (We require m and n to be even so that neither a nor b is central.) Then use Table (4.1) to define arbitrary products of the form ap bq and hence multiplication in B. Specifically, define (ap1 bq1 sα1 tβ1 )(ap2 bq2 sα2 tβ2 ) = ap1 +p2 bq1 +q2 sα1 +α2 +p2 (q1 +p1 q2 ) tβ1 +β2 +q1 p2 q2 reducing the exponents of s and t modulo 2 and those of a and b modulo m and n, respectively. It is tedious, but straightforward, to see that B is a Bol loop that is not Moufang. Moreover, one can check that each of the non-Moufang Bol loops of order 8 (there are six) can be obtained by this construction. For instance, the Bol loop of order eight and exponent two is obtained by setting a2 = b2 = t = 1. Acknowledgments The research of Dr. Goodaire was supported by a Discovery Grant from the Natural Sciences and Engineering Research Council of Canada. References [CG01] O. Chein and E.G. Goodaire, Minimally nonassociative commutative Moufang loops, Results Math. 39 (2001), no. 1–2, 11–17. [CG02] , Minimally nonassociative Moufang loops with a unique nonidentity commutator are ring alternative, Comment. Math. Univ. Carolin. 43 (2002), no. 1, 1–8. [CG03] , Minimally nonassociative nilpotent Moufang loops, J. Algebra 268 (2003), no. 1, 327–343. [Goo83] E.G. Goodaire, Alternative loop rings, Publ. Math. Debrecen 30 (1983), 31–38. [GR95] E.G. Goodaire and D.A. Robinson, A class of loops with right alternative loop rings, Comm. Algebra 22 (1995), no. 14, 5623–5634. [GR96] , A construction of loops which admit right alternative loop rings, Results Math. 59 (1996), 56–62. [Kun98] K.Kunen, Alternative loop rings, Comm. Algebra 26 (1998), 557–564. [MM03] G.A. Miller and H.C. Moreno, Nonabelian groups in which every subgroup is abelian, Trans. Amer. Math. Soc. 4 (1903), 398–404. [Pai55] L.J. Paige, A theorem on commutative power associative loop algebras, Proc. Amer. Math. Soc. 6 (1955), 279–280. [Pfl90] H.O. Pflugfelder, Quasigroups and Loops: Introduction, Heldermann Verlag, Berlin, 1990. [Sco64] W.R. Scott, Group Theory, Prentice-Hall, Inc., 1964.

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Weil representations of symplectic groups Gerald Cliff and David McNeilly University of Alberta, Dept. of Mathematical and Statistical Sciences, Edmonton, Alberta, T6G 2G1, Canada [email protected] [email protected]

We discuss certain questions dealing with Weil representations of the symplectic group Sp(2n, K), where K is a finite or a nonarchimedean local field. We also consider the case where K is a finite ring. In the case that K is a local field of characteristic 0, we consider the problem of realizing the Weil representation over a number field. References for the local field case are [MVW], [K], [RR], [W]. For the case of a finite fields, see [I], [G]. For the case of a finite ring see [CMS1], [CMS2]. We first define the symplectic and Heisenberg groups, which can be defined for any field K. For our purposes in studying the Weil representation, K will be one of: the real numbers R, a finite field of characteristic not 2, or a local field. In the case that K is a local field, it is the field of fractions of a complete discrete valuation ring O having maximal ideal P and finite residue class field k = GF (q) where q is a power of a prime p 6= 2. Let V be a vector space over a field K, of even dimension 2n. We suppose that h , i is a nondegenerate alternating bilinear form on V . The symplectic group is by definition Sp = Sp(V ) = {g ∈ GL(V ) : hvg, wgi = hv, wi , for all v, w ∈ V }. Let V have K-basis {x1 , x2 , . . . , xn , y1 , y2 , . . . , yn }. We define our alternating form by hxi , xj i = hyi , yj i = 0, 1 ≤ i, j ≤ n,

hxi , yj i = δij , 1 ≤ i < j ≤ n.

We define X and Y to be the K-span of the sets {x1 , x2 , . . . , xn } and {y1 , y2 , . . . , yn } respectively, so V = X ⊕ Y , and hx, yi = 0 for x ∈ X, y ∈ Y . For some of the results mentioned below, K is taken to be a commutative ring; then we assume that V is a free K-module of even rank 2n. To discuss the Weil representation, one starts with the Heisenberg group H = H(V ) which as a set is given by H =K ×V 81

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with multiplication given by (c1 , v1 )(c2 , v2 ) = (c1 + c2 + hv1 , v2 i , v1 + v2 ),

c1 , c2 ∈ V, v1 , v2 ∈ K.

It is not difficult to see that H is a nilpotent group of class 2, whose center is Z(H) = {(c, 0) : c ∈ K}, so Z(H) is isomorphic to the additive group of K, denoted K + . The symplectic group Sp acts on the Heisenberg group H by (c, v)g = (c, vg),

g ∈ Sp, (c, v) ∈ H.

Note that under this action, each element of the center Z(H) is fixed by Sp. Fix a continuous character λ ∈ Hom(K + , C∗ ). If K is a finite field or a local field or the real numbers, there is an irreducible representation, the Schr¨ odinger representation, S of H with the property that S(c, 0) is scalar multiplication by λ(c); one says that S has central character λ. We describe how S can be constructed. One can also make this construction in the case that K is a finite ring. In this case we assume that K is Z/p` Z for some odd prime p, or more generally, K = O/P ` where O is a discrete valuation ring having 2 as a unit, where P is the maximal ideal of O. One takes V = K 2n as above. We assume that the character λ is primitive, in the sense that its kernel contains no nontrivial ideal of K. Since hx, yi = 0 for x, y ∈ X, then the subset A of H given by A = K × X = {(c, x) : x ∈ X, c ∈ K} is an abelian normal subgroup of H. We extend the character λ to a character λ0 ∈ Hom(A, C∗ ) by defining λ0 (c, x) = λ(c), x ∈ X. In the case that K is a finite field, or more generally O/P ` , we define 0 ∗ indH A λ to be the set of all functions φ from H to C such that φ(ah) = λ0 (a)φ(h), for all a ∈ A, h ∈ H. 0 Given h ∈ H, we have the operator S(h) on indH A λ defined by

(1)

(S(h)φ)(h0 ) = φ(h0 h),

h, h0 ∈ H.

0 We can find functions in indH A λ by taking coset representatives T of A in H, taking any function φ defined on T , and extending φ to H by φ(at) = λ0 (a)φ(t). A natural choice of coset representatives is to take T to be the 0 subset of H given by {(0, y) : y ∈ Y }. Then indH A λ is identified with the set S(Y ) of all complex functions on Y . With this identification, the action of the operator S(h), given in equation (1) on φ ∈ indH A (λ), is now given, if h = (c, x + y), by ® (2) S(c, x + y)φ(y 0 ) = λ(c + hy, xi + 2 y 0 , x )φ(y + y 0 ), y 0 ∈ Y, φ ∈ S(Y ).

Then S(Y ) gives us an irreducible representation of H with central character λ, and is the unique irreducible representation, up to equivalence, with this property.

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Historically, the Schr¨ odinger representation was first defined in the case that K is the field of real numbers R. The character λ is usually defined by λ(r) = eir . One takes the set of functions S(Y ) to be all L2 complex-valued functions on Y = Rn , with the action of H given by equation (2). The Stone-von Neumann Theorem says that this gives us an irreducible unitary representation of H acting on the Hilbert space S(Y ), with central character λ, and is unique up to equivalence. In the case that K is a local field, this situation was first considered by Weil [W]. One picks a Haar measure on K, which gives a Haar measure on Y , and one can again take S(Y ) to be the set of all L2 complex-valued functions on S(Y ). It is convenient to take S(Y ), as we will now do, to be the Bruhat-Schwarz space consisting of all locally constant functions on Y of compact support. One can describe these functions more explicitly as follows. We identify Y with K n , and we let L = On . Then L is compact. One can show that a function φ on Y has compact support precisely when φ vanishes on P k L for some (possibly negative) integer k, and that such a function φ is locally constant when φ takes the same value on each coset in P k L/P h L for some integer h ≥ k. The action of H on S(Y ) is given as in equation (2), and a version of the Stone-von Neumann theorem ([MVW]) again tells us that S(Y ) gives us an irreducible representation of H, and is the unique one up to equivalence with central character λ. Given g ∈ Sp, there is the conjugate representation, which we will denote S g , given by S g (h) = S(hg),

g ∈ Sp, h ∈ H.

For g ∈ Sp, c ∈ K, φ ∈ S(Y ), S(c, 0)φ = λ(c)φ,

S g (c, 0)φ = S((c, 0)g)φ = S(c, 0)φ = λ(c)φ.

Since there is a unique S (up to equivalence) with central character λ, then S g is equivalent to S. So there is an operator W (g) on X such that (3)

W (g)−1 S(h)W (g) = S(hg),

h ∈ H, g ∈ Sp,

Using Schur’s Lemma, one sees that for some complex scalar α(g1 , g2 ) 6= 0 W (g1 g2 ) = α(g1 , g2 )W (g1 )W (g2 ),

g1 , g2 ∈ Sp.

Then W gives us a projective representation of Sp, and α is a 2-cocycle on G, with values in C∗ . The functions S(Y ) can be decomposed as the direct sum of even functions S(Y )e and odd functions S(Yo ): (4)

S(Y ) = S(Y )e ⊕ S(Y )o .

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It is clear that each of S(Y )e and S(Y )o are invariant under the operators S(h), h ∈ H. It follows that the Weil representation, considered as a projective representation of Sp decomposes as the direct sum of two projective representations, which we call We and Wo . If we replace each W (g) by a nonzero scalar multiple W 0 (g) = c(g)W (g),

g ∈ Sp

W 0 (g)

we get new operators that still satisfy (3); the cocycle for the new W 0 (g), say α0 , is related to the orginal α by α0 (g1 , g2 ) = α(g1 , g2 )c(g1 )c(g2 )/c(g1 g2 ). It is known (see [G], [I]) that if K is a finite field, the cocycle α is a coboundary, which means that for each g ∈ G there is a nonzero scalar c(g) ∈ C∗ so that if we define W 0 (g) = c(g)W (g), then W 0 (g1 g2 ) = W 0 (g1 )W 0 (g2 ) for g1 , g2 ∈ Sp; thus the Weil representation is not just a projective representation but an ordinary representation of Sp. We now give a proof of this fact, which also is valid if K is a finite ring having an odd number of elements. Proposition. If K is a finite commutative ring having an odd number of elements, it is possible to define the Weil representation W of Sp acting on S(Y ) so that W (g1 g2 ) = W (g1 )W (g2 ) for any g1 , g2 ∈ Sp. Proof. Define W as above (3), having cocycle α. Let We and Wo be the restrictions of W to the even and odd subspaces of S, respectively. For y ∈ Y , define φy ∈ S(Y ) by φy (y) = 1,

φy (y 0 ) = 0, y 0 6= y.

Then S(Y ) has basis given by {φy : y ∈ Y }. Pick a subset P of Y consisting of half the nonzero elements of Y , such that −y ∈ / Y if y ∈ Y . Then S(Y )o has basis {φy − φ−y : y ∈ P} and S(Y )e has basis {φy + φ−y : y ∈ P} ∪ {φ0 }. It follows that the dimension d of S(Y )o is one less than the dimension of S(Y )e . From the equation Wo (gg 0 ) = α(g, g 0 )Wo (g)Wo (g 0 ) we get det Wo (gg 0 ) = α(g, g 0 )d det Wo (g) det Wo (g 0 ). For the even functions, we get det We (gg 0 ) = α(g, g 0 )d+1 det We (g) det Wo (g 0 ). Define γ(g) = det We (g)/ det Wo (g). It follows that α(g, g 0 ) = γ(g)γ(g 0 )/γ(gg 0 ), g ∈ Sp which shows that α is indeed a coboundary.

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In the case that K is a finite field, it is known that each of the summands appearing in (4) gives us an irreducible representation of Sp. In [CMS2] it is shown that if K = O/P ` O then the Weil representation breaks up into a direct sum of ` + 1 irreducible components. If K is a local field, it is known [W] that α is not a coboundary, but each W (g) can be multipliplied by a suitable scalar so that the cocycle α0 satisfies α0 (g, g 0 ) = ±1 for all g, g 0 ∈ Sp. Then W can be viewed as an ordinary representation of the double cover M p of Sp. The group M p = M p(2n, K) is defined as the set of all pairs (², g),

g ∈ Sp,

² = ±1

with multiplication given by (²1 , g1 )(²2 , g2 ) = (α0 (g1 , g2 )²1 ²2 , g1 g2 ). f , of M Sp is given by The Weil representation, which we call W (5)

f (², g) = ²W (g), W

g ∈ Sp,

² = ±1.

f acts on the vector space S(Y ). The representation W Question (Raised by Dipendra Prasad [P]). Can the Weil representation be realized over a number field? That is, is there a field F of finite degree over Q and a representation W 0 of M Sp acting on an F -vector space S 0 such that there is an M Sp-isomorphism S 0 ⊗F C ∼ = S(Y ) where M Sp acts f on S(Y ) via the Weil representation W ? If K has characteristic 0, the character λ(c) can be an arbitrary p-power root of unity, the Schr¨ odinger representation S cannot be realized over a number field. For example, if K = Qp , then there is a character λ ∈ Hom(Q+ p , C) given as follows: if a ∈ Z is a nonnegative integer, then −a λ(p ) = µpa , a primitive pa -th root of 1, whereas λ(pa ) = 1. Theorem. Considered as a representation of M p(2n, K), the Weil representation can be realized over the field Q[µp , i] where µp is a p-th root of 1. We will outline a proof of this result assuming that K is a local field of characteristic 0. Define E to be the field generated over Q by the character values λ(c), c ∈ K; then E is the field obtained by adjoining to Q all p-power roots of unity µpk . Let S(Y, E) be the E-subspace of S(Y ) consisting of all φ ∈ S(Y ) such that φ(y) ∈ E for all y ∈ Y . Equation (2) shows that S(Y, E) is left invariant under the action of the Schr¨ odinger representation of H. We want to show that the Weil representation can be defined over E. To do this, we use an explicit formula for the Weil operator W (g), [K], Proposition 2.3. (See also [RR]). Viewing elements of V = X ⊕ Y as row

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vectors (x, y) where x ∈ X, y ∈ Y , an element g ∈ Sp can be written in the form µ ¶ a b g= c d where a : X → X, d : Y → Y , b : X → Y , c : Y → X. Then there is a Haar measure dµg on X/ ker c such that W (g) as in (3) is given by Z (6) W (g)φ(y) = λ(hxa, xbi − 2 hxb, yci + hyc, ydi)φ(xa + yc)dµg (x). X/ ker c

The space X can be identified with K n ; we pick the Haar measure dµg on X/ ker c such that the measure of the image of On in X/ ker c is 1. Since the functions φ ∈ S(Y, E) are locally constant of compact support with values in E, they are E-linear combinations of characteristic functions on the sets P k L/P h L. It follows that the integral (6) is a finite sum, and since λ takes values in E, this sum lies in E. So our Weil operators W (g) can be chosen to leave S(Y, E) invariant. We have constructed S and W using a fixed character λ. There are others: if a ∈ K ∗ define λ[a](b) = λ(ab). To show the dependence of S and W on λ, write Sλ and Wλ . Let G be the Galois group of E over Q. Since G acts on E, it acts on the E-valued functions S(Y, E), by σ

φ(y) = σ(φ(y)),

φ ∈ S(Y, E), σ ∈ G.

There is an associated G-action on End S(Y, E): if σ ∈ G and ψ ∈ End S(Y, E) then ³ ´ −1 σ ψ(φ) = σ ψ(σ φ) , φ ∈ S(Y, E). The group G is isomorphic to the multiplicative units Z∗p of the p-adic P i integers, as follows. If t ∈ Z∗p , t = ∞ i=0 ti p , 0 ≤ ti < p, then σ(t) ∈ G is given by ∞ Y i σ(t) µ= µti p i=0

if µ is a p-power root of unity. For the action of σ(t) on the Schr¨ odinger representation, one shows that (7)

σ(t)

Sλ = Sλ[t] ,

t ∈ Z∗p .

Since we are assuming that K has characteristic 0, then Zp ⊂ K. For t ∈ Z∗p we have the element ft ∈ GL(V ) given by scalar multiplication by t on Y and the identity on X. It is easy to see that conjugation by ft leaves Sp invariant. Also ft acts on the Heisenberg group by (c, v)ft = (tc, vft ),

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(c, v) ∈ H.

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It follows that there is a conjugate Sλft of Sλ , and and a conjugate Wλft and one shows that (8)

Sλft = Sλ[t] ,

Wλft = ct Wλ[t] .

for some nonzero scalar ct ∈ E. In order to deal with the representation f of M Sp, we must replace the cocycle α by a cocycle α0 whose values W are ±1. There is an explicit way to do this, given in [RR]. If the operators W (g) is given as in (6), then we replace each W (g) by c(g)W (g) as in [RR, Theorem 5.3]. The elements c(g) do not lie in E, but they do lie in the field √ E 0 = E[ q, i], so we replace the field E in our construction of S(Y, E) by E 0 . Note that the group G acts on E 0 . Then we have the Weil representation f acting on S(Y, E 0 ). As above, we let α0 be the cocycle coming from the W f (g) as in (5). new operators c(g)W (g) and construct W ∗ For t ∈ Zp , define gt ∈ GL(V ) by gt (x + y) = tx + t−1 y,

x ∈ X, y ∈ Y.

Then gt ∈ Sp; indeed, gt ∈ Sp(2n, O). It is known (see [MVW] or [P]) that the cocycle α0 (g, g 0 ) is equal to 1 if g, g 0 ∈ Sp(2n, O). For t ∈ Zp∗ , the operator W (gt ) defined by W (gt )φ(y) = φ(ty),

φ ∈ S(Y ), y ∈ Y

satisfies equation (3) where g = gt . After normalizing by the appropriate c(gt ), we have a homomorphism δ : G → GL(S(Y, E 0 )),

f (gt ), δ(σ(t)) = W

t ∈ Zp∗

such that δ(gt ) is fixed by all σ ∈ G. For our purposes, we want to consider the analogue γ of this homomorphism defined by f (gt ), t ∈ Zp∗ . γ(t2 ) = W This is well defined at least for t in the subgroup U1 (Z∗p ) of elements t ∈ Z∗p such that t ≡ 1 mod p. So γ can be viewed as a 1-cocycle on U1 (Z∗p ) with values in GL(S(Y, E 0 )). A result of Speiser (see [S, Proposition 3, Chapter X]) says that if G is a group acting on a field F and X is a finite-dimensional vector space over F , then H 1 (G, GL(X)) is trivial. In our case, S(Y, E 0 ) is not finitedimensional over the field E 0 , but it is the increasing union of G-invariant finite dimensional subspaces Sk (Y, E 0 ) consisting of E 0 -valued functions φ on Y such that φ is 0 outside P −k L and φ takes the same value on each coset in P −k L/P k L. One then shows that our 1-cocyle γ is trivial, which means that there is an operator M on S(Y, E 0 ) such that (9)

γ(σ) = M −1 (σ (t)M ),

t ∈ U1 (Zp ∗).

We can write ft as the compostion ft2 = gt ◦ (t1)

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where 1 is the identity on V . It follows from this and equations (7) and (8) that σ(t2 ) f fλ (gt )W fλ (g)W fλ (gt )−1 . (10) Wλ (g) = W f M −1 Using equations (9) and (10) it follows that the representation M W ∗ −1 fM is left fixed by all elements σ(t), t ∈ U1 (Zp ). Then M W gives us a model of the Weil representation defined over the fixed subfield of E 0 under σ(U1 (Z∗p )). The fixed field of E under σ(U1 (Z∗p )) is Q[µp ] so the fixed field E 0 under σ(U1 (Z∗p )) is Q[µp , i], as desired. f can be realized over the field With some more work, we can show that W √ √ Q[ p, −p]. We now discuss the theory of dual reductive pairs, and the local theta, or Howe, correspondence. (See [MVW]). Suppose that G1 and G2 are reductive subgroups of Sp(2n, K), such that each is the centralizer in Sp of the other. The standard example is the case that V = V1 ⊗K V2 where V1 has a nondegenerate symmetric bilinear form, and V2 has a nondegenerate alternating bilinear form. Then V1 ⊗ V2 has a nondegenerate alternating bilinear form. The isometry group of the form on V1 is the orthogonal group O(n1 , K) and can be identified with a subgroup G1 of Sp(V ), whereas the the isometry group of the form on V2 is the symplectic group Sp(n2 , K) that can also be identified with a subgroup G2 of Sp(V ). e i be the inverse image in M p(V ) of Gi . The Weil For i = 1, 2, let G f of M p(V ) can be restricted to G e1 ×G e 2 . Let π1 be a smooth representation W e 1 . Let A(π1 ) be the maximal quotient of S(Y ) irreducible representation of G f e f (G e 2 ) acts on A(π1 ), so we on which W (G1 ) acts as a multiple of π1 . Then W e 1 ×G e 2 , acting on A(π1 ). A theorem of Waldspurger have a representation of G [Wa], confirming a conjecture of Howe, is that if the residue characteristic p 6= 2, then A(π1 ) is equivalent to π1 ⊗ π2 for a unique smooth irreducible e 2 . The representation π2 is called the theta lift, or representation π2 of G the Howe correspondent, of π1 . Using our result on the realizablity of the Weil representation in the field √ √ Q[ p, −p], the theta correspondence can be used to give a model of π2 over a number field if one had such a model for π1 . References [CMS1] G. Cliff, D. McNeilly, and F. Szechtman, Weil representations of symplectic groups over rings, J. London Math. Soc. (2) 62 (2000), 423–436. [CMS2] G. Cliff, D. McNeilly, and F. Szechtman, Character fields and Schur indices of irreducible Weil characters, J. Group Theory 7 (2004), 39–64. [G] P. G´erardin, Weil representations associated to finite fields, J. Algebra 46 (1977), 54–101. [I] I.M. Isaacs, Characters of solvable and symplectic groups, Amer. J. Math. 95 (1973), 594–635.

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S. Kudla, Notes on the local theta correspondence, http://www.math.umd.edu/ ~ssk/castle.pdf [MVW] C. Moeglin, M.-F. Vign´eras, and J.-L. Waldspurger, Correspondances de Howe sur un Corps p-adique, Lecture Notes in Math., Vol. 1291, Springer-Verlag, Berlin, 1987. [P] D. Prasad, A brief survey on the theta correspondence, Number theory (Tiruchirapalli, 1996) 171–193, Contemp. Math., 210, Amer. Math. Soc., Providence, RI, 1998. [RR] R. Ranga Rao, On some explicit formulas in the theory of Weil representations, Pacific J. Math. 157 (1993), 335–371. [S] J.-P. Serre, Local fields, Springer-Verlag, Berlin, 1979. [Wa] J.-L. Waldspurger, Demonstration d’une conjecture de duality de Howe dans le case p-adiques, p 6= 2, Festschrift in Honor of Piatetski-Shapiro, vol. 2, Israel Math. Conf. Proc., 1990, pp. 267–324. [W] A. Weil, Sur certaines groupes d’op´erateurs unitaires, Acta Math., 11 (1964), 143–211.

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Gradings and graded identities for the upper triangular matrices over an infinite field Onofrio M. Di Vincenzo Dipartimento di Matematica, Universit`a degli Studi di Bari, 70125 Bari, Italy [email protected] Plamen Koshlukov IMECC, UNICAMP, Caixa Postal 6065, Campinas, SP, 13083-970, Brasil [email protected] Angela Valenti Dipartimento di Metodi e Modelli Matematici, Universit`a di Palermo, Viale delle Scienze, 90128 Palermo, Italy [email protected] Abstract. The algebras U Tn (K) of the upper triangular matrices over a field K and their polynomial identities play an important role in PI theory. Gradings, graded identities, involutions, and identities with involution are of significant importance for the study of identities of a given algebra. We discuss here several recent results concerning the gradings and the graded polynomial identities that these algebras satisfy. We discuss also the involutions on U Tn (K).

Preliminaries Graded polynomial identities are one of the principal tools in PI theory. On the other hand, gradings on algebras and their graded identities are interesting and important on their own. This importance was revealed by the celebrated work of Kemer (see for example [20]) on the structure theory of T-ideals that led him to the positive solution of the famous Specht problem. Graded identities provide quite a lot of information about the ordinary identities satisfied by a given algebra. It is well known that if A is Key words and phrases. Graded identities, Upper triangular matrices, Graded algebra, Elementary grading. 2000 AMS MSC: 16P90, 16R10, 16R20, 16R50, 16W50. 91

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a G-graded algebra where G is a finite group then A is PI if and only if its neutral component in the grading is PI, see [4, 7]. Furthermore one may use graded identities in order to describe the concrete ordinary identities satisfied by a given algebra. In Kemer’s theory graded identities were essential in obtaining the PI equivalence of classes of verbally prime PI algebras, the so called Tensor product theorem. Later on Regev (see [26]) gave another proof of this theorem, and his proof was based once again on graded identities. Several cases of the tensor product theorem were handled in the papers [9, 13, 14, 22, 2, 3], once more using graded identities. Given an algebra (or a class of algebras) it is of importance to describe all possible gradings on it by a fixed group. In [32] all Z2 -graded simple algebras were described. In recent years the gradings on matrix algebras by finite groups were described, see for example the bibliography of [6]. Most of these result have also been transferred to the case of simple Jordan and Lie algebras, see for example the references of [29]. The algebras U Tn (K) of the upper triangular matrices of order n over a field K are of particular interest for PI theory. It is well known that the behavior of their identities determines the subvarieties of the variety of algebras generated by the matrix algebra of order two, when the base field is of characteristic 0. The identities of U Tn (K) are known for every field K, see for example [31]. The corresponding T-ideal is finitely generated as a T-ideal for every field K. In particular, if K is infinite then the T-ideal of U Tn (K) is generated by the polynomial [x1 , x2 ][x3 , x4 ] . . . [x2n−1 , x2n ] where [a, b] = ab − ba is the usual commutator of a and b. All Z2 -gradings on U T2 (K) when charK = 0 were described in [28]. Furthermore in the latter paper a variety of numerical characteristics of these graded identities were described as well. In [29] the gradings on U Tn (K) by finite abelian groups were given provided that K is algebraically closed of characteristic 0. It turns out that such gradings are isomorphic to elementary ones, that is, gradings where the matrix units eij are all homogeneous. Recall that according to [5], when G is abelian, every G-grading on Mn (K) is isomorphic to the tensor product of an elementary and a fine grading. The grading is fine if every homogeneous component is of dimension at most 1. It turns out that algebras of block-triangular matrices are quite important in the description of extremal varieties of algebras with respect to their codimension growth, see [17, 18, 19]. Since upper triangular matrices are the simplest such algebras we believe that studying their gradings and graded identities will give new information about the behavior of such extremal algebras. When one studies Brauer groups one is led naturally to the concept of involution ∗ on an algebra A, that is, an automorphism of order 2 of the additive group of A such that (ab)∗ = b∗ a∗ for every a, b ∈ A. The involutions on central simple algebras are well understood, see for example

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[21]. An involution is of the first kind if it is a linear automorphism, that is, if it preserves pointwise the centre of A, otherwise the involution is of the second kind. Concerning the algebras Mn (K), there exist two classes of nonequivalent involutions of the first kind on them. The first is generated by the usual matrix transpose t and the second is generated by the usual symplectic involution (the latter case is possible only when n is even). Identities with involution have long been studied in PI theory, see for example [27] and its bibliography. Concerning matrix algebras little is known about the identities with involution satisfied by them The only situation where one knows the exact identities with involution for Mn (K) is n = 2, see [24, 25] for the cases of characteristic 0 and of finite field, and [8] for infinite fields of characteristic different from 2. When one studies involutions on U Tn (K) a major problem is that this algebra is rather far from being even semisimple, its radical is ample. Nevertheless one can say quite a lot about the involutions on U Tn (K). Below we discuss briefly this point. 1. The natural grading on U Tn (K) Suppose G is a group and let Xg , g ∈ G, be infinite (countable) disjoint sets, X = ∪g∈G Xg . The free algebra K(X) that is freely generated over K by the set of variables X can be given a G-grading assuming that the variables of Xg are homogeneous of homogeneous degree g, and then extending this to the monomials on X. Write Xg = {xg1 , xg2 , . . .}. Let A = ⊕g∈G Ag be a Ggraded K-algebra. A polynomial f (xg1 ,1 , . . . , xgk ,k ) ∈ K(X) is a G-graded identity for A (or simply a graded identity) if f (ag1 , . . . , agk ) = 0 in A for every agi ∈ Agi . We shall use the notation f = 0 as well to indicate that f is a graded identity. Denote by I = IdG (A) the set of all graded identities of A. Then I is an ideal of K(X) that is closed with respect to all G-graded endomorphisms of K(X). One defines a consequence of a graded identity, variety of graded algebras and so on, in analogy with the case of ordinary identities. In a similar manner a relatively free graded algebra in a variety of graded algebras can be defined. Let G = Zn be the cyclic group of order n. The matrix algebra Mn (K) has a natural G-grading, Mn (K) = V0 ⊕ V1 ⊕ · · · ⊕ Vn−1 where Vi is the span of the matrix units eab with b − a = i (mod n). The G-graded identities of Mn (K) with this natural grading were described in [30] when charK = 0 and in [1] when K is an arbitrary infinite field. This elementary grading induces one on U Tn (K) where the i-th homogeneous component Vi equals Vi = {a1,i+1 e1,i+1 + a2,i+2 e2,i+2 + · · · + an−i,n en−i,n | at,s ∈ K}. Here obviously one has that Vi Vj ⊆ Vi+j (the sum i + j is taken modulo n), and Vi Vj = 0 if i + j > n.

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The algebra U Tn (K) satisfies the graded identities x01 x02 − x02 x01 = 0,

xi1 xj1 = 0,

i+j ≥n

(1)

The first of these holds since diagonal matrices commute, and the second follows from the rules of the matrix multiplication. The following theorem was proved in [23]. Theorem 1. If K is infinite then the graded identities (1) generate the ideal of Zn -graded polynomial identities for U Tn (K). Recall that under the hypotheses of the above theorem, the monomials of the type u = w0 xk1 i1 w1 xk2 i2 . . . wt−1 xkt it wt with k1 + k2 + · · · + kt < n and w0 , w1 , . . . , wt being monomials in the neutral variables x0i (that is the variables of homogeneous degree 0), written in ascending order in each wj , form a linear basis of the corresponding relatively free algebra. There are several proofs of the above theorem. One of them is direct and first shows the linear independence of the monomials of the above type by means of suitable evaluations on the matrix units. Another uses generic graded matrices and although more involved it admits various extensions and applications. We recall that the generic graded matrices are defined in the same way as in the nongraded case, one has to take care only of the grading. Let us suppose that yijk are commuting variables and consider the tensor product U Tn (K)⊗K K[yijk ] ∼ = U Tn (K[yijk ]) where K[yijk ] is the polynomial algebra in the variables yijk with i, j, k ≥ 0. The matrices Yik = y1,i+1,k e1,i+1 + y2,i+2,k e2,i+2 + · · · + yn−i,n,k en−i,n where 0 ≤ i ≤ n − 1, k ≥ 1 are the generic graded matrices in our case, and they generate a subalgebra Gn of U Tn (K[yijk ]). Then Gn is n-graded in a natural way. Furthermore one has the following, see [23, Lemma 3.1]. Proposition 2. The algebra Gn is relatively free in the variety of graded algebras generated by U Tn (K). Consider two monomials m1 and m2 in K(X), and suppose that their difference m = m1 − m2 vanishes when evaluated on some set of generic graded matrices. Then m is a graded identity for U Tn (K). In fact one may prove that m is a graded identity for U Tn (K) if and only if the evaluations of m1 and m2 have, in the same positions on their first rows, the same nonzero entries. Equivalently, the evaluation of m has zeros on its first row. The above observation is quite useful when one studies the numerical invariants of the graded T-ideal of U Tn (K). One of the most important such invariants is given by the codimensions. If I is the T-ideal of the algebra A then the n-th codimension cn (A) = dim Pn /(Pn ∩ I) gives information

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about the growth of the identities of A. Here Pn stands for the span of all monomials xσ(1) xσ(2) . . . xσ(n) where σ runs over the permutations of {1, 2, . . . , n}. Recall that in recent years it was proved that if A is a PI algebra over a field K of characteristic 0 then the limit limn→∞ (cn (A))1/n always exists and is an integer. It is called the exponent of A, see [17, 18] for the proofs and further applications of this very important result. When one deals with graded identities one may define various codimensions. We consider two of these for the algebra U Tn . The first generalization of codimensions to the graded case is the natural one. Let us fix the distinct 0-variables x0,j1 , x0,j2 , . . . , x0,js and the distinct variables xz1 ,r1 , xz2 ,r2 , . . . , xzg ,rg where g + s = m are also fixed. Suppose further that 1 ≤ zi ≤ n − 1 for every i and that z1 + z2 + · · · + zg < n. Then one computes that the span of all multilinear monomials on these variables modulo the graded identities of U Tn (K) is of dimension (m − s)(m − s + 1)s , see [23, Proposition 4.1]. While the above-mentioned graded codimensions are a straightforward generalization of the ungraded ones, it turns out that much more useful are some other graded codimensions. We introduce them now, and we refer to [4, 10] for more details. Let xij be (graded) variables in K(X), 0 ≤ i ≤ n, 1 ≤ j ≤ m and fix the integer m. Now consider the multilinear monomials in these variables: xi1 ,j1 xi2 ,j2 . . . xim ,jm where {j1 , j2 , . . . , jm } = {1, 2, . . . , m}, 0 ≤ is ≤ n − 1. This means that we do not admit, in a monomial, repeating indices on the second position. In this way one obtains a generalization of the usual multilinear polynomials since our graded polynomials can be obtained from xi1 xj2 . . . xjm by means of giving all possible gradings on its variables. n the span of all such monomials in K(X), it is immediate Now set Pm (n) n that dim Pm = cm = nm m!. If A is a graded algebra with ideal of graded n (A) = P n /(P n ∩ I) is the multilinear component of the identities I then Pm m m corresponding relatively free graded algebra. Now define the codimension (n) n (A). cm (A) = dimK Pm One may compute these graded codimensions for the algebra U Tn (K). The following theorem was proved in [23, Theorem 4.1]. Theorem 3. Let M = min{m, n − 1}, then (n) cm (U Tn (K))

¶ M µ ¶µ X m n−1 q!(q + 1)m−q . = q q q=0

The proof of this theorem is combinatorial using compositions of a number into several parts. As a direct corollary of the last theorem one obtains the asymptotic behavior of these codimensions. Namely it was proved in [23,

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Corollary 4.1] that cm (U Tn (K)) ' n1−n mn−1 nm . Here for the functions f (x) and g(x) we write f ' g if limx→∞ f (x)/g(x) = 1. 2. Gradings on U Tn (K): The generic case Let G be a group and let A = U Tn (K). Depending on G and on n there might be quite a lot of gradings on A. The case n = 1 is trivial and we omit it. Suppose now n = 2. The following theorem was proved in [28]. Theorem 4. Let K be an arbitrary field. If U T2 (K) is G-graded then either the grading is trivial (that is, only the neutral component is nonzero) or the grading is isomorphic to canonic one (that is, the neutral component consists of the diagonal matrices, and the only other nonzero component is the Jacobson radical). Hence if one wants to study the graded identities for the algebra U T2 (K) one has to consider only the canonic grading (or the trivial one). A far-reaching generalization of this last theorem was obtained in [29]. The main result of [29] states the following. Theorem 5. Let G be a finite abelian group and let K be an algebraically closed field of characteristic 0. Then U Tn (K) is isomorphic as a G-graded algebra to U Tn (K) equipped with some elementary G-grading. Hence when studying gradings by a finite abelian group on U Tn (K) and when the field is “good” one can consider elementary gradings. Recall that elementary are the gradings where the matrix units are all homogeneous. One may think of possible extensions of this theorem. One such extension was considered in [15]. In that paper the authors considered the elementary G-gradings on U Tn (K) when the group G is finite, under the (mild) restriction on the field to be infinite. One of the key technical tools in [15] was the following analogue of the classical Specht reduction to proper (or commutator) polynomials. Proposition 6. [15, Proposition 1.2]Let A be G-graded and let 1 ∈ A. If T is the ideal of graded identities of A then T is generated as an ideal of graded identities by its elements such that every neutral variable appears in commutators only. Now we recall a convenient way to describe the elementary gradings on an algebra of linear transformations. Let dimK V = n and let the vectors v1 , . . . , vn form a basis of V . Suppose g = (g1 , . . . , gn ) ∈ Gn is an ntuple of elements of G. Set vi to be of homogeneous degree gi and induce a G-grading on the algebra of all linear transformations on V . In this way Mn (K) is G-graded and if eij are the matrix units then deg(eij ) = gi−1 gj . It is well known that a G-grading on Mn (K) is elementary if and only if it can be obtained by the above procedure.

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Analogously one may describe the elementary G-gradings on U Tn (K). As an example we note that the grading considered initially and called natural may be obtained if G is the cyclic group of order n, G = {1, g, g 2 , . . . , g n−1 } and if g = (1, g, g 2 , . . . , g n−1 ). Let U Tn (K) be G-graded. If the grading is elementary one may recover it knowing only the homogeneous degrees of relatively few elements of U Tn (K). Lemma 7. An elementary G-grading on U Tn (K) is uniquely determined by the homogeneous degrees of: (i) The matrices e12 , e13 , . . . , e1n ; (ii) Alternatively, of the matrices e12 , e23 , . . . , en−1,n . We observe that the matrices eii always belong to the neutral component of the grading. Thus the proof of both assertions of the lemma is an easy exercise of linear algebra. Next we describe the elementary gradings on U Tn (K) and their graded identities. Fix an n-tuple ε = (ε1 , ε2 , . . . , εn ) ∈ Gm . Let Y = X1 be the set of the neutral variables (assuming that G is multiplicative), and set Z = ∪g6=1 Xg . Hence we shall use the symbols y for the variables in Y , and z for these in Z, with or without indices. We call the variables in Y even, and those in Z odd variables. A polynomial in K(X) is Y -proper if every even variable appears in commutators only. We assume that all z ∈ Z are Y -proper. In order to proceed we need the following notion. Definition. Let η = (η1 , η2 , . . . , ηm ) ∈ Gm . We call η a good sequence in the elementary G-grading ε if there exist matrix units r1 , r2 , . . . , rm ∈ U Tn (K) none of which is idempotent, such that r1 r2 . . . rm 6= 0 and ri is of homogeneous degree ηi for all i = 1, 2, . . . , m. Otherwise η is called ε-bad sequence. If η ∈ Gm is a sequence we denote fη = fη,1 fη,2 . . . fη,n where fη,i = [y2i−1 , y2i ] when ηi = 1, and fη,i = xηi ,i if ηi 6= 1. Here the lower index ηi means that the respective variable belongs to Xηi . The ε-bad sequences are related to the graded identities of U Tn (K). Proposition 8. Let fη be the multilinear polynomial defined above. Then fη is a graded identity for U Tn (K) if and only if η is an ε-bad sequence. Using the above description of the multilinear graded identities one has, as in [15, Theorem 2.3]: Theorem 9. If G is a finite group then there exist |G|n−1 different elementary G-gradings on the algebra U Tn (K). Furthermore different gradings satisfy different graded polynomial identities. One proves this theorem by observing that the n-tuples (g1 , g2 , . . . , gn ) and (1, g1−1 g2 , . . . , g1−1 gn ) of Gn define the same grading on U Tn (K). In order to show that different gradings satisfy different identities one proceeds as follows. The unique good sequence of nonidempotent matrix units

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of length n − 1 is (e12 , e23 , . . . , en−1,n ). Hence if ε = (ε1 , ε2 , . . . , εn ) is a −1 fixed grading then the sequence d(ε) = (ε1−1 ε2 , ε−1 1 ε3 , . . . , ε1 εn ) is the only ε-good sequence of length n − 1. But this sequence determines the homogeneous degrees of the first diagonal above the main one and it determines the grading uniquely. In this way fd(ε) will be a graded identity for U Tn (K) with respect to the grading defined by ε0 , for every ε0 6= ε, and fd(ε) is not a graded identity for the grading ε. An immediate and important corollary of this theorem is the following. Corollary 10. If G is finite then there exist |G|n−1 nonisomorphic elementary G-gradings on U Tn (K). In order to describe the graded identities for some given elementary grading on U Tn (K) we will need detailed information about the Y -proper polynomials. We list some of their properties that will be used afterwards, and introduce some notation. 1. Let f be Y -proper, deg f ≥ 1. Then f is a linear combination of products of Y -commutators whose Z-degree is at most 1. If some z ∈ Z appear in some Y -commutator c then one may assume that z is the first variable in the commutator. That is, either c = z or c = [z, y1 , . . . , yt ] for some t. Here and in the sequel we assume that the commutators are left normed, that is [a, b] = ab − ba and [a, b, c] = [[a, b], c]. 2. If c is Y -commutator of Z-degree ≤ 1 and if c is written in accordance with the Property 1 above, we call c normal. Now if c is normal and its Z-degree is 1, then its homogeneous degree equals that of z, the unique odd variable that appears in c. 3. Let c = c1 c2 . . . cm be a product of normal commutators. Denote ηc = (η1 , η2 , . . . , ηm ) ∈ Gm where ηi is the homogeneous degree of ci . If the sequence ηc is ε-bad then c is a graded identity for U Tn (K) (with respect to the grading defined by ε). 4. Fix an elementary G-grading ε on U Tn (K), and denote by W (ε) the vector space of all Y -proper polynomials in the relatively free (graded) algebra defined by the identities of ε. Set I(ε) the ideal of the graded identities for the grading ε. Then the relatively free algebra above is K(X)/I(ε). The normal commutator [yj1 , yj2 , . . . , yjp ] of Z-degree 0, is semistandard if j1 > j2 ≤ j3 ≤ · · · ≤ jp . The normal commutator [zj1 , yj2 , . . . , yjp ] of Z-degree 1, and with p ≥ 1, is semistandard if j2 ≤ j3 ≤ · · · ≤ jp . 5. In the notation of (4), every Y -proper polynomial f ∈ K(X)/I(ε), deg f ≥ 1, is a linear combination of products of the type c1 c2 . . . cm where every ci is semistandard commutator and the sequence η = (η1 , η2 , . . . , ηm ) is ε-good. Here ηi is the homogeneous degree of the commutator ci . The proofs of the above assertions are quite straightforward, with a (probable) exception of the last one. In it, one uses the fact that if c is a normal

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commutator, say of Z-degree 1, then one may permute the Y -variables in c at the expense of adding a linear combination of products of two normal commutators. We give a sketch of how to derive this. Assume for simplicity that c = [c0 , yt , yt+1 , . . .] where c0 = [z, y1 , . . . , yt−1 ]. Then by the Jacobi identity c = [yt+1 , yt , c0 , . . .] + [c0 , yt+1 , yt , . . .] = [c0 , yt+1 , yt , . . .] + [[yt+1 , yt ]c0 , . . .] − [c0 [yt+1 , yt ], . . .]. Now [ab, c] = [a, c]b + a[b, c] and this shows how one can reorder the entries of c. The case of Z-degree 0 is similar. Theorem 11. [15, Theorem 2.8] Assume that the field K is infinite and that U Tn (K) is graded by the group G, and let ε = (ε1 , ε2 , . . . , εn ) be an elementary grading on U Tn (K). (1) The ideal Iε of the graded identities for U Tn (K) is generated as such ideal by the multilinear polynomials fη where η = (η1 , η2 , . . . , ηm ) runs over all ε-bad sequences with m ≤ n. (2) The vector space of the Y -proper polynomials in the relatively free algebra K(X)/Iε has a basis consisting of 1 and all polynomials c1 c2 . . . cm where ci are semistandard commutators and the sequence ηc = (η1 , η2 , . . . , ηm ) is ε-good. Here ηi stands for the homogeneous degree of ci . Now let us observe that when G = {1} is the trivial group then Iε coincides with the ideal of the ordinary polynomial identities for U Tn (K). In this case all sequences of length m ≤ n − 1 are good, and we obtain once more the following well-known result. Corollary 12. When the field K is infinite, the polynomial [x1 , x2 ][x3 , x4 ] . . . [x2n−1 , x2n ] is a basis of the ordinary polynomial identities for the algebra U Tn (K). The proof of the above theorem is quite explicit and it allows us to construct a basis of the graded identities for U Tn (K) for a given elementary grading ε. We omit here the details but we refer the reader to [15, Section 3]. 3. Related topics We already mentioned that the algebras of upper triangular matrices are the simplest objects inside the important class of block triangular matrices. The identities of the latter are closely related to the so-called minimal varieties of algebras, see [19]. Since the minimal varieties of algebras are described in terms of Z2 -graded algebras one is led to study the Z2 -graded identities for block-triangular matrices as well.

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Let A, B be Z2 -graded algebras and W be a· Z2 -graded A–B bimodule. ¸ A W Then the block-triangular matrix algebra R = is graded in a nat0 B ural way by putting, for any i ∈ Z2 : · ¸ Ai Wi Ri = . 0 Bi Let us denote by T2 (S) the ideal of the graded polynomial identities for a Z2 graded algebra S. Then we have T2 (A)T2 (B) ⊆ T2 (R). In [12], the authors considered the case when A, B are graded subalgebras of the matrix algebras (Mm , g) and (Mn , h) equipped with the elementary Z2 -gradings induced by n the vectors g = (g1 , . . . , gm ) ∈ Zm 2 and h = (h1 , . . . , hn ) ∈ Z2 . In this case, if W = Mm×n is the vector space of m × n rectangular matrices, then R is a graded subalgebra of Mm+n endowed with the Z2 -grading defined by the vector g ∗ h = (g1 , . . . , gm , h1 , . . . , hn ). One of the main results of that paper consists in describing suitable conditions for the structures of A, B, W so that T2 (R) = T2 (A)T2 (B). We remark that the authors studied the more general situation when the grading is given by any abelian group. In our case, the notion of Z2 -regularity for any graded subalgebra S of (Mm , g) is the following. First, we consider the generic graded matrices associated to S. More precisely, let E = {a1 , . . . , ah } ∪ {b1 , . . . , bk } be a linear basis of S given by the disjoint union of the bases for the even and the odd component of S. Let us assume that yij , zil are commuting variables, and define P = P (S) = K[yij , zil | i ≥ 1, j ≤ h, l ≤ k ]. In the tensor product Mm ⊗K P ∼ = Mm (P ) we consider the matrices Yi = yi1 a1 + . . . + yih ah ,

Zi = zi1 b1 + . . . + zik bk .

They are the generic graded matrices in this case, and generate a graded subalgebra S 0 of (Mn (P ), g) that is relatively free in the variety of graded algebras generated by S. For any fixed element g ∈ Z2 , we define a K-linear map πg : Mm (P ) → Mm (P ) as follows: X X cij eij 7→ cij eij i,j

i,j,gi =g

where 1 ≤ i, j ≤ m. Moreover, we can define π ˆg : S 0 → Mm (P ) as the 0 restriction of πg to S . We have: Definition. The Z2 -graded subalgebra S of (Mm , g) is said to be Z2 -regular if the maps π ˆg are injective for any g ∈ Z2 . In the same way, we can define also the F -linear map πg∗ : Mm (P ) → Mm (P ) X X cij eij 7→ cij eij i,j

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i,j,gj =g

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and its restriction π ˆg∗ : S 0 → Mm (P ). It is proved in [12, Proposition 4.2], that the maps π ˆg are all injective if and only if the maps π ˆg∗ are such, for all g ∈ Z2 . Finally, as in [12, Theorem 4.5], we have Theorem 13. Let R be the Z2 -graded subalgebra of (Mm+n , g ∗ h) defined as follows: · ¸ A W R= 0 B where A ⊆ (Mm , g) B ⊆ (Mn , h) are graded subalgebras and W = Mm×n . If one of such subalgebras is Z2 -regular, then the T2 -ideal T2 (R) factorizes as: T2 (R) = T2 (A)T2 (B). The main tools in the proof are the graded version of the Lewin’s Theorem and the construction of a countable free set of homogeneous elements in a graded A0 –B 0 -bimodule related to W . We refer to [12, Sections 3, 4] for the complete description of this results. We remark also that the authors give an effective characterization of the elementary gradings which make the full matrix algebra Z2 -regular. More precisely, we have ([12, Theorem 5.4]): Proposition 14. Let | | : {1, 2, . . . , m} → Z2 be the natural map induced by the vector g = (g1 , . . . , gm ) ∈ Zm 2 . Then the full matrix algebra (Mm , g) is Z2 -regular if and only if the map | | is surjective and its fibers are equipotent. Using this characterization, it is possible to prove that, when A = Mm , B = Mn then the regularity of A or B is a necessary condition for the ideal T2 (R) to be factorable (see Theorem 5.4 of [11] or Theorem 5.8 of [12] when the grading is given by an arbitrary group of prime order). Another topic not touched upon yet is the identities with involution on the algebras of upper triangular matrices. Very little is known about them. One of the reasons is that the upper triangular matrices are not “nice” from the point of view of involutions: U Tn (K) is very far from being even semisimple, and its radical is huge. In a recent work [16] the involutions on U Tn (K) were described. Namely, it was shown that up to equivalence there exist at most two types of involutions on U Tn (K), the first being the analogue of the transpose (orthogonal) involution on Mn (K). It is represented by the symmetry with respect to the second diagonal of the upper triangular matrices. Denote it by t. The other, s, is possible only when n is even, and is the analogue of the symplectic involution on Mn (K). One divides the matrices in U T2m (K) into four square blocks of size m × m, and then ¶ µ ¶s µ t a b c −bt −→ . 0 at 0 c

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Here a, c ∈ U Tm (K), b ∈ Mm (K) and t is the reflection with respect to the second diagonal. More precisely, in [16] it was proved the following proposition: Proposition 15. Let ∗ be an involution on U Tn (K), then either (U Tn (K), ∗) ≈ (U Tn (K), t) or (U Tn (K), ∗) ≈ (U Tn (K), s). Moreover, we can extend ∗ to an involution on the full matrix algebra Mn (K) in a unique way. Clearly, algebras with involutions that are isomorphic satisfy the same ∗ polynomial identities. However, the problem of finding the ∗-polynomial identities in (U Tn (K), ∗) is still open. A complete solution for this problem is given in [16] only for n = 2. We hope that having the description of the involutions on U Tn (K) it would be easier to describe these identities. Acknowledgments Dr. Di Vincenzo’s work was partially supported by MIUR Cofin 2003 and Universit` a di Bari. Dr. Koshlukov’s work was partially supported by CNPq and by FAPESP, and Dr. Valenti’s work was partially supported by MIUR and Universit` a di Palermo. References [1] S. Azevedo, Graded identities for the matrix algebra of order n over an infinite field, Commun. Algebra 30, no. 12 (2002), 5849–5860. [2] S. Azevedo, M. Fidelis, and P. Koshlukov, Tensor product theorems in positive characteristic, J. Algebra 276 (2004), 836–845. [3] S. Azevedo, M. Fidelis, and P. Koshlukov, Graded identities and PI equivalence of algebras in positive characteristic, Commun. Algebra 33, (2005), 1011–1022. [4] Yu. Bahturin, A. Giambruno, and D. Riley, Group graded algebras satisfying a polynomial identity, Israel J. Math. 104 (1998), 145–155. [5] Yu. Bahturin, S.K. Sehgal, and M. Zaicev, Groups gradings on associative algebras, J. Algebra 241 (2001), 677–698. [6] Yu. Bahturin and M. Zaicev, Graded algebras and graded identities. Polynomial identities and combinatorial methods (Pantelleria, 2001), 101–139, Lecture Notes in Pure and Appl. Math., 235, Dekker, New York, 2003. [7] J. Bergen and M. Cohen, Actions of commutative Hopf algebras, Bull. London Math. Soc. 18, no. 2 (1986), 159–164. [8] J. Colombo and P. Koshlukov, Identities with involution for the matrix algebra of order two in characteristic p, Israel J. Math. 146, 2005, 337–356. [9] O.M. Di Vincenzo, On the graded identities of M1,1 (E), Israel J. Math. 80, no. 3 (1992), 323–335. [10] O.M. Di Vincenzo, Cocharacters of G-graded algebras, Commun. Algebra 24, no. 10 (1996), 323–335. [11] O.M. Di Vincenzo, Z2 -graded polynomial identities of superalgebras of blocktriangular matrices, Serdica Math. J. 30, (2004), 111–134. [12] O.M. Di Vincenzo and R. La Scala, Block-triangular matrix algebras and factorable ideals of graded polynomial identities, J. Algebra 279, no. 1 (2004), 260–279.

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[13] O.M. Di Vincenzo and V. Nardozza, Zk+l × Z2 -graded polynomial identities for Mk,l (E) ⊗ E, Rend. Sem. Mat. Univ. Padova 108 (2002), 27–39. [14] O.M. Di Vincenzo and V. Nardozza, Graded polynomial identities for tensor products by the Grassmann algebra, Commun. Algebra 31, no. 3 (2003), 1453–1474. [15] O.M. Di Vincenzo, P. Koshlukov, and A. Valenti, Gradings on the algebra of upper triangular matrices and their graded identities, J. Algebra 275, no. 2 (2004), 550–566. [16] O.M. Di Vincenzo, P. Koshlukov, and R. La Scala, Involutions on the algebra of upper triangular matrices, submitted, 2005. [17] A. Giambruno and M. Zaicev, On codimension growth of finitely generated associative algebras, Adv. Math. 140 (1998), 221–243. [18] A. Giambruno and M. Zaicev, Exponential codimension growth of PI algebras: an exact estimate, Adv. Math. 142 (1999), 145–155. [19] A. Giambruno and M. Zaicev, Codimension growth and minimal superalgebras, Trans. Amer. Math. Soc. 355 (2003), 5091–5117. [20] A. Kemer, Ideals of identities of associative algebras, Translations Math. Monographs 87, AMS, 1991. [21] M.-A. Knus, A. Merkurjev, M. Rost, and J.-P. Tignol, The Book of Involutions, Amer. Math. Soc. Colloq. Publ. vol. 44, AMS, 1998. [22] P. Koshlukov and S. Azevedo, Graded identities for T-prime algebras over fields of positive characteristic, Israel J. Math. 128 (2002), 157–176. [23] P. Koshlukov and A. Valenti, Graded identities for the algebra of n × n upper triangular matrices over an infinite field, Int. J. Algebra Comput. 13, no. 5 (2003), 517–526. [24] D. Levchenko, Finite basis property of identities with involution of a second-order matrix algebra, Serdica 8 (1982), no. 1, 42–56 (Russian). [25] D. Levchenko, Bases of identities with involution of second-order matrix algebras over finite fields, Serdica 10 (1984), no. 1, 55–67 (Russian). [26] A. Regev, Tensor products of matrix algebras over the Grassmann algebra, J. Algebra 133, no. 2 (1990), 512–526. [27] L. H. Rowen, Polynomial Identities in Ring Ttheory, Acad. Press, 1980. [28] A. Valenti, On graded identities of upper triangular matrices of size two, J. Pure Appl. Algebra 172 (2002), 325–335. [29] A. Valenti and M. Zaicev, Abelian gradings on upper-triangular matrices, Arch. Math. 80 (2003), 12–17. [30] S.Yu. Vasilovsky, Zn -graded polynomial identities of the full matrix algebra of order n, Proc. Amer. Math. Soc. 127, no. 12 (1999), 3517–3524. [31] S.M. Vovsi, Triangular Products of Group Representations and Their Applications, Progress Math., vol. 17, Birkh¨ auser, Boston, MA, 1981. [32] C.T.C. Wall, Graded Brauer groups, J. Reine Angew. Math. 213 (1963/1964), 187– 199.

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Structure of some classes of repeated-root constacyclic codes over integers modulo 2m Hai Q. Dinh Department of Mathematical Sciences, Kent State University, Trumbull, 4314 Mahoning Avenue, Warren, OH 44483 [email protected] Abstract. We study the classes of (2θ − 1)-constacyclic codes of length 2t over Z2m , for positive integers t, m, θ such that 2 6 θ 6 m, which are generalizations of repeated-root negacyclic codes. It will be shownd that the ring 2Zt2m [x] θ is a finite chain ring, and such codes form a hx +1−2 i chain as ideals of this ring. Dual codes and sizes of such constacyclic codes will also be obtained.

1. Introduction While the algebraic theory of error-correcting codes has traditionally taken place in the setting of vector spaces over finite fields, codes over finite rings have been studied since the early seventies. However, the papers on the subject during the seventies and eighties were scarse and may have been considered mostly as a mere mathematical curiosity since they did not seem to be aimed at solving any of the pressing open problems that were considered of utmost importance at the time by coding theorists. There are well-known families of nonlinear codes over finite fields, such as Kerdock, Preparata, Nordstrom-Robinson, Goethals, and Delsarte-Goethals codes [12], [14], [15], [23], [25], [30], [34], [7], that have more codewords than every comparable linear codes known to date. They have great errorcorrecting capabilities as well as remarkable structure, for example, the weight distributions of Kerdock and Preparata codes are MacWilliams transform of each other. Several researchers have investigated these codes and have shown that they are not unique, and large numbers of codes exist with the same weight distributions [19], [20], [21], [22], [2], [10], [37]. Key words and phrases. Codes over finite rings, linear codes, cyclic codes, negacyclic codes, constacyclic codes, chain rings, Galois rings. 105

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It was only recently that the study of linear codes over finite rings gained prominence, due to the discovery that these codes are actually equivalent to linear codes over the ring of integers modulo four, the so-called Quaternary codes (cf. [8], [16], [27], [28], [32], [33]). Nechaev pointed out that the Kerdock codes are, in fact, cyclic codes over Z4 in [28]. Furthermore, the intriguing relationship between the weight distributions of Kerdock and Preparata codes, a relation that is akin to that between the weight distributions of a linear code and its dual, was explained by Calderbank, Hammons, Kumar, Sloane and Sol´e [8], [16] when they showed in 1993 that these wellknown codes are in fact equivalent to linear codes over the ring Z4 that are dual to one another. The families of Kerdock and Preparata codes exist for all length n = 4k > 16, and at length 16, they coincide, providing the Nordstrom-Robison code [30], [35], [15], which is the unique binary code of length 16, consisting of 256 codewords, and minimum distance 6. In [8], [16], it has also been shown that the Nordstrom-Robison code is equivalent to a quaternary code that is self-dual. From that point on, codes over finite rings in general and over Z4 in particular, have gained considerable prominence in the literature. There are now numerous research papers on this subject and at least one book devoted to the study of Quaternary Codes [39]. The structure of cyclic and negacyclic codes over Z4 has been well studied. Furthermore, the structure of such codes over more general rings, such as Zpm , Galois rings, and chain rings R has been obtained for the case when ¯ the code length n is not divisible by the characteristic of the residue field R (cf. [16], [9], [32], [33], [31], [13]). Attention has also been given to cyclic and negacyclic codes of length n over a finite chain ring R when n is divisible by the characteristic of the ¯ When R is a field, such cyclic codes are called repeated-root residue field R. cyclic codes, and have been studied by G. Castagnoli, J.L. Massey, P.A. Schoeller, and N. von Seemann [11], and by van Lint [38] among others (see for example, [41], [36], [29]). Recently, repeated-root cyclic codes over Z4 with length a power of 2 were discussed by T. Abualrub and R. Oehmke in [1]. Blackford used a transform approach to characterize repeated-root cyclic codes of oddly even length [6], and repeated-root negacyclic codes of even length [5] over Z4 . Dinh and L´opez-Permouth [13] gave the structure of repeated-root negacyclic codes of length a power of 2 over Z2m . Both cyclic and negacyclic are subclasses of the class of constacyclic codes, whose definition and some basic properties are recalled in Section 2. In this paper, we continue to study repeated-root codes in a more general setting. We will obtain the structure of (2θ − 1)-constacyclic codes of length 2t over Z2m , for positive integers t, m, θ such that 2 6 θ 6 m. In the case θ = m, (2θ −1)-constacyclic codes are precisely negacyclic codes, so our results imply the previous results of repeated-root negacyclic codes in [13].

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The rest of this paper is arranged as follows. Section 2 gives basic back ground on chain rings and Galois rings, as well as constacyclic codes over m finite rings. In Section 3, we will show that the ring 2Zt2 [x] θ is a chain hx +1−2 i

ring, and henceforth providing structure of all (2θ − 1)-constacyclic codes of length 2t over Z2m , for positive integers t, m, θ such that 2 6 θ 6 m. The dual and the sizes of such codes are also obtained. Section 4 shows a one-to-one correspondence between subclasses of constacyclic codes over m Z2m and constacyclic codes over 2Zt2 [x] θ . hx +1−2 i

2. Preliminaries All rings are finite commutative associate rings with identity. An ideal I of a ring R is called principal if it is generated by one element. A ring R is a principal ideal ring if its ideals are principal. R is called a local ring if R/radR is a field, or equivalently, if R has a unique maximal ideal. Moreover, R is called a chain ring if the set of all ideals of R is a chain under set-theoretic inclusion. In the class of finite rings, a ring is a right chain ring if and only if it is a left chain ring (e.g., [26, XX.9, p. 394]), and hence we only use the term chain ring for the class of finite rings. However, a finite chain ring need not be commutative. The smallest noncommutative chain ring has order 16 [24], that can be represented as R = GF(4) ⊕ GF(4), where the operations +, · are (a1 , b1 ) + (a2 , b2 ) = (a1 + a2 , b1 + b2 ), (a1 , b1 ) · (a2 , b2 ) = (a1 a2 , a1 b2 + b1 a22 ). Consider finite commutative chain rings R. Let γ be a fixed generator of the maximal ideal M of R. Then γ is nilpotent and we denote its nilpotency index by $. The ideals of R form a chain: R = hγ 0 i ) hγ 1 i ) · · · ) hγ $−1 i ) hγ $ i = h0i. For the class of finite commutative chain rings, we have the following equivalent conditions (see, for example, [13]). Proposition 2.1. For a finite commutative ring R the following conditions are equivalent: (i) R is a local ring and the maximal ideal M of R is principal, (ii) R is a local principal ideal ring, (iii) R is a chain ring. In the class of integer residue rings Zpa , a polynomial in Zpa [x] is a basic irreducible polynomial if and only if its reduction modulo p is irreducible in Zp [x]. The Galois ring of characteristic pa and dimension m, denoted by

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GR(pa , m), is the Galois extension of degree m of the ring Zpa . Equivalently, GR(pa , m) =

Zpa [u] , hh(u)i

where h(u) is a monic basic irreducible polynomial of degree m in Z2a [u]. Note that if a = 1, then GR(p, m) = GF(pm ), and if m = 1, then GR(pa , 1) = Zpa . The following facts about Galois rings are well known (cf. [26, 16, 18]). Proposition 2.2. Let GR(pa , m) =

Zpa [u] be a Galois ring, then the folhh(u)i

lowing conditions hold: (i) Each ideal of GR(pa , m) is of the form hpk i = pk GR(pa , m), for 0 6 k 6 a. In particular, GR(pa , m) is a chain ring with maximal ideal hpi = p GR(pa , m), and residue field GF(pm ). (ii) For 0 6 i 6 a, |pi GR(pa , m)| = pm(a−i) . (iii) Each element of GR(pa , m) can be represented as upk , where u is a unit and 0 6 k 6 a, in this representation k is unique and u is unique modulo hpn−k i. (iv) h(u) has a root ξ, which is also a primitive (pm − 1)th root of unity. The set m Tm = {0, 1, ξ, ξ 2 , . . . , ξ p −2 } GR(pa , m) is a complete set of representatives of the cosets = p GR(pa , m) m a a GF(p ) in GR(p , m). Each element r ∈ GR(p , m) can be written uniquely as r = ξ0 + ξ1 p + · · · + ξa−1 pa−1 ,

with ξi ∈ Tm , 0 6 i 6 a − 1. (v) For each positive integer d, there is a natural injective ring homomorphism GR(pa , m) → GR(pa , md). (vi) There is a natural surjective ring homomorphism GR(pa , m) → GR(pa−1 , m) with kernel hpa−1 i. (vii) Each subring of GR(pa , m) is a Galois ring of the form GR(pa , l), where l divides m. Conversely, if l divides m then GR(pa , m) contains a unique copy of GR(pa , l). That means, the number of subrings of GR(pa , m) is the number of positive divisors of m. A polynomial f ∈ R[x] is irreducible if f is not a unit and whenever f = gh then either g or h is a unit. g is called a divisor of f if hf i ⊆ hgi, and a proper divisor if hf i ( hgi. It is well known and easy to verify that for a finite commutative local ring R with the maximal ideal M , and residue ¯ = R/M , g is a proper divisor of f if and only if g is a divisor of f field R ¯ and g¯ is a proper divisor of f¯ in R[x] (cf. [26, Chp. XIII]). A polynomial is called reducible if it is not irreducible. Of course, the irreducibility of a polynomial depends on the ring, for example, x2 + 1 is irreducible over

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Z, but it is reducible over Z2 . The following so-called Eisenstein criterion makes it easier to check the irreducibility of a certain class of polynomials in Z[x] over Q (cf. [17, Theorem 3.10.2]). Theorem 2.3 (Eisenstein criterion). Let n > 1 and f (x) = a0 + a1 x + · · · + an xn ∈ Z[x]. If there is a prime number p such that p|a0 , . . . , p|an−1 , p6 |an , p2 6 |a0 , then f (x) is irreducible in Q(x). However, the Eisenstein criterion does not hold for Zk [x]. For example, consider f (x) = x2 + 3 ∈ Z6 [x], then p = 3 satisfies the conditions of Eisenstein criterion for f (x), but f (x) is reducible in Z6 [x], as f (x) = x2 + 3 = (x+3)2 . The following criterion allows us to check irreducibility of some polynomials over the ring Zpk , k > 2. Proposition 2.4 (cf. [13, Lemma 6.2]). Let p be a prime, and n, k be positive integers with k > 2, and f (x) = a0 + a1 x + · · · + an xn ∈ Zpk [x] (a0 , . . . , an ∈ {0, . . . , pk −1}, an 6=0). If in Z, p|a0 , . . . , p|an−1 , p 6 | an , p2 6 | a0 , then f (x) is irreducible in Zpk [x]. Let R be a finite commutative ring. The cyclic shift τ and negashift ν on Rn are defined as usual, i.e., for any word (x0 , x1 , . . . , xn−1 ) ∈ Rn , τ (x0 , x1 , . . . , xn−1 ) = (xn−1 , x0 , x1 , · · · , xn−2 ), and ν(x0 , x1 , . . . , xn−1 ) = (−xn−1 , x0 , x1 , · · · , xn−2 ). A code C is called cyclic if it is closed under the cyclic shift τ , while C is called negacyclic if it is closed under the negashift ν. Negacyclic codes over finite fields were initiated by Berlekamp in the late 1960s (cf. [3, 4]). Recently, negacyclic codes over finite rings have been investigated by Wolfmann [40], Blackford [5], and Dinh and L´opez-Permouth [13], among others. Moreover, if λ is a unit of the ring R, then the λ-constacyclic (λ-twisted) shift τλ on Rn is the shift τλ (x0 , x1 , . . . , xn−1 ) = (λxn−1 , x0 , x1 , · · · , xn−2 ), and a code C is said to be λ-constacyclic if C is closed under the the λconstacyclic shift τλ . By this definition, when λ = 1 (or −1) then the λ-constacyclic code is a cyclic (or negacyclic) code. We identify each codeword c = (c0 , c1 , . . . , cn−1 ) with its polynomial representation c(x) = c0 + c1 x + · · · + cn−1 xn−1 , and the code C with the set of all polynomial representations of its codewords. Then in hxR[x] n −λi , xc(x) corresponds to a λ-constacyclic shift of c(x). From that, the following fact is well known and straightforward: Proposition 2.5. A linear code C of length n is λ-constacyclic over R if and only if C is an ideal of hxR[x] n −λi .

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Given x = (x0 , x1 , . . . , xn−1 ), y = (y0 , y1 , . . . , yn−1 ) ∈ Rn , their inner product (or dot product) is x · y = x0 y0 + x1 y1 + · · · + xn−1 yn−1 (evaluated in R). Two words x, y are called orthogonal if x · y = 0. For a linear code C over R, its dual code C ⊥ is the set of words over R that are orthogonal to all codewords of C, i.e., C ⊥ = {x | x · y = 0, ∀y ∈ C}. A code C is called self-dual if C = C ⊥ . The following result is well known (cf. [9], [13]). Proposition 2.6. Let R be a finite ring of order pα . The number of codewords in any linear code C of length n over R is pk , for some integer k ∈ {0, 1, . . . , αn}. Moreover, the dual code C ⊥ has pl codewords, where k + l = αn. In this paper, we study the classes of linear λ-constacyclic codes of length over Z2m for λ = 2θ − 1, 2 6 θ 6 m. By Proposition 2.5, such (2θ − 1)m constacyclic codes are precisely the ideals of 2Zt2 [x] θ . Notice that, if θ = m

2t

hx +1−2 i

then 2θ − 1 = −1 (in Z2m ), and therefore (2θ − 1)-constacyclic codes are just negacyclic codes in this case. 3. Structure of (2θ − 1)-constacyclic codes Lemma 3.1. Let n, m, θ be positive integers such that 2 6 θ 6 m. Then n n there exists a polynomial αn (x) ∈ Z[x] such that (x+1)2 = x2 +1+2αn (x), m and viewing as a polynomial in 2Zt2 [x] θ , αn (x) is invertible. hx +1−2 i

Proof. We proceed by induction on n. For n = 1, (x + 1)2 = x2 + 1 + 2x, and m α1 (x) = x is invertible in 2Zt2 [x] θ . Now assume n > 1 and the statement hx +1−2 i is true for all positive integer less than n. Then n

(x + 1)2 = [(x + 1)2 n−1

= [x2

n−1

]2

+ 1 + 2αn−1 (x)]2

n

2 = x2 + 1 + 4αn−1 (x) + 2x2 2n

=x

n−1

+ 4αn−1 (x) + 4x2

n−1

αn−1 (x)

+ 1 + 2αn (x), n−1

n−1

2 + 2αn−1 (x) + 2αn−1 (x) + 2x2 αn−1 (x). Since x is where αn (x) = x2 n−1 m m invertible in 2Zt2 [x] θ , x2 is also invertible in 2Zt2 [x] θ . Therefore, as

2 is

hx +1−2 i m nilpotent in 2Zt2 [x] θ , hx +1−2 i

© 2006 by Taylor & Francis Group, LLC

hx +1−2 i

n−1

we can write αn (x) as αn (x) = x2

(1 + z),

Structure of some classes of repeated-root constacyclic codes over integers

where z is a nilpotent element in such that

zξ

Z2m [x] . t hx2 +1−2θ i

111

Now let ξ be an odd integer

= 0, we have 1 = 1 + z ξ = (1 + z)(1 − z + z 2 − · · · + z ξ−1 ).

That means, 1+z is invertible in

Z2m [x] , hx2t +1−2θ i

n−1

and hence, αn (x) = x2

(1+z)

Z2m [x] . hx2t +1−2θ i

is invertible in

¤

Proposition 3.2. Let k be a nonnegative integer and m, θ be integers such k that 2 6 θ 6 m. Then φ(x) = x2 + 1 − 2θ is irreducible in Z2m [x]. ¡ k ¢ k k k Proof. Let ϕ(x) = φ(x + 1) = (x + 1)2 + 1 − 2θ = x2 + 2k2−1 x2 −1 + · · · + ¡2k ¢ θ−1 ). 1 x + 2(1 − 2 Denote   − 2θ−1 ) mod 2m , if i = 0, 2(1 ¡2k ¢ m ai = if 1 6 i 6 2k − 1, i mod 2 ,   1, if i = 2k . k

Then ϕ(x) = a0 + a1 x + · · · + a2k x2 , ai ∈ {0, 1, . . . , 2m − 1} for 0 6 ¡ k¢ i 6 2k . Furthermore, for 1 6 i 6 2k − 1, 2i are even integers, thus, 2|a0 , . . . , 2|a2k −1 , 26 |a2k , 22 6 |a0 . Hence, by Proposition 2.4, ϕ(x) is irreducible in Z2m [x]. Therefore, φ(x) is also irreducible in Z2m [x]. ¤ Z2m [x] , hx2t +1−2θ i

Proposition 3.3. x + 1 is nilpotent in

with nilpotency 2t m.

Proof. By Lemma 3.1, there exists an invertible polynomial αt (x) in t t m Z2m [x] such that (x + 1)2 = x2 + 1 + 2αn (x). That means in 2Zt2 [x] θ t hx2 +1−2θ i hx +1−2 i we get t

t

(x + 1)2 = (x2 + 1 − 2θ ) + 2θ + 2αt (x) = 2[2θ−1 + αt (x)] = 2[αt (x)]−1 (1 + y), where y = 2θ−1 [αt (x)]−1 is a nilpotent element in odd integer such that y ι = 0, it follows that

Z2m [x] . hx2t +1−2θ i

Let ι be an

1 = 1 + y ι = (1 + y)(1 − y + y 2 − · · · + y ι−1 ). Thus 1 + y is invertible in Z2m [x] . t hx2 +1−2θ i

x + 1 in

Z2m [x] , hx2t +1−2θ i

Since the nilpotency of 2 in

Z2m [x] hx2t +1−2θ i

t

and therefore h(x + 1)2 i = h2i in Z2m [x] t hx2 +1−2θ i

is m, the nilpotency of

is 2t m.

¤

We now consider alternative representations of the elements of the ring m As usual, every element f (x) ∈ 2Zt2 [x] θ can be viewed as a

Z2m [x] . hx2t +1−2θ i

© 2006 by Taylor & Francis Group, LLC

hx +1−2 i

112

H.Q. Dinh

polynomial with degree less than or equal to 2t − 1 in Z2m [x]. In the other words, there exist b0 , b1 , . . . , b2t −1 ∈ Z2m such that t −1

f (x) = b0 + b1 x + · · · + b2t −1 x2

.

For i = 0, . . . , 2t − 1, each bi has a unique binary representation: bi = bi0 + 2bi1 + ... + 2m−1 bi,m−1 , where bik ∈ {0, 1} (k = 0, . . . , m − 1). Hence, f (x) can be written as f (x) = f0 (x) + 2f1 (x) + · · · + 2m−1 fm−1 (x), where fj (x) ∈ Z2 [x], 0 6 j 6 m − 1. Furthermore, x can be represented as x = (x + 1) + (2m − 1) = (x + 1) + (1 + 2 + 22 + · · · + 2m−1 ). Consequently, f (x) can be written (uniquely) as f (x) =

t −1 2X

j

a0j (x + 1) + 2

j=0

t −1 2X

j

m−1

a1j (x + 1) + · · · + 2

j=0

t −1 2X

am−1,j (x + 1)j ,

j=0

where aij ∈ Z2 , 0 6 i 6 m − 1, 0 6 j 6 2t − 1. Proposition 3.4. Let f (x) ∈ f (x) =

t −1 2X

j

a0j (x + 1) + 2

j=0

Z2m [x] hx2t +1−2θ i

t −1 2X

be written as j

m−1

a1j (x + 1) + · · · + 2

j=0

t −1 2X

am−1,j (x + 1)j ,

j=0

where aij ∈ Z2 , 0 6 i 6 m − 1, 0 6 j 6 2t − 1. Then f (x) is invertible if and only if a00 = 1. Proof. If a00 = 0, then f (x) = (x + 1)g(x) + 2h(x). By Lemma 3.3, (x + 1) m and 2 are nilpotent in 2Zt2 [x] θ , hence (x + 1)g(x) and 2h(x) are nilpotent hx +1−2 i

Z2m [x] . hx2t +1−2θ i

in Therefore, f (x) is nilpotent in 2Zt2 [x] θ , i.e., f (x) is not hx +1−2 i invertible. On the other hand, if a00 = 1, then f (x) = 1 + (x + 1)e g (x) + 2h(x). As e above f = (x+1)e g (x)+2h(x) is nilpotent. Thus, there exists an odd positive integer k such that fek = 0. Now 1 = 1+ fek = (1+ fe)(1− fe+ fe2 −· · ·+ fek−1 ). Hence, f (x) = 1 + fe is invertible. ¤ Theorem 3.5. For any integers t > 1, 2 6 θ 6 m, with the maximal ideal hx + 1i.

© 2006 by Taylor & Francis Group, LLC

m

Z2m [x] t hx2 +1−2θ i

is a local ring

Structure of some classes of repeated-root constacyclic codes over integers t

t

113 t

Proof. We have x2 + 1 − 2θ = (x2 − 1) + 2(1 − 2θ−1 ). Hence, x2 + 1 − m 2θ = 2(1 − 2θ−1 ) + (x + 1)β(x). Thus, in 2Zt2 [x] θ , 2 ∈ hx + 1i. Let hx +1−2 i

f (x) ∈ f (x) =

Z2m [x] , hx2t +1−2θ i t −1 2X

and write f (x) as j

a0j (x + 1) + 2

j=0

t −1 2X

j

m−1

a1j (x + 1) + · · · + 2

j=0

t −1 2X

am−1,j (x + 1)j ,

j=0

where aij ∈ Z2 , 0 6 i 6 m − 1, 0 6 j 6 2t − 1. Then by Proposition 3.4, either f (x) ∈ hx + 1i (if a00 = 0), or f (x) is invertible (if a00 = 1). m Consequently, 2Zt2 [x] θ is a local ring with the maximal ideal hx + 1i. ¤ hx +1−2 i

In light of Proposition 2.2, Theorem 3.5 implies that the ring

Z2m [x] t hx2 +1−2θ i

is a chain ring with the maximal ideal hx + 1i. That gives us a complete list of the (2θ − 1)-constacyclic codes of length 2t over Z2m . Theorem 3.6. The (2θ −1)-constacyclic codes of length 2t over Z2m , i.e., the m m ideals of 2Zt2 [x] θ , are precisely h(x+1)i i ⊆ 2Zt2 [x] θ , for i = 0, 1, . . . , 2t m. hx +1−2 i

hx +1−2 i

t m−i

Proposition 3.7. For i = 0, 1, . . . , 2t m, there are 22 (2θ − 1)-constacyclic code h(x + 1)i i.

codewords in the

Proof. The chain of 2t m + 1 ideals h(x + 1)i i (i = 0, 1, . . . , 2t m) of is a strict chain:

Z2m [x] hx2t +1−2θ i

h1i = h(x + 1)0 i ) h(x + 1)i ) h(x + 1)2 i ) . . . t t ) h(x + 1)2 m−1 i ) h(x + 1)2 m i = h0i. By Proposition 2.6, each of these ideals has order 2k for some integer k ∈ {0, 1, . . . , 2t m}. Clearly, there are precisely 2t m + 1 such orders and furthermore t t 22 m > 22 m−1 > · · · > 21 > 20 . t m−i

Therefore, for i ∈ {0, 1, . . . , 2t m}, |h(x + 1)i i| = 22

.

¤

Theorem 3.8. Let C be a (2θ − 1)-constacyclic code of length 2t over Z2m . t Then for some i ∈ {0, . . . , 2t m} and (x + 1)i is calculated modulo (x2 + 1 − 2θ ), the following hold: Z2m [x] , |C| = hx2t +1−2θ i t m 1)2 m−i i ⊆ 2Zt2 [x] θ , hx +1−2 i

t m−i

(a) C = h(x + 1)i i ⊆

22

(b) C ⊥ = h(x +

|C ⊥ | = 2i .

.

Proof. (a) follows from Theorem 3.6 and Proposition 3.7. For (b), by Proposition 2.6, |C ⊥ | = 2i . Since C ⊥ is also a (2θ − 1)-constacyclic code, Propot sition 3.7 implies that C ⊥ = h(x + 1)2 m−i i. ¤

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114

H.Q. Dinh

Theorem 3.8 implies that a (2θ − 1)-constacyclic code C is self-dual if and only if t h(x + 1)2 m−i i = C = C ⊥ = h(x + 1)i i, if and only if i = 2t−1 m. Hence we get: t−1 m

Corollary 3.9. C = h(x+1)2

Z2m [x] hx2t +1−2θ i

i⊆

is the only self-dual (2θ −1)-

constacyclic code of length 2t over Z2m .

4. A correspondence between constacyclic codes In this section, to simplify notations, we denote Z2m [x] . hx + 1 − 2θ i

R2θ −1 =

2t

By Theorem 3.5, R2θ −1 is a chain ring with the maximal ideal hx + 1i, and the element x is a unit in R2θ −1 . Therefore, if we consider x as a constant R

θ

[z]

in the polynomial ring R2θ −1 [z], then the ideals of hz2n−1 −xi can be viewed as linear x-constacyclic codes of length n over R2θ −1 . We now provide a relation between x-constacyclic codes over R2θ −1 and (2θ − 1)-constacyclic code over Z2m . Each element r ∈ R2θ −1 has an x-adic representation over Z2m r = r0 + r1 x + · · · + r2t −1 x2

t −1

,

where r0 , r1 , . . . , r2t −1 ∈ Z2m . This x-adic representation can be extended naturally to an x-adic representation over Z2m of n-tuple v ∈ Rn2θ −1 : (0)

(2t −1) 2t −1 x ,

(1)

(0)

v = (v0 + v0 x + · · · + v0 (0)

(2t −1)

(1)

t −1

. . . , vn−1 + vn−1 x + · · · + vn−1 x2 where

(2t −1) 2t −1 x ,...

(1)

v1 + v1 x + · · · + v1

t −1)

) = v (0) + v (1) x + · · · + v (2

t −1

x2

t v (0) , v (1) , . . . , v (2 −1)

v (0) =

∈ Zn2m with (0) (0) (0) (v0 , v1 , . . . , vn−1 ), v (1) t −1)

v (2 Let Φ be the map

(2t −1)

= (v0

(2t −1)

, v1

(1)

(1)

(1)

= (v0 , v1 , . . . , vn−1 ), . . . (2t −1)

, . . . , vn−1 ). t

Φ : Rn2θ −1 −→ Z22mn defined as Φ(v) = (v (0) |v (1) | · · · |v (2 (0)

(0)

(0)

(1)

(1)

(1)

t −1)

)=

(2t −1)

(2t −1)

(2t −1)

(v0 , v1 , . . . , vn−1 , v0 , v1 , . . . , vn−1 , . . . , v0 , v1 , . . . , vn−1 ) where “|” denotes the usual vector concatenation. We get

© 2006 by Taylor & Francis Group, LLC

,

Structure of some classes of repeated-root constacyclic codes over integers

115

Φ(τx (v)) (0)

(2t −1)

t −1

(2t −1)

t −1

(1)

= Φ(τx (v0 +v0 x+· · ·+v0 x2 (0) (1) (2t −1) 2t −1 vn−1 + vn−1 x + · · · + vn−1 x )) (0)

(1)

= Φ(x[vn−1 +vn−1 x+· · ·+vn−1 x2

(0)

(2t −1)

(1)

vn−2 + vn−2 x + · · · + vn−2 x2 (2t −1)

= Φ((2θ − 1)vn−1

(0) v0

+

(1) v0 x

(1)

(2t −1) 2t −1 x ,...,

(0)

(1)

(2t −1) 2t −1 x ,...,

], v0 +v0 x+· · ·+v0

)

(0)

(2s −2)

(1)

+ vn−1 x + vn−1 x2 + · · · + vn−1 x2 (2t −1)

+ · · · + v0 (2t −1)

t −1

(0)

, v1 +v1 x+· · ·+v1

(0)

x

2t −1

, ...

(0)

(0) , vn−2

(0)

+

(0)

(1) vn−2 x

(1)

(1)

t −1

+ ··· +

,

(2t −1) t vn−2 x2 −1 )

(1)

= ((2θ − 1)vn−1 , v0 , v1 , . . . , vn−2 , vn−1 , v0 , v1 , . . . , vn−1 , . . . ,

(2t −1)

v0

(2t −1)

, v1

(2s −2)

, . . . , vn−1 )

(0)

(0)

(0)

(1)

(1)

(1)

= τ2θ −1 (v0 , v1 , . . . , vn−1 , v0 , v1 , . . . , vn−1 , . . . ,

(2t −1)

v0

(2t −1)

, v1

(2t −1)

, . . . , vn−1 )

= τ2θ −1 (Φ(v)) Hence, for each code C of length n over R2θ −1 , i.e., C ⊆ Rn2θ −1 , Φ(C) is a (2θ − 1)-constacyclic code if and only if for any codeword v ∈ C, Φ(τx (v)) = τ2θ −1 (Φ(v)) ∈ Φ(C), if and only if τx (v) ∈ C, which is equivalent to the condition that C is x-constacyclic. Therefore, x-constacyclic codes of length n over Rn2θ −1 correspond to (2θ − 1)-constacyclic codes of length 2s n over Z2m by the following commutative diagram: R2θ −1 [z] hz n −xi

 

Φ

−−−−→

Z2m [z] hz 2t n +1−2θ i

Φ

Z22mn

Ψy

Rn2θ −1 −−−−→

  yΨ t

where Ψ denotes the usual bijection between polynomials and vectors. We summerize this discussion in the following proposition. Proposition 4.1. C is an ideal of Z2m [x] . hx2t n +1−2θ i

R2θ −1 [z] hz n −xi

if and only if Φ(C) is an ideal of

Equivalently, C is a x-constacyclic code of length n over R2θ −1

if and only if Φ(C) is a (2θ − 1)-constacyclic code of length 2t n over Z2m . If C is linear then Φ(C) is also linear, whence our results in Section 3 about linear (2θ − 1)-constacyclic codes over Z2m having length a power of 2 can be used for linear x-constacyclic codes of length n = 2s over R2θ −1 .

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Proposition 4.2. C is an x-constacyclic code of length 2s over R2θ −1 if and only if Φ(C) is a (2θ − 1)-constacyclic code of length 2s+t over Z2m , if and only if Φ(C) = h(x + 1)i i, the ideal generated by (x + 1)i of the ring Z2m [x] , for some integer i ∈ {0, 1, 2, . . . , 2s+t m}. 2s+t θ hx

+1−2 i

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Units in noncommutative orders Ann Dooms and Eric Jespers Department of Mathematics, Vrije Universiteit Brussel, Pleinlaan 2, 1050 Brussels, Belgium [email protected] [email protected] Dedicated to our dear friend C´esar Polcino Milies on the occasion of his 60th birthday

For a ring R with unity 1, we denote by U(R) the unit group of R, i.e., the group of invertible elements in R. For decades one has investigated the unit group of a group ring R[G] of a finite group G over a (commutative) ring R. This is a very important ring, mainly because the representations of G on R-modules can be considered as R[G]-modules. The unit group U(R[G]) plays an important role in the study of the algebraic structure of G versus the ring structure of its group ring R[G]. This for example is shown via the study of the isomorphism problem for integral group rings. Hence integral group rings are very interesting algebraic structures because of their obvious relationship with group theory and ring theory and because the investigations in the structure also involve, for example, the theory of fields, linear algebra, algebraic topology, algebraic number theory, and algebraic Ktheory. Thus the research in group rings is a subject where many branches of algebra meet and the knowledge of the unit group is crucial herein. A reasonable approach to study this problem is considering Z[G] as an order (we give a formal definition later in the text) in the rational group algebra Q[G]. This idea comes from the important commutative case where the ring of algebraic integers Ok in an algebraic number field k is an order for which the unit group is described in the following theorem (see for example [31, Theorem IV.3.13]). Theorem 1 (Dirichlet Unit Theorem). Let k be an algebraic number field of degree [k : Q] = r1 + 2r2 , where k has r1 real and 2r2 complex embeddings. Then ∼ F × C, U(Ok ) = where F is a free abelian group of rank r1 + r2 − 1, and C is a finite cyclic group of roots of unity in k. 119

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The generators of F are called fundamental units. In general, one does not know generic constructions for these units. However, in the special case of the nth cyclotomic field k = Q(ξn ), where ξn denotes a primitive nth root of unity, the cyclotomic units (1 − ξni ) , (1 − ξn ) where i is a natural number greater than one and relatively prime to n, generate a subgroup of finite index in U(Ok ). Note that in this case we have that Ok = Z[ξn ]. The Dirichlet Unit Theorem was generalized to integral group rings of finite abelian groups by Higman (see for example [51, Theorem 2.9]). Theorem 2 (Higman). Let G be a finite abelian group, then U(Z[G]) = ±G × F, where F is a finitely generated free abelian group, the rank of which is described by the order of G and its cyclic subgroups. Unfortunately, there does not exist a general structure theorem covering the group ring case of arbitrary finite groups. Even for specific examples of finite groups G it is a hard problem to describe the full unit group U(Z[G]). Only for few examples one has managed to do this, although we know, by a theorem of Borel and Harish Chandra [6], that for any finite group G the unit group U(Z[G]) is finitely generated (even finitely presented). In fact, the unit group of any order Γ in a finite-dimensional Q-algebra is finitely generated. This follows from the fact that U(Γ) is an arithmetic group (an easy proof can be found in [17, Lemma 2.1]). For a rigorous definition of this notion we refer the reader to an excellent survey of Kleinert [32]. As mentioned above, orders turn out to be very helpful in the investigations of the unit group of an integral group ring. A useful fact in the study of the unit group of general orders Γ is that two Z-orders in a finite-dimensional Q-algebra A have a common subgroup of finite index [51, Lemma 4.2-4.6] in their respective unit groups. Naturally, the algebraic structure of the Q-algebra A plays an important role in the study of the unit group U(Γ). On one hand, one can try to extend known group ring results to this much larger class of rings and, on the other hand, one can apply results obtained for general orders to several classes of important rings, such as matrix rings over group rings and, more generally, semigroup rings. Let us recall the notion of orders in a finite-dimensional Q-algebra. Definition. [51] Let A be a (finite-dimensional) Q-algebra. A subring Γ with the same unity is called an order (or a Z-order) if Γ is a finitely generated Z-submodule so that Γ contains a Q-basis of A.

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As mentioned before, the ring of algebraic integers Ok is an order in an algebraic number field k and the integral group ring Z[G] of a finite group G is an order in the semisimple algebra Q[G]. Of course a matrix ring Mn (Z[G]) is an order in Mn (Q[G]). However the latter fits in the more general context of rings of matrix type. Let R be a ring, I, M nonempty sets and let P be an M × I matrix with entries in R. By M(R, I, M, P ) one denotes the set consisting of all I × M matrices over R with finitely many nonzero entries (by θ we denote the zero matrix). This becomes a ring for the component-wise addition of matrices and multiplication of the elements A, B ∈ M(R, I, M, P ) is defined as A ◦ P ◦ B, where ◦ denotes the ordinary matrix multiplication. The ring T = M(R, I, M, P ) is said to be of matrix type over R and P is called the sandwich matrix . If every row and column of P contains a unit of R, then T is called a Munn ring over R. Note that if i ∈ I, m ∈ M and pmi a unit of R, ∼ then e = (p−1 mi )im is an idempotent of T such that eT e = R. When the sets I and M are finite, say |I| = n and |M | = m, we will denote M(R, I, M, P ) simply as M(R, n, m, P ). If R is the integral group ring Z[G], m = n and P is the n × n-identity matrix, then T = Mn (Z[G]). Denote by eij the elementary matrix with 1 on the (i, j)-entry and zeroes elsewhere, then S = {eij | 1 ≤ i, j ≤ n} ∪ {θ} is a subsemigroup of T and T = Z[S]/Zθ, the quotient of the integral semigroup ring Z[S] of S by the ideal Zθ. The standard notation for the quotient ring Z[S]/Zθ is Z0 [S] and this is called the (integral) contracted semigroup ring of S. For more details we refer the reader to [9]. Actually, a Munn ring with R equal to an integral group ring Z[G] and sandwich matrix P of which the nonzero entries are in G is a contracted semigroup ring Z0 [M(G, I, M, P )] = Z[M(G, I, M, P )]/Zθ, where M(G, I, M, P ) is the subsemigroup of T consisting of the matrices with at most one nonzero entry, and furthermore this entry must belong to G. Semigroups of this type are called completely 0-simple. Let S be a semigroup. An element θ of S is said to be a zero if θs = sθ = θ for all s ∈ S. By S 0 one denotes the smallest semigroup with a zero that contains S. So S 0 = S ∪ {θ} if θ 6∈ S. Similarly, S 1 denotes the smallest semigroup with an identity that contains S. µ ¶ 1 Example. Let G be a group with identity 1 and let P = , then the 1 matrix semigroup M(G, 1, 2, P ) is isomorphic to the semigroup G1 ∪G2 ∪{θ}, where G1 and G2 are isomorphic copies of G such that G1 G2 ⊆ G2 and G2 G1 ⊆ G1 . Integral semigroup rings of finite semigroups form a large class of orders, but contrary to integral group rings, they are not necessarily semiprime. We

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now recall an important result showing that completely 0-simple semigroups are in fact essential building blocks of finite semigroups. For an ideal I of a semigroup S we denote by S/I the Rees factor , that is, the set-theoretic difference S \ I with a zero θ adjoined. The multiplication is naturally induced from S but so that st = θ if st ∈ I. A strictly decreasing series S 0 = S1 ⊃ S2 ⊃ · · · ⊃ Sn ⊃ Sn+1 = {θ} is a principal series if each Si is an ideal of S and there are no ideals of S strictly between Si and Si+1 . µ ¶ Z Z ∼ Example. If S = {e11 , e12 , e22 , θ}, then = Z0 [S] and S has a 0 Z principal series S ⊃ {e22 , e12 , θ} ⊃ {e12 , θ} ⊃ {θ} with Rees factors S/{e22 , e12 , θ} ∼ = {e11 }0 , {e22 , e12 , θ}/{e12 , θ} ∼ = {e22 }0 ∼ and {e12 , θ}/{θ} = {e12 , θ}, a null semigroup. Theorem 3. [9, Theorem 2.40] A finite semigroup always has a principal series and the principal factors are either completely 0-simple or null, that is, a semigroup with zero in which the product of any two elements is zero. Although, for an arbitrary order Γ, the unit group U(Γ) is finitely generated, determining a finite set of (generic) generators seems to be a problem beyond reach. In this paper we give a survey on recent accomplishments on the following topics concerning U(Γ): (1) special subgroups, (2) generators for a subgroup of finite index, (3) orders in quaternion algebras. For a survey on results on the unit group of an integral group ring we refer the reader to [19, 22, 51]. 1. Special subgroups In recent years one has investigated the centre of U(Z[G]) (see for example [29]), the normalizer of the group G in U(Z[G]) (see for example [18, 19, 23, 25, 26, 39, 41, 43]), the finite conjugacy centre (see for example [7, 8, 19, 24, 52, 53]), the second centre and the hypercentre (see for example [1, 19, 20, 21, 35, 36, 37, 38]) of U(Z[G]). It turns out that these subgroups are closely related and that they are connected with the Isomorphism Problem for integral group rings. Recall that the latter asks whether G and H are isomorphic finite groups when Z[G] is isomorphic to Z[H]. Thus to what extent does the integral group ring reflect the properties of the group? Since ±G is a subgroup of U(Z[G]), called the trivial units, one might expect that G can be deduced

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from the unit group of Z[G]. Due to Higman’s theorem we know that this is the case for finite abelian groups. Indeed, the trivial units are the torsion subgroup of the unit group of the integral group ring. Roggenkamp and Scott in [50] gave an affirmative answer to the Isomorphism Problem for the class of finite nilpotent groups. However Hertweck in [18] constructed a counter example using the normalizer of G in U(Z[G]). Hence we indicated the importance of the previously mentioned subgroups of the unit group and thus it is interesting to investigate these in the more general framework of the unit group of orders. In [13] we together with Juriaans investigated the finite conjugacy centre, second centre, and hypercentre of the unit group of a general order Γ in a finite-dimensional Q-algebra A. For a group G, we denote by ∆(G) the finite conjugacy centre of G, that is the characteristic subgroup of G consisting of all elements g with finitely many conjugates, i.e., [G : CG (g)] < ∞. The upper central series of a group G 1 = Z0 (G) ≤ Z1 (G) ≤ Z2 (G) ≤ . . . is defined by Zn+1 (G)/Zn (G) = Z (G/Zn (G)) . The subgroup Zn (G) is called the nth centre of G and this is a characteristic subgroup of G. The centre Z1 (G) we simply denote by Z(G). The union [ Z∞ (G) = Zi (G) i

is called the hypercentre of G. If there exists m ∈ N such that Z∞ (G) = Zm (G) and m is the smallest possible number with this property, then m is called the central height of G. For a finite-dimensional Q-algebra A, the finite conjugacy centre of U(A) always is central [11], while for orders the situation is different, as shown for example in [24]. Indeed, Jespers and Juriaans proved that for a periodic group G, ∆(U(Z[G])) is central except if G is a Q∗ -group. Recall that G is a Q∗ -group if G has an abelian subgroup A of index 2 and if there exists an a ∈ A of order 4 such that for all h ∈ A and for all g ∈ G \ A, g 2 = a2 and g −1 hg = h−1 . For a Q∗ -group G the finite conjugacy centre of U(Z[G]) is central-bycyclic. Motivated by this result, we investigated what causes the noncentrality of the finite conjugacy centre of the unit group of an order Γ in a finite-dimensional Q-algebra A. Because of the following result of

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Dokuchaev, Juriaans, Polcino Milies and Sobral Singer [11] the obstacle comes from homomorphic images of A that are division rings. Lemma 4. [11] Let K be the field of fractions of an infinite domain D. Let Γ be a subring of a K-algebra A so that Γ is a finitely generated D-module and KΓ = A. Then the elements of ∆(U(Γ)) commute with nilpotent elements of A. In order to formulate the next result we first need to define some terminology and notation. The Wedderburn-Malcev Theorem states that A decomposes, as a vector space, as the sum of a semisimple subring S(A) and the Jacobson radical J (A) of A. We can write A = T (A) × I(A),

Q

with T (A) = e S(A)e, where e runs through the primitive central idempotents of S(A) so that eJ (A) = {0} = J (A)e, and J (A) ⊆ I(A). For a ∈ A = S(A) ⊕ J (A) we denote by aS the natural image of a in S(A). Hence, for an order Γ in A one can write Γ ⊆ ΓS + J (A), a sum of Z-modules, and ΓS = {γS | γ ∈ Γ}. Furthermore, Γ ⊆ ΓT × ΓI , a direct product of orders, where ΓT (respectively ΓI ) is the natural projection of Γ in T (A) (respectively I(A)). For a primitive central idempotent e of S(A) we denote by ae the element (aS )e and Γe = {(γS )e | γ ∈ Γ}. Note that if Γ is a Z-order in A, then ∆(U(Γ)) ⊆ ∆(U(ΓT )) × ∆(U(ΓI )). Hence to determine, up to finite index, the finite conjugacy centre of U(Γ) it is sufficient to deal separately with the cases that Γ is an order in a semisimple Q-algebra and Γ = ΓI , that is J (Γ) 6= {0}. Because of the lemma the finite conjugacy centre of U(Γ) commutes with matrix rings of S(A) and with the Jacobson radical J (A). It turns out that, in case A is semisimple, the existence of homomorphic images of A that are totally definite quaternion algebras is crucial. We recall the definition. Definition. Let R be a commutative ring and let a, b ∈ R. The algebra µ ¶ a, b = R[i, j | i2 = a, j 2 = b, ji = −ij] R is called a quaternion algebra over R. When a = b = −1 we denote this algebra by H(R) and call it the classical quaternion algebra over R.

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³ A quaternion algebra definite if

a,b k

µ

125

´ over an algebraic number field k is totally a, b k

¶ ⊗k R = H(R).

In the nonsemisimple case things turn out to be different. Theorem 5. [13] (1) Let A be a semisimple finite-dimensional Q-algebra and let Γ be an order in A. (a) If A does not have homomorphic images that are totally definite quaternion algebras, then ∆(U(Γ)) = Z(U(Γ)). (b) If e1 , · · · , en are the primitive central idempotents of A, then ∆(U(Γ)) ⊆

n Y

∆(U(Γ))ei

i=1

and if ∆(U(Γ))ei is not central, then it has a subgroup of finite index that is central in the totally definite quaternion algebra Aei . (c) In particular, ∆(U(Γ)) is finitely generated central-by-finite. (2) Let A be a finite-dimensional Q-algebra so that T (A) = {0} (and thus J (A) 6= {0}) and let Γ be an order in A. (a) ∆(U(Γ)) is finitely generated abelian. (b) ∆(U(Γ)) is not central in U(Γ) if and only if there exists a j ∈ Z(J (Γ)) such that 1 < [U(ΓS ) : CU (ΓS ) (j)] < ∞. (c) Γ1 , Γ2 orders in A, then ∆(U(Γ1 )) = Z(U(Γ1 )) if and only if ∆(U(Γ2 )) = Z(U(Γ2 )). The proof of this theorem makes essential use of the following result of Sehgal and Zassenhaus [53]. Proposition 6. [51, Lemma 21.3] Let Γ be an order in a finitedimensional noncommutative division Q-algebra D. Let S be a subgroup of finite index in U(Γ). (1) If the Q-subalgebra QS generated by S is properly contained in D, then D is a totally definite quaternion algebra. (2) Z(U(Γ)) is of finite index in U(Γ) if and only if D is a totally definite quaternion algebra.

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Concerning the second centre and hypercentre of the unit group of the integral group ring of a periodic group G, we have that the results of Jespers and Juriaans, and Li and Parmenter [24, 36] imply that these groups equal the finite conjugacy centre of the unit group. For general orders this no longer remains valid. Although Z2 (U(Γ)) behaves quite similarly to ∆(U(Γ)) (it also commutes with matrix rings of S(A)), it does not necessarily commute with J (A). Example. Consider the semigroup S = {e12 , e13 , e23 } ∪ {θ}, where e12 , e13 , e23 are matrix units in M3 (Q). Then the semigroup ring Γ = Z0 [S 1 ] is an order in Q0 [S 1 ] with J (Γ) = Z0 [S]. Note that J (Γ)3 = {0}. Furthermore, U(Γ) = 1 + J (Γ) and Z(U(Γ)) = 1 + Ze13 ⊇ 1 + J (Γ)2 . As 1 + J (Γ)/1 + J (Γ)2 is abelian, we have that U(Γ) = Z2 (U(Γ)). Hence the elements of Z2 (U(Γ)) do not commute with J (Γ). Therefore, under the assumptions stated, Lemma 4.1 and Theorem 4.3 in [13] are not valid. In the proof given one makes use of [13, Lemma 2.2]. However, in order to apply the latter one needs that Z(A) is semiprime. We now state the corrected version of the mentioned lemma and theorem. As a consequence some modifications also have to be made to Corollary 5.2 and Corollary 5.3 in [13] (see Theorem 10 and Corollary 11). Lemma 7. [13] Let k be a field of characteristic zero and let A be an algebraic k-algebra with Z(A) semiprime. Let u ∈ U(A). If (u, 1+a) ∈ Z(A) for all unipotent elements 1 + a in A, then u commutes with nilpotent elements of A. It follows that if A is a finite-dimensional Q-algebra so that Z(A) is semiprime and if Γ is a Z-order in A, then the elements of Z2 (U(Γ)) commute with nilpotent elements of A. In case A is generated by U(Γ) as a Q-algebra, one obtains that the obstruction for an element of Z2 (U(Γ)) to be central in U(Γ) again comes from homomorphic images of A that are totally definite quaternion algebras. Theorem 8. [13] Let Γ be a Z-order in a finite-dimensional Q-algebra A so that Z(A) is semiprime. Assume that A is generated as a Q-algebra by U(Γ). Then (1) if T (A) does not have epimorphic images that are totally definite quaternion algebras, then Z2 (U(Γ)) = Z(U(Γ)).

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Q Q (2) Z2 (U(Γ)) ⊆ e Z2 (U(Γe )) × f Z(U(Γf )) × Z(U(ΓI )), where e runs through all primitive central idempotents of T (A) so that T (A)e is a totally definite quaternion algebra and where f runs through all primitive central idempotents of T (A) so that T (A)f is not a totally definite quaternion algebra. In particular, the central height of U(Γ) is bounded by the central height of all U(Γe ). (3) Z2 (U(Γ)) ⊆ ∆(U(Γ)) and in particular Z2 (U(Γ)) is a finitely generated central-by-finite group (central in U(Γ)). (4) if Z2 (U(Γ)) = ∆(U(Γ)), then Z2 (U(Γ)) = Z3 (U(Γ)) and ∆(U(ΓI )) is central. We now apply our results on the finite conjugacy centre and the hypercentre to semigroup algebras. Let S be a finite semigroup. Assume that Z[S] has a unity 1 and that the finite semigroup S 0 has a principal series S 0 = S1 ⊃ S2 ⊃ · · · ⊃ Sn ⊃ Sn+1 = {θ}, so that if Sj /Sj+1 is completely 0-simple, then it is isomorphic to M(Gj , nj , nj , Pj ), a square completely 0-simple semigroup, and Gj a finite group so that Pj is invertible in Mnj (Q[Gj ]). The background given below shows that the latter holds for example if Q[S] is semisimple or if S is Malcev nilpotent. Theorem 9. [45, Theorem 14.24] Let k be a field and S a finite semigroup. The following are equivalent: (1) k[S] is semisimple, (2) S 0 has a series of ideals S 0 = S1 ⊃ S2 ⊃ · · · ⊃ Sn ⊃ Sn+1 = {θ}, with each principal factor Si /Si+1 ∼ = M(Gi , ni , ni , Pi ), a square completely 0-simple semigroup and Gi a finite group so that char(k) |Gi | and Pi is invertible in Mni (k[Gi ]). Definition. [40, 44, 45] Let S be a semigroup. Consider for x, y in S and w1 , w2 , . . . in S 1 the sequence defined inductively as follows: x0 = x, y0 = y, and for n ≥ 0 xn+1 = xn wn+1 yn , yn+1 = yn wn+1 xn . If xn = yn for all x, y ∈ S and w1 , w2 , . . . in S 1 and n is the least positive integer with this property, then S is said to be Malcev nilpotent of class n. Finite Malcev nilpotent semigroups have been classified by Okni´ nski in [46]. Also note that a group is Malcev nilpotent of class n if and only if it is nilpotent of class n in the classical sense [45, Theorem 7.2]. The only Malcev nilpotent completely 0-simple semigroups [28, Lemma 2.1] are the Brandt semigroups, i.e., semigroups of the form M(G, M, M, I|M | ), where I|M | denotes the M × M -identity matrix.

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Theorem 10. [13] Let S be a finite semigroup with principal factors either null semigroups or completely 0-simple semigroups of the type M(G, n, n, P ) with sandwich matrix P invertible in Mn (Q[G]). Suppose Z[S] has a unity 1. Then ∆(U(Z[S])) ⊆ HCA, where H, C, and A are subgroups of the unit group of an order in Q[S] with H central in Q[S], C the direct product of cyclic groups of order 4 (one for each principal factor that is a Q∗ -group that is not an Hamiltonian 2-group) and Hamiltonian 2-groups (one for each principal factor that is an Hamiltonian 2-group) and A a finitely generated abelian group that is contained in ∆(U(Z[S]I )). In particular, these results imply the following extension of the group ring result [24, 36]. Corollary 11. [13] Let S be a finite semigroup with principal factors either null semigroups or completely 0-simple semigroups of the type M(G, n, n, P ) with sandwich matrix P invertible in Mn (Q[G]). Suppose Z[S] has a unity 1 and that Z(Q[S]) is semiprime. Then (with notations as in Theorem 10) Z2 (U(Z[S])) ⊆ HC(A ∩ Z(Q[S])) and the central height of U(Z[S]) is at most two. If ∆(U((Z[S])I )) is central (for example if Q[S] is semisimple), then ∆(U(Z[S])) = Z2 (U(Z[S])) = Z3 (U(Z[S])). Moreover, these groups are central if and only if S contains no principal factors that are Q∗ -groups. 2. Generators for a subgroup of finite index Since the construction of finitely many generators for U(Γ) seems out of reach today, one might compromise by searching for a finite set of generators of a subgroup of finite index in the unit group. This question was stated for integral group rings as Problem 23 in [51]. When G is a finite abelian group, Bass and Milnor [3] constructed such a set of generators, called the Bass cyclic units (we recall the definition later). This construction is based on the idea of the cyclotomic units of Z[ξn ], where ξn denotes a primitive nth root of unity. Recently, Marciniak and Sehgal in [42] constructed new generic units in abelian group rings. Unfortunately, there are not many recipes for constructing units. Apart from the Bass cyclic units, there are the bicyclic units (we recall the definition later). Ritter and Sehgal [49] have shown that the Bass cyclic and the bicyclic units (of the first type) generate a subgroup of finite index in U(Z[G]) for many nilpotent groups G. More generally, Jespers and Leal [27]

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showed that if G does not have a nonabelian fixed-point free homomorphic image and if Q[G] does not have a simple Wedderburn-Artin component that is a 2 × 2-matrix ring over the rationals, over an imaginary quadratic extension of the rationals or over a noncommutative division algebra, then the Bass cyclic and the bicyclic units generate a subgroup of finite index in U(Z[G]). These components (together with noncommutative division algebras other than a totally definite quaternion algebra) are called exceptional components and the ultimate reason for excluding the matrices is that the Congruence Subgroup Theorem [2, 4, 55] fails for 2 × 2-matrices over maximal orders in the mentioned division rings. Recall that a group G is said to be fixed-point free if it has a complex irreducible representation ρ such that for every nonidentity element g of G, ρ(g) has all eigenvalues different from one. Note that if G is a finite group such that it does not have a nonabelian homomorphic image that is fixed-point free, then we have that Q[G] has no noncommutative division ring as a simple component [48, p. 333]. We recall some general procedures to obtain generators for a subgroup of finite index in the unit group of an order Γ of a semisimple Q-algebra A. For each nonabelian simple component of A that is not a division algebra a noncentral idempotent f is needed and one considers the following units vf,h = 1 + n2f (1 − f )hf

0 and vf,h = 1 + n2f f h(1 − f ),

where nf is a positive integer so that nf f ∈ Γ. Based on the ideas of Jespers and Leal [27] and the earlier results of Ritter and Sehgal [49], Bhandari and Ritter [5] showed that for an order Γ in a finite-dimensional semisimple Q-algebra A the central units together 0 with the finite set of units vf,h and vf,h (where f runs through a finite set of noncentral idempotents of A and h runs through a finite set of additive generators of Γ) generate a subgroup of finite index in U(Γ) in case A has no exceptional components. Instead of using all central units, for some important classes of orders Γ in semisimple Q-algebras A, one can work with units of the following type  ϕ(n) i−1 X 1 − iϕ(n) + u(x, i) = uΓ (x, i) = 1 + xj  x ˆ n j=1

where x ∈ Γ is such P that the cyclic semigroup generated by x is a group n−1 i hxi of order n, x b = i=0 x , (i, n) = 1, 1 < i < n and ϕ is the Euler phi function. Note that the identity e of hxi is an idempotent that is not necessarily the unity 1 of the order Γ. So for x one can take for example any idempotent of Γ. With such an element x there always is associated a natural idempotent x e = n1 x ˆ. Ritter and Bhandari [5] have shown that one always can construct noncentral idempotents f of such form. Jespers and Leal [27] have shown

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that for the integral group ring of a finite group G the idempotent f can be chosen of this type but with x ∈ G, provided that G does not have a nonabelian fixed-point free homomorphic image and Q[G] does not have simple components that are 2 × 2-matrix rings of the type mentioned above. If x ∈ Γ and f = x e, then vf,t = 1 + (1 − x)βtˆ x,

0 vf,t =1+x ˆtβ(1 − x),

for some β ∈ Zhxi and t ∈ Γ. Put ux,t = 1 + (1 − x)tˆ x,

u0x,t = 1 + x ˆt(1 − x).

Assume now that Γ is the integral semigroup ring Z[S] (with unity 1) of a finite semigroup S. Definition. (1) The elements u(s, i), with s ∈ S and hsi a group of order n, (i, n) = 1, 1 < i < n, are called the Bass cyclic units of Z[S]. (2) The elements us,t and u0s,t with s, t ∈ S and hsi a group are called the bicyclic units of first respectively second type of Z[S]. Example. Let Γ = Mm (Z[G]), where G is a finite group. Then Γ is the contracted semigroup ring Z0 [S], where S = {eij | 1 ≤ i, j ≤ m} ∪ {θ}. Take s = ge11 and t = he11 , with g, h ∈ G, i such that (i, n) = 1, 1 < i < n, where n is the order of g in G. Then     u(g, i) 0 . . . 0 ug,h 0 . . . 0   0 0 1 ... 0  1 ... 0      u(s, i) =  and u =  .. .. .. ..  .. ..  . s,t    . . . . . .  0

0 ...

1

0

0 ...

1

Theorem 12. [12, 27, 30] If Q[S] is semisimple and has no exceptional components, then the group generated by the Bass cyclic units and the units vfi ,h , vf0 i ,h , where h ∈ S and fi a noncentral idempotent in every simple component that is not a division ring, is of finite index in U(Z[S]). Furthermore, suppose that any maximal subgroup G of S does not have a nonabelian fixed-point free homomorphic image, then the unit group U(Z[S]) contains a subgroup of finite index generated by the Bass cyclic and bicyclic units. This theorem immediately can be applied to Mm (Z[G]) and in this way one recovers a result of Lee and Sehgal [34]. In [12] explicit generators for a subgroup of finite index in the unit group of an integral semigroup ring Z[S] are given and this for a large class of finite semigroups for which Q[S] is not necessarily semisimple. Of course, throughout the section we assume

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that Z[S] has a unity 1. In order to construct such a set one needs additive generators for an additive subgroup of finite index in the Jacobson radical J (Z[S]). This can be done in an algorithmic manner by a method of Drazin [14]. However for Malcev nilpotent semigroups this can be done in a generic way. Theorem 13. [12] Let S be a finite semigroup. Assume that Q[S] does not have a homomorphic image that is a 2 × 2-matrix ring over either the rationals, an imaginary quadratic extension of the rationals or a noncommutative division ring. Further, assume that if G is a maximal subgroup of S so that Q[G] and Q[S] have an isomorphic noncommutative simple homomorphic image, then G does not have a nonabelian homomorphic image that is fixed-point free. Then the group generated by the following units is of finite index in U(Z[S]): (1) the Bass cyclic units, (2) the bicyclic units, and (3) the units of the form 1 + x, where x runs through a finite multiplicatively closed set of additive generators of the Jacobson radical J (Z[S]) of Z[S] (or of some additive subgroup of finite index). As mentioned above, for Malcev nilpotent semigroups the additive generators of the Jacobson radical can be explicitly stated. To do so, we introduce some notation. Let S be a finite semigroup and let S 0 = S0 ⊃ S1 ⊃ · · · ⊃ Sn ⊃ {θ} = Sn+1 be a descending chain of ideals so that Sj /Sj+1 is either power nilpotent (by joining consecutive null semigroups) or a completely 0-simple semigroup. In the latter case we put X ej = f, f ∈Sj \Sj+1 , f =f 2

the sum of all idempotents in Sj \ Sj+1 . When Sj /Sj+1 is a power nilpotent semigroup we put ej = 0; also put en+1 = 0. Theorem 14. [12] Let S be a finite Malcev nilpotent semigroup. Assume that Q[S] does not have a homomorphic image that is a totally definite quaternion algebra or a 2×2-matrix ring over either the rationals, an imaginary quadratic extension of the rationals or a noncommutative division ring. Then the group generated by the following units is of finite index in U(Z[S]): (1) the Bass cyclic units, (2) the bicyclic units, and (3) the units 1 + (1 − en+1 )(1 − en )(1 − en−1 ) . . . (1 − ej )s with s ∈ Sj−1 \ Sj where Sj−1 /Sj is power nilpotent.

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3. Orders in quaternion algebras As the results in the previous sections indicate, describing the unit group of an order in a finite-dimensional noncommutative division Q-algebra D is a fundamental problem. Surprisingly, very little seems to be known, even in case D is a classical order such as a classical quaternion algebra H(Ok ), where Ok is an order in a number field k. Even describing some generic classes of units in these groups is hard. We refer the reader to an excellent book and article of Kleinert [32, 33], where the author mainly concentrates on the case where the noncommutative division algebra D splits over R. In this section we explain the first example obtained in [10] of a characterization of the unit group of a group ring that has an epimorphic image that is an order in a noncommutative division algebra that is not a totally definite quaternion algebra. Let Q8 denote the quaternion group of order 8. Clearly √ √ √ Q( −7)[Q8 ] = 4Q( −7) ⊕ H(Q( −7)) and thus

√ ¸ √ ¸ √ ¸¶ · µ · −7 1 + −7 1 + −7 Z [Q8 ] ⊆ 4Z ⊕H Z . 2 2 2 ³ h √ i´ As U Z 1+ 2 −7 = {1, −1} it follows that √ ¸ √ ¸¶¶ µ · ¶ µ µ · 1 + −7 1 + −7 4 U Z [Q8 ] ⊆ {1, −1} × U H Z . 2 2 h √ i Hence determining the unit group of the group ring Z 1+ 2 −7 [Q8 ] is re·

1+

√

duced to describing the unit group of H(Z[ 1+ 2 −7 ]). √

√

Let ω = 1+ 2 −7 , ω ¯ = 1− 2 −7 , R = Z[ω] a unique factorization domain with field of fractions k = Q(ω). Let n : H(k) −→ k denote the classical norm on the quaternion algebra. By H0 (k) we denote the imaginary part of H(k). Let SL1 (H(R)) = {x ∈ H(R) | n(x) = 1}. Clearly U(H(R))/SL1 (H(R)) is a subgroup of U(R) = {1, −1}. Hence to describe U(H(R)) it is sufficient to describe SL1 (H(R)). To accomplish this one makes use of the fact that · ¸ x y H(k) → M2 (C) : x + yj 7→ −σ(y) σ(x) is an embedding, with x, y in the maximal subfield k[i] of H(k) and σ(x) = jxj −1 . Hence SL1 (H(R)) ⊆ SL2 (C) and thus SL1 (H(R)) acts on the threedimensional hyperbolic space H 3 = {z + rj | z ∈ C, r ∈ R+ }. Because k has exactly one pair of complex embeddings it follows from [15, Theorem 10.1.2] that SL1 (H(R)) is a discrete group with a compact

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fundamental domain that has finite volume. Therefore the Poincar´e method [47] gives a theoretical way for obtaining a presentation for SL1 (H(R)). In [10] a finite algorithm is given to execute this procedure. This resulted in the following presentation. √

Proposition 15. Let R = Z[ 1+ 2 −7 ]. The group SL1 (H(R)) has the presentation defined by the generators J, P, R, Cl , Cr , Sl , Sr , Vl , Vr , Wl , Wr , Chl , Chr , Dl , Dr and the relations P 2 = R2 = Cl2 = Cr2 = Sl2 = Sr2 = Vl2 = Vr2 = Wl2 = Wr2 = Ch2l = Ch2r = Dl2 = Dr2 = J, Sl Cl Vl Chl Wl Dl = Sr Cr Vr Chr Wr Dr = P Dr RDl = Vl Wl Sl = Sr Wr Vr = Cr P Cl = Chr RChl = 1, (Sl P Sr P )2 = (Wl RWr R)2 = (Vl Vr )2 = (P R)2 = J

and

J 2 = 1.

For y ∈ U(H(k)), let τ (y) denote the orthogonal 3×3-matrix (with respect to the norm n and the standard basis i, j, k of H0 (k)) associated to the isometry τy : H0 (k) −→ H0 (k) defined by τy (x) = yxy −1 . A well-known theorem of Cartan-Dieudonn´e says that the sequence τ

1 → k ∗ → U(H(k)) → SO3 (k) → 1 is exact, where k ∗ = k \ {0}. Restricting this sequence to R[1/2], using that the Picard group of R[1/2] is trivial, using [16, 7.2.20] and then further restricting to R one obtains the sequence τ

1 → U(R) → U(H(R)) → SO3 (R). Note that because of a general theorem on arithmetic groups (see for example [54]) one knows that τ (U(H(R))) is a subgroup of finite index in SO3 (R). As a consequence one obtains a description for this orthogonal group. √

Proposition 16. [10] Let R = Z[ 1+ 2 −7 ] = Z[ω] and let x = τ ((ω + i)U(H(R))), y = τ ((¯ ω + i)U(H(R))), 1 u = τ (( (1 + i + j + k))U(H(R))) 2 and v = x−1 ux. Then τ (U(H(R))) is a normal subgroup of SO3 (R) and SO3 (R)/τ (U(H(R))) = hx, y, ui = hu, vi o hx, yi ' Z23 o Z22 and the action of hx, yi on hu, vi is given by ux = v, v x = u, uy = v 2 and v y = u2 .

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The direct product of Q8 with the cyclic group C7 of order 7, Q8 × C7 , is the group of least order for which no finite set of generators for a subgroup of finite index in the unit group of its integral group ring is known (see [22, 51]). It is readily verified that, U(Z[Q8 × C7 ]) ⊆ {1, −1}4 × ±Q8 × U(Z[ξ7 ]) × U(H(Z[ξ7 ])), where ξ7 is a primitive 7th root of unity. Hence the study of U(Z[Q8 × C7 ]) √ is reduced to that of U(H(Z[ξ7 ])). Clearly U(H(Z[ 1+ 2 −7 ])) ⊆ U(H(Z[ξ7 ])). Since there are 6 complex embeddings of Q(ξ7 ) the calculations in case of U(H(Z[ξ7 ])) remain a challenge. Acknowledgments Research partially supported by the Onderzoeksraad of Vrije Universiteit Brussel, Fonds voor Wetenschappelijk Onderzoek-Vlaanderen. Dr. Dooms is Postdoctoral Fellow of the Fonds voor Wetenschappelijk Onderzoek-Vlaanderen. References [1] S.R. Arora and I.B.S. Passi, Central height of the unit group of an integral group ring, Comm. Algebra 21 (1993), no. 10, 3673–3683. [2] A. Bak and U. Rehmann, The congruence subgroup and metacyclic problems for SLn (n ≥ 2) of division algebras, J. Algebra 78 (1982), 475–547. [3] H. Bass, The Dirichlet unit theorem, induced characters and Whitehead groups of finite groups, Topology 4 (1966), 391–410. [4] H. Bass, J. Milnor, and J.P. Serre, Solution of the congruence subgroup problem for SLn (n ≥ 3) and Sp2n (n ≥ 2), Publ. Math. I.H.E.S. 33 (1967), 59–137. [5] A. Bhandari and J. Ritter, Large subgroups in the unit groups of arithmetic orders, J. Algebra 178 (1995), no. 2, 512–529. [6] A.A. Borel and H. Chandra, Arithmetic subgroups of algebraic groups, Ann. of Math. 75 (1962), 485–535. [7] A.A. Bovdi, The periodic normal divisors of the multiplicative group of a group ring I, Sibirsk Mat. Z. 9 (1968), 495–498. [8] A.A. Bovdi, The periodic normal divisors of the multiplicative group af a group ring II, Sibirsk Mat. Z. 11 (1970), 492–511. [9] A.H. Clifford and G.B. Preston, The Algebraic Theory Of Semigroups. Vol. I, Mathematical Surveys, no. 7, American Mathematical Society, Providence, R.I. 1961. ´ del R´ıo, Presentations of the unit group of [10] C. Corrales, E. Jespers, G. Leal, and A. an order in a non-split quaternion algebra over the ring of integers of an imaginary quadratic number field, Adv. Math. 186 (2004), 498–524. [11] M.A. Dokuchaev, S.O. Juriaans, C. Polcino Milies, and M.L. Sobral Singer, Finite conjugacy in algebras and orders, Proc. Edinb. Math. Soc. (2) 44 (2001), no. 1, 201– 213. [12] A. Dooms and E. Jespers, Generators for a subgroup of finite index in the unit group of an integral semigroup ring, J. Group Theory 7 (2004), no.4, 543–553. [13] A. Dooms, E. Jespers, and S.O. Juriaans, Units in orders and integral semigroup rings, J. Algebra 265 (2003), no. 2, 675–689. [14] M. P. Drazin, Structure matrices of algebras, J. Algebra 87, (1984), no. 1, 247–260.

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[15] J. Elstrodt, F. Grunewald, and J. Mennicke, Groups Acting On Hyperbolic Space (Harmonic Analysis and Number Theory), Springer, 1998. [16] A.J. Hahn and O.T. O’Meara, The classical groups and K-theory, Grundlehren der mathematischen Wissenschaften 291, Springer-Verlag, Heidelberg, 1989. [17] B. Hartley, Free groups in normal subgroups of unit groups and arithmetic groups, Contemporary Mathematics, 93 (1989), 173–177. [18] M. Hertweck, A counterexample to the isomorphism problem for integral group rings, Ann. of Math. (2) 154 (2001), no. 1, 115–138. [19] M. Hertweck, Contributions to the integral representation theory of groups, (http://elib.uni-stuttgart.de/opus), 2003, Habilitationsschrift. [20] M. Hertweck, Hypercentral units in integral group rings, preprint. [21] E. Iwaki, E. Jespers, and S.O. Juriaans, The hypercentre of the unit group of an integral group ring, J. Algebra Appl., to appear. [22] E. Jespers, Units In Integral Group Rings: A Survey, Methods in ring theory (Levico Terme, 1997), 141–169, Lecture Notes in Pure and Appl. Math., 198, Dekker, New York, 1998. [23] E. Jespers and S.O. Juriaans, Isomorphisms of integral group rings of infinite groups, J. Algebra 223 (2000), no. 1, 171–189. [24] E. Jespers and S.O. Juriaans, The finite conjugacy centre of the unit group of orders in algebras, J. Group Theory 6 (2003), no. 1, 93–102. [25] E. Jespers, S.O. Juriaans, J.M. de Miranda, and J.R. Rogerio, A note on the normalizer problem, Mat. Contemporanea 21 (2001), 117–130. [26] E. Jespers, S.O. Juriaans, J.M. de Miranda, and J.R. Rogerio, On the normalizer problem, J. Algebra 247 (2002), no. 1, 24–36. [27] E. Jespers and G. Leal, Generators of large subgroups of the unit group of integral group rings, Manuscripta Math. 78 (1993), 303-315. [28] E. Jespers and J. Okni´ nski, Nilpotent semigroups and semigroup algebras, J. Algebra 169 (1994), no. 3, 984–1011. [29] E. Jespers, M.M. Parmenter, and S.K. Sehgal, Central units of integral group rings of nilpotent groups, Proc. Amer. Math. Soc. 124 (1996), no. 4, 1007–1012. [30] E. Jespers and D. Wang, Units of integral semigroup rings, J. Algebra 181 (1996), 395–413. [31] G. Karpilovsky, Unit Groups Of Group Rings, Longman Scientific and Technical Press, 1988. [32] E. Kleinert, Units of classical orders: a survey, Enseign. Math. (2) 40 (1994), no. 3-4, 205–248. [33] E. Kleinert, Units In Skew Fields, Progress in Math. 186, Birkh¨ auser Verlag, Basel, 2000. [34] G. Lee and S.K. Sehgal, Generators of subgroups of finite index in GLm (ZG), Advances in ring theory (Granville, OH, 1996), 211–219, Trends Math., Birkh¨ auser Boston, Boston, MA, 1997. [35] Y. Li, The hypercentre and the n-centre of the unit group of an integral group ring, Canad. J. Math. 50 (1998), no. 2, 401–411. [36] Y. Li and M.M. Parmenter, Hypercentral units in integral group rings, Proc. Amer. Math. Soc. 129 (2001), no. 8, 2235-2238 (electronic). [37] Y. Li and M.M. Parmenter, Some results on hypercentral units in integral group rings, Comm. Algebra, 31 (2003), no.7, 3207-3217. [38] Y. Li and M.M. Parmenter, The upper central series of the unit group of an integral group ring, preprint. [39] Y. Li, M.M. Parmenter and S.K. Sehgal, On the normalizer property for integral group rings, Comm. Algebra, 27 (1999), no.9, 4217-4223.

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[40] A.I. Malcev, Nilpotent semigroups, Uc. Zap. Ivanovsk. Ped. Inst. 4 (1953), 107–111. (In Russian) [41] Z.S. Marciniak and K.W. Roggenkamp, The normalizer of a finite group in its integral group ring and C˘ ech cohomology, Algebra-Representation theory (Constanta 2000), 159–188, NATO Sci. Ser. II Math. Phys. Chem. 28 Kluwer Acad. Publ., Dordrecht 2001. [42] Z.S. Marciniak and S.K. Sehgal, Generic units in abelian group rings, preprint 2004. [43] M. Mazur, The normalizer of a group in the unit group of its group ring, J. Algebra 212 (1999), no. 1, 175–189. [44] B.H. Neumann and T. Taylor, Subsemigroups of nilpotent groups, Proc. Roy. Soc, Ser.A 274 (1963), 1–4. [45] J. Okni´ nski, Semigroup Algebras, Marcel Dekker, 1991. [46] J. Okni´ nski, Nilpotent semigroup of matrices, Math. Proc. Camb. Phil. Soc. 120 (1996), no. 4, 617–630. [47] H. Poincar´e, M´emoire sur les groupes klein´ees, Acta. Math. 3 (1883) 49–92. [48] J. Ritter and S.K. Sehgal, Generators of subgroups of U(ZG), Contemp. Math. 93 (1989), 331–347. [49] J. Ritter and S.K. Sehgal, Construction of units in integral group rings of finite nilpotent groups, Trans. Amer. Math. Soc. 324 (1991), 603–621. [50] K.W. Roggenkamp and L.L. Scott, Isomorphisms for p-adic group tings, Ann. Math. 126 (1987), 593–647. [51] S.K. Sehgal, Units In Integral Group Rings, Longman Scientific and Technical Press, Harlow, 1993. [52] S.K. Sehgal and H.J. Zassenhaus, Group rings whose units form an FC-group, Math. Z. 153 (1977), 29–35. [53] S.K. Sehgal and H.J. Zassenhaus, On the supercentre of a group and its ring theoretic generalization, Lecture Notes in Mathematics no. 882, Springer-Verlag, Berlin, 1981, in Integral Representations and its Applications, 117–144. [54] J.P. Serre, Arithmetic groups in homological group theory (Proc. Sympos., Durham, 1977), pp.105–136, London Math. Soc. Lecture Note Ser., 36, Cambridge Univ.Press, Cambridge-New York, 1979. [55] L.N. Vaserstein, The structure of classic arithmetic groups of rank greater than One, Math. USSR Sbornik 20 (1973), 465–492.

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Idempotents in group algebras and coding theory Raul A. Ferraz, Val´ eria O. Luchetta, and C´ esar Polcino Milies Instituto de Matem´atica e Estat´ıstica, Universidade de S˜ao Paulo, Caixa Postal 66281, S˜ao Paulo, SP, 05315-970, Brasil [email protected] [email protected] [email protected] Abstract. In this survey, we consider the semisimple group algebra of a finite abelian group A over a finite field Fq and we give conditions for the number of simple components to be minimum. Under these conditions, we show how to compute the set of primitive idempotents directly, from the subgroup structure of A. Similar results are obtained for group algebras of Dihedral groups. In the case of cyclic codes, a simple way of computing generating polynomials and dimensions is given.

1. Introduction The structure of group rings provides an efficient setting for compact storage of error-correcting codes and for developing fast encoding and decoding algorithms. It is well known that all classical codes can be regarded as ideals in several classes of group rings. The reader may consult, for example, [7], [15], [16], [18], [19], [20], [22], [24], etc. In this paper, we shall consider abelian codes in the semisimple case, survey some known results, and show how to determine all minimal abelian codes, their dimensions, and weights in an interesting situation: when the number of simple components of the group algebra is minimal. In the last section we consider similar results in the case of finite dihedral groups. 2. Basic facts In a general setting, we can define a code in the following way: a q-ary code C is a subset of sequences of length n of symbols taken from a finite set A with q elements, which is called the alphabet of the code. The sequences in C are called its words. 137

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The main objective of the theory of error-correcting codes is to detect when a word, transmitted through a channel, has been received with some error and, more important, to be able to correct that error. To do so, one needs to formalize the notion of proximity among words. So, we define the Hamming distance between two code words v = (v0 , v1 , . . . , vn−1 ) and w = (w0 , w1 , . . . , wn−1 ) to be the number of coordinates in which they differ; i.e., d(v, w) = |{i | vi 6= wi , 0 ≤ i ≤ n − 1 }|. The weight or minimum distance of a C is the number: w(C) = min{d(v, w) | v, w ∈ C}. As we are interested in having some auxiliary algebraic structure, the alphabet is chosen to be a finite field Fq , so the number of its elements q is then a power of a certain prime p. Also, the set of code words is taken to be a linear subspace of the vector space Fnq and, in this case, the code is referred to as being a linear code. An important class of codes is defined as follows. A linear code C is called a cyclic code if for every word v = (v0 , v1 , . . . , vn−1 ) ∈ C, the vector (vn−1 , v0 , . . . , vn−2 ) obtained from v by the cyclic shift of coordinates i 7→ i + 1, taken modulo n, is also in C. Cyclic codes are some of the most useful codes. In particular, Golay codes, Reed-Solomon codes and the binary Hamming codes can all be realized as cyclic codes. We remark that the vector space Fnq can be constructed in different ways. In particular, the quotient ring Rn =

Fq [X] hX n − 1i

is an n-dimensional vector space over Fq and it is easy to see that the obvious isomorphism ψ : Fnq → Rn maps cyclic codes to ideals of Rn . Similarly, if G is a cyclic group of order n, then the group algebra Fq G is also an n-dimensional vector space over Fq and it is well known that it is isomorphic to Rn ; thus, cyclic codes can also be realized as ideals in Fq G. Moreover, several types of cyclic codes can be constructed as ideals in other group algebras. Given an ideal I of Rn , it is clear that there is a unique monic polynomial g ∈ Fq [X], which is a divisor of X n − 1, such that I is generated by the class g in Rn . We shall refer to it as the generating polynomial of I. Also, the polynomial h = (X n −1)/g is such that the ideal annihilated by h is precisely I and will be called the check polynomial of I. Also, if gcd(q, n) = 1, then Rn is semisimple and there is an idempotent ε ∈ I, which is the identity element in I such that I = Rn ε. The polynomial e ∈ Fq [X] of minimum degree such that e = ε will be called the idempotent generator of I. The corresponding elements in Fq G are given by g(a), h(a), and e(a).

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Since g and h are relatively prime, there exist polynomials r, s ∈ Fq [X] such that rg + sh = 1. If one of the polynomials g, e is known, then the other can be found according to the following formulas: g = gcd(X n − 1, e).

e = rg,

We recall, for later use, the following definition. Let g be an element of the finite abelian group A. The q-cyclotomic class of g is the set j Cg = {g q |0 ≤ j ≤ tg − 1}, where tg is the smallest positive integer, such that q tg ≡ 1( mod o(g)), and o(g) denotes the order of g. Since (q, o(g)) = 1, such a number tg always exists. It follows easily that if Sg 6= Sh , then Sg ∩ Sh = ∅. If G = hai is a cyclic group of order n, then an element g ∈ G is of the form g = as and we can define the q-cyclotomic class of s as the set of integers Ωs = {s, qs, q 2 s, . . . , q ts −1 s} where ts is the smallest positive integer such that q ts s ≡ s (mod n) 3. Some cyclic codes In two recent papers, S.K. Arora and M. Pruthi [1], [25] characterized cyclic codes under certain conditions, which we briefly survey in this section, considering several cases. I. Codes of length n, when q = λn + 1. Notice that, when q = λn + 1, the field Fq is a splitting field for the polynomial X n − 1 and, hence, also a splitting field for the cyclic group G = hai of order n. Consequently, the primitive idempotents of Fq G are given by 1 X εi = χi (1)χi (x−1 )x |G| x∈G th n -root

and, if ζ denotes a primitive of unity in Fq , then the irreducible characters of G over Fq are the maps χi : G −→ F given by aj 7−→ ζ ij ,

1 ≤ i ≤ n.

So, we immediately get the following. Theorem 3.1. ([25, Theorem 2.1]) If q = nλ + 1 then Fq G has precisely n primitive idempotents given by n−1 1 X i −j j εi = (ζ ) a , n j = 0

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0 ≤ i ≤ n − 1.

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where ζ is a primitive nth -root of unity in Fq . These idempotents can be used, for example, to determine all BCH codes of length n over a field Fq , in the given conditions. This class of codes was first considered by A. Hocquenghem [14] in 1959 and independently by R. C. Bose and D. K. Ray-Chaudhuri [4] and [5] in 1960. Originally, these were binary codes and the idea was extended in 1961 by D. Gorenstein and N. Zierler [13] to define BCH codes over Galois fields. Let m be the multiplicative order of q modulo n and let K denote the field with q m elements. Then K contains the splitting field of X n − 1 and we denote by ζ a primitive nth -root of unity. We recall that given positive integers b and δ, with 2 ≤ δ ≤ n, the BCH code with designed distance δ is the cyclic code generated in Rn by the least common multiple of the minimal polynomials of the elements ζ b , ζ b+1 , . . . , ζ b+δ−2 . The following result is easy to prove. Lemma 3.2. If K is a splitting field for the polynomial X n −1, then a BCH code has designed distance δ if and only if its dimension is n − δ + 1. Hence, we shall determine all BCH codes of a given dimension k. In this case, the designed distance is δ = n − k + 1. The corresponding idempotent, the check polynomial, and the generating polynomial are: b+n−1 X

eb =

ej ,

j = b+n−k

hb (X) =

b+n−1 Y

(X − ζ i )

and gb (X) =

j = b+n−k

Xn − 1 , hb (X)

1 ≤ b ≤ n,

where indeces are taken modulo n. II. Codes of length n, when q has order ϕ(pm ) modulo pm . Let p be a prime not dividing q = |Fq |. We wish to study the cyclic codes of length n = pm . Let G = hai be a cyclic group of order n. Since Fq G is semisimple, to determine all the ideals of Fq G it suffices to describe the minimal ones and to do so, in turn, it suffices to find the set of primitive idempotents. Given a cyclotomic class Ci in G we consider the element X Ci = g ∈ Fq G. g∈Ci

Lemma 3.3. [25, Lemma 3.1] Let p and q be as above and assume that q has order ϕ(pm ) modulo pm . Then, G contains m + 1 cyclotomic classes given by i−1

Ci = {ap

, ap

i−1 q

i−1 q ϕ(pm−i+1 )−1

, . . . , ap

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},

1 ≤ i ≤ m + 1.

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After studying the products of the form Ci Cj for 1 ≤ i, j ≤ m + 1 and taking into account the previous lemma, one can obtain the following. Theorem 3.4. [25, Theorem 3.5] Under the conditions above, the group algebra Fq G has m + 1 primitive idempotents given by e0 =

m+1 1 X Ci , pm i=1

and ei =

1 [(p − 1)(1 + Ci+1 + · · · + Cm ) − Ci ], 1 ≤ i ≤ m. pm−i+1

III. Codes of length 2pn , when q has order ϕ(pm ) modulo pm , p odd. This case was treated at length in [1], describing the minimal idempotents of Rn in this case. However, if we think in terms of group algebras, it is easy to obtain the computation of the idempotents as a corollary of the previous one. Let G be a cyclic group of order 2pn , p an odd prime. Write G = C × A where A is the p-Sylow subgroup of G and C = {1, t} is its 2-Sylow subgroup. Then, we have that ∼ F (C × A) = ∼ (F C)A = ∼ (F ⊕ F )A. FG = Since the two primitive idempotents of F C are (1 + t)/2 and (1 − t)/2 and the idempotents of F A were computed in Theorem 3.4 above, we immediately have the following. Theorem 3.5. [1, Theorem 2.6] Let F be a field with q elements and G a cyclic group of order 2pn , p an odd prime, such that o(q) = ϕ(pn ) in U (Z2pn ). With the notations above, if ei , 0 ≤ i ≤ n, denote the primitive idempotents of F A, then the primitive idempotents of F G are (1 + t) (1 − t) · ei and · ei 0 ≤ i ≤ n. 2 2 There is a natural way to produce central idempotents in group algebras. Given a field K and a finite normal subgroup H of a group G such that char(K) 6 | |H|, then the element X b = 1 H h |H| h∈H

is a central idempotent of KG ([23, Lemma 3.6.6]). In the case of a cyclic group of order pm , the lattice of subgroups is a chain: G = A0 ⊃ A1 ⊃ · · · ⊃ An = {1}.

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In this case, the elements b and ei = A ci − A [ e0 = A i−1 , 1 ≤ i ≤ n, form a set of orthogonal idempotents such that e0 + e1 + · · · + en = 1 ([12, Lemma VII.1.2]). With the additional hypothesis that q has order ϕ(pm ) modulo pm , it can be shown that these idempotents coincide with those obtained in Theorem 3.4. In fact, note that 1 e0 = (1 + C 1 + C 2 + · · · + C n ) pn   X X X 1  = 1+ as + as + · · · + as  pn s∈Ω1 s∈Ω2 s∈Ωn 1 X b = a = G. pn a∈G

and ei = =

1 pn−i+1

[(p − 1)(1 + C i+1 + · · · + C n ) − C i ]

p 1 (1 + C i+1 + · · · + C n ) − n−i+1 (1 + C i + C i+1 + · · · + C n ) pn−i+1 p

where



1 + C i+1 + · · · + C n = 1 +

 X s∈Ωi+1

X

as +

as + · · · +

s∈Ωi+2

X

as  .

s∈Ωn

Since the q-cyclotomic classes modulo pn−i are: Ω1 Ω2 Ωn−i+1

= = ··· =

n−i+1

)−1 } {1, q, . . . , q φ(p n−i {p, pq, . . . , pq φ(p )−1 }

{pn−i , pn−i q, . . . , pn−i q φ(p)−1 },

we have that 1 + C i + C i+1 + · · · + C n =

X

a.

a∈Ai−1

So ci − A [ ei = A i−1 . It is then natural to ask when is it possible to compute idempotents as done above, i.e., directly from the subgroup structure of G, without reference to the cyclotomic classes. We shall discuss this question in the next section.

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When idempotents can be obtained in this way, it is easy to give a direct computation of the corresponding generating polynomials. If ei (X) ∈ F [X] is any polynomial such that ei (a) = ei , then the generating polynomial of Ii is given by n

gi (X) = gcd(ei (X), X p − 1),

0 ≤ i ≤ n.

We compute ei (X) =

pn−i X−1

1 pn−i

X

1  p− pn−i+1

Also X

pn

− 1 = (X

pi−1

−

j=0



=

jpi

p−1 X

pn−i+1 X−1

1 pn−i+1

 i−1

X jp

j=0

pn−i X−1



j=0

X jp

i−1

 i X jp 

j=0

   n−i  p−1 pX −1 X i−1 i − 1)  X jp   X jp  j=0

and it is easy to check that

j=0

 i−1

gi (X) = (X p

− 1) 

pn−i X−1

 i X jp  .

j=0

Since deg(gi (X)) = pn − pi + pi−1 we obtain that dim(Ii ) = pn − deg(gi (X)) = pi − pi−1 = ϕ(pi ). 4. Semisimple group algebras with minimum number of simple components In [10], Ferraz gave a general method to compute the number of simple components of a semisimple group algebra without using characters. In the case of finite group algebras of abelian groups, there is a simpler way to determine such a number. Let F be a finite field, with |F | = q elements, and let A be a finite abelian group such that (q, |A|) = 1. Then F A is semisimple and, if {e1 , . . . , er } is the set of primitive idempotents of F A, we have that F A = ⊕ri=1 (F A)ei ' ⊕ri=1 Fi , where Fi ' (F A)ei , 1 ≤ i ≤ r are fields that are finite extensions of F . Set A = ⊕ri=1 F ei . Notice that F ei ' F as fields in a natural way and that the number r of simple components is also the dimension of A as a vector space over F . Computing this dimension, we have the following.

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Theorem 4.1. ([11, Theorem 2.2]) Let F be a finite field, with |F | = q, and let A be a finite abelian group, such that (q, |A|) = 1. Then, the number of simple components of F A is equal to the number of q-cyclotomic classes of A. A well-known theorem, due to Perlis and Walker [21], (see [23, Corollary 3.5.5]) shows that the number of simple components of the rational group algebra of a finite abelian group A is equal to both the number of cyclic subgroups of A and the number of its cyclic factors. j Notice that, if h is an element in a cyclotomic class Cg , then h = g q for some j. As (q, o(g)) = 1, it follows that hgi = hhi. So each q-cyclotomic class Cg is a subset of the set Gg of all generators of the cyclic group hgi. So, it is clear that the number of cyclic subgroups of A is a lower bound for the number of simple components and that this bound is attained if and only if Sg = Gg , for all g ∈ A. For positive integers r and m, we shall denote by r ∈ Zm the image of r in the ring of integer modulo m. Then, Gg = {g r | (r, o(g)) = 1} = {g r | r¯ ∈ U (Zo(g) } and we have the following. Theorem 4.2. [11, Theorem 2.3] Let F be a finite field with |F | = q, and let A be a finite abelian group, of exponent e, such that (q, |A|) = 1. Then Sg = Gg , for all g ∈ A if and only if U (Ze ) is a cyclic group generated by q¯ ∈ Ze . As an easy consequence, we have the following. Corollary 4.3. Let F be a finite field with |F | = q, and let A be a finite abelian group, of exponent e. Then Gg = Sg for all g ∈ G if and only if one of the following holds: (i) e = 2 and q is odd. (ii) e = 4 and q ≡ 3 (mod 4 ). (iii) e = pn and o(q) = ϕ(pn ) in U (Zpn ). (iv) e = 2pn and o(q) = ϕ(pn ) in U (Z2pn ). 5. Minimal abelian codes We wish to extend the construction of primitive idempotents given in section 3 to the case of a finite abelian group A. According to Corollary 4.3, to be able to do so A must be either a p-group or a group of exponent e = 2pm . We shall first consider the case of p-groups. Let A be an abelian p-group. For each subgroup H of A such that A/H 6= {1} is cyclic we shall construct an idempotent of F A. We remark that, since A/H is a cyclic group of pth-power order, there exists a unique subgroup

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b −H c∗ . H ∗ of A containing H, such that |H ∗ /H| = p. We define eH = H Clearly eH 6= 0 and we have the following. b form Lemma 5.1. The elements eH , defined as above together with eA = A a set of pairwise orthogonal idempotents of F A whose sum is equal to 1. The following is an immediate consequence of the lemma above and Corollary 4.3. Theorem 5.2. Let p be an odd prime and let A be an abelian p-group of exponent pr . Then, the set of idempotents above is the set of primitive idempotents of F A if and only if one of the following holds: (i) pr = 2, and q is odd. (ii) pr = 4 and q ≡ 3 (mod 4 ). (iii) p is an odd prime and o(q) = Φ(pn ) in U (Zpn ). Also, we have the following. Theorem 5.3. Let p be an odd prime and let A be an abelian group of exponent 2pr . Write A = E × B, where E is an elementary abelian 2-group and B a p-group. Then the primitive idempotents of F A are products of the form e.f , where e is a primitive idempotent of F E and f a primitive idempotent of F B. Notice that the primitive idempotents of F B are given by Theorem 5.2 above and, writing E = ha1 i × · · · × han i, a product of cyclic groups of order 2, then the primitive idempotents of F E are all products of the form e = e1 e2 · · · en , where 1 + ai 1 − ai ei = or ei = , 1 ≤ i ≤ n. 2 2 It should be noted that, in view of Corollary 4.3, these are the only cases where primitive idempotents of finite abelian group algebras can be computed in this way. The dimensions of minimal abelian codes, in these cases, can easily be obtained from general results on group rings (see [11, Section 5]). 6. Dihedral codes Let Dn = ha, b | an = b2 = 1, bab = a−1 i be the dihedral group of order 2n, let Fq be a field with q elements, and assume that char(Fq ) does not divide 2n. We wish to determine the values of n for which the set of primitive central idempotents can be obtained in a natural way, as before. The results in this section are due to F.S. Dutra and proofs will appear in [9]. First, we remark that it can be shown that the number of simple components of Fq Dn is greater than or equal to the number of simple components

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of the rational group algebra QDn . As before, our methods will produce the set of central simple idempotents when both these numbers are equal. Theorem 6.1. The number of simple components of Fq Dn and QDn are equal if and only if one of the following conditions holds: (i) n = 2 or 4 and q is odd. (ii) n = 2m , with m ≥ 3 and q is congruent to either 3 or 5, modulo 8. (iii) n = pm with p an odd prime and the class q is a generator of the group U(Zpm ). (iv) n = pm with p an odd prime, the class q is a generator of the group U 2 (Zpm ) = {x2 | x ∈ U(Zpm }) and −1 is not a square modulo pm . (v) n = 2pm with p an odd prime and the class q is a generator of the group U(Z2pm ). (vi) n = 2pm with p an odd prime, the class q is a generator of the group U 2 (Zpm ) = {x2 | x ∈ U(Z2pm }) and −1 is not a square modulo 2pm . (vii) n = 4pm with p an odd prime and either q or −q has order ϕ(pm ) modulo 4pm . m1 m2 1 m2 (viii) n = pm 1 p2 with p1 , p2 odd primes, (ϕ(p1 ), ϕ(p2 )) = 2 and either m1 m2 m1 m2 q or −q has order ϕ(p1 )ϕ(p2 )/2 modulo p1 p2 . m2 1 m2 1 (ix) n = 2pm with p1 , p2 odd primes, (ϕ(pm 1 p2 1 ), ϕ(p2 )) = 2 and eim1 m2 m1 m2 ther q or −q has order ϕ(p1 )ϕ(p2 )/2 modulo p1 p2 .

To describe the idempotents in these cases, let us remark first that, under the hypotheses of the theorem, the idempotents of Fq Dn are given by the same formulas as those of QDn , except for the fact that the coefficients should be understood as elements of Fq instead of Q. Hence, we shall describe how to obtain the formula of primitive central idempotents of QDn . Denote A = hai and let {em }m | n be the set of idempotents of QA. Then all these are also primitive central idempotents of QDn , except for n−1 1X i b e1 = a = A, n i=0

and, if n is even, setting

A2

=

ha2 i,

also for

c2 − A. b e2 = A When n is odd, the idempotent e1 splits into: e11 =

1+b e1 2

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and e12 =

1−b e1 2

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and, when n is even, both e1 and e2 split into: 1+b 1−b e11 = e1 , e12 = e1 2 2 1+b 1−b e21 = e2 and e22 = e2 2 2 Acknowledgments Research supported by FAPESP, Procs. 02/02933-0 and 00/07291-0 and CNPq Proc. 300243/79-0 (RN). References [1] S.K. Arora and M. Pruthi, Minimal cyclic codes of length 2pn , Finite Field Appl., 5 (1999), 177–187. [2] S.D. Berman, Semisimple cyclic and abelian code II, Cybernetics, 3, 3 (1967), 17–23. [3] I.F. Blake and R.C. Mullin, The Mathematical Theory of Coding, Academic Press, New York, 1975. [4] R.C. Bose and D.K. Ray-Chaudhuri, On a class of error correcting binary group codes, Information and Control, 3 (1960), 68–79. [5] R.C. Bose and D.K. Ray-Chaudhuri, Further results on error correcting binary group codes, Information and Control, 3 (1960), 279–290. [6] C.W. Curtis and I. Reiner, Methods of Representation Theory, Vol I, Wiley Interscience, New York, 1981. [7] V. Drensky and P. Lakatos, Monomial ideals, group algebras and error-correcting codes, Applied Algebra, Algebraic Algorithms and error-correcting codes, Lecture Notes in Computer Sci., 357 (1989), 181–188. [8] L. Dornhoff, Group Representation Theory, Part B, Marcel Dekker Inc., New York, 1971. [9] F.S. Dutra, C´ odigos dihedrais e quat´ernios, Tese de Doutoramento, UFMG, to appear (in Portuguese). [10] R. Ferraz, Simple components and central units in group algebras, J. Algebra, 279 (2004), 191–203. [11] R. Ferraz and C. Polcino Milies, Idempotents in group algebras and minimal abelian codes, preprint. [12] E.G. Goodaire, E. Jespers, and C. Polcino Milies, Alternative Loop Rings, North Holland Math. Studies N. 184, Elsevier, Amsterdam, 1996. [13] D. Gorenstein and N. Zierler, A class of error correcting codes in pm symbols, SIAM J. Appl. Math., 9 (1961), 207–214. [14] A. Hocquenghem, Codes correcteurs d´erreures, Chiffres, 2 (1959), 147–156. [15] A.V. Keralev, Ring Constructions and Applications, World Scientific, 2002. [16] A.V. Keralev and P. Sol´e, Error-correcting codes as ideals in group rings, Contemporary Math., 273 (2001), 11–18. [17] B. Kh¨ ulshammer, Bemerkugen u ¨ber die Gruppenalgebra als symmetrische Algebra III, J. of Algebra, 88 (1984), 279–291. [18] P. Landrock and O. Manz, Classical codes as ideals in group rings, Designs, Codes and Cryptography, 2 (1992), 273–285. [19] P. Langevin, Weights of Abelian codes, Designs, Codes and Cryptography, 14 (1998), 239–245. [20] R.L. Miller, Minimal codes in Abelian group algebras, J. Combin. Theory, Ser. A, 26 (1979), 166–178.

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[21] S. Perlis and G. Walker, Abelian group algebras, Trans. Amer. Math. Soc., 68 (1950), 420–426. [22] V. Pless, W.C. Huffman, and R.A. Brualdi, Handbook of Coding Theory, Elsevier, New York, 1998. [23] C. Polcino Milies and S.K. Sehgal, An introduction to Group Rings, Kluwer Academic Publishers, Dordrecht, The Netherlands, 2002. [24] A. Poli and M. Ventou, Nilpotent principal codes of maximum dimension in the Fq of an elementary abelian p-group, C.R. Acad. Sci. Paris Ser. I Math., 296 (1983), 283–285. [25] M. Pruthi and S.K. Arora, Minimal codes of prime power length, Finite Field and Appl., 3 (1997), 99–113. [26] J.J. Rotman, An introduction to the theory of groups, fourth ed., Graduate texts in Math., vol 148, Springer-Verlag, New York, 1995.

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Finitely generated constants of free algebras Vitor O. Ferreira and Lucia S.I. Murakami Instituto de Matem´atica e Estat´ıstica, Universidade de S˜ao Paulo, Caixa Postal 66281, S˜ao Paulo, SP, 05315-970, Brasil [email protected] [email protected] Dedicated to Professor C´esar Polcino Milies on the occasion of his sixtieth birthday. Abstract. It is shown that the subalgebra of constants of a free associative algebra of finite rank under the action of a set of linear derivations is finitely generated if and only if the derivations are all scalar. The approach of the proof follows W. Dicks’ ideas for the proof of a similar result for invariants of linear automorphisms.

It has been known that given a Lie algebra of linear derivations of a free associative algebra, the subalgebra of constants is free (see [Jo78] and [Kh81]). A natural question that arises is to determine when this subalgebra is finitely generated. An analogous problem was considered for automorphisms. Invariants of a free algebra under the action of a group of linear automorphisms form a free subalgebra (cf. [La76] and [Kh78]). In [DF82] and independently in [Kh84] it was proved that if G is a finite group of linear automorphisms of a free associative algebra of finite rank over a field, then the subalgebra of invariants is finitely generated if and only if the action of G is scalar. Somewhat later, W. Dicks gave a simplified proof of this fact. It is this proof that appears in [Co85, Theorem 10.4] and in [Pa89, Theorem 32.7]. The aim of this note is to show that Dicks’ proof can be adapted to the case of derivations giving rise to the same conclusion. We shall then present a self-contained proof that the subalgebra of constants of a free algebra of finite rank over a field of positive characteristic under the action of a Lie algebra of linear derivations is finitely generated exactly when the derivations act scalarly. Our proof follows that of Dicks very closely for 2000 Mathematics Subject Classification. 16S10, 16W25. Key words and phrases. Free associative algebra, constants of derivations. 149

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the case of automorphisms with appropriate modifications where necessary. This result also follows from the work of A. Koryukin [Ko94]. Let k be a field and let R be a k-algebra. By a derivation of R we understand a map δ : R → R such that (λr + s)δ = λrδ + sδ and (rs)δ = rδ s + rsδ , for all r, s ∈ R and λ ∈ k. Let khXi be a free associative algebra on a set X over k. The elements of khXi are k-linear P combinations of monomials of the free monoid hXi on X. Given f = w∈hXi λw w, with λw ∈ k, we call the subset {w ∈ hXi : λw 6= 0} of hXi the support of f . If w = x1 · · · xn , with xi ∈ X, then we say that w has degree n. The degree of an element f of khXi is the maximal degree of the monomials occurring in the support of f and is denoted by deg f . The L free algebra khXi can then be graded by the degree, that is, khXi = n∈N Rn , where Rn = {f ∈ khXi : f has degree n} is called the homogeneous component of khXi of degree n. A derivation of khXi is completely determined by its image on X, that is, for every choice of elements {fx ∈ khXi : x ∈ X} there exists a unique derivation δ of khXi such that xδ = fx , for all x ∈ X. We say that a derivation δ of khXi is linear if for each x ∈ X, there exist λxy ∈ k, with P y ∈ X, such that xδ = y∈X λxy y. A linear derivation δ is said to be scalar (or to act scalarly on khXi) if there exists λ ∈ k such that xδ = λx, for all x ∈ X. If δ is a linear derivation of khXi, then its restriction to the homogeneous component R1 of degree 1 of khXi is a linear operator on the k-vector space with basis X. We can, therefore, speak of the trace (tr δ) of δ, when X is finite. Note that the matrix of a scalar derivation of a free algebra of finite rank is just a diagonal matrix with the same element occurring on each diagonal entry, that is, a scalar matrix. For each positive integer n, let Symn denote the symmetric group on n letters. The elements of Symn act on the homogeneous component Rn of khXi in the following way. Given σ ∈ Symn and x1 , . . . , xn ∈ X, we define (x1 · · · xn )σ = x1σ · · · xnσ and extend this action by linearity. If f is an element of Rn and σ ∈ Symn , we say that f σ is a place permutation of f . We start with the following trivial facts. Lemma 1. Let δ be a linear derivation of khXi and let f ∈ khXi be a homogeneous element of degree n. Then (f δ )σ = (f σ )δ , for all σ ∈ Symn . ¤ Lemma 2. Let d be a positive integer and let X S= (sgn σ)x1σ · · · xdσ σ∈Symd

be the standard polynomial of degree d on x1 , . . . , xd . Let δ be a linear derivation of khx1 , . . . , xd i. Then S δ = (tr δ)S. ¤

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We can now state and prove our main result. Theorem 3. Let k be a field of positive characteristic and let L be a set of linear derivations of a free associative k-algebra R of finite rank. Then the subalgebra of constants RL is finitely generated if and only if the action of each element of L on R is scalar. Proof. Let p be the characteristic of k and let d be the rank of R, so that R = khx1 , . . . , xd i. Suppose that each derivation in L acts scalarly on R. Thus for each δ ∈ L, there exists λδ ∈ k such that xδi = λδ xi , for all i = 1, . . . , d. Therefore, given a monomial v ∈ hx1 , . . . , xd i, we have v δ = (deg v)λδ v. It follows that the elements of RL are linear combinations of monomials of degree divisible by p. Hence RL is generated by the dp monomials of hx1 , . . . , xd i of degree p. Conversely, suppose that RL is finitely generated. We may assume that k is algebraically closed, for if k denotes the algebraic closure of k, then each element of L defines a linear derivation of R = R ⊗k k in such a way that L R = RL ⊗k k. Take δ ∈ L. Since the action of δ on R is linear, it restricts to a linear operator on the d-dimensional vector space V with basis {x1 , . . . , xd } over k. Since k is algebraically closed, there exists a basis B of V with respect to which the matrix of δ is in Jordan normal form. The subalgebra of R generated by B coincides with R and is freely generated by B. We may therefore assume that B = {x1 , . . . , xd }. Our task is to prove that the matrix of δ with respect to B is diagonal. Let S denote the standard polynomial of degree d in x1 , . . . , xd . By Lemma 2, S δ = (tr δ)S and, therefore, (S n )δ = n(tr δ)S n , for all n ≥ 1. In particular, it follows that S p ∈ RL . Let A be the set of all monomials different from 1 that occur in the support of the elements of a finite set of generators for RL . (Note that since S p ∈ RL , A is a nonempty finite set.) We have that RL is contained in the subalgebra of R generated by A. Let q be a multiple of p that is greater than the degrees of the elements of A. On the one hand, S q lies in RL . On the other hand, S q is homogeneous of degree dq, hence the elements of the symmetric group Symdq act on S q as place permutations. Thus, by Lemma 1, every place permutation of S q lies in RL . In particular, every place permutation of (x1 x2 · · · xd )q is a product of elements of A. Suppose that the matrix of δ is not diagonal. Then, after relabeling the indeterminates, if necessary, we have xδ1 = λx1 + x2 , xδ2 = λx2 + εx3 , for some λ, ε ∈ k and ε = 0 or 1, and x1 and x2 do not occur in the support of xδi if i ≥ 3. Now xq1 xq2 · · · xqd is a place permutation of (x1 x2 · · · xd )q and

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so is a product of elements of A. Since the elements of A have degree not exceeding q − 1, there must exist an integer m with 1 ≤ m < q such that L xm 1 ∈ A. Hence there exists a generator r in R , which we may assume to be homogeneous of degree m, since the action of δ is linear, such that xm 1 m−1 occurs in the support of r with coefficient 1, that is, r = xm x2 + f , 1 + µx1 where µ ∈ k and f is a k-linear combination of monomials of degree m that m−1 are different from xm x2 . So 1 and x1 rδ = mλxm 1 + +

m−1 X

xi1 x2 xm−1−i + (m − 1)λµxm−1 x2 1 1

i=0 m−2 X

µxi1 x2 xm−2−i x2 + λµxm−1 x2 + εµxm−1 x3 + f δ . 1 1 1

i=0

rδ

Since = 0, by comparing coefficients, we get mλ = 0 and 1 + mλµ = 0, a contradiction. Therefore, the matrix of δ must be diagonal, that is, for each i = 1, . . . , d, there exists λi ∈ k such that xδi = λi xi . Now, there exists a monomial w ∈ hx1 , . . . , xd i such that wxi ∈ A for all i = 1, . . . , d, for if it were not so, we could construct a monomial of degree q whose initial segments would all lie outside A, but this would be a contradiction with the fact that all place permutations of (x1 x2 · · · xd )q are products of elements of A. Write w = xi1 xi2 · · · xit . Then wδ = ηw, where η = λi1 +λi2 +· · ·+λit ∈ k. So, for each i = 1, . . . , d, we have ηwxi +λi wxi = (wxi )δ = 0. Hence λi = −η for all i = 1, . . . , d and so the action of δ is scalar. ¤ Over a field of characteristic zero, nonzero scalar derivations have trivial constants, but the theorem above does not hold. For instance, consider the derivation δ of the free algebra khx, yi of rank 2 over a field k of characteristic zero defined by xδ = x and y δ = 0. In this case, the subalgebra of constants is k[y], which is finitely generated, but δ is not scalar. Corollary 4. Let k be a field of positive characteristic and let L be a finite dimensional restricted Lie k-algebra of linear derivations of a free associative algebra R = khXi on a set X over k. Then the subalgebra of constants RL is finitely generated if and only if X is finite and the action of each element of L on R is scalar. In this case if L 6= {0}, then L has dimension 1 over k. Proof. We have seen above that if X is finite and the action of the elements of L is scalar, then RL is finitely generated. Suppose, now, that RL is finitely generated. If X were infinite, then, since RL is finitely generated, RL would be contained in khx1 , . . . , xd i for some finite subset {x1 , . . . , xd } of X. This would give rise to an infinite chain of intermediate free subalgebras of R containing RL : RL ⊆ khx1 , . . . , xd i ⊆ khx1 , . . . , xd , xd+1 i ⊆ khx1 , . . . , xd , xd+1 , xd+2 i ⊆ · · · ⊆ R,

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where xd+1 , xd+2 , . . . ∈ X \ {x1 , . . . , xd }. By [Kh81, Theorem 9], there exists an inclusion inverting one-to-one correspondence between the restricted Lie subalgebras of L and the free subalgebras of R containing RL . Therefore we would have an infinite chain of Lie subalgebras of a finite dimensional Lie algebra, which is impossible. Hence X is finite and, by Theorem 3, the action of the elements of L on R is scalar. If L 6= {0}, then L = kδ, for any 0 6= δ ∈ L. ¤ It is worth pointing out that it has been shown recently that the subalgebra of invariants of a free algebra under a linear action of a Hopf algebra is free (see [FMP04, Corollary 3.2]). In [Ko94], A. N. Koryukin exhibits necessary and sufficient conditions for this subalgebra to be finitely generated. It would be interesting to know if one could obtain a result of the same nature as Dicks-Formanek-Kharchenko’s and Theorem 3 above for the subalgebra of invariants of a free algebra under a linear action of a finite dimensional pointed Hopf algebra. The Galois correspondence that exists for this case (see [FMP04]) suggests that this might indeed be possible. Acknowledgments Dr. Ferreira was partially supported by FAPESP (Projeto No. 02/120655). References [Co85]

P.M. Cohn, Free Rings and Their Relations, 2nd. Ed., Academic Press, London, 1985. [DF82] W. Dicks and E. Formanek, Poincar´e series and a problem of S. Montgomery, Linear and Multilinear Algebra 12 (1982/83), 21–30. [FMP04] V.O. Ferreira, L.S.I. Murakami, and A. Paques, A Hopf-Galois correspondence for free algebras, J. Algebra 276 (2004), 407–416. [Jo78] T. de W. Jooste, Primitive derivations in free associative algebras, Math. Z. 164 (1978), 15–23. [Kh78] V.K. Kharchenko, Algebras of invariants of free algebras, Algebra i Logika 17 (1978), 478–487. (English translation in Algebra and Logic 17 (1979), 208–258.) [Kh81] V.K. Kharchenko, Constants of derivations of prime rings, Izv. Akad. Nauk SSSR Ser. Mat. 45 (1981), 435–461, 464. (English translation in Math USSR-Izv. 18 (1982), 381–401.) [Kh84] V.K. Kharchenko, Noncommutative invariants of finite groups and Noetherian varieties, J. Pure Appl. Algebra 31 (1984), 83–90. [Ko94] A.N. Koryukin, On noncommutative invariants of bialgebras, Algebra i Logika 33 (1994), 654–680, 716. (English translation in Algebra and Logic 33 (1994), 366–380 (1995).) [La76] D.R. Lane, Free Algebras of Rank Two and Their Automorphisms, PhD Thesis, London, 1976. [Pa89] D.S. Passman, Infinite Crossed Products, Academic Press, Boston, MA, 1989.

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Partial actions of groups on semiprime rings Miguel Ferrero Instituto de Matem´atica, Universidade Federal do Rio Grande do Sul, Porto Alegre, 91509-900, Brasil [email protected] Dedicated to my friend Francisco C´esar Polcino Milies on his 60th birthday. Abstract. In this note we introduce the notion of weak enveloping action and prove that any proper partial action of a group on a semiprime ring possesses a weak enveloping action.

Introduction As it is well known a group G acts on a set X if for any g ∈ G there exists a bijection αg of X such α1 = idR and αg ◦αh = αgh , for any g, h ∈ G, where 1 denotes the identity element of G. In the case that R is a ring G acts on R if the above holds for R = X and, in addition, any αg is an automorphism of R, for every g ∈ G. Partial actions of groups appeared independently in various areas of mathematics, in particular, in the theory of operator algebras as a powerful tool in their study (see [4], [5], [6], [9], [12]). In the most general setting of a partial action of a group on a set X the definition is as follows: Definition. Let G be a group with identity element 1 and X be a set. A partial action α of G on X is a collection of subsets Sg ⊆ X (g ∈ G) and bijections αg : Sg−1 → Sg such that (i) S1 = X and α1 is the identity map of X; (ii) S(gh)−1 ⊇ αh−1 (Sh ∩ Sg−1 ), for any g, h ∈ G; (iii) αg ◦ αh (x) = αgh (x) for any x ∈ αh−1 (Sh ∩ Sg−1 ). Note that conditions (ii) and (iii) mean that the function αgh is an extension of the function αg ◦ αh . It is easy to see that condition (ii) can be replaced by the apparently stronger condition: αh−1 (Sh ∩ Sg−1 ) = Sh−1 ∩ Sh−1 g−1 . So, conditions (i), (ii) and (iii) are equivalent to the following: (i0 ) S1 = X and α1 is the identity map on X; 155

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(ii0 ) αg (Sg−1 ∩ Sh ) = Sg ∩ Sgh , for any g, h ∈ G; (iii0 ) αg (αh (x)) = αgh (x), for all x ∈ Sh−1 ∩ S(gh)−1 . Recently, in a pure algebraic context, partial representations and partial actions of groups on algebras have been defined and studied in [1] and [2]. Also a Galois Theory of commutative rings with partial actions has been developed [3]. In what follows, R is a ring (equivalently, a nonnecessarily unital associative algebra over a commutative ring k) and α is a partial action of a group G on R. This means, according to [2], that for any g ∈ G there exists an isomorphism of algebras without identity αg : Sg−1 → Sg , where {Sg | g ∈ G} is a family of ideals of R, and the above conditions (i), (ii) and (iii) are satisfied. In this note a partial action is said to be proper if all the ideals Sg are nonzero. Natural examples of partial actions can be obtained by restricting a global action to an ideal. More precisely, suppose that a group G acts on an algebra T by automorphisms βg : T → T and let R be an ideal of T . Assume that Sg = R ∩ βg (R) 6= 0, for every g ∈ G. Define αg as the restriction of βg to Sg−1 . Then it is easy to verify that α = {αg : Sg−1 → Sg | g ∈ G} is a partial action of G on R. We say, in this case, that α is the restriction of β to R. Thus it is natural to ask when a given partial action can be obtained as the restriction of a global action. In [2] the authors defined enveloping action of a partial action α on R as a global action β of G on an extension T of R such that R is an ideal of T and α is the restriction of β to R, with some additional conditions. In this note we say that a global action β on T is a weak enveloping action for α if for any g ∈ G the restriction of the map βg to Sg−1 coincides with αg . In the first part of this note we survey some results on partial actions and Martindale rings of quotients. In the last section we prove that any proper partial action of a group G on a semiprime ring R possesses a weak enveloping action. Throughout this paper R is always an associative ring not necessarily with identity element. When we say ideal we mean two-sided ideal. 1. Enveloping actions Assume that R is a ring and α is a partial action of G on R. Suppose that G acts on T by automorphisms βg and R is an ideal of T . The subalgebra T1 of T generated by ∪g∈G βg (R) is invariant with respect to βg , for all g ∈ G. Hence, if α can be obtained as a restriction of β, then it can also be obtained as a restriction of the action of G on T1 . Thus in order to have uniqueness it is reasonable to require that T1 = T and in this case α is called an admissible restriction of β.

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We recall the definition of equivalent partial actions ([2], Definition 4.1): Definition 1.1. Given two partial actions α = {αg : Sg−1 → Sg | g ∈ G} and α0 = {αg0 : Sg0 −1 → Sg0 | g ∈ G} of a group G on rings R and R0 , respectively, we say that α and α0 are equivalent if there exists an algebra isomorphism ϕ : R → R0 such that for every g ∈ G the following conditions hold: (ι) ϕ(Sg ) = Sg0 ; (ιι) αg0 ◦ ϕ(x) = ϕ ◦ αg (x), for all x ∈ Sg−1 . The definition of enveloping action is the following ([2], Definition 4.2): Definition 1.2. An action β of G on an algebra T is said to be an enveloping action for the partial action α of G on R if α is equivalent to an admissible restriction of β to an ideal of T . In other words, the action β of G on T is said to be an enveloping action for α if there exists an algebra isomorphism ϕ of R onto an ideal of T such that for all g ∈ G the following three properties are satisfied: (ι0 ) ϕ(Sg ) = ϕ(R) ∩ βg (ϕ(R)); (ιι0 ) ϕ ◦ αg (x) = βg ◦ ϕ(x) for any x ∈ Sg−1 ; (ιιι0 ) T is generated by ∪g∈G βg (ϕ(R)). It is clear that if the partial action α of G on R has an enveloping action (T, β) we may assume, unless equivalence, that R is an ideal of T and β is an action of G on T as automorphisms {βg | g ∈ G} such that T is generated by ∪g∈G βg (R), Sg = R ∩ βg (R) and βg (x) = αg (x), for any g ∈ G and x ∈ Sg−1 . The existence of enveloping actions has been considered in [2], where the authors proved the following: Theorem 1.3. ([2], Theorem 4.5) Let R be an algebra with identity element. Then a partial action α of a group G on R admits an enveloping action if and only if each ideal Sg (g ∈ G) is a unital algebra. Moreover, if such an enveloping action exists, it is unique up to equivalence. Note that an ideal I of a unital algebra R has an identity element if and only if there exists a central idempotent e ∈ R such that I = eR. Thus the condition of the above theorem says that any of the ideals Sg is generated by a central idempotent of R. For a ring R and partial action α = {αg : Sg−1 → Sg | g ∈ G}, the partial skew group ring R ∗α G is defined as the set of all the formal sums P a g∈G g ug , ag ∈ Sg , with the usual addition and multiplication determined by (ag ug ).(bh uh ) = αg (αg−1 (ag )bh )ugh ([2], Definition 1.2). As it was pointed out in [2] this gives a well-defined structure of algebra over k that, however, is not always associative ([2], Example 3.5). So a main question is to find conditions for the associativity of the partial skew group

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ring. This question was considered in [2] and some conditions have been found. A way of proving associativity of the partial skew group ring is, of course, to find an associative ring containing R∗α G. The following is clearly true: if the partial action α possesses an enveloping action (T, β), the partial skew group ring R ∗α G can be embedded into the skew group ring T ∗ G. In particular, R ∗α G is an associative ring. Note that for the associativity question of the partial skew group ring it is not essential to have uniqueness of the enveloping action and a weaker enveloping action would be enough. We give the following: Definition 1.4. An action β of G on T is said to be a weak enveloping action for a partial action α of G on R if there exists a monomorphism of rings φ : R → T such that βg ◦ φ|Sg−1 = φ ◦ αg , for any g ∈ G. Hence, by the argument given above we have: Proposition 1.5. Assume that the partial action α of G on R admits a weak enveloping action (T, β). Then the partial skew group ring R ∗α G can be embedded in the skew group ring T ∗G. In particular, R∗α G is associative. In their paper the authors also proved that if R is a semiprime ring, then for any partial action α of a group G on R the partial skew group ring R∗α G is associative (see Corollary 3.4 of [2]). This result was obtained without the use of enveloping actions. As we said in the introduction, a partial action is called here proper if all the ideals Sg are nonzero. The main purpose of this note is to prove the following: Theorem 1.6. Assume that R is a semiprime ring and α is a proper partial action of G on R. Then α possesses a weak enveloping action. As an immediate consequence of Theorem 1.6 we have another proof of Corollary 3.4 of [2] for proper partial actions. 2. Rings of quotients In this section we recall some information on rings of quotients of a semiprime ring of R that we will need in the next section. For background on this subject we refer the reader to [10, Chapter 3], [11, Section 24], [13, Chapter 9]. Hereafter R is always semiprime. An ideal H of a semiprime ring R is essential as a two-sided ideal if AnnR (H) = 0, where AnnR (H) = {a ∈ R : Ha = 0} = {a ∈ R : aH = 0}. The set of all essential ideals of R will be denoted by E = E(R). Note that E is a filter that is closed under multiplication. If I is an ideal of R, then I + AnnR (I) = I ⊕ AnnR (I) ∈ E. Denote by Q = QE the ring of right quotients of R with respect to the filter E, i.e., the Martindale ring of right quotients of R. Recall that elements of Q

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arise from right R-homomorphisms of some H ∈ E to R: for any H ∈ E and right homomorphism f : H → R there exists q ∈ Q such that qh = f (h), for any h ∈ H, and conversely, if q ∈ Q there exists F ∈ E such that qF ⊆ R. Also, elements q, p ∈ Q are equal if and only if they coincide on some essential ideal of R. Consequently, if qH = 0, for some q ∈ Q and H ∈ E, then q = 0. The extended centroid of R is the center of Q and will be denoted by C. The elements of C are precisely those elements of Q arising from R-bimodule mappings. Recall that the ring Q is semiprime and C is a von Neumann regular ring. We can easily describe idempotent elements of C: let H be a nonzero ideal of R and put KH = H ⊕ AnnR (H) ∈ E. Thus the mapping f : KH → R defined by f (a + b) = a, for a ∈ H and b ∈ AnnR (H), is an R-bimodule homomorphism and so there exists eH ∈ C such that eH (a + b) = eH a = a. The element eH is clearly an idempotent of C. Conversely, if e ∈ C is an idempotent element there exists an essential ideal I of R such that H = eI ¢ R. Thus we have e = eH . Note that eH = 1 if and only if H ∈ E. We denote by Ip(C) the set of all the idempotent elements of C. Given an ideal I of R, the closure of I in R as defined in [7] is [I] = {x ∈ R | there exists H ∈ E(R) with xH ⊆ I} = {x ∈ R | there exists H ∈ E(R) with Hx ⊆ I}. It is easy to see that [I] = AnnR (AnnR (I)) and so the closed ideals of R, i.e, the ideals I with [I] = I, are the annihilator ideals. By the results in [7] we know that there is a one-to-one correspondence between the set of all the closed ideals of R, the set of all the Q-closed ideals of Q, and the set of all the C-closed ideals of C. More precisely, this correspondence if given as follows: let A be a closed ideal of R, A∗ a Q-closed ideal of Q, and A0 a C-closed ideal of C. Then these ideals are corresponding closed ideals if and only if A∗ ∩ R = A (equivalently A∗ = {q ∈ Q | qH ⊆ A, for some H ∈ E}) and A∗ ∩ C = A0 (equivalently A∗ = A0 C). From this it follows that for any ideal I of R the closed ideal I ∗ of Q that corresponds to the closure [I] of I in R can be obtained as the set of all the elements q ∈ Q such that there exists H ∈ E with qH ⊆ I. It is well known and easy to check that every closed ideal of C is a direct summand and so is generated by an idempotent. Hence the same is true for any Q-closed ideal of Q. In particular, if K is a closed ideal of Q, then K ⊕ AnnQ (K) = Q and K = eQ for some e ∈ Ip(C). We include the proof of the following well-known result for the sake of completeness. Lemma 2.1. Any nonzero ideal of a semiprime ring is a semiprime ring. Proof. Assume that R is semiprime and A is a nonzero ideal of R. If I ¢ A and I 2 = 0, then (R# IR# )3 ⊆ R# IAIR# ⊆ R# I 2 R# = 0, where R#

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denotes the ring R with an identity adjoined. Hence R# IR# = 0 and so I = 0. ¤ Now we will give some propositions that are certainly known by specialists, but we will include short proofs here for the sake of completeness and because we could not find a proper reference for them in the way we will use later on. Let A be a nonzero ideal of the semiprime ring R. Then A is a semiprime ring and so it has a Martindale ring of right quotients Q(A). Let I be an ideal of R that is maximal with respect to the condition I ∩ A = 0. Then A+I = A⊕I and denote by [A] and [I] the closure of A and I in R (actually [I] = I), and by A∗ and I ∗ the closed ideals of Q corresponding to [A] and [I], respectively. It is clear that A∗ ⊕ I ∗ = Q. Proposition 2.2. Q(A) ' A∗ = {q ∈ Q | qI ∗ = 0} = {q ∈ Q | qI = 0}. Proof. Take q ∈ Q. Clearly if qI ∗ = 0, then qI = 0. Conversely, assume that qI = 0 and take any p ∈ I ∗ . Then there exists H ∈ E with pH ∈ I. Hence qpH = 0 and so qp = 0. Therefore qI = 0 if and only if qI ∗ = 0. Also, since Q = A∗ ⊕ I ∗ it follows that q ∈ A∗ if and only if qI ∗ = 0. Assume that q ∈ Q(A) and let H ∈ E(A) with qH ⊆ A. It is clear that H ⊕ I is an essential ideal of R. Define f : H ⊕ I → R by f (h + i) = qh, for all h ∈ H and i ∈ I. Thus f determines an element p ∈ Q such that pI = 0, i.e., p ∈ A∗ , which coincides with q in H. Conversely, suppose that p ∈ A∗ and let H be an essential ideal of R with pH ⊆ A. Then clearly p can be seen as an application of H ∩ A in A and so defines an element of Q(A), since H ∩ A is an essential ideal of A. Using these remarks the proof can easily be completed. ¤ Proposition 2.3. Assume that A and B are ideals of R and φ : A → B is an isomorphism of rings. Then φ can be extended to a ring isomorphism φ∗ : Q(A) → Q(B). Proof. Assume that q ∈ Q(A) and let f : H → A such that qh = f (h), for any h ∈ H, where H ∈ E(A). Then φ(H) is an essential ideal of B and φ∗ (f ) : φ(H) → B given by φ∗ (f )(φ(h)) = φ ◦ f (h), for any h ∈ H, defines an element of Q(B). So we have an application φ∗ : Q(A) → Q(B). If a ∈ A, then a can be seen as an element of Q(A) that is realized by the left multiplication al : A → A. For b = φ(c) ∈ B, c ∈ A, we have φ∗ (al )(b) = φ ◦ al (c) = φ(ac) = φ(a)φ(c) = φ(a)l (b). Hence φ∗ (a) = φ(a)l , for any a ∈ A ⊆ Q(A), and thus φ∗ |A = φ. The proof can easily be completed. ¤ The extension φ∗ of the isomorphism φ of the above proposition is unique. This is a particular case of the following:

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Proposition 2.4. Assume that A is an ideal of R, φ : A → R is a monomorphism of rings and f, f 0 : Q(A) → Q(φ(A)) are isomorphisms of rings such that f |A = f 0 |A = φ. Then f = f 0 . Proof. Take q ∈ Q(A) and let H be an essential ideal of A with qH ⊆ A. Then f (qh) = f 0 (qh), for any h ∈ H. Then (f (q) − f 0 (q))φ(h) = 0, for any h ∈ H. Hence f (q) = f 0 (q) since φ(H) is an essential ideal of φ(A). ¤ 3. Main result In this section we assume that R is a semiprime ring, G is a group, α = {αg : Sg−1 → Sg | g ∈ G} is a partial action of G on R, and Sg 6= 0 for any g ∈ G. We denote by Q the Martindale ring of right quotients of R and by Sg∗ the closed ideal of Q corresponding to the closure [Sg ] of Sg in R. The main result of this note is a consequence of the following: Theorem 3.1. Under the above assumptions there exists a partial action α∗ = {αg∗ : Sg∗−1 → Sg∗ | g ∈ G} of G on Q such that αg∗ |Sg−1 = αg , for any g ∈ G. Proof. By Propositions 2.2 and 2.3 αg can be extended to an isomorphism αg∗ : Sg∗−1 → Sg∗ . It remains to prove that {αg∗ | g ∈ G} defines a partial action of G on Q. Since the first condition of the definition of partial actions is trivially satisfied we prove the second one. ∗ ), i.e., α∗ (q) ∈ S ∗ ∩ S ∗ . Then there Assume that q ∈ αh∗ −1 (Sh∗ ∩ Sg−1 h h g −1 exists an essential ideal F1 of R such that αh∗ (q)F1 ⊆ Sh ∩ Sg−1 . Also, since q ∈ Sh∗−1 there exists an essential ideal F2 of R with qF2 ⊆ Sh−1 . Thus H = F1 ∩ F2 is an essential ideal of R with αh∗ (q)H ⊆ Sh ∩ Sg−1 and qH ⊆ Sh−1 . For h ∈ H we have αh∗ (qh) ∈ Sh ∩ Sg−1 and qh ∈ Sh−1 . Thus αh (qh) ∈ ∗ Sh ∩ Sg−1 and so qh ∈ αh−1 (Sh ∩ Sg−1 ) ⊆ S(gh)−1 . Consequently q ∈ S(gh) −1 and hence condition (ii) of the definition of partial actions is fulfilled. Therefore as we said in Section 1 condition (ii’) is satisfied, i.e., we have ∗ ∗ αh∗ −1 (Sh∗ ∩ Sg∗−1 ) = Sh∗−1 ∩ S(gh) −1 = (Sh−1 ∩ S(gh)−1 ) ∗ are defined in (S ∗ and both mappings αg∗ ◦ αh∗ and αgh h−1 ∩ S(gh)−1 ) . Also αg ◦ αh and αgh are defined in αh−1 (Sh ∩ Sg−1 ) = Sh−1 ∩ S(gh)−1 and they ∗ are equal coincide on this domain. Thus by Proposition 2.4 αg∗ ◦ αh∗ and αgh ∗ on (Sh−1 ∩ S(gh)−1 ) and so condition (iii’) is also satisfied. The proof is complete. ¤

Proof of Theorem 1.6. By Theorem 3.1 the partial action α can be extended to a partial action α∗ on the Martindale ring of quotients Q of R. All the ideals Sg∗ are closed ideals of Q and so have identity element. Hence the partial action α∗ has an enveloping action (T, β). It follows that (T, β) is a weak enveloping action for α. ¤

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Acknowledgments This research was partially supported by Conselho Nacional de Desenvolvimento Cient´ıfico e Tecnol´ ogico (CNPq., Brasil). References [1] M. Dokuchaev, R. Exel, and P. Piccione, Partial representations and partial group algebras, J. Algebra 226(1) (2000), 251–268. [2] M. Dokuchaev and R. Exel, Associativity of crossed products by partial actions, enveloping actions and partial representations, Trans. Amer. Math. Soc., 357. n. 5 (2005), 1931–1952. [3] M. Dokuchaev, M. Ferrero, and A. Paques, Partial Galois theory of commutative rings, to appear. [4] R. Exel, Twisted partial actions: a classification of regular C ∗ -algebraic bundles, Proc. London Math. Soc. 74(3) (1997), 417–443. [5] R. Exel, Partial actions of groups and actions of semigroups, Proc. Am. Math. Soc. 126(12) (1998), 3481–3494. [6] R. Exel, M. Laca, and J. Quigg, Partial dynamical systems and C ∗ -algebras generated by partial isometries, J. Operator Theory 47(1) (2002), 169–186. [7] M. Ferrero, Closed submodules of centred bimodules over semiprime rings, Nova J. Math., Game Theory and Alg. 5 (1996), 309–345. [8] P.A. Fillmore, A User’s Guide to Operator Algebras, Wiley-Interscience, 1996. [9] K. McClanahan, K-theory for partial crossed products by discrete groups, J. Funct. Anal. 130(1) (1995), 77–117. [10] S. Montgomery, Fixed rings of finite automorphism groups of associative rings, Lecture Notes in Math. 818, Springer-Verlag, Berlin, 1980. [11] D.S. Passman, A course in ring theory, Wadsworth and Brooks, Pacific Grove, 1991. [12] J.C. Quigg and I. Raeburn, Characterizations of crossed products by partial actions, J. Operator Theory 37 (1997), 311–340. [13] B. Stenstr¨ om, Rings of Quotients, Springer-Verlag, Berlin, 1975.

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Representations of affine Lie superalgebras Vyacheslav Futorny Instituto de Matem´atica e Estat´ıstica, Universidade de S˜ao Paulo, Caixa Postal 66281, S˜ao Paulo, SP, 05315-970, Brasil [email protected] Dedicated to C´esar Polcino Milies on the occasion of his 60th birthday. Abstract. We discuss the classification problem of irreducible modules with a nonzero action of the central element over nontwisted affine Lie superalgebras, comparing it with the cases of affine and finite-dimensional Lie algebras. The discussion is based on recent joint results with S. Rao. The question of integrability of irreducible modules is considered in detail.

1. Introduction Let G be a complex Lie (super)algebra with a Cartan subalgebra H. The problem of classification of irreducible G-modules is extremely difficult and is only known for the case of G = sl(2) ([3]). A G-module V is called H-diagonalizable (weight) if V = ⊕λ∈H∗ Vλ , where Vλ = {v ∈ V |hv = λ(h)v, ∀h ∈ H}. Denote by K(G) the category of all such modules with finite-dimensional weight spaces (dim Vλ < ∞, ∀λ ∈ H∗ ). We give an overview of the classification problem of irreducible modules in the category K(G) for the following types of G: • Reductive finite-dimensional Lie algebras. • Basic classical Lie superalgebras. • Affine Lie algebras. • Affine Lie superalgebras. In the affine cases we will assume in addition that the center of the algebra acts nontrivially on the irreducible modules. There are two key concepts in our considerations: parabolic partitions of the root system and cuspidality. 163

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Let ∆ be the root system of G with respect to H. Definition: A parabolic partition P is a closed subset of the root system ∆ such that P ∪ (−P ) = ∆. Note that every parabolic partition P defines a parabolic subalgebra P of G, P = P 0 ⊕ P + , where P 0 corresponds to P ∩ (−P ) and P + corresponds to P \ (−P ). If P ∩ (−P ) = ∅ then P is called a Borel subalgebra. If N ∈ K(P 0 ) is a module in the category K(P 0 ) then it can be viewed as a P-module, where P + acts trivially on N . Hence one can construct a generalized Verma G-module MP (N ) = ind(P, G; N ), an induced module associated with G, P and N . If N is irreducible P 0 module then MP (N ) has a unique irreducible quotient LP (N ). Definition: A G-module M in K(G) is not cuspidal if M ' LP (N ) for some proper P ⊂P∆ and some P 0 -module N ; M is cuspidal otherwise. Let G = H ⊕ α∈∆ Gα be the root space decomposition with respect to H. 2. Reductive Lie algebras Assume first that G is a finite-dimensional reductive Lie algebra. It is well known that in this case all Borel subalgebras are conjugate and each of them corresponds to a choice of positive roots in ∆. In general, parabolic partitions are determined by the choice of a basis π of ∆ and a subset S of π. The cuspidality of irreducible modules in K(G) was studied by Fernando [9], who showed that an irreducible module V in K(G) is cuspidal if and only if V is torsion free, i.e., the nonzero elements of Gα act injectively on V for all α ∈ ∆. Moreover, for any irreducible V ∈ K(G) there exist π, S ⊂ π, and a cuspidal P 0 -module N such that V ' LS (N ). Therefore the study of irreducible objects in K(G) was reduced to the study of cuspidal modules. Denote by U (G) the universal enveloping algebra of G. Let V ∈ K(G) be an irreducible module, AnnV the annihilator of V , and UV = U (G)/AnnV . Then AnnV is a primitive ideal. As it was shown by Gabber and Joseph the Gelfand-Kirillov dimension GK(UV ) of UV (or the dimension of the corresponding associated variety) is no more than 2GK(V ). If V is cuspidal then one can see that GK(V ) = rk[G, G] = dim H. By [13] the dimension of the minimal nilpotent coadjoint orbit in G∗ with respect to the coadjoint action of the group G, G = Lie(G), equals GK(UV ) for some V . Thus the minimal nilpotent coadjoint orbit in G∗ has dimension ≤ 2rk[G, G]. But the only simple Lie algebras G with such property are those of type A and C. Example 2.1. Denote by m(G) the dimension of the minimal nilpotent coadjoint orbit in G∗ . Then m(G) = 2n if G = sl(n + 1) (type An ) or

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G = sp(2n) (type Cn ). On the other hand m(G) = 8 for G = so(7) (type B3 , rk[G, G] = 3) and m(G) = 6 for G = G2 of rank 2. The final step in the study of irreducible modules in the category K(G) was accomplished by Mathieu [17], who gave a classification of irreducible cuspidal modules of types A and C. 3. Basic classical Lie superalgebras A Lie superalgebra is a Z2 -graded vector space G = G0 ⊕ G1 with a Lie bracket satisfying [a, b] = −(−1)|a||b| [b, a], [a, [b, c]] = [[a, b], c] + (−1)|a||b| [b, [a, c]]. A Lie superalgebra is basic classical if it is simple, finite-dimensional, the even part G0 is reductive, and G admits a nondegenerate invariant bilinear form. Let V = V0 ⊕ V1 be a Z2 -graded vector space, dim V0 = m, dim V1 = n. Then the endomorphism algebra EndV has a natural Z2 -gradation and becomes an associative superalgebra. Defining the Lie bracket as [A, B] = AB − (−1)(degA)(degB) BA we make EndV into a Lie superalgebra l(m, n). A linear function str : l(m, n) → C such that stridV = m−n and str[A, B] = 0, A, B ∈ l(m, n), is called the supertrace. Denote sl(m, n) = {A ∈ l(m, n)|strA = 0}. If m 6= n this is the superalgebra of type A(m, n). In case m = n, sl(n, n) has a one-dimensional ideal consisting of scalar matrices. Its quotient psl(n, n) is the Lie superalgebra of type A(n, n). ³ ´ ³ ´ Im 0 C1 0 Let F = ,C= , C1t = C1 , C2t = −C2 ⇒ 0 − In 0 C2 osp(m, 2n)a = {A ∈ l(m, n)a |F a At C + CA = 0}, a = 0, 1. Let B(m, n) = osp(2m + 1, 2n), C(n) = osp(2, 2n), D(m, n) = osp(2m, 2n), m > 1. The series A(m, n), B(m, n), D(m, n), C(n) and certain special Lie superalgebras D(2, 1; a), G(3), F (4) form a complete list of basic classical Lie superalgebras. They have been classified by Kac [14]. Unlike the Lie algebra case, Lie superalgebra in general contains several classes of conjugacy of Borel subsuperalgebras. Conjugacy classes of Borel subsuperalgebras were describe in [14]. The concept of cuspidality in the setting of Lie superalgebras was studied by Dimitrov, Mathieu, and Penkov [6], [7].

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P Denote by Q the roots lattice of G, which is a free abelian group: Q = α∈∆ Zα. If V is a G-module then denote by injV the subset in ∆ consisting of those roots for which the corresponding root elements act injectively on V . Let MV =< injV >+ be the monoid generated by injV . Note that if G is a reductive finite-dimensional Lie algebra then V is cuspidal if and only if MV = Q. The situation is different for Lie superalgebras ([6]). Theorem 3.1. A G-module V is cuspidal if and only if MV is a subgroup of finite index in Q. As we saw in the previous section the only simple finite-dimensional Lie algebras that admit cuspidal irreducible modules are those of type A and C. A Lie superalgebra G is called cuspidal superalgebra if it admits cuspidal modules. A complete classification of cuspidal Lie superalgebras was obtained in papers [6], [7]. Theorem 3.2. The basic classical cuspidal osp(m, 2n), m = 1, 3, 4, 5, 6 and D(2, 1; a).

Lie

superalgebras

are

This result immediately implies the complete classification of irreducible modules in the category K(G) for G of types A(m, n) and C(n). Indeed, it is enough to classify cuspidal modules over cuspidal Lie subsuperalgebras of G. Using the theorem above we see that the only such subsuperalgebras are reductive Lie subalgebras with simple components of type A and C, for which the classification of irreducible modules is known. Remark 3.3. For G = osp(1, 2) the classification of irreducible modules in K(G) is due to Bavula and van Oystaeyen [1]. In this case the universal enveloping algebra U (G) is a generalized Weyl algebra and hence a general technique from this theory can be applied. If the basic classical Lie superalgebra G is not of type A(m, n) or C(n) then [6] and [7] provide a classification of irreducible modules in K(G) up to a finite indeterminacy. Indeed, by Proposition 6.3 in [6], the module induced from an irreducible cuspidal module over the even part G0 of G has only finitely many nonisomorphic cuspidal submodules; moreover, any irreducible cuspidal G-module appears as a submodule of finitely many such nonisomorphic induced modules. 4. Affine Lie algebras Let g be a simple complex finite-dimensional Lie algebra with the Killing form <, > and G = g ⊗ C[t, t−1 ] ⊕ Cc ⊕ Cd a nontwisted affine Lie algebra, where [x ⊗ tn , y ⊗ tm ] = [x, y] ⊗ tm+n + mδm,−n < x, y > c,

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[c, z] = 0 and [d, x ⊗ tn ] = n(x ⊗ tn ) for all x, y ∈ g, z ∈ G. A classification of all Borel subalgebras of G was obtained in [10]. In this case not all of them are conjugate but there exists a finite number of conjugacy classes. Essentially the representatives of the conjugacy classes can be described as follows. Let B be a Borel subalgebra of g, ρ = ρ0 ⊕ ρ+ a parabolic subalgebra of g containing B. Then B = ρ0 ⊗ C[t]t ⊕ ρ+ ⊗ C[t, t−1 ] ⊕ B ⊕ Cc ⊕ Cd is a Borel subalgebra of G associated with B and ρ. Moreover, the parabolic subalgebras correspond to subsets of a fixed basis of the root system of g and a choice of a Borel subalgebra B [10]. It is natural to expect that the cuspidality of an object V in K(G) is equivalent to the fact that all nonzero real root elements of G are injective on V . There are good indications that this conjecture is true but the question remains open in general. ˜ We are interested in a full subcategory K(G) of K(G) consisting of modules with a nonzero action of the central element c. Irreducible modules in ˜ the category K(G) were described in [12]. Namely we have the following: ˜ Theorem 4.1. If V is an irreducible module in K(G), then there exists a basis π of the root system ∆, a subset S π and an irreducible cuspidal P 0 -module such that V ' LP (N ), where P is the parabolic subalgebra associated with π and S and P 0 is the finite-dimensional reductive Lie subalgebra determined by S. Since P 0 is a finite-dimensional reductive Lie algebra then any cuspidal irreducible P 0 -module is torsion free and simple components of P 0 are of type A and C. Hence we have a complete classification of irreducible modules ˜ ˜ in K(G). Note that S is a proper subset of π and thus the category K(G) has no cuspidal G-modules. Let W (G) be the category of all weight G-modules and K0 (G) the full subcategory of K(G) of modules of zero level. A G-module V ∈ W (G) is dense if the set of weights is a coset of H ∗ /Q. The following conjecture was formulated in [10]: Conjecture. An irreducible module V ∈ K(G) is dense if and only if it is cuspidal. Note that this conjecture does not hold in the category W (G) even in the case of finite-dimensional G. On the other hand the conjecture is true in ˜ the category K(G) by Theorem 4.1. The conjecture would reduce the classification problem to the classification of irreducible weight modules with infinite-dimensional weight spaces over the Heisenberg subalgebra G = h ⊗ C[t, t−1 ] ⊕ Cc ⊕ Cd

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of G, where h is a Cartan subalgebra of g, and irreducible dense modules over affine Lie subalgebras of G. Irreducible modules with infinite-dimensional weight spaces over the Heisenberg algebra necessarily have a nonzero level. Examples of such modules were constructed in [2]. First examples of dense G-modules with infinite-dimensional weight spaces were constructed in [5]. In [18] all irreducible modules in the category K0 (G), which are no longer irreducible as modules over the derived subalgebra [G, G], were classified. It is very likely that these modules, together with the modules of type LP (N ) with a trivial action of c, exhaust all irreducible modules in the category K0 (G). This has been conjectured in [11]. This conjecture with Theorem 4.1 would complete a classification of irreducible modules in K(G). 5. Affine Lie superalgebras Finally, we consider the case of affine Lie superalgebras. Let G = g ⊗ C[t, t−1 ] ⊕ Cc ⊕ Cd be a nontwisted affine Lie superalgebra, where g is a basic P classical finite-dimensional Lie superalgebra. Then G = G0 ⊕ G1 , G0 = N j=0 G0j (N ≤ 3), g = g0 ⊕ g1 , g0j ⊂ g0 , g00 is abelian, g0j is simple (j 6= 0) Lie algebra. ˜ The following description of irreducible modules in the category K(G) was recently obtained in [11]. ˜ Theorem 5.1. Let V be an irreducible module in K(G). Then there exists a parabolic subset P ⊂ ∆ and an irreducible cuspidal module N over the Levi subsuperalgebra P 0 of the corresponding parabolic subsuperalgebra P ⊂ G such that V ' LP (N ). ˜ The theorem reduces the classification of irreducible modules in K(G) to the classification of cuspidal modules. Now we need to establish cuspidal Levi subsuperalgebras of affine Lie superalgebras, i.e., those that admit cuspidal modules. Cuspidal Levi subsuperalgebras were described in [11]. It turns out that simple cuspidal superalgebras are exactly those obtained by Dimitrov, Mathieu, and Penkov in the finite-dimensional case [6], [7] (cf. Theorem 3.2). Consider now some examples. Examples: 1. Let G be the affinization of basic classical Lie superalgebra of type A(m, n) with the diagram ³⊗a ³³ α aaa 0 ³³ a ³³

³³ g

α1

³

³ ³³

³

g

αn

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⊗ γ

g

β1

aa aa aa a

g

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In this case any cuspidal subsuperalgebra is a subalgebra of sl(m) ⊕ sl(n) and, hence, we get a complete classification of irreducible modules in the ˜ category K(G). 2. Let now G be the affinization of the Lie superalgebra of type C(n), n ≥ 3, with the following diagram α0 ⊗XXX XX g » » »

⊗»»

α2

α1

" g " b b

g

αn−1

α3

g

αn

Any cuspidal subsuperalgebra is a subalgebra of sp(2n − 2k) ⊕ sp(2k − 2) (k = 2, . . . , n), which again means that we have a complete classification of irreducible G-modules. 3. Consider the affinization G of the Lie superalgebra of type D(2, 1; a) with the diagram α0 g

g

⊗

g

α2

α1

α3

Any noncuspidal module in this case is induced from a subalgebra of sl(2) ⊕ sl(2) ⊕ sl(2). Hence we have a complete classification of noncuspi˜ dal irreducible modules in K(G). Classification of cuspidal modules for this affine Lie superalgebra is an open question. 4. Let G be the affinization of G(3) with the diagram

g

⊗

α0

α1

" g" b b

α2

g

α3

Subsuperalgebras of the Lie superalgebras sl(2) ⊕ sl(3), sl(2) ⊕ osp(3, 2), sl(3) ⊕ osp(1, 2), osp(4, 2) exhaust all possible cuspidal Levi subsuperalgebra in this case.

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Now we make the final remarks. If basic classical Lie superalgebra g has type X we will say that affine Lie superalgebra G has type Xˆ. Remark 5.2. (1) Theorem 5.1 together with Mathieu’s classification of cuspidal modules over simple Lie algebras of type A and C gives a complete clas˜ sification of irreducible modules in K(G) when G is of type A(m, n)ˆ ˆ and C(n) . (2) If G is not of type A(m, n)ˆ and C(n)ˆ then the classification of ir˜ reducible modules in K(G) is reduced to a finite indeterminacy by [6],Proposition 6.3. ˜ (3) Classification of noncuspidal irreducible modules in K(G) for D(2, 1; a) is based on the classification of cuspidal sl(2)-modules and hence complete. ˜ (4) Classification of irreducible modules in K(G) will be complete after the classification of cuspidal irreducible modules over Lie superalgebras osp(m, 2n), m = 1, 3, 4, 5, 6 and D(2, 1; a). 6. Integrable modules ˜ Next we address a question of integrability in the category K(G). Consider first the case of affine Lie algebras. We use the notations from Section 4. A standard definition of the integrability is the following. Definition. A weight G-module V is called integrable if all nonzero real root elements act locally nilpotently on V . Equivalently, a weight G-module V is integrable if every real roots subspace Gα is locally finite on V , i.e., dim U (Gα )v ≤ ∞ for all v ∈ V . In particular, g is locally finite on V . Integrable modules is a natural generalization of finite-dimensional representations of finite-dimensional Lie algebras. For such modules, the action of the affine Lie algebra can be integrated to an action of the corresponding affine Lie group. All irreducible integrable modules with finite-dimensional weight spaces over affine Lie algebras were classified by Chari [4]. Depending on the action of the central element they are divided into two groups: integrable loop modules (when the central element c acts trivially) and highest weight modules (when c acts by positive integer). Loop modules are irreducible quotients of modules induced from the Heisenberg subalgebra G. Integrable irreducible highest weight modules have dominant integral highest weights. From Chari’s classification we immediately obtain that any irreducible inte˜ grable module in K(G) is a highest weight module with a dominant integral highest weight. In the case of affine Lie superalgebras loop modules were constructed by Dimitrov and Penkov [8]. For G different from A(m, n)ˆ and C(n)ˆ the

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irreducible integrable loop modules were classified by Rao and Zhao in [20, Theorem 5.6]. Irreducible integrable highest weight modules were classified by Kac and Wakimoto [15], [16]. An interesting phenomenon occurs in the case of affine Lie superalgebras as only trivial integrable modules exist for almost all affine superalgebras if the central element acts nontrivially. Namely, we have the following theorem [20, Theorems 7.1, 7.2]. Theorem 6.1. Let V be an irreducible integrable module over an affine Lie ˜ superalgebra G in the category K(G). (i) If G is not of type A(0, n)ˆ, B(0, n)ˆ, n ≥ 1 and C(n)ˆ, n ≥ 3, then V is a trivial module. (ii) If G is of type A(0, n)ˆ, B(0, n)ˆ, n ≥ 1 and C(n)ˆ, n ≥ 3, then V is a highest weight module with respect to a certain choice of a Borel subalgebra. Combining Theorem 6.1 with the results of [15], [16] we obtain a complete ˜ classification of irreducible integrable modules in K(G). Rao [19] showed a ˜ complete reducibility of integrable modules in the category K(G). Hence we have a classification of all such modules in this category. In the category K0 (G) the classification of irreducible integrable modules is known for affine Lie superalgebras different from A(m, n)ˆ and C(n)ˆ [20, Theorem 7.4]. The cases of superalgebras of type A(m, n)ˆ and C(n)ˆ are open. The existence of mostly trivial integrable modules for affine Lie superalgebras led to the concept of partial integrability in the paper [16], which was called a weak integrability in [11]. Definition. A module V over an affine Lie superalgebra is weakly integrable if it is integrable over an affine subalgebra of the even part G0 and the underlined finite-dimensional superalgebra g is locally finite on V . Thus weak integrability requires that only a part of G0 (G0j for some j ≥ 1) acts locally nilpotent and the whole g is locally finite. For C(n)ˆ a weak integrability coincides with the usual integrability since N = 1 in this case. A complete classification of irreducible weakly integrable highest weight modules was obtained by Kac and Wakimoto in [16]. Moreover, the following result obtained in [11] shows that they exhaust all weakly ˜ integrable irreducible modules in the category K(G). Theorem 6.2. ˜ (i) Let V be an irreducible module in K(G) such that g0 is locally finite on V . Then V is a highest weight module. ˜ (ii) Any weakly integrable irreducible module in K(G) is a highest weight module.

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Note that in Theorem 6.2 one only needs to use the fact that g0 is locally ˜ finite on an irreducible module in K(G) to conclude that this module is a highest weight module. References [1] V. Bavula and F. van Oystaeyen, The simple modules of the Lie supealgebra osp(1, 2), J. Pure and Applied Algebra. 150 (2000), 41–52. [2] V. Bekkert, G.Benkart, and V.Futorny, Weyl algebra modules, Contemporary Mathematics, 343 (2004), 17–42. [3] R. Block, The irreducible representations of the Lie algebra sl(2) and of the Weyl algebra, Adv. Math. 39 (1981), 69–110. [4] V. Chari, Integrable representations of affine Lie algebras, Invent. Math. 85 (1986), no.2, 317–335. [5] V. Chari and A.Pressley, New unitary representations of loop groups, Math. Ann. 275 (1986), 87–104. [6] I. Dimitrov, O. Mathieu, and I.Penkov, On the structure of weight modules, Trans. Amer. Math. Soc. 352 (2000), 2857–2869. [7] I. Dimitrov, O. Mathieu, and I.Penkov, Errata, Trans. Amer. Math. Soc. (to appear). [8] I. Dimitrov and I. Penkov, Partially and fully integrable modules over Lie superalgebras, Studies in Advanced Mathematics (series editor S.-T. Yau), 4, AMS and Internatl. Press 1997, 49–67. [9] S. Fernando, Lie algebra modules with finite dimensional weight spaces I, Trans. Amer. Math. Soc. 322 (1990), 757–781. [10] V. Futorny, Representations of Affine Lie algebras, Queen’s Papers in Pure and Applied Mathematics, v. 106, 1997, Kingston, Canada. [11] V. Futorny and S. Rao, Classification of integrable modules for affine Lie superalgebras, RT-MAT 2004-11. [12] V. Futorny and A. Tsylke, Classification of irreducible nonzero level modules with finite-dimensional weight spaces for affine Lie algebras, J. Algebra 238 (2001), 426– 441. [13] A. Joseph, The minimal orbit in a simple Lie algebra and its associated maximal ´ Norm. Sup. 9 (1976), 1–30. ideal, Ann. Scient. Ec. [14] V. Kac, Lie superalgebras, Adv. Math. 26 (1977), 8–96. [15] V. Kac and M. Wakimoto, Integrable highest weight modules over affine Lie superalgebras and number theory, Lie Theory and Geometry, Birkhauser, Progress in Mathematics 123 (1994), 415–456. [16] V. Kac and M. Wakimoto, Integrable highest weight modules over affine Lie superalgebras and Appell’s function, Comm. Math. Phys. 215 (2001), 631–682. [17] O. Mathieu, Classification of irreducible weight modules, Ann. Inst. Fourier 50 (2000), 537–592. [18] S. Rao, Classification of loop modules with finite-dimensional weight spaces, Math. Ann. 305 (1996), 651–663. [19] S. Rao, Complete reducibility of integrable modules for the affine Lie (super)algebras, J. Algebra 264 (2003), 269–278. [20] S. Rao and K. Zhao, On integrable representations for toroidal Lie superalgebras, Contemporary Mathematics 343 (2004), 243–261.

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On algebras and superalgebras with linear codimension growth Antonio Giambruno, Daniela La Mattina, and Paola Misso Dipartimento di Matematica e Applicazioni, Universit`a di Palermo, Via Archirafi 34, 90123 Palermo, Italy [email protected] [email protected] [email protected] Abstract. We present the classification, up to PI-equivalence, of the algebras over a field of characteristic zero whose sequence of codimensions is linearly bounded. We also describe the generalization of this result in the setting of superalgebras and their graded identities. As a consequence we determine all linear functions describing the ordinary codimensions and the graded codimensions of a given algebra.

1. Introduction The polynomial identities satisfied by a given algebra A over a field of characteristic zero can be measured through the sequence of codimensions cn (A), n = 1, 2, . . . , of the algebra. This sequence was introduced by Regev in [21] and its n-th term measures the dimension of the space of multilinear polynomials in n variables in the corresponding relatively free algebra of countable rank. Since in characteristic zero, by the multilinearization process, every identity is equivalent to a system of multilinear ones, the sequence of codimensions of A gives a quantitative measure of the identities satisfied by the given algebra. Maybe the most important feature of this sequence proved in [21] is that in case A is an associative algebra satisfying a polynomial identity (PI-algebra), then cn (A) is exponentially bounded. A primary tool for studying the identities of an algebra is provided by the representation theory of the symmetric group Sn . By the above reasons and also since the representation theory of Sn is well developed in characteristic zero, most attention has been focused on algebras over a field of characteristic zero and we shall make such assumption here. The exponential rate of growth of the sequence of codimensions was determined in [9] and [10] by Giambruno and Zaicev who proved that for any 173

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p PI-algebra A, the limit limn→∞ n cn (A) exists and is a nonnegative integer. Such integer denoted exp(A) is called the PI-exponent of the algebra A. Having at hand a scale provided by the PI-exponent most work has been concentrated in recent years in trying to characterize PI-algebras, or their identities, having the same PI-exponent (see [13], [14], [12]). Here we shall be dealing with algebras A for which exp(A) ≤ 1, i.e., whose sequence of codimensions is polynomially bounded. In [16], [17] Kemer gave a characterization of the algebras (or of their identities) whose sequence of codimensions is polynomially bounded. He proved that A is such an algebra if and only if G and U T2 do not satisfy all the identities of the algebra A, where G is the Grassmann algebra and U T2 is the algebra of 2 × 2 upper triangular matrices. As a consequence of this result he observed that the sequence of codimensions of any PI-algebra A, is either polynomially bounded or grows exponentially. Hence no intermediate growth is allowed. A refinement of the result of Kemer was recently achieved in [4]. In fact, there the authors completely classified all T-ideals whose sequence of codimensions is linearly bounded. As a consequence it turned out that the allowed linear functions can be classified. The purpose of this paper is to present these results and a generalization in the setting of superalgebras. Since the ordinary algebras are superalgebras with trivial grading we shall be able to recover the classification obtained in [4]. We should mention that the superalgebras and their graded identities play a relevant role in the structure theory of varieties developed by Kemer (see [18]). In the case of superalgebras, one defines similar invariants measuring the growth of the graded identities ([8]). In particular in the sequence of graded codimensions cgr n (A), n = 1, 2, . . . , of a superalgebra A, the n-th term measures the dimension of the space of multilinear elements of the relatively free superalgebra of countable rank of A. It turns out that if a superalgebra satisfies an ordinary identity, then its sequence of graded codimensions is exponentially bounded ([8]). Moreover the hyperoctahedral group Z2 o Sn and its representation theory are a natural tool for studying the graded identities of a superalgebra in characteristic zero. The problem of characterizing the graded identities of a superalgebra whose sequence of graded codimensions is polynomially bounded was studied in [7]. It was proved that a superalgebra A has such property if and only if its graded identities are not a consequence of the graded identities of five explicit superalgebras. Four of these algebras are the algebras G and U T2 endowed with suitable Z2 -gradings. In particular these results show that also for the superalgebras no intermediate growth is allowed. For finitely generated superalgebras satisfying anpordinary polynomial identity, it was shown in [1] that the limit limn→∞ n cgr n (A) exists and is

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a nonnegative integer, and a theory extending the ordinary case is being developed. Here we shall present a complete classification of the ideals of identities and graded identities whose sequence of codimensions and graded codimensions is linearly bounded. Moreover for each such ideal I we shall exhibit an algebra or superalgebra having I as ideal of identities or graded identities. 2. Generalities Throughout this paper, we shall denote by F a field of characteristic zero and by A an associative algebra over F. Let F hXi be the free associative algebra on the countable set X = {x1 , x2 , . . .} and let Id(A) = {f ∈ F hXi | f ≡ 0 on A} be the set of polynomial identities of A. Id(A) is a T -ideal of F hXi, i.e., an ideal invariant under all endomorphisms of F hXi. It is well known that in characteristic zero Id(A) is completely determined by its multilinear polynomials and we denote by Vn = spanF {xσ(1) · · · xσ(n) | σ ∈ Sn } the space of multilinear polynomials in the indeterminates x1 , . . . , xn . The symmetric group Sn acts on the left on Vn as follows: if σ ∈ Sn and f (x1 , . . . , xn ) ∈ Vn , then σf (x1 , . . . , xn ) = f (xσ(1) , . . . , xσ(n) ). Since the subspace Vn ∩ Id(A) is invariant under this action, Vn Vn (A) = Vn ∩ Id(A) has a structure of Sn -module and cn (A) = dimF Vn (A) is the n-th codimension of A. Given an algebra A, let var(A) denote the variety of algebras generated by A. It is clear that if A and B are F -algebras and B ∈ var(A) then var(B) ⊆ var(A) and cn (B) ≤ cn (A). Moreover if A and B are F -algebras, then Id(A⊕B) = Id(A)∩Id(B) and cn (A⊕B) ≤ cn (A)+cn (B). In case B is a nilpotent algebra and B k = 0, then for all n ≥ k we have cn (A ⊕ B) = cn (A) in this case. In case of superalgebras one defines similar notions. Recall that A is Z2 -graded with grading (A(0) , A(1) ) if A has a vector space decomposition A = A(0) ⊕ A(1) such that A(0) A(0) + A(1) A(1) ⊆ A(0) and A(0) A(1) + A(1) A(0) ⊆ A(1) . In the sequel we shall usually write A = A(0) ⊕A(1) and also A = (A(0) , A(1) ) to indicate that A is such a Z2 -graded algebra. The elements of A(0) and of A(1) are called homogeneous of degree zero (or even elements) and of degree one (or odd elements), respectively. Moreover, a subalgebra B ⊆ A is Z2 -graded or homogeneous if B = (B ∩ A(0) ) ⊕ (B ∩ A(1) ).

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The free associative algebra F hXi can be made into a superalgebra as follows. Write X = Y ∪ Z as the disjoint union of two sets Y and Z. Then, if we denote by F 0 the subspace of F hXi spanned by all monomials in the variables of X having even degree in the variables of Z and by F 1 the subspace spanned by all monomials of odd degree in Z, then F hXi = F hY, Zi = F 0 ⊕ F 1 is a Z2 -graded algebra with grading (F 0 , F 1 ). F hY, Zi is the free superalgebra on Y and Z over F. Recall that a polynomial f (y1 , . . . , yn , z1 , . . . , zm ) ∈ F hY, Zi is a graded identity for the superalgebra A = A(0) ⊕ A(1) if f (a1 , . . . , an , b1 , . . . , bm ) = 0 for all a1 , . . . , an ∈ A(0) , b1 , . . . , bm ∈ A(1) . In this case we also write f ≡ 0 in A. Let Idgr (A) = {f ∈ F hY, Zi | f ≡ 0 in A} be the set of graded identities of A. It is readily seen that Idgr (A) is a T2 -ideal of F hY, Zi, i.e., an ideal invariant under all endomorphisms η of F hY, Zi such that η(F 0 ) ⊆ F 0 and η(F 1 ) ⊆ F 1 . It is well known that, in characteristic zero, every graded identity is equivalent to a system of multilinear graded identities. Hence Idgr (A) is completely determined by its multilinear polynomials. Let Vngr = span{wσ(1) · · · wσ(n) | wi = yi or zi , i = 1, . . . , n, σ ∈ Sn } be the space of multilinear graded polynomials in y1 , z1 , . . . , yn , zn . We act on Vngr via the hyperoctahedral group Z2 o Sn . Recall that if Z2 = {±1} is the multiplicative group of order 2 and Sn is the symmetric group on {1, . . . , n}, then Z2 o Sn = {(a1 , . . . , an ; σ) | ai ∈ Z2 , σ ∈ Sn } with multiplication given by (a1 , . . . , an ; σ) (b1 , . . . , bn ; τ ) = (a1 bσ−1 (1) , . . . , an bσ−1 (n) ; στ ). The action of the group Z2 o Sn on Vngr is defined as follows: aσ(i) for h = (a1 , . . . , an ; σ) ∈ Z2 o Sn , we set hyi = yσ(i) and hzi = zσ(i) = zσ(i) or −zσ(i) depending on whether aσ(i) = 1 or −1, respectively. For every Z2 -graded algebra A, the vector space Vngr ∩ Idgr (A) is invariant under this action. Hence the space Vngr (A) = Vngr /(Vngr ∩ Id(A)) has a structure of gr left Z2 o Sn -module and cgr n (A) = dimF Vn (A) is called the nth graded codimension of A. Given a superalgebra A let vargr (A) denote the variety of superalgebras generated by A. Notice that any F -algebra A can be regarded as a superalgebra with trivial grading, i.e., A = A(0) ⊕A(1) where A = A(0) and A(1) = 0. Hence the theory of graded identities generalizes the ordinary theory of polynomial identities. The relation between ordinary codimensions and graded codimensions is given in [8]: given a superalgebra A, cn (A) ≤ cgr n (A) for all n ≥ 1 and, in n case A satisfies an ordinary polynomial identity, cgr n (A) ≤ 2 cn (A).

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3. Varieties of polynomial growth Let G denote the infinite dimensional Grassmann algebra over F . Recall that G is the algebra generated by the elements e1 , e2 , . . . , subject to the condition ei ej = −ej ei i, j ≥ 1. Let also U T2 be the algebra of 2 × 2 upper triangular matrices over F . The identities of G were determined by Krakowski and Regev in [19], whereas the identities of U T2 were determined by Malcev in [20]. We state this in the following lemma. For f1 , . . . , fm ∈ F hXi, let hf1 , . . . , fm iT denote the T -ideal generated by the polynomials f1 , . . . , fm . Lemma 1. - Id(G) = h[x1 , x2 , x3 ]iT and cn (G) = 2n−1 . - Id(U T2 ) = h[x1 , x2 ][x3 , x4 ]iT and cn (U T2 ) = 2n−1 (n − 2) + 2. As we mentioned in the introduction, Kemer ([16], [17]) characterized the varieties of algebras of polynomial growth. He proved the following: Theorem 2. Let A be an algebra. Then the sequence of codimensions cn (A), n = 1, 2, . . . , is polynomially bounded if and only if G, U T2 6∈ var(A). One can try to refine Kemer’s result relating the polynomial growth of the codimensions to a list of algebras not belonging to the variety. This project has been carried out in case of bounded or linear growth and we present these results here. We start by describing the identities of some special algebras. In [15] Guterman and Regev determined the T-ideal of identities of the subalgebras P of the n × n upper triangular matrix algebra of the type F ekk + i<j F eij , 1 ≤ k ≤ n. Here we state only a special case of this theorem. µ ¶ µ ¶ 0 F F F Lemma 3. Let M1 = and M2 = . Then 0 F 0 0 - Id(M1 ) = hx1 [x2 , x3 ]iT and cn (M1 ) = n. - Id(M2 ) = h[x1 , x2 ]x3 iT and cn (M2 ) = n.   a b c Lemma 4. Let M3 = { 0 a d  | a, b, c, d ∈ F }. Then Id(M3 ) = 0 0 a h[x1 , x2 , x3 ], [x1 , x2 ][x3 , x4 ]iT and for all n > 3, cn (M3 ) =

n(n−1) 2

+ 1.

Proof. Let Q = h[x1 , x2 , x3 ], [x1 , x2 ][x3 , x4 ]iT . Since [M3 , M3 ] ⊆ span{e13 } it is clear that [x1 , x2 , x3 ] and [x1 , x2 ][x3 , x4 ] are identities of M3 and Q ⊆ Id(M3 ). Before proving the opposite inclusion, we find a generating set of Vn modulo Vn ∩ Q. It is well known (see for instance [2, Theorem 5.2.1]) that any multilinear polynomial of degree n can be written, modulo h[x1 , x2 ][x3 , x4 ]iT , as a linear combination of polynomials of the type xi1 · · · xim [xk , xj1 , . . . , xjn−m−1 ]

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where i1 < · · · < im , j1 < · · · < jn−m−1 , k > j1 , m 6= n − 1. Thus, because of the identity [x1 , x2 , x3 ], the elements (1)

x1 · · · xn , xi1 · · · xin−2 [xi , xj ], i1 < · · · < in−2 , i > j

span Vn modulo Vn ∩ Q. We next prove that these elements are linearly independent modulo Id(M3 ). Suppose that n X f= αij xi1 · · · xin−2 [xi , xj ] + βx1 · · · xn ≡ 0 (mod Vn ∩ Id(M3 )). i,j=1 i>j

By making the evaluation xk = e11 + e22 + e33 , for all k = 1, . . . , n, we get β = 0. Also, for fixed i and j, the evaluation xi = e12 , xj = e23 and xk = e11 + e22 + e33 for k ∈ / {i, j} gives αij = 0. Thus the elements in (1) are linearly independent modulo Vn ∩ Id(M3 ). Since Vn ∩ Q ⊆ Vn ∩ Id(M3 ), this proves that Id(M3 ) = Q and the elements in (1) are a basis of Vn modulo Vn ∩Id(M3 ). By counting them we obtain cn (M3 ) = dim Vn /(Vn ∩Id(M3 )) = n(n − 1)/2 + 1. ¤ We remark that there is a precise relation between the algebra M3 and the algebras G and U T2 . In fact an inspection of the identities of these algebras shows that M3 ∈ var(G) ∩ var(U T2 ). Recall that two algebras A and B are PI-equivalent and we write A ∼P I B if Id(A) = Id(B). We can now state the classification, up to PI-equivalence, of the algebras whose sequence of codimensions is bounded by a constant ([4]). Theorem 5. For an F -algebra A, the following conditions are equivalent. 1) M1 , M2 , M3 ∈ / var(A). 2) Either A ∼P I N or A ∼P I C ⊕ N where N is a nilpotent algebra and C is a commutative algebra. 3) cn (A) ≤ k for some constant k ≥ 0, for all n ≥ 1. 4) There exists n0 > 0 such that cn (A) ≤ 1, for all n ≥ n0 . The next step is to classify the identities of the algebras with linearly bounded codimension growth. We start by stating another special case of the result in [15].     F F F 0 F F Lemma 6. Let M4 =  0 0 F  , M5 =  0 0 F  and M6 = 0 0 0 0 0 F   0 F F  0 F F  . Then for all n > 3, 0 0 0 - Id(M4 ) = h[x1 , x2 ]x3 x4 iT , Id(M5 ) = hx1 x2 [x3 , x4 ]iT , Id(M6 ) = hx1 [x2 , x3 ]x4 iT .

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- cn (M4 ) = cn (M5 ) = cn (M6 ) = n(n − 1). We next introduce the following  a b M7 = { 0 0 0 0

algebra  c d  |a, b, c, d ∈ F }. a

It is not hard to see, using the representation theory of Sn (see [4]) that the sequence of codimensions of M7 is bounded from below by n2 . We now state the classification theorem concerning linear growth of the codimensions ([4]). Theorem 7. Let A be an F -algebra. Then the following conditions are equivalent. 1) cn (A) ≤ kn for all n ≥ 1, for some constant k. 2) M3 , M4 , M5 , M6 , M7 6∈ var(A). 3) A is PI-equivalent to either N or C ⊕ N or M1 ⊕ N or M2 ⊕ N or M1 ⊕ M2 ⊕ N where N is a nilpotent algebra and C is a commutative algebra. As in the previous lemmas, an analysis of the identities of the algebra M1 ⊕ M2 shows that cn (M1 ⊕ M2 ) = 2n − 1. Hence the following corollary can be easily deduced. Corollary 8. Let A be an F -algebra such that cn (A) ≤ kn for all n ≥ 0. Then there exists n0 such that for all n > n0 we must have either cn (A) = 0 or 1 or n or 2n − 1. 4. Supervarieties of polynomial growth Kemer’s theorem on the characterization of varieties of polynomial growth has been generalized in [7] in the setting of varieties of superalgebras and we shall describe this result below. Recall that the Grassmann algebra G has a natural Z2 -grading G = G(0) ⊕ (1) G where G(0) is the span of all monomials in the ei s of even length and G(1) is the span of all monomials in the ei s of odd length. We shall write G to mean G with the trivial grading and Ggr to mean G with the above grading (G(0) , G(1) ). Also it can be easily checked that the algebraµ U T2 has¶ only F 0 (0) (1) (0) one nontrivial grading U T2 = U T2 ⊕ U T2 where U T2 = and 0 F µ ¶ 0 F (1) U T2 = (see [22]). The algebra U T2 regarded as a superalgebra 0 0 with trivial grading will be denoted U T2 and endowed with the above grading will be denoted U T2gr . Finally we let F ⊕ tF be the commutative algebra with grading (F, tF ) where t2 = 1. The graded identities of Ggr , U T2gr and F ⊕ tF were studied in [7], [22], and [6], respectively. Given polynomials

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f1 , . . . , fm ∈ F hY, Zi, let hf1 , . . . , fm iT2 denote the T2 -ideal generated by f1 , . . . , fm . We have Lemma 9. gr n - Idgr (Ggr ) = h[y1 , y2 ], [y, z], z1 z2 + z2 z1 iT2 and cgr n (G ) = 2 . gr n−1 . - Idgr (U T2gr ) = h[y1 , y2 ], z1 z2 iT2 . and cgr n (U T2 ) = 1 + n2 gr gr - Id (F ⊕ tF ) = h[y1 , y2 ], [y, z], [z1 , z2 ]iT2 and cn (F ⊕ tF ) = 2n . The main result of [7] says that the above five superalgebras characterize the graded varieties of polynomial growth. Theorem 10. Let A be a superalgebra. Then the sequence of graded codimensions cgr n (A), n = 1, 2, . . . , is polynomially bounded if and only if G, Ggr , U T2 , U T2gr , F ⊕ tF 6∈ vargr (A). Recall that given superalgebras A and B, we say that A is T2 -equivalent to B and we write A ∼T2 B if Idgr (A) = Idgr (B). In order to obtain a classification up to PI-equivalence of the superalgebras whose sequence of graded codimensions is linearly bounded we need to study the algebras M1 and M2 with all possible Z2 -gradings. Below we denote by M1,1 and M2,1 the algebras M1 and M2 , respectively, endowed with the trivial grading. µ ¶ µ ¶ µ ¶ 0 F 0 0 0 F Lemma 11. Let M1,1 = ( , 0), M1,2 = ( , ), 0 F µ 0 ¶F 0 0 ¶ ¶ µ µ F 0 0 F F F , 0), M2,2 = ( , ). Then M2,1 = ( 0 0 0 0 0 0 - Idgr (M1,1 ) = hy1 [y2 , y3 ], ziT2 , Idgr (M2,1 ) = h[y1 , y2 ]y3 , ziT2 . - Idgr (M1,2 ) = h[y1 , y2 ], yz, z1 z2 iT2 and Idgr (M2,2 ) = h[y1 , y2 ], zy, z1 z2 iT2 . gr - cgr n (M1,1 ) = cn (M2,1 ) = n. gr gr - cn (M1,2 ) = cn (M2,2 ) = n + 1. Proof. The graded identities of M1,1 and M2,1 are the ordinary identities of M1 and M2 respectively. Hence the first part of the lemma is Lemma 3 above. If Q is the T2 -ideal generated by the polynomials [y1 , y2 ], yz, z1 z2 , then it is easily checked that Q ⊆ Idgr (M1,2 ). Also the polynomials y1 · · · yn , zi yi1 · · · yin−1 , i1 < · · · < in−1 gr (mod Vn ∩ Q) and we claim that they are linearly Idgr (M1,2 ). In fact, let f ∈ Vngr ∩ Idgr (M1,2 ) be a linear

Vngr

span independent modulo combination of these polynomials. By multihomogeneity of T2 -ideals we may assume that either f = αy1 · · · yn or f = βzn y1 · · · yn−1 . By making the evaluations y1 = · · · = yn = e22 and zn = e12 , y1 = · · · = yn−1 = e22 , we get α = β = 0. Thus the above polynomials are linearly independent modulo Vngr ∩ Idgr (M1,2 ). Since Vngr ∩ Q ⊆ Vngr ∩ Idgr (M1,2 ), it follows that Idgr (M1,2 ) = Q and

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the above polynomials are a basis of Vngr (mod Vngr ∩ Idgr (M1,2 )). Clearly cgr ¤ n (M1,2 ) = n + 1. A similar proof holds for the algebra M2,2 . We also have the following: ¶ µ ¶ µ a b a 0 Lemma 12. Let M = { | a, b ∈ F } with grading ( , 0 a 0 a µ ¶ 0 b ). Then Idgr (M ) = h[y1 , y2 ], [y, z], z1 z2 iT2 and cgr n (M ) = n + 1. 0 0 We remark that if M3,1 denotes the algebra M3 with trivial grading, the relation between the algebras classifying polynomial growth given in Theorem 10 and the above algebras is given in the following Remark 13. M3,1 ∈ vargr (G) ∩ vargr (U T2 ) and M ∈ vargr (Ggr ) ∩ vargr (U T2gr ) ∩ vargr (F ⊕ tF ). We can now state the result classifying, up to PI-equivalence, the superalgebras whose sequence of codimensions is bounded by a constant ([5]). Theorem 14. For a superalgebra A, the following conditions are equivalent. 1) M1,1 , M1,2 , M2,1 , M2,2 , M, M3,1 ∈ / vargr (A). 2) A is T2 -equivalent to either N or C ⊕ N where N is a nilpotent superalgebra and C is a commutative algebra with trivial grading. 3) cgr n (A) ≤ k for some constant k ≥ 0, for all n ≥ 1. 4) cgr n (A) = cn (A) ≤ 1 for n large enough. The superalgebras whose sequence of codimensions is linearly bounded have been classified up to PI-equivalence in [5]. In there the authors also found a list of 24 superalgebras Ai , 1 ≤ i ≤ 24, characterizing the supervarieties of linear growth, that is, cgr / n (A) ≤ kn if and only if A1 , . . . , A24 ∈ vargr (A). The classification is the following. Theorem 15. For a superalgebra A the following conditions are equivalent. 1) A is T2 -equivalent to either N, a nilpotent superalgebra, or C ⊕ N, where C is a commutative algebra with trivial grading, or M ⊕ N or B ⊕ N or B ⊕ M ⊕ N or B1 ⊕ B2 ⊕ N or B1 ⊕ B2 ⊕ M ⊕ N where B ∈ C1 ∪C2 , B1 ∈ C1 and B2 ∈ C2 with C1 = {M1,1 , M1,2 , M1,1 ⊕M1,2 }, C2 = {M2,1 , M2,2 , M2,1 ⊕ M2,2 }. 2) cgr n (A) ≤ kn for all n ≥ 1, for some constant k. As a consequence it can be proved that the only allowed sequences of graded codimensions linearly bounded are the following: 0, 1, n, n + 1, 2n − 1, 2n, 2n + 1, 3n − 1, 3n, 3n + 1, 4n − 1, 4n, 5n − 1. Acknowledgments Research partially supported by MIUR of Italy.

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References [1] F. Benanti, A. Giambruno, and M. Pipitone, Polynomial identities on superalgebras and exponential growth, J. Algebra 269 (2003), no. 2, 422–438. [2] V. Drensky, Free Algebras and PI-Algebras, Graduate course in algebra, SpringerVerlag Singapore, Singapore, 2000. [3] V. Drensky and A. Giambruno, Cocharacters, codimensions and Hilbert series of the polynomial identities for 2 × 2 matrices with involution, Canadian J. Math., 46 (1994), 718–733. [4] A. Giambruno and D. La Mattina, PI-algebras with slow codimension growth, J. Algebra 284 (2005), no. 1, 371–391. [5] A. Giambruno, D. La Mattina, and P. Misso, Polynomial identities on superalgebras: classifying linear growth, preprint. [6] A. Giambruno and S. Mishchenko, Polynomial growth of the ∗-codimensions and Young diagrams, Comm. Algebra 29 (2001), 277–284. [7] A. Giambruno, S. Mishchenko, and M. Zaicev, Polynomial identities on superalgebras and almost polynomial growth, Comm. Algebra 29 (2001), no. 9, 3787–3800. [8] A. Giambruno and A. Regev, Wreath products and P.I. algebras, J. Pure Applied Algebra, 35 (1985), 133–149. [9] A. Giambruno and M. Zaicev, On codimension growth of finitely generated associative algebras, Adv. Math. 140 (1998), 145–155. [10] A. Giambruno and M. Zaicev, Exponential codimension growth of P.I. algebras: an exact estimate, Adv. Math. 142 (1999), 221–243. [11] A. Giambruno and M. Zaicev, A characterization of algebras with polynomial growth of the codimensions, Proc. Amer. Math. Soc. 129 (2000), 59–67. [12] A. Giambruno and M. Zaicev, Asymptotics for the standard and the Capelli identities, Israel J. Math. 135 (2003), 125–145. [13] A. Giambruno and M. Zaicev, Minimal varieties of exponential growth, Adv. Math. 174 (2003), 310—323. [14] A. Giambruno and M. Zaicev, Codimension growth and minimal superalgebras, Trans. Amer. Math. Soc. 355 (2003), no. 12, 5091–5117. [15] A. Guterman and A. Regev, On the growth of identities, In: Algebra (Moscow, 1998), 319–330, de Gruyter, Berlin, 2000. [16] A.R. Kemer, T-ideals with power growth of the codimensions are Specht (Russian), Sibirskii Matematicheskii Zhurnal 19 (1978), 54–69; English translation: Siberian Math. J. 19 (1978), 37–48. [17] A.R. Kemer, Varieties of finite rank, Proc. 15-th All the Union Algebraic Conf., Krasnoyarsk, Vol 2, p. 73, 1979 (Russian). [18] A.R. Kemer, Ideals of Identities of Associative Algebras, AMS Translations of Mathematical Monograph, Vol. 87, 1988. [19] D. Krakowski and A. Regev, The polynomial identities of the Grassmann algebra, Trans. Amer. Math. Soc. 181 (1973), 429–438. [20] Yu.N. Maltsev, A basis for the identities of the algebra of upper triangular matrices, (Russian) Algebra i Logika 10 (1971), 242–247. [21] A. Regev, Existence of identities in A ⊗ B, Israel J. Math. 11 (1972), 131–152. [22] A. Valenti, The graded identities of upper triangular matrices of size two, J. Pure Appl. Algebra 172 (2002), no. 2-3, 325–335.

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On spectra of group rings of finite abelian groups Andr´ e Gimenez Bueno and Michael Dokuchaev Instituto de Matem´atica e Estat´ıstica, Universidade de S˜ao Paulo, Caixa Postal 66281, S˜ao Paulo, SP, 05315-970, Brasil [email protected] [email protected] Dedicated to Professor C´esar Polcino Milies on his sixtieth birthday. Abstract. In this paper we study the spectrum of integral group rings Z[G] of finite abelian groups G from the point of view of the theory of schemes. The irreducible components consist of spectra of rings of integers of certain cyclotomic fields, whose corresponding fraction fields are the fields that appear in the Wedderburn decomposition of Q[G], with the same multiplicities. The singular points of Spec(Z[G]) are found by calculating the module of K¨ ahler differentials of Z[G] over Z. As a simple byproduct of these calculations, we obtain the ramified points of the morphism Spec(Z[G]) → Spec(Z). It is shown that this set of points is the same as the set of singular points. This is the set of maximal primes p ∈ Z[G] that lie over those rational primes p such that p divides | G | .

1. Introduction It is usual in studying questions involving group rings to pay attention to their structure as rings and algebras, and to investigate modules over them as well as their groups of units. It is well known that from this point of view they have been and are being intensively studied (see [8], [9], [11], [12]). But if the group is commutative, these rings are commutative rings and as such have their naturally associated affine schemes. This was already in some sense studied before: the prime ideals of the spectrum of the representationring of G was considered in [2], when G is finite, and in [10], when G is a compact Lie group. On the other hand, such problems as the determination of the singular points, ramification points and K¨ahler differentials, seemingly have not been treated yet. Let G be a finite abelian group, A := Z[G] its integral group ring, and Spec A its spectrum, that is, the set of all prime ideals in A, considered 183

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as a topological space with the Zariski (spectral) topology (also denoted by Spec A) and endowed with the structural sheaf. Then A is a noetherian commutative ring and Spec A is a noetherian affine scheme. In Classical Algebraic Geometry the points of a variety correspond to the maximal ideals of its coordinate ring and the tangent space at a point x is the dual of the Ox /mx -vector space mx /mx 2 , where Ox is the local ring of x and mx is its maximal ideal. The point x is regular if and only if the ring Ox is regular. These concepts extend to arbitrary locally noetherian schemes, in particular to the spectrum of a commutative noetherian ring. We remind that in the spectral topology the closed points are exactly the maximal ideals. In the case of groups of prime order Cp the underlying topological space Spec Z[Cp ], consists of two irreducible components, one of which is homeomorphic to Spec (Z) and the other to Spec OK , where OK is the ring of integers of the cyclotomic field Q(ζp ) of p-th roots of unity. As it is well known, OK = Z[ζp ]. The components intersect at a single point, which is the only singular point of Spec Z[Cp ]. In this paper we describe Spec Z[G], for finite abelian G; in particular we determine its components and singularities. We shall use the following notations. For an ideal I in A we denote by V (I) the set of prime ideals in A containing I. These are the closed sets of the spectral topology. By dim A we mean the Krull dimension of A.

2. The modules of differentials of group rings For later use we shall make some remarks on K¨ahler differentials for group rings and for convenience we recall first some basic facts. For proofs, see, for example, [5, Ch. 0, §20], [1, Ch. 5,6,7], [7, Ch. 1], and [13, Ch. 11]. Let A and B be commutative rings, and A → B a ring map by means of which B becomes an A-algebra. An A-derivation from B to a B-module M is a map d : B → M that is A-linear and satisfies the Leibniz rule from Calculus. We define the B-module ΩB/A as the object representing the (covariant) functor M 7→ DerA (B, M ), and canonically attached to it there is the universal A-derivation dB/A : B → ΩB/A . In other words, any A-derivation d : B → M factors through the universal dB/A . The pair (dB/A , ΩB/A ) always exists and is unique up to unique isomorphism by its very definition. It can be constructed as follows. Let B ⊗A B → B be the map b ⊗ b0 7→ bb0 and I its kernel. Then the pair (dB/A , ΩB/A ) is (d, I/I2 ), where d : B → I/I2 is d(b) = 1 ⊗ b − b ⊗ 1 mod I2 . This construction carries naturally over to schemes. Let X be an S-scheme and ∆X/S : X → X ×S X the diagonal morphism. Then we define the sheaf of differentials ΩX/S = ∆∗ (I/I 2 ), where I is the sheaf of ideals defining the closed immersion of ∆(X) in some open set W of X ×S X. This gives ΩX/S a natural structure of an OX -module. When X = Spec(B) and S = Spec(A), we have ΩX/S = (ΩB/A )e.

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We shall need some fundamental functorial properties of the module ΩB/A . Given ring maps A → B → C, we have the exact sequence of Cmodules: (1)

ΩB/A ⊗B C → ΩC/A → ΩC/B → 0 .

The maps in the exact sequence come from the universal property of the module of differentials and the fact that the funtor extension of scalars is left adjoint to the functor restriction of scalars (For more details on this and the following properties see, for example, [1, Chap. 6, §1].) If now C = B/K, where K is any ideal of B, we have another exact sequence of C-modules: (2)

K/K2 → ΩB/A ⊗B C → ΩC/A → 0 .

The first map is k + K2 7→ dB/A (k) ⊗ ¯ 1. Furthermore, the module of differentials is compatible with extension of the ring of scalars, A → A0 , i.e., (B 0 = B ⊗A A0 is the base-extended algebra) (3)

ΩB 0 /A0 = ΩB/A ⊗B B 0 = ΩB/A ⊗A A0

with localization, i.e., if S ⊂ A is a multiplicative set mapping into a multiplicative set T ⊂ B, (4)

ΩT −1 B/S −1 A = T −1 ΩB/A

and with the tensor product B1 ⊗A B2 of two A-algebras, (5)

ΩB1 ⊗A B2 /A = j1∗ (ΩB1 /A ) ⊕ j2∗ (ΩB2 /A ) .

Here jα (α = 1, 2,) denotes the canonical morphism Bα → B1 ⊗A B2 and jα∗ the functor extending the scalars from Bα to B1 ⊗A B2 , that is, jα∗ (M ) = M ⊗Bα (B1 ⊗A B2 ). Observe that when passing to a localization the canonical derivation is obtained by the usual quotient rule. Remark 2.1. The module of differentials is also compatible with direct products ΩB1 ×B2 /A = ΩB1 /A ⊕ ΩB2 /A . It follows at once from this and the fact that ΩL/K = 0 for any finite separable field extension L/K, that for a finite separable K-algebra A (i.e., A := A ⊗ K has zero Jacobson radical) ΩA/K = 0. This is the case of the group algebra KG of a finite abelian group G over a perfect field K whose characteristic does not divide | G | . We shall now compute the modules of differentials of abelian group rings. It is well-known (and easy to see) that the polynomial ring B = A[T1 , . . . , Tn ], viewed as an algebra over A, has as its module of differentials ΩB/A the free module of rank n over B, and the universal derivation dB/A is the usual differential of polynomials in terms of partial derivatives. Since any finitely generated A-algebra C is a quotient A[T1 , . . . , Tn ]/K of the polynomial algebra A[T1 , . . . , Tn ] (for some n), the exact sequence (2)

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shows that the module ΩC/A can be described as the C-module with generators dT1 , . . . , dTn and relations dPi = 0, 1 ≤ i ≤ m, where {P1 , . . . , Pm } are generators of the ideal K. For example, if C is generated by one element, C = A[T ]/(P (T )), we have ΩC/A = C/(P 0 (t)), where t is the image of T in C. This is the case of the group algebra Z[Cn ] = Z[T ]/(T n − 1) of the cyclic group of order n, over the ring Z (in view of the compatibility (3), this is the essential case to be considered). Thus we have (6)

ΩZ[Cn ]/Z = Z[Cn ]/(n) = (Z/nZ)[Cn ] .

More generally, consider the integral group algebra of an arbitrary finite abelian group G. Since G is the direct product of cyclic p-groups, formula (5) reduces the computation of ΩZ[G]/Z to that of (6) with n = ph . Hence we have the formula M (7) ΩZ[G]/Z = (Z/pαi i )[Cpαi ] ⊗Z [C αi ] Z[G] , i

i

p i

in which the direct sum is over the cyclic p-groups in the decomposition of G. Another important example in what follows is the case of a group algebra over a field. In fact, to study the ramification of the morphism Spec(Z[G]) → Spec(Z), we can reduce to analyzing the ramification of Spec(Fp [G]) → Spec(Fp ), which in its turn can be seen by looking at ΩFp [G]/Fp . So consider Fp [G] as an algebra over Fp . Again, using (5), we reduce to computing ΩFp [G]/Fp when G is a cyclic p-group. In view of (2.1) and the semisimplicity of F` [G] for ` 6= p, (Maschke’s theorem) the module ΩF` [G]/F` 6= 0 at most when ` = p. Indeed, in this case (i.e., G = Cpn ) ΩFp [G]/Fp = Fp [G], i.e., it is free of rank 1. In general, in view of formula (5) and the result for the cyclic case, we obtain at once the following: Proposition 2.2. Let G be a finite abelian group. Then (8)

ΩFp [G]/Fp = (Fp [G])⊕lp ,

where lp is the number of p-cyclic factors of G. The question of regularity for some schemes over fields can be decided by looking at the K¨ahler differentials: Proposition 2.3. Let (A, m) be a local ring containing a field k isomorphic to its residue field A/m. Assume that A is the localization of a finitely generated k-algebra, and the field k is perfect. Then A is regular if and only if the module ΩA/k is free of rank dim(A). Proof. See [6, Ch. 2, Prop 8.13].

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187

Using the above two propositions we have the next fact: Proposition 2.4. Let G = Gp × Gp0 be a finite abelian group with Sylow p-subgroup Gp . Then Spec Fp [G] is a disjoint union of Spec Fp [Gp ] and Spec ∆Fp (G, Gp0 ), where ∆Fp (G, Gp0 ) stands for the kernel of the natural map Fp [G] → Fp [Gp ]. Moreover, Spec Fp [G] is singular if and only if Gp 6= 1. Proof. The decomposition of Spec Fp [G] into the disjoint union comes from P the idempotent |G1 0 | σ∈G 0 σ. If Gp 6= 1 then the rank of ΩFp [Gp ]/Fp is f > 0, p

p

while the local ring Fp [Gp ], being artinian, has dimension zero.

¤

Remark 2.5. Obviously Spec Fp [Gp0 ] is a disjoint union of one-point schemes that are spectra of fields, and, more generally, the same holds for Spec K[G] in which K is a field whose characteristic does not divide the order of the finite abelian group G. In particular, Spec K[G] is regular whenever char K does not divide | G | . Proposition 2.6. Let G be a finite abelian group. (i) The fiber of Spec(Z[G]) → Spec(Z) over a prime pZ is Spec(Fp [G]), and over the generic point of Spec(Z) is Spec(Q[G]). (ii) The only ramified fibres of Spec(Z[G]) (over the corresponding residue fields) are those lying over the primes dividing the order of G. (iii) The ramification points of Spec(Z[G]) → Spec(Z) are precisely those primes lying over the prime divisors of | G |, in particular, Spec(Z[G]) → Spec(Z) is not ´etale. Proof. (i) The fiber Xy of X = Spec(Z[G]) over a prime y = pZ, by definition, is X ×SpecZ Spec(κ(y)), where κ(y) is the residue field of y, which in our case is the prime field Fp (for more details on fibered products, see [5, Ch. 1, §3]). Hence this fibre is Spec(Fp [G]). The case of the generic point is similar. (ii) We know that a morphism of schemes X → S, which is locally of finite type, is unramified at x ∈ X if and only if the stalk (ΩX/S )x is 0 (see [1, Ch. 6, Prop. 3.3]). Thus (ii) follows immediately from Proposition 2.2. By [7, Ch.1, Prop.3.2] a morphism of schemes φ : X → S, which is locally of finite type, is unramified at x ∈ X if and only if the fibre over φ(x) is unramified over the residue field of φ(x). This implies (iii). ¤ Remark 2.7. Since Z[G] is flat over Z, and all the closed fibres have dimension 0, the scheme Spec(Z[G]) is one-dimensional (see [7, Ch.1, Remark 2.6]). Remark 2.8. The generic fibre allows one to have a coarse description of Spec(Z[G]) : Spec(Q[G]) has as many connected components (the fields in the Wedderburn decomposition of Q[G]) as Spec(Z[G]) has irreducible components. Indeed, obviously, the number of connected components of

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∼ φ−1 (0), Spec(Q[G]) equals the number of its points. Since Spec(Q[G]) = with φ : Spec(Z[G]) → Spec(Z) corresponding to the immersion Z → Z[G], this is the number of primes of Spec(Z[G]) lying over 0. On the other hand, since dim Spec(Z[G]) = 1, the primes of Spec(Z[G]) lying over 0 are exactly those that determine the irreducible components of Spec(Z[G]). 3. Integral group rings of finite abelian groups For the rest of this paper, G denotes a finite abelian group and H a subgroup G of order h. Theorem 3.1. (i) There is a one-to-one correspondence between the cyclic subgroups H of G and the minimal prime ideals aH of Z[G] such that [ (9) Spec Z[G] = V (aH ), H

is the decomposition of Spec Z[G] into irreducible components. Moreover, (10) V (aH ) ∼ = Spec Z[ζh ], where ζh is a primitive h-th root of unity. For a fixed h dividing the exponent of G, the number of irreducible components satisfying (10) is equal to the number of cyclic subgroups of G of order h. (ii) If q ∈ Spec Z[G] belongs to only one component of Spec Z[G], say V ∼ = Spec Z[ζh ], then Z[G]q ∼ = (Z[ζh ])˜q , where ˜q ∈ Spec Z[ζh ] is the prime corresponding to q. In particular, q is regular and dim Z[G]q = 1 if q is closed and 0 otherwise. (iii) The singular points of Z[G] are exactly those primes p ∈ Spec ZG that lie over the prime divisors of | G | . Q Proof. (i) The Wedderburn decomposition of Q[G] is Q[G] = H Q(ζh ) (see [9, p. 148]), and corresponding to it we have the canonical epimorphisms ψ0 : G → H, ψ1 : Z[G] → Z[ζh ], and ψ2 : Q[G] → Q[ζh ]. Defining aH = Ker(ψ1 ), we have the exact sequence 0 → aH → Z[G] → Z[ζh ] → 0. On tensoring it with Q (over Z), we obtain the exact sequence 0 → aH ⊗ Q → Q[G] → Q[ζh ] → 0, since Q is flat over Z. We have the bijection aH ←→ aH ⊗ Q, which gives the homeomorphism between the fiber over 0 and the scheme Spec Q[G], as explained in Remark 2.8. Note that aH ⊗ Q can be seen as the (maximal) ideal of Q[G] generated by aH . (ii) Evidently, since q is not an intersection point, removing from Spec A all components but V we obtain an open neighborhood in Spec Z[ζh ]. The

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189

latter scheme is regular. Because the local ring of a point x is determined in any open neighborhood of x, (ii) follows. (iii) Suppose that p ∈ Spec ZG is maximal and lies over the prime divisor p of | G | . Let m = mp be the maximal ideal of (ZG)p and κ = κ(p) = (ZG)p /m, the residue field of p. We shall derive the singularity of p from the exact sequence (2), which in our case becomes (11)

δ

m/m2 −→ Ω(ZG)p /Z ⊗(ZG)p κ → Ωκ/Z → 0.

Because p lies over p and δ annihilates the image of Z ∩ p in m/m2 , we see that p ∈ Ker(δ). It is easily seen that p ∈ / p2 . For the image of p under the augmentation homomorphism ε : ZG → Z is a prime ideal in Z, which must be pZ, as p ∈ p. Hence ε(p2 ) = p2 Z and thus p cannot be in p2 . Since p is maximal, one has a canonical isomorphism p/p2 ∼ = m/m2 given by 2 2 x + p 7→ x/1 + m , x ∈ p (see [4, Chap.II, §3.3, Prop. 9]). Consequently, p/1 ∈ / m2 , and thus Ker(δ) 6= 0. Applying (1) to Z → Fp → κ, we have the exact sequence (12)

ΩFp /Z ⊗Fp κ → Ωκ/Z → Ωκ/Fp → 0.

An appropriate version of the Nullstlellensatz (see [3, p. 70, Ex. 18]) implies that κ ⊇ Fp is a finite algebraic extension. Since Fp is perfect, it is also separable. It follows that the module Ωκ/Fp is 0, and so is ΩFp /Z , in view of the epimorphism Z → Fp . Hence Ωκ/Z = 0 and (11) gives (13)

δ

m/m2 −→ Ω(ZG)p /Z ⊗(ZG)p κ → 0.

By (7), (14)

M

ΩZ[G]/Z =

(Z/o(σ)Z) hσi ⊗Zhσi Z[G].

q σ∈Σq

where q runs over the prime divisors of | G |, and Σq is a minimal set of generators of the Sylow q-subgroup of G. Taking S = {1} and T = ZG \ p in (4), one has Ω(ZG)p /Z = (ΩZG/Z )p = ΩZG/Z ⊗ZG (ZG)p . It follows from (14) that

M

Ω(ZG)p /Z =

(Z/o(σ)Z) hσi ⊗Zhσi (ZG)p ,

σ∈Σp

where (ZG)p is a Zhσi-module via Zhσi → ZG → (ZG)p . Consequently, M Ω(ZG)p /Z ⊗(ZG)p κ = (Z/o(σ)Z) hσi ⊗Zhσi (ZG)p ⊗(ZG)p κ = σ∈Σp

M

σ∈Σp

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(Z/o(σ)Z) hσi ⊗Zhσi κ,

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where κ is a Zhσi-module by means of Zhσi → ZG → (ZG)p → (ZG)p /m = κ. Since σ is a p-element, o(σ) is zero in κ and thus o(σ)Z · κ = 0 and (Z/o(σ)Z) hσi ⊗Zhσi κ ∼ = Z hσi/(o(σ)Zhσi) ⊗Zhσi κ ∼ = κ/(o(σ)Zhσi) · κ = κ. Hence

Ω(ZG)p /Z ⊗(ZG)p κ ∼ = κlp , where lp is the number of elements in Σp . Therefore, the exact sequence (13) becomes δ

m/m2 −→ κlp → 0.

(15)

As Ker(δ) 6= 0, we conclude that dimκ (m/m2 ) ≥ lp + 1 = lp + dim ZG. This means, in particular, that p is singular.

¤

Acknowledgments Dr. Gimenez Bueno supported by FAPESP of Brazil, and Dr. Dokuchaev was partially supported by CNPq of Brazil. References [1] A. Altman and S. Kleiman Introduction to Grothendieck Duality Theory, Lecture Notes in Mathematics 146, Springer, Heidelberg, 1970. ´ 9, 1961, [2] M.F. Atiyah, Characters and cohomology of finite groups, Publ. Math. IHES, p. 23–64. [3] M.F. Atiyah and I.G. Macdonald, Introduction to Commutative Algebra, AddisonWesley, New York, 1994. [4] N. Bourbaki, Commutative Algebra, Addison-Wesley, Reading, Mass., 1972. ´ ements de G´eom´etrie Alg´ebrique IV, [5] A. Grothendieck and J. Dieudonn´e, El´ Grundlehren der Mathematischen Wissenschaften, Springer-Verlag, BerlinHeidelberg, 1971. [6] R. Hartshorne, Algebraic Geometry, Springer-Verlag, New York, 1977. ´ [7] J. Milne, Etale Cohomology, Princeton University Press, 1980. [8] D.S. Passman, The Algebraic Structure of Group Rings, Interscience, New York, 1977. [9] C. Polcino Milies and S.K. Sehgal, An Introduction to Group Rings, Kluwer, Dordrecht, 2002. ´ 34, [10] G. Segal, The representation-ring of a compact Lie group, Publ. Math. IHES, 1968, p. 113–128. [11] S.K. Sehgal, Topics in Group Rings, Marcel Dekker, New York, 1978. [12] S.K. Sehgal, Units in Integral Group Rings, Longman, Essex, 1983. [13] W. Waterhouse, Introduction to Affine Group Schemes, Graduate Texts in Mathematics 66, Springer-Verlag, New York, 1979.

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Wedderburn decomposition of small rational group algebras Carmen Rosa Giraldo Vergara Faculdade de Filosofia, Ciˆencias e Letras, FEMM, Av. Mal. Castelo Branco, 2765, Sete Lagoas, MG, CEP 35701-242, Brasil [email protected] Abstract. In this paper we classify the rational group algebras of dimension ≤ 32. For that, we use the classification of groups of order ≤ 32 and compute the Wedderburn decomposition of its rational group algebras.

1. Introduction Let G be a finite group and QG the rational group algebra. It is well known by Maschke’s theorem that QG is a semisimple algebra, then in order to classify these algebras, we compute the Wedderburn decomposition of its rational group algebras. This decomposition became important because of its applications in the theory of group representations, and to determine the units group of ZG. For this classification we use the group table notation of Thomas and Wood (see [8]). It is easy to prove that if G is decomposable, i.e., G = H1 × H2 , then QG = QH1 ⊗ QH2 , and if G = ha : an = 1i is a cyclic group, then M Q[x] M QG ' ' Q(ξd ), hϕd (x)i d|n

d|n

dth

where ϕd (x) is a cyclotomic polynomial. Thus, we are only going to show the Wedderburn decomposition of the group algebras over non abelian indecomposable groups of order greater than or equal to 16 and less than or equal to 32. In general, this decomposition is known for some group families like permutation, dihedral, semidihedral, and quaternion groups (see [1]). In fact, P if G = Sn then QG = Mn1 (Q) ⊕ Mn2 (Q) ⊕ · · · ⊕ Mnk (Q) with ki=1 n2i = n! and k is the number of conjugacy classes, if G is the dihedral group of order 2n, i.e., G = D2n = hx, y : xn = 1, y 2 = 1, xy = yx−1 i 191

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C.R. Giraldo Vergara

then

QG ∼ =

M

Ad

d|n

where Ad ∼ = Q ⊕ Q if d = 1, 2 and Ad ∼ = M2 (Q[ξd + ξd−1 ]) if d > 2, if G is the semidihedral group of order 16n, i.e., − G = D16n = hx, y : x8n = 1, y 2 = 1, xy = yx4n−1 i

then

QG ∼ = D8n

M

M2 (Q[ξd − ξd−1 ]),

d=2k+2 r r|m

where k and m are nonnegative integers with m odd, such that n = 2k m, and if G is the quaternion group of order 4n, i.e., G = Q4n = hx, y : x2n = 1, y 2 = xn , xy = yx−1 i then

QG ∼ = QDn

M

Q[ξ2d , j],

d=2k r r|m

where k and m are nonnegative integers with m odd, such that n = 2k m, j 2 = −1 and αj = j α, for all α ∈ Q[ξ2d ]. Remark 1. If d = 1, then Q[ξ2d , j] ∼ = Q(i) and if d 6= 1, denoting w2d = −1 ξ2d + ξ2d , we have Ã ! µ −1 2 2 − 4, −1 ¶ (ξ − ξ ) , −1 w2d 2d −1 2d ∼ Q[ξ2d , j] ∼ Q[w ][ξ − ξ , j] = . = = 2d 2d 2d Q[w2d ] Q[w2d ] In the following tables we exhibit the Wedderburn decomposition of the rational symmetric, quaternion, dihedral, and semidihedral algebras where dimensions are in the range comprised between 16 and 32. Group type Wedderburn Decomposition 24/12 QS4 ∼ = 2Q ⊕ M2 (Q) ⊕ 2M3 (Q)√ 16/12 QD16 ∼ = 4Q ⊕ M2 (Q) ⊕ M2 (Q[ 2]) −1 18/4 QD18 ∼ = 2Q ⊕ M2 (Q) ⊕ √M2 (Q[ξ9 + ξ9 ]) 20/3 QD20 ∼ = 4Q ⊕ 2M2 (Q[ 5]) −1 22/2 QD22 ∼ = Q ⊕ Q ⊕ M2 (Q[ξ11 + ξ11 √]) ∼ 24/10 QD24 = 4Q ⊕ 3M2 (Q) ⊕ M2 (Q[ 3]) −1 26/2 QD26 ∼ ]) = Q ⊕ Q ⊕ M2 (Q[ξ13 + ξ13 −1 −1 ∼ 28/3 QD28 = 4Q ⊕ M2 (Q[ξ7 + ξ7 ])√ ⊕ M2 (Q[ξ14 + ξ14 ]) −1 30/4 QD30 ∼ 2Q ⊕ M (Q) ⊕ M (Q[ 5]) ⊕ M (Q[ξ + ξ15 ]) = 2 2 2 15 p √ √ 32/49 QD32 ∼ = 4Q ⊕ M2 (Q) ⊕ M2 (Q[ √2]) ⊕ M2 (Q[ 2 + 2]) − ∼ 16/13 QD16 = 4Q ⊕ M2 (Q) ⊕ M2 (Q( −2)) p √ √ − ∼ 32/50 QD32 = 4Q ⊕ M2 (Q) ⊕ M2 (Q[ 2]) ⊕ M2 ( −2 + 2)

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Group type Wedderburn Decomposition ³ ´ −1,−1 √ 16/14 QQ16 ∼ 4Q ⊕ M (Q) ⊕ = 2 Q[ 2]

20/4

QQ20

24/11

QQ24

28/4

QQ28

32/51

QQ32

µ ¶ √ −5+ 5 √ ,−1 2 ∼ √ = 2Q ⊕ 2Q ⊕ Q(i) ⊕ M2 ( 5]) ⊕ Q[ 5] ³ ´ ³ ´ ∼ √ ⊕ −1,−1 = 4Q ⊕ 2M2 (Q) ⊕ −2,−1 Q Q[ ³3] ´ −1 2 (ξ14 −ξ14 ) ,−1 −1 ∼ 2Q ⊕ Q(i) ⊕ M (Q[ξ + ξ ]) ⊕ = 2 7 −1 7 Q[ξ14 +ξ14 ]! Ã √ √ 2,−1 ∼ 4Q ⊕ M2 (Q) ⊕ M2 (Q[ 2]) ⊕ q2− √ = Q[

2+

√ 2+ 2]

2. Normal subgroups and components Observe that if θ : G → H is a group homomorphism, then Q(θ(G)) is isomorphic to a direct component in the decomposition of QG. A proof of this fact, and every proposition in the following section can be found in [5], and we cite without any proof. Thus, in order to find components of QG, we find normal subgroups N of G G and therefore Q N is in the decomposition of QG. The following proposition guarantees when two normal subgroups produce different components. Proposition 1. Let G be a finite group, N a normal subgroup of G, and b = 1 P n ∈ QG. Then denote N |N | n∈N

b and 1 − N b are orthogonal idempotents in QG such that QGN b ' (1) N G b ⊕ QG(1 − N b ). Q N and QG = QGN (2) If M1 and M2 are normal subgroups of G such that N M1 M2 = b )M c1 and (1 − N b )M c2 are orthogonal idempotents M1 M2 then (1 − N b in QG(1 − N ). For some groups, this technique is enough to obtain every component of the decomposition as the following example illustrates. Group type 18/5 G = ha, b, c : a3 = 1, b3 = 1, c2 = 1, ab = ba, ac = ca−1 , bc = cb−1 i Observe that G0 = hai × hbi. In addition, hai, hbi, habi, and ha2 bi are c hbi, c habi, d and ha [ 2 bi normal subgroups in G, the idempotent elements hai, G ∼ c0 ) and are orthogonal in such component. Since survive in QG(1− G hai = S3 , G ∼ G ∼ G ∼ = S3 , = S3 and 2 = S3 , it follows that hbi

habi

ha bi

QG ∼ = 2Q ⊕ 4M2 (Q). G0

2 G G0

In the table below, we take N = the commutator subgroup, thus is G an abelian group and therefore Q G0 is easy to calculate, and M1 , M2 , ... are

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C.R. Giraldo Vergara

c1 , M c2 , . . . are orthogonal in QG(1 − G b 0 ) and normal subgroups such that M survive in that component. Case Group Type

G G0

' C2 × C2 × C2 × C2

Group Representation ha, b, c, d, e|a2 = b2 = c2 = d2 = e2 = 1

32/33

ce = eac, de = ebd, comute in other casesi ha, b, c|a4 = 1, b4 = 1, c2 = 1,

32/34

ab = ba, ac = ca−1 , bc = cb−1 i ha, b, c|a4 = b4 = 1, c2 = a2 ,

32/35

ab = ba, ac = ca−1 , bc = cb−1 i ha, b, c, d|a2 = b2 = c4 = d2 = 1, bd = dab, cd = dc−1 ,

32/36

comute in other casesi ha, b, c, d|a2 = b2 = c4 = 1, d2 = c2 , bd = dab, cd = dc−1 ,

32/37

comute in other casesi ha, b, c, d : a4 = b2 = c2 = d2 = 1, ad = dab, cd = da2 c,

32/38

comute in other casesi 32/39

ha, b, c|a4 = b4 = c2 = 1, ab = ba, ac =

ca−1 , bc

=

ca2 b−1 i

ha, b, c : a4 = 1, b4 = 1, c2 = a2 b2 ,

32/40

ab = ba, ac = ca−1 , bc = ca2 b−1 i 32/41

ha, b, c : a4 = b4 = c2 = 1, ab = ba, ac = cb2 a−1 , bc = ca2 bi

Case Group Type 32/18

G G0

Normal subgroups Mi G ∼ 16/6 habi = G ∼ = 16/6 hai G hbi G ha2 b2 i G ha2 i G hb2 i G ha2 b2 i G ha2 i G hb2 i G hac2 i G hai G hc2 i G hac2 i G hai G hc2 i G hba2 i G hbi G ha2 i G ha2 b2 i G ha2 i G hb2 i G hc2 i G ha2 i G hb2 i G ha2 b2 i G ha2 i G hb2 i

∼ = 16/6 ∼ = 16/7 ∼ = 16/6

Wedderburn Decomposition 8Q ⊕ 6M2 (Q)

8Q ⊕ 4M2 (Q) ⊕ 2H(Q)

∼ = 16/6 ∼ = 16/7 ∼ = 16/6 ∼ = 16/7 ∼ = 16/8

8Q ⊕ 2M2 (Q) ⊕ 4H(Q)

∼ = 16/6 ∼ = 16/6 ∼ = 16/8

8Q ⊕ 4M2 (Q)⊕ √ M2 (Q( −1))

∼ = 16/7 ∼ 16/6 = ∼ = 16/8

8Q ⊕ 2M2 (Q) ⊕ 2H(Q)⊕ √ M2 (Q( −1))

∼ = 16/8 ∼ = 16/6 ∼ = 16/8

8Q ⊕ 2M2 (Q)⊕ √ 2M2 (Q( −1))

∼ = 16/6 ∼ = 16/8 ∼ = 16/8

8Q ⊕ 2M2 (Q)⊕ √ 2M2 (Q( −1))

∼ = 16/7 ∼ 16/8 = ∼ = 16/8

8Q ⊕ 2H(Q)⊕ √ 2M2 (Q( −1))

∼ = 16/8 ∼ = 16/8

√ 8Q ⊕ 3M2 (Q( −1))

' C4 × C4

Group Representation ha, b, c|a2 = 1, b4 = 1, c2 = 1, ab = ba, ac = ca, bc = cabi

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Normal subgroups Mi G ∼ 16/9 hc2 i = G ∼ 16/10 = 2 hac i

Wedderburn Decomposition 4Q ⊕ 6Q(i) ⊕ 3M2 (Q) ⊕ H(Q)

Wedderburn decomposition of small rational group algebras

Case Group Type 16/9

G G0

' C2 × C4

Group Representation ha, b| a4 = 1, b2 = 1, c2 = 1, ab = ba, ac = ca3 b, bc = cbi

16/10

ha, b|a4 = 1, b4 = 1, ab = ba−1 i ha, b, c|a2 = 1, b8 = 1,

32/27

c2 = 1, ab = ba, ac = ca, bc = cab−1 i ha, b, c|a2 = 1, b8 = 1,

32/28

195

c2 = b4 , ab = ba, ac = ca, bc =

cab−1 i

32/29

ha, b|a8 = b4 = 1, ab = ba−1 i

32/30

ha, b|a8 = b4 = 1, ab = ba3 i

Normal subgroups Mi G ∼ 8/4 ha2 i = G ∼ = 8/4

hbi G hb2 i G ha2 b2 i G hab4 i G hb4 i G hai G hab4 i G hb4 i G hai G ha4 b2 i G ha4 i G hb2 i G ha4 b2 i G ha4 i G hb2 i

∼ = 8/4 ∼ 8/5 = ∼ = 16/13 ∼ = 16/9 ∼ = 16/12 ∼ = 16/13 ∼ = 16/9 ∼ = 16/14 ∼ = 16/14 ∼ = 16/10 ∼ 16/12 = ∼ = 16/14 ∼ = 16/10 ∼ = 16/13

Wedderburn Decomposition 4Q ⊕ 2Q(i) ⊕ 2M2 (Q) 4Q ⊕ 2Q(i) ⊕ H(Q) ⊕ M2 (Q)

√ 4Q ⊕ 2Q( −1) ⊕ 2M2 (Q)⊕ √ √ M2 (Q[ 2]) ⊕ M2 (Q[ −2]) 4Q ⊕ 2Q(i) ⊕ 2M2 (Q)⊕ √ √ H(Q[ 2]) ⊕ M2 (Q[ −2]) √ 4Q ⊕ 2Q( −1) ⊕ M2 (Q)⊕ √ √ M2 (Q([ 2]) ⊕ H(Q) ⊕ H(Q[ 2]) 4Q ⊕ 2Q(i) ⊕ M2 (Q) ⊕ H(Q)⊕ √ √ M2 (Q([ −2]) ⊕ H(Q[ 2])

3. Finding a new component In the section before, we determined every component analyzing the normal subgroup of G, but sometimes, this is not enough. We are going to use the following proposition in order to determine a new component. Proposition 2. Let G be a finite subgroup. Then the dimension of the center of QG is exactly the conjugation classes number of G. Proposition 3. Let A be a noncommutative algebra of dimension 4n2 over k whose center is 1-dimensional. Then A ∼ = M2n (k) or A ∼ = Mn (D), where D is a quaternion algebra over k. In addition, if A has n + 1 orthogonal idempotents then A ∼ = M2n (k). Remark 2. If A is a 4-dimensional over k and A has a zero divisor then A∼ = M2 (k). Thus, if we find enough idempotents or zero divisors, then A has to be isomorphic to a matrix over k. The following examples illustrate this strategic. Group type 16/8 G = ha, b, c : a4 = 1, b2 = 1, c2 = 1, ab = ba, ac = ca, bc = ca2 bi Observe that G0 = hai, G/G0 ∼ = C2 × C2 × C2 , and G has 10 conjugation c classes. It follows that QG(1 − G0 ) is 8-dimensional over Q and this center

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has dimension 2. In addition a is in the center of G and · µ ¶¸2 µ ¶ 2 1 − a2 c0 ))2 = a 1 − a c0 ), (a(1 − G =− = −(1 − G 2 2 c0 ) is 4-dimensional over thus a2 = −1 in this component, therefore QG(1 − G Q(i) and this center has dimension 1 over Q(i). Since (ab + c)2 is a divisor of zero, we conclude that QG ∼ 2 = 8Q ⊕ M2 (Q(i)). Group type 20/5 G = ha, b : a5 = 1, b4 = 1, ab = ba3 i Observe that G/G0 ∼ = C4 and G has 5 conjugation classes. It follows that 0 c c0 )) is 1-dimensional QG(1 − G ) is 16-dimensional over Q and Z(QG(1 − G d d c 1 − hb c are noncentral orthogonal 2 i and hb 2 i − hbi over Q. In addition, hbi, c0 ), thus idempotent of QG(1 − G QG ∼ = 2Q ⊕ Q(i) ⊕ M4 (Q).

2

Group type 32/31 G = ha, b, c : a4 = b4 = 1, c2 = 1, ab = ba, ac = ca, bc = cab−1 i G ∼ Observe that H = ha2 i is a normal subgroup of G and H =16/9, then b Q(1 − H) has dimension 16 over Q and its center has dimension 4 over Q. 2

3

Let n = 1−b+b2 −b and m = n + cnc. It is easy to see that n commutates with a and b, then by construction m is central. Besides, (1 − b)(1 + b2 ) (1 − b2 )(1 + ab) n2 = n = and cnc = 4 4 ³ ´ 1−a2 in QG . It follows that n is idempotent and ncnc = cncn = 0 in 2 ³ ´ ³ ´ 2 2 QG 1−a , then m 1−a is a central idempotent that survives in that 2 2 component. Thus µ ¶ µ ¶ µ ¶ 1 − a2 ∼ 1 − a2 1 − a2 QG ⊕ QG(1 − m) = QGm 2 2 2 and

h ³ ´i2 ³ ´ 2 1−a2 am 1−a = −m 2 2

and

h ³ ´i ³ ´ 2 1−a2 a(1 − m) 1−a = −(1 − m) . 2 2

³ ´ 2 So, Q 1−a has two 4-dimensional components over Q(i) . Since 2 noncentral idempotent in those components, it follows that QG ∼ = 4Q ⊕ 2Q(i) ⊕ 2M2 (Q) ⊕ 2M2 (Q(i)).

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1+c 2

is a 2

Wedderburn decomposition of small rational group algebras

197

∼ A ⊕ B where A is determined In the following tables, we have QG = by the normal subgroups H1 , H2 . . . Hl as in the previous section, B = b 1 ) · · · (1− H b 2 ) has dimension n over Q and its center has dimension QG(1− H m over Q and dimension one over k = Q(α1 , . . . , αk ) a finite field extension of Q, where αj are central elements in the component B. Group Type 12/4

16/8

16/11

27/4

27/5

32/19

32/21

32/32

Group Representation ha, b : a2 = 1, b3 = (ba)3 = 1i H = hai, G/H ∼ = C3 ha, b, c : a4 = 1, b2 = c2 = 1, ab = ba, ac = ca, bc = ca2 bi H = ha2 i, ∼ G/H = C2 × C2 × C2 ha, b : a8 = b2 = 1, ab = ba5 i H = ha4 i, G/H ∼ = C2 × C4 ha, b, c : a3 = b3 = 1, c3 = 1, ab = ba, ac = cab−1 , bc = cbi H = hbi, G/H ∼ = C3 × C3 ha, b : a9 = 1, b3 = 1, ab = ba4 i H = ha3 i, G/H ∼ = C3 × C3 ha, b : a8 = b4 = 1, ab = ba5 i H1 = ha4 i, G/H1 ∼ = C4 × C4 H2 = hb2 i, G/H2 ∼ =16/11 ha, b : a4 = b8 = 1, ab = ba−1 i H1 = ha2 i, G/H1 ∼ = C2 × C4 H2 = hb4 i, G/H2 ∼ =16/10 ha, b : a8 = 1, b 4 = a4 , ab = ba−1 i H = hb4 i, G/H ∼ =16/10

n-m

k/Q

9-1

Q

8-2

Q(a) ' Q(i) a2 = −1

ab + c

8Q⊕ M2 (Q(i))

8-2

Q(a2 ) ' Q(i) (a2 )2 = −1

b+1

4Q ⊕ 2Q(i)⊕ M2 (Q(i))

Zero divisor

Wedderburn Decomposition Q ⊕ Q(ξ3 )⊕ M3 (Q)

18-2

√ Q(b) ' Q( −3) b2 + b + 1 = 0

√ Q ⊕ 4Q(√ 3i)⊕ M3 (Q[ 3i])

18-2

√ Q(a3 ) ' Q( −3) (a3 )2 + a3 + 1 = 0

√ Q ⊕ 4Q(√ 3i)⊕ M3 (Q[ 3i])

8-2

8-2

16-4

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Q(a) ' Q(i) a2 = −1

Q(b2 ) ' Q(i) (b2 )2 = −1

√ Q(b2 , a − a3 ) ' Q( 2, i) 2 2 (b ) = −1 (a − a3 )2 = 2

a2 − b

a2 − b

a2 + b2

4Q ⊕ 6Q(i)⊕ 2M2 (Q(i))

4Q ⊕√2Q(i)⊕ 2Q[ 2, i]⊕ M2 (Q)⊕ H(Q)⊕ M2 (Q(i)) 4Q ⊕ 2Q(i)⊕ M2 (Q)⊕ H(Q)⊕ √ M2 (Q(i, 2))

198 Group Type 20/5

24/15

32/16

32/17

32/20

32/22

32/26

C.R. Giraldo Vergara Group Representation ha, b : a5 = b4 = 1, ab = ba3 i H = hai, G/H ∼ = C4 ha, b, c : a3 = b4 = 1, c2 = 1, ab = ba−1 , ac = ca, bc = cb−1 i H1 = hai, G/H1 ∼ = D8 H2 = hb2 i, G/H2 ∼ = D12 ha, b, c : a4 = b4 = 1, c2 = 1, ab = ba, ac = ca, bc = ca2 bi H1 = ha2 i, G/H1 ∼ = C2 × C2 × C4 H2 = hb2 i, G/H2 ∼ =16/8 ha, b, c : a8 = b2 = 1, c2 = 1, ab = ba, ac = ca, bc = ca4 bi H = ha4 i, G/H ∼ = C2 × C2 × C4 ha, b, c : a2 = b8 = 1 c2 = 1, ab = ba, ac = ca, bc = cabi H1 = hai, G/H1 ∼ = C2 × C8 H2 = hb4 i, G/H2 ∼ =16/9 ha, b : a16 = b2 = 1, ab = ba9 i H = ha8 i, G/H ∼ = C2 × C8 ha, b, c : a8 = 1, b2 = 1, c2 = 1, ab = ba, ac = ca−1 , bc = ca4 bi H = ha4 i, G/H ∼ = C2 × D8

n-m

k/Q

Orthogonal Idempotents

Wedderburn Decomposition

16-1

Q

c hbi d 2i 1 − hb d c 2 hb i − hbi

2Q ⊕ Q(i)⊕ M4 (Q)

8-2

√ Q( −3) ((2a + 1)c)2 = −3

1+c 2

4Q ⊕ 3M√ 2 (Q)⊕ M2 (Q( 3i))

1+c 2

8Q ⊕ 4Q(i)⊕ 2M2 (Q(i))

8-2

16-4

8-2

16-4

16-4

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Q(i) a2 = −1

√ Q( 2, i) (a2 )2 = −1 (a + a−1 )2 = 2

Q(i) (b2 )2 = −1

√ Q( 2, i) 2 (a )4 = −1 √ Q( 2, i) (a2 b)2 = −1 (a − a3 )2 = 2

1+b 2

1+c 2

1+b 2

1+c 2

8Q ⊕ 4Q(i)⊕ √ M2 (Q[ 2, i])

4Q ⊕√2Q(i)⊕ 2Q[ 2, i]⊕ 2M2 (Q)⊕ M2 (Q(i))

4Q ⊕ 2Q(i)⊕ √ 2Q[ 2, √ i]⊕ M2 (Q[ 2, i]) 8Q ⊕ 2M √2 (Q)⊕ M2 (Q[ 2, i])

Wedderburn decomposition of small rational group algebras

199

Case n = 16, m = 1, and k = Q Group Type

Group Representation

Orthogonal Idempotents

32/42

ha, b, c, d : a4 = 1, b2 = c2 = d2 = a2 ,

( 1+ac )( 1−a ) 2 2

ab = ba−1 , cd = dc−1 , other case comutei H = ha2 i, G/H ∼ = C2 × C2 × C2 × C2 32/44

ha, b, c : a8 = 1, b2 = 1, c2 = 1, ab = ba7 , ac = ca5 , bc = cbi H = ha4 i, G/H ∼ = C2 × D8

32/46

ha, b, c, d : a2 = b2 = c2 = 1, d4 = 1, bd = dab, cd = dabc, other case comutei H = hai, G/H ∼ =16/9

32/47

ha, b, c : a8 = b2 = c2 = 1, ab =

ba5 , ac

= cba, bc = cbi ∼16/9 H = ha4 i, G/H =

Wedderburn Decomposition

2

2

( 1−ac )( 1+bd )( 1−a ) 2 2 2 2

16Q ⊕ M4 (Q)

( 1−ac )( 1−bd )( 1−a ) 2 2 2 4

( 1−c )( 1−a ) 2 2

4

( 1+b )( 1+c )( 1−a ) 2 2 2 4

( 1−b )( 1+c )( 1−a ) 2 2 2 1−a ( 1+c )( ) 2 2 ( 1+b )( 1−c )( 1−a ) 2 2 2 1−b 1−c 1−a ( 2 )( 2 )( 2 )

4 ( 1+b )( 1−a ) 2 2 1−b 1+c 1−a4 ( 2 )( 2 )( 2 ) 4 ( 1−b )( 1−c )( 1−a ) 2 2 2

8Q ⊕ 2M2 (Q)⊕ M4 (Q) 4Q ⊕ 2Q(i)⊕ 2M2 (Q)⊕ M4 (Q) 4Q ⊕ 2Q(i)⊕ 2M2 (Q)⊕ M4 (Q)

4. A new quaternion component In this section, we are going to use the same argument as in the section before. Next, we show that the algebra has a quaternion component, determining an algebra homomorphism between QG and an appropriate quaternion algebra. Now, we show a special example. Group type 24/13 G = ha, b, c : a4 = 1, b2 = a2 , c3 = 1, ab = ba−1 , c−1 ac = b, c−1 bc = abi G ∼ Since H = ha2 i is a normal subgroup such that Q H = QA4 ∼ = Q ⊕ Q(ξ3 ) ⊕ M3 (Q). Consider the mapping

τ : QG −→ H(Q) a 7−→ i b 7−→ j 1 c 7−→ (−1 + i + j + k). 2 It follows that τ is a well-defined endomorphism, thus H(Q) ⊂ QG(1 − d 2 i) is a component in the decomposition of QG. Now, we need to find ha d 2 a component in QG(1 ³ − 2ha´ i) of dimension 8 and center of dimension 2. (1+a+b+ab) d 2 i)). Observe that d3 = 1 in Let d = c 1−a ∈ Z(QG(1 − ha 2 2 √ d and d = 1 in the quaternion component, then −3 is in the QG(1 − ZG) ³ ´ d c 1−a2 is a 2 i). In addition, hci center of the other component in QG(1 − ha 2

d Therefore noncentral idempotent in QG(1 − ZG). √ QG ∼ = Q ⊕ Q(ξ3 ) ⊕ M3 (Q) ⊕ M2 (Q( −3)) ⊕ H(Q).

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2

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C.R. Giraldo Vergara

Group Type

Group Representation ha, b : a3 = 1, b8 = 1,

24/14

ab = ba−1 i

Algebra Homomorphim ³√

H1 = hai, G/H1 ∼ = C8

j2 =

H2 = hb4 i, G/H2 ∼ = Q12 a2

32/43

b4

ha, b, c, d : = = = 1, d2 = b2 , ac = cb2 a, bc = cb3 , H = hb2 i, G/H ∼ = C2 × C2 × C2 × C2 ha, b, c : a8 = 1, b2 = 1, c2 = a4 , ab = ba5 , ac = ca−1 , bc = cbi H = ha4 i, G/H ∼ = C2 × D8

32/48

a 7→

j−1 ,b 2

7→ k

c2

other case commutei

32/45

´

−1,−3 √ Q( −1) √ −3, k2 = −1

QG →

Wedderburn Decomposition 2Q ⊕ Q(i)⊕ √ √ Q[ 2, −1] ⊕ M2 (Q) ³ ´ ⊕ −1,−3 Q ³√ ´ √ ⊕ Q(−1,−3 −1)

ha, b, c : a8 = 1, b2 = 1, c2 = a4 , ab = ba5 , ac = cba, bc = cbi H = ha4 i, G/H ∼ =16/9

QG ³ → ´M2 (H(Q)) ³ ´ a 7→ 01 10 , b 7→ 0i 0i ³ ´ ³ ´ 0 0 c 7→ 10 −1 , d 7→ 0j −j QG ³ →´M2 (H(Q)) ³ ´ 0 i 0 , b 7→ 10 −1 j 0 ³ ´ 0 c 7→ 0i −j

a 7→

QG ³ →´M2 (H(Q)) ³ ´ 0 a 7→ 0j 0i , b 7→ 10 −1 ³ ´ c 7→ 0i 0i

16Q ⊕ M2 (H(Q))

8Q⊕ 2M2 (Q)⊕ M2 (H(Q)) 4Q ⊕ 2Q(i)⊕ 2M2 (Q)⊕ M2 (H(Q))

Acknowledgments The author is grateful to Prof. Guilherme Leal for many valuable discussions and suggestions. Also, my thanks to Prof. Fabio E. Brochero for pointing out some mistakes in a previous version of this paper. References [1] C.R. Giraldo V., Algebra de grupos racionais, Diserta¸ca ˜o de Mestrado, UFRJ, 1997. [2] C.W. Curtis and I. Reiner, Representation Theory of Finite Groups and Associative Algebras, Wiley, New York 1962. [3] N. Jacobson, Basic Algebra II. W.H Freman and Company. New York, 1989. [4] E. Jespers and M.M Parmenter, Units of group rings of groups of order 16, Glasgow Math J. 35, (1993) 367-379. [5] D.S. Passman, The Algebraic Structure of Group Rings. Wiley Interscience. New York, 1977. [6] R. Pierce, Associative Algebras. Springer-Verlag. New York, 1980. [7] C. Polcino Miles, An´eis de Grupos. SBM. S˜ ao Paulo, 1976. [8] A.D. Thomas and G.V. Wood, Group Table. Shiva Publishing Limited. Great Britain, 1980.

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Some questions on skewfields Jairo Z. Gon¸ calves Instituto de Matem´atica e Estat´ıstica, Universidade de S˜ao Paulo, Caixa Postal 66281, S˜ao Paulo, SP, 05315-970, Brasil [email protected] Dedicated to C´esar Polcino on his 60th birthday. Abstract. We give an overview on Lichtman’s Conjecture, and for a quaternion skewfield of characteristic 6= 2 we establish sufficient conditions on two elements a and b, such that 1+ma and 1+mb, m a positive integer, generate a free noncyclic subgroup.

1. Introduction Let D be a skewfield with center F , and let D∗ be its multiplicative group. In [8], answering in the negative a question posed by Bachmut, if a linear group over D satisfies Tits’ Alternative, A. Lichtman raises the following. Conjecture. D∗ contains a free (noncyclic) group. If D is finite dimensional over F it is easy to see that the claim is true (see [1]). But this proof has a drawback, since it is existential and does not exhibit the free group. Its argument relies heavily on Tits’ Alternative which, on the other hand, is quite nonconstructive. The first real breakthrough on the subject is [9], where the author uses valuation theory to produce a specialization of the skewfield on a finite dimensional skewfield. He then uses the specialization to lift the free group from the finite dimensional image. This technique is pushed even further in [10], where the Conjecture is proved for skewfields that are generated over its center by a polycyclic-byfinite group. Now, there is a close resemblance between the group of units of a group algebra and the multiplicative group of a skewfield. So, it is interesting to investigate if some facts that are true in the former are also true in the latter. For this see [3], [6] and [7]. 2000 Mathematics Subject Classification. 16S36. 201

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Let A be an algebra over a nonabsolute field F of characteristic 6= 2. We know (see [2] and [5]) how to produce free products of the form h1 + α · ai ∗ h1 + α · bi, when a2 = b2 = 0, ab is not nilpotent and α is a suitable element of F . One can ask what would be the analogous result for skewfields. We saw in [3], Theorem 5, that if G is residually torsion free nilpotent, if x and y are noncommuting elements of G, and if c and d are nonzero elements of F then the elements 1 + cx and 1 + dy freely generate a group in the Malcev-Neumann field of fractions of F G. In this note we investigate this problem for the quaternion case. Let us point out that some seemingly obvious conjectures are not true. For example, Dr. Diego Rattagi, from ETHZ obtained (personal communication) a group relation of length 106, that shows that the elements u = 1 + 2i and v = 1 + 4k in the rational quaternion algebra do not generate a free group. Nevertheless, by [3], the group generated by u and v contains a free subgroup. We start establishing some notation. Let D = {α + βi + γj + δk|i2 = a, j 2 = b, ij = −ji, ij = k, a, b ∈ F } be the quaternion skewfield over a field F of characteristic 6= 2. Let us recall that every element d of D not in F has a minimal polynomial of the form f (X) = X 2 − t(d)X + n(d), where t(d) is the reduced trace of d and n(d) is the reduced norm of d. We can now offer the following: Theorem 1. Let r and s be two nonzero elements of D such that t(r) = t(s) = 0, n(r) = n(s) and rs = −sr. (1) If char F > 0 then h1 + r, 1 + si = h1 + ri ∗ h1 + si ∼ = Z ∗ Z. (2) If char F = 0 there are infinitely many integers m such that h1 + mr, 1 + msi = h1 + mri ∗ h1 + msi ∼ = Z ∗ Z. From the result above some questions immediately arise: Problem 1 : What is the analogous theorem for skewfields of higher dimension? Problem 2 : Is there a positive integer m such that, in the rational quaternion algebra, (1+2i)m and (1+4k)m generate a free group? And for 1+2mi and 1 + 4mk? 2. Proof of the theorem Proof. We note, first, that the elements 1,r,s and rs are F -linearly independent, since they generate an F -algebra of dimension greater than 2. So, refreshing our notation, we can assume that our skewfield has the relations i2 = j 2 = a, ij = −ji. We now consider two cases:

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203

1. charF = p > 0. Let L = GF (p) be the prime subfield of F . If a were algebraic over L, then [i, j]L , the subalgebra of D generated by i and j over L, would be a finite noncommutative skewfield. Therefore a is transcendental over L, and the first claim follows from [3], Theorem 2. 2. charF = 0. If a is transcendental over Q then we argue as before. So, from now on, we assume that a is algebraic over Q. The number a can be written as a = α/q, where α is an algebraic integer over Q and q is a positive integer. Substituting i by qi and j by qj we can assume that our quaternion skewfield satisfies the relations i2 = β = qα, j 2 = β, ij = −ji. Let G be the Galois group of the splitting field N of f (X) = irrat(β/Q). We will look for a nonarchimedean valuation ν of N (i), and for a positive integer m such that ν(β) = ν(m) = 0 and ν(1+mi) 6= ν(1−mi). This valuation and Theorem 2 of [3] will give the result. Let m be a positive integer to be chosen later. We haveQ (1 − mi)(1 + mi) = 1 − m2 β, and let us consider the polynomial h(X) = σ∈G (−β σ X 2 + 1). Since the coefficients of h(X) are invariant under G it follows that h(X) ∈ Z[X]. Also, by a simple computation, it follows that h(X) is not a power of a linear polynomial in Z[X]. By Siegel’s Theorem [11] there are infinitely many positive integers m ∈ N such that the primes that occur in h(m) are infinite. Take one of these odd primes, and let P be a prime ideal in the ring of integers of N (i) that lies over it. Let ν be the nonarchimedean valuation determined by P. Then 1 − m2 β ∈ P, but not both 1 − mi and 1 + mi can belong to P. Otherwise 2 ∈ P, and since P contains an odd prime, it follows that 1 ∈ P. By the same reason m ∈ / P, otherwise 1 ∈ P, a contradiction again. This proves the second case. ¤ Acknowledgments The author’s research was supported in part by CNPq Grant 303.756/82-5 and Fapesp-Brazil, Proj. Tematico 00/07.291-0. References [1] J.Z. Gon¸calves, Free subgroup of units in group rings, Canadian Math. Bull., 27 (1984), 309–312. [2] J.Z. Gon¸calves and D.S. Passman, Construction of free subgroups in the group of units of modular group algebras, Comm. Algebra 24 (1996), 4211–4215. [3] J.Z. Gon¸calves, A. Mandel, and M. Shirvani, Free products of units in algebras I. Quaternion algebras, J. Algebra 214 (1999), 301–316.

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[4] J.Z. Gon¸calves, A. Mandel, and M. Shirvani, Free products of units in algebras II. Crossed products, J. Algebra 233 (2000), 567–593. [5] J.Z. Gon¸calves and A. Mandel, Free subgroups in the group of units of a twisted group algebra, Comm. Algebra 29 (2001), 2231–2238. [6] J.Z. Gon¸calves and D.S. Passman, Free unit groups in algebras, Comm. Algebra 31 (2003), 2219–2227. [7] J.Z. Gon¸calves and M. Shirvani, Free products arising from elements of finite order in simple rings, Proceedings AMS 133, 7, (2005), 1917–1923. [8] A.I. Lichtman, On subgroups of the multiplicative group of skewfields, Proceedings AMS 63 (1977), 15–16. [9] A.I. Lichtman, Free subgroups of normal subgroups of the multiplicative group of skewfields, Proceedings AMS 71 (1978), 174–178. [10] A.I. Lichtman, On normal subgroups of the multiplicative group of skewfields generated by a polycyclic-by-finite group, J. Algebra 78 (1982), 548–577. [11] T. Shorey and R. Tijdeman, Exponential Diophantine Equations, Cambridge University Press, Cambridge, 1986.

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On the role of rings and modules in algebraic coding theory Marcus Greferath Department of Mathematics, University College Dublin, Belfield, Dublin 4, Ireland [email protected] Sergio R. L´ opez-Permouth Department of Mathematics, 321 Morton Hall, Ohio University, Athens, OH 45701, USA [email protected] Abstract. Foundational and theoretical aspects of coding theory over rings and modules are considered. Topics discussed include the role of rings and modules as alphabets in coding theory and the necessity of exploring metrics other than the traditional Hamming weight on such alphabets. We survey attempts to extend the classical results of coding theory over finite fields to this new setting and consider, in particular, the extension of the MacWilliams equivalence theorem and of the MacWilliams duality identities to the context of codes over finite Frobenius rings. We also discuss arguments that seem to point at finite Frobenius rings or to character modules over arbitrary rings as the natural alphabets for an extended coding theory that would still include such theoretical staples. Work on the establishment of bounds on the parameters of ring-linear codes is also sampled as are constructions of codes and the design of decoding algorithms.

1. Introduction Through the pioneering work of Shannon, Hamming, and others, the theory of Error Correcting Codes arose out of the need for more reliable communication and computer data storage. It has rapidly evolved into a sophisticated mathematical theory with strong links to algebra, combinatorics, number theory, and geometry. Algebraic coding theory over rings emerged at the end of the 1980s, when it was discovered that certain very good codes that were nonlinear in the traditional binary sense were actually linear when considered as codes over Z4 . This paper is devoted to giving an introduction to this chapter of applied algebra. It is not our intention to be encyclopedic; such a goal would be beyond the scope of this work. Clearly, 205

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206

M. Greferath and S.R. L´ opez-Permouth

the inclusion of topics is bound to reflect our own research interests. Considering that the likely readership of this volume will be of group, ring, and group-ring theorists, we will focus particularly on several foundational and theoretical aspects of this discipline that are likely to be appealing to such an audience. Our intent is to provide algebraists with a snapshot of a corner of coding theory that may be attractive to them as we give insight into the current state of the art. 1.1. What is coding theory? Messages that are sent via a noisy channel may be altered during transmission; mistakes in the interpretation of the message may have many negative consequences. For this reason, when a message travels from an information source to a destination, both the sender and the receiver would like to ensure that the message received is free of error.

Sender

Encode

Codeword

Noisy

Channel

Decode Receiver

Corrupted Word

Figure 1. Discrete Channel If it contains errors, these errors need to be detected and, ideally, corrected. This can be achieved by encoding: adding redundancy to the information message in an optimal way, such that any two encoded messages differ substantially. In this way, a relatively small number of errors will not change one encoded message into another (for the sake of detection) and possibly keep the erroneous version of the message closest to the original message (for the sake of correction.) The general scheme of a communication system is illustrated in Figure 1. The source information may be a picture, a sound pattern, a piece of text, etc., that needs to be stored on a hard drive or a cd-rom, or transmitted reliably over a phone line, radio link, or a fiber optic cable. In his celebrated 1948 result, Shannon [34] showed that every communication channel has an associated capacity such that the decoded error probability can be made arbitrarily small at transmission rates below the given capacity. Shannon’s proof was nonconstructive, involving the use of random codes, without practical encoding and decoding procedures. A major

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challenge for coding researchers was to construct classes of codes containing sequences of increasingly long ‘very good’ codes that, when optimally decoded, could reach information rates close to Shannon’s limit while achieving acceptably low bit error rates. In addition, the desired classes of codes should have practical encoding and decoding algorithms. 1.2. Why rings and modules in algebraic coding theory? Practice has shown that considering linear codes (codes with the additional structure of being vector spaces over finite fields) often yields significant advantages over the nonlinear counterparts when it comes to complex tasks like encoding and decoding. Still, it was recognized early [32, 26] that the class of binary block codes contained “very good” codes, which were not linear. Such is the case, for example with the notoriously nonlinear families of Preparata, Kerdock codes, Goethals, and Goethals-Delsarte codes. Surprisingly, however, the mentioned families exhibit formal duality properties in terms of their distance enumerators that mimic those among linear codes and their duals. This phenomenon went unexplained for a long time. A breakthrough in the understanding of this behavior came in the 1990s when Nechaev [30] and, independently, Hammons et al. [22] discovered that these families allow a representation in terms of Z4 -linear codes. 0 00 3

10

01

1

11 2

Figure 2. Gray Isometry between Z4 and Z2 A central aspect of this ring-theoretic representation is that Z4 is equipped with an alternative metric, the Lee weight, rather than with the Hamming weight. The Hamming weight only distinguishes whether an element is zero or nonzero, but the Lee weight is finer, setting 2 apart from the other nonzero elements of the ring. Figure 2 depicts the so-called Gray isometry, which is a bijection between (Z4 , wLee ) and (Z22 , wH ) that preserves the weight. In fact, respectively assigning 1 and 3 to (0, 1) and (1, 0), whereas (1, 1) is mapped to 2, is a weight preserving mapping. Extending this in a coordinatewise manner to Zn4 we obtain a binary code of length 2n from each Z4 -code of length n.

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It is exactly this way that the above code families found their Z4 -linear representations. Expressing the nonlinear binary codes mentioned above as Z4 codes raises the opportunity for decoding the nonlinear binary code via an algebraic Lee weight decoder. The fact that the usual algebraic and metric setting of linear coding theory (namely finite fields equipped with the Hamming metric) is actually too narrow prompts us to expand the current theory in at least two directions. On the one hand, it seems obvious that the next more general algebraic structure that might serve as an alphabet for linear coding is that of finite rings. In fact, we shall actually see that there are advantages to going a little further and considering appropriate modules, instead of rings, as the natural choice for an alphabet. On the other hand, the appropriateness of the Lee weight for Z4 -linear coding suggests that the metric component of a generalized coding theory also requires reconsideration. 2. The foundations: code equivalence Let R be a finite ring. By a left R-linear code of length n we mean any submodule C of R Rn . If we think of R equipped with a weight function w : R −→ R and extend this function additively to the Rn , then a natural question arising is that for a characterization of all R-linear w-isometries ϕ : C −→ Rn , i.e., of all those linear mappings for which w(ϕ(c)) = w(c) for all c ∈ C. A basic theorem characterizing these isometries in the context of traditional finite-field linear coding theory is MacWilliams’ equivalence theorem. She proved it in her doctoral thesis [28]. To understand its terms, let’s agree that a Hamming isometry of C into Rn is an R-linear map that preserves the Hamming weight. A monomial transformation of R Rn is a mapping ψ : Rn −→ Rn with ϕ(x) = xP D for all x ∈ Rn , where P is a (coordinate) permutation matrix and D is an invertible diagonal matrix in Mn (R). MacWilliams’ equivalence theorem then can be stated as follows: Theorem 2.1. If F is a finite field and C ≤ F n a linear code, then every Hamming isometry C −→ F n can be extended to a monomial transformation of F n . Wood [37] showed that regarding the concept of code equivalence finite Frobenius rings are an appropriate class of rings for generalized linear coding theory. In the paper mentioned, he proved that MacWilliams’ equivalence theorem holds for linear codes over finite Frobenius rings. Theorem 2.2. (a) If R is a finite Frobenius ring and C ≤ R Rn a linear code, then every Hamming isometry C −→ Rn can be extended to a monomial transformation of R Rn .

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(b) If a finite commutative ring R satisfies, that all its Hamming isometries between linear codes allow for monomial extensions, then R is a Frobenius ring. Remark 2.3. Recent work by M. Greferath, A.A. Nechaev, and R. Wisbauer ˆ := Hom(R, Q/Z) is used as [19] shows that when the character module R alphabet in lieu of the ring R itself, then an appropriate version of the above theorem holds without imposing any hypotheses on the underlying ring R. In light of theorem 2.2(b), it is desirable to characterize the class of those finite rings and modules that allow for appropriate versions of the equivalence theorem of algebraic coding theory to hold. As mentioned, Wood’s work [37] answers this question for finite commutative rings and comes to the conclusion that the class of finite rings that allow for the fundamental extension and duality theorems of coding theory consists precisely of the finite Frobenius rings. In the noncommutative case this question is still open. Recent work by Hai Q. Dinh and S. L´opez-Permouth has shown that the MacWilliams’ equivalence theorem comes close to giving a general characterization of finite Frobenius rings in coding theoretic terms. For convenience, let us say that a ring that allows for MacWilliams’ equivalence theorem on its right linear codes is a right MacWilliams ring. More generally, one refers to a right module M satisfying an analog property as a MacWilliams module. Dinh and L´opez-Permouth’s results show that MacWilliams rings are Frobenius under several hypotheses including when the ring is homogeneous semilocal or basic [12, 13]. More generally, their results in [13] also deal, over arbitrary rings, with the potential centrality of the character module as a MacWilliams module. They have conjectured and provide some evidence (over specific rings) that a MacWilliams module must be a direct summand of the character module. Theorem 2.4. Let R be a right MacWilliams finite ring then R is Frobenius under any of the following hypothesis: (a) When R is a direct sum of homogeneous semilocal rings [12]. (b) When R is a basic ring [13]. An important technical tool in [13] is a theorem that translates certain aspects of these problems into one that deals only with matrices over finite fields. The statement of this theorem, as it appears in [13], is more complicated than what we state here. However, one can actually extend that proof to get the following result. Theorem 2.5. The following two statements are equivalent. (a) For any finite field F and for any n ∈ Z, the right M(n+1)×(n+1) (F )module Mn×(n+1) (F ) of n × (n + 1) matrices over F is not a MacWilliams module.

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(b) For any finite ring R and for any right R-modules M and S, with S simple, the uniform dimension of the homogeneous component of S in the socle of M is bounded above by the uniform dimension of the homogeneous component of S in R/J (where J is the Jacobson radical of R). 3. A technical tool: weight enumerators and duality One of the most important computational tools in traditional algebraic coding theory is based on a theorem by MacWilliams that establishes a connection between the weight enumerators of a linear code and its dual. To be more precise, let C ≤ R Rn be a left linear code. The integer polynomial WC (x) =

n X

Ai xi where Ai := #{c ∈ C | w(c) = i} for i ∈ {1, . . . , n}

i=0

is called the weight enumerator of C. The dual code of C is defined as n X ⊥ n C := {x ∈ R | ci xi = 0 for all c ∈ C}. i=1

This is obviously a right R-linear code, and there rises the question in how far the weight distribution of C determines the weight distribution of C ⊥ . Now the MacWilliams’ theorem on weight enumerators gives an answer and may be stated as follows: Theorem 3.1. Let F be the finite field of q elements, and let C ≤ F n be a linear code with Hamming weight enumerator WC (x) =

n X

Ai xi .

i=0

Then the Hamming weight enumerator of C ⊥ is given by n 1 X WC ⊥ (x) = Ai (x − 1)i (x + q − 1)n−i . |C| i=0

It has been essentially J. Wood’s pioneering work to observe that this theorem is true for F exchanged by any finite Frobenius ring R, and where consequently q = |R|. This stems from the fact that the statement allows for a formulation in a much more general context. For this denote by R× the group of invertible elements of R, and let U be a subgroup of R× of R that is central in R. This means ur = ru for all u ∈ U and r ∈ R. Let S := R/U denote the set of all U -associate classes of elements in R. We consider the polynomial ring Z[x] where x = (xs )s∈S , and

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define the symmetrized weight enumerator of the left R-linear code C ≤ R Rn as XY WC (x) = xsns (c) , where ns (c) = #{1 ≤ i ≤ n | ci ∈ s}. c∈C s∈S

Then we have the following theorem: Theorem 3.2. Let R be a finite Frobenius ring and let C ≤ R Rn be a left linear code with symmetrized weight enumerator WC (x). Then the symmetrized weight enumerator of C ⊥ is given as 1 WC ⊥ (x) = WC (M T x) |C| P where M is the matrix with entry Ms,t = u∈U χ(uαβ) where α and β are chosen from the classes s and t respectively, and χ is a so-called generating character of the ring R. Note that the mentioned work [19] again reveals that using the character ˆ as alphabet for C ⊥ rather than R itself, then the above theorem module R holds without imposing any hypotheses on the underlying ring R. 4. Code quality: bounds on the parameters of linear codes Among the traditional bounds on the parameters of block codes over finite fields the most important are probably the sphere-packing bound, the Singleton bound, the Gilbert-Varshamov bound, the linear programming bound, the Plotkin and Elias bounds, and the Griesmer bound. With the exception of the Gilbert-Varshamov bound, which states the existence of codes having parameters above a certain bound, all these bounds are upper bounds restricting the existence of codes with given parameters. These bounds give rise to several notions of code optimality. Let F be the finite field of q elements, and let Aq (n, d) denote the maximal number of words a (not necessarily linear) code of length n over F that has minimum Hamming distance d. It is easy to see that the sphere of radius t in the Hamming space F n has the volume t µ ¶ X n Vq (n, t) = (q − 1)i . i i=0

Theorem 4.1.

(a) (Sphere-Packing Bound) For every n, d there holds Aq (n, d) ≤

qn . Vq (n, b d−1 c) 2

(b) (Singleton Bound) For every n, d there holds Aq (n, d) ≤ q n+1−d .

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(c) (Plotkin Bound) For every n, d with Aq (n, d) ≤

q−1 q

n < d there holds

d . d−γn

(d) (Elias Bound) For every n, d, t with t ≤ γ n and t2 − 2 tγ n + dγ n > 0 there holds Aq (n, d) ≤

γ nd qn · . t2 − 2 tγ n + dγ n Vq (n, t)

Regarding existence bounds for ring-linear codes we need to deal with two questions that point in different directions: the first question is that for a most general ring or module class for which generalizations of these bounds might hold. It will not surprise that the class of all finite Frobenius rings is a class subject to most investigations. The second question aims at a generalization of most traditional bounds to weight functions (and metrics) on a ring that differ from the Hamming weight. Regarding the first question we like to mention an approach in [35] for codes over finite commutative Frobenius rings (Singleton bound). In [36] there can be found a generalization of the Griesmer bound for codes over finite commutative Frobenius rings. As indicated, however, these results are restricted to the Hamming weight on the ring in question. One of the most important contributions to the analysis of more general weight functions on rings is due to I. Constantinescu and W. Heise [11], who introduced the concept of a homogeneous weight. The homogeneous weight on a finite ring shares many combinatorial properties with the Hamming weight on a finite field. Observing its properties it seems to be an appropriate generalization of Hamming weight to finite rings. Homogeneous weights on a finite ring R are defined as functions w : R −→ Q with the following three properties: (i) w(0) = 0. (ii) Whenever R× x = R× y then w(x) = w(y). (iii) There is a constant γ ∈ Q such that 1 X w(y) = γ for all x ∈ R \ {0}. |R| y∈Rx

This number γ might be called the average value of w, and (iii) states that the average weight of every principal (left) ideal of R is the same. There also have been studied weights with smaller symmetry groups, hence weakening the above property (ii). These have been called egalitarian (Heise, Honold) or prehomogeneous weights (Wood, Greferath). They often may be analyzed with similar tools as homogeneous weights, and results suggest that they have a finer spectral structure.

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In the context of ring-linear coding theory we expect that most of the classical bounds have natural counterparts, at least as far as the homogeneous weight on a given finite ring is concerned. The sphere-packing bound for example requires no modification, except that its version for a finite ring no longer involves a simple expression in terms of a binomial distribution. Recent work of Greferath and O’Sullivan [20] in this area provides an exponential sum technique to derive homogeneous versions of the Plotkin and Elias bounds. Surprisingly, these are entirely analogous to those in the traditional context. The mentioned paper also gives asymptotic versions for these bounds. More detailed, let R be a finite Frobenius ring and let w be a homogeneous weight on R of average value γ. Let AR (n, d) denote the maximal number of words a (not necessarily linear) code of length n over R that has minimum homogeneous distance d. Finally let Vw (n, t) be the volume of the sphere of (homogeneous) radius t in the space Rn . Theorem 4.2. holds

(a) (Plotkin Bound) For every n, d with γ n < d there AR (n, d) ≤

d . d−γn

(b) (Elias Bound) For every n, d, t with t ≤ γ n and t2 − 2 tγ n + dγ n > 0 there holds γ nd |R|n AR (n, d) ≤ 2 · . t − 2 tγ n + dγ n Vw (n, t) 5. Good examples: constructions and code design Apart from the Z4 -linear series of codes that led to a rediscovery of the role of rings in algebraic coding theory, it might be worth mentioning the so-called Calderbank-McGuire code [10], a sporadic Z4 -linear example of a code whose Gray image is a binary code that has more codewords than any other known code of the same length and minimum distance. Using the rings Z8 and Z9 equipped with the homogeneous weights the papers [17, 16] yield a similar type of outperforming examples by Hensellifting the extended binary Golay code and the ternary [24, 12, 9]3 quadratic residue code. More specifically, over Z8 , there is a lift of the binary Golay code that has good minimum distance properties with respect to the homogeneous weight (cf. [17]). Over Z9 there is a lift of the ternary Golay code and also the ternary [24, 12, 9]3 QR-code that provides examples of superior codes (cf. [16]). A further example is that the GR(4, 2)-linear code generated by a generator matrix of the [8, 4]4 Octacode can be used to produce an example of a nonlinear code of length 32 over F4 (cf. [29]), that has four times the number of codewords of any previously known F4 -code of same length and distance.

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Being the first instance when a ring alphabet and a non-Hamming metric surfaced in a leading role, it might not be surprising that there is a strong emphasis on Z4 -linear coding with respect to the Lee weight (cf. [15, 27, 4, 6]). Clearly, Z4 is the ring-linear coding analog of Z2 in finite-field linear coding theory. Another argument for the role of Z4 is a strong impact of Z4 -linear coding to the theory of lattices (cf. [23, 5, 1]). It is worthwhile mentioning that linear codes over other 4-element rings have also been studied in detail (see [18, 14]). From the engineering aspect, Z4 -linear coding on the basis of the Lee weight has been emphasized (cf. [15, 27, 4, 6]), partly because many communications systems use so-called QPSK modulation which is well suited to the Lee weight. Linear codes over other types of 4-element rings have also been studied in detail (see [18, 14]). What is desirable in practice is to have a weight function that matches the error patterns of the modulation scheme, that is, higher weight elements of the ring should correspond to errors that are less likely to occur. In general, a weight function taking a variety of values would be preferable. Inspired by the success in providing an algebraic representation of good families of codes in terms of ring-linearity, there is a natural interest in further examples of good ring-linear codes. Although the results are not numerous, they are promising. 6. Practical questions: designing of decoding algorithms As to the decoding question the paper [22] gives a decoding algorithm for the Z4 -linear Preparata and Kerdock codes. Further decoders have been developed by Helleseth and others (cf. [24, 33]) for the Goethals codes and the Calderbank-McGuire code. There are several contributions (cf. [7, 9]) that mainly deal with algorithms decoding a given ring-linear code up to half of its Hamming minimum distance. Greferath and Vellbinger’s earlier work [21] (extended in [2]) yields a successive decoding for linear codes over Zpk . More precisely, a decoding scheme A for a Zp -linear code can be lifted to decode any of its Zpk -linear free preimages under the natural surjection. It corrects any error pattern for which each component of its p-adic representation is correctable by A. If Zpk is equipped with the Hamming weight and A allows to decode up to half of the minimum Hamming distance, then the resulting decoder is a bounded distance decoder as well. The number of error patterns that can be corrected by this decoder is, however, much larger. Some decoding algorithms of finite-field coding theory have been extended to decode ring-linear codes but mostly with respect to the Hamming weight (cf. [25, 31, 8]).

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In the 1980s Berlekamp designed a decoding algorithm for negacyclic codes with the Lee metric (cf. [3]). Its basic technique is to encode error values by multiplicities of zeros of the error locator polynomial. This decoder has strict limitations in the finite field context due to the fact that it only works for prime fields of odd order and is restricted to errors of low weight. It might be interesting to see if this limitation can be avoided when considering negacyclic codes over Z4 and more generally constacyclic codes over Z2k . We will close by saying that even though contemporary coding theory is mostly interested in decoders that perform beyond half of the minimum distance of a given code, it seems reasonable, as a first step, to have decoders realizing half of the minimum distance with respect to a given metric on the underlying alphabet.

References [1] T. Aoki, P. Gaborit, M. Harada, M. Ozeki, and P. Sol´e, On the covering radius of Z4 -codes and their lattices, IEEE Trans. Inform. Theory 45 (1999), no. 6, 2162–2168. [2] N. Suresh Babu and K.-H. Zimmermann, Decoding of linear codes over Galois rings, IEEE Trans. Inform. Theory 47 (2001), no. 4, 1599–1603. [3] E.R. Berlekamp, Algebraic Coding Theory, Aegean Park Press, Laguna Hills, 1984. [4] A. Bonnecaze, E. Rains, and P. Sol´e, 3-colored 5-designs and Z4 -codes, J. Statist. Plann. Inference 86 (2000), no. 2, 349–368, Special issue in honor of Professor Ralph Stanton. [5] A. Bonnecaze, P. Gaborit, M. Harada, M. Kitazume, and P. Sol´e, Niemeier lattices and type II codes over Z4 , Discrete Math. 205 (1999), no. 1-3, 1–21. [6] A. Bonnecaze, P. Sol´e, C. Bachoc, and B. Mourrain, Type II codes over Z4 , IEEE Trans. Inform. Theory 43 (1997), no. 3, 969–976. [7] E. Byrne and P. Fitzpatrick, Gr¨ obner bases over Galois rings with an application to decoding alternant codes, J. Symbolic Comput. 31 (2001), no. 5, 565–584. [8] , Gr¨ obner bases over Galois rings with an application to decoding alternant codes, J. Symbolic Comput. 31 (2001), no. 5, 565–584. [9] , Hamming metric decoding of alternant codes over Galois rings, IEEE Trans. Inform. Theory 48 (2002), no. 3, 683–694. [10] A.R. Calderbank and Gary M. McGuire, Construction of a (64, 237 , 12) code via Galois rings, Des. Codes Cryptogr. 10 (1997), no. 2, 157–165. [11] I. Constantinescu and W. Heise, A metric for codes over residue class rings of integers, Problemy Peredachi Informatsii 33 (1997), no. 3, 22–28. [12] H. Dinh and S. L´ opez-Permouth, On the equivalence of codes over finite rings, Appl. Algebra Engrg. Comm. Comput. 15 (2004), 37–50. [13] , On the equivalence of codes over rings and modules, Finite Fields and their Applications 10 (2004), 615–625. [14] S.T. Dougherty, P. Gaborit, M. Harada, and P. Sol´e, Type II codes over F2 + uF2 , IEEE Trans. Inform. Theory 45 (1999), no. 1, 32–45. [15] S.T. Dougherty, M. Harada, and P. Sol´e, Shadow codes over Z4 , Finite Fields Appl. 7 (2001), no. 4, 507–529. [16] I.M. Duursma, M. Greferath, S. Litsyn, and S.E. Schmidt, A Z9 -linear code inducing a ternary (72, 325 , 24)-code, Proceedings of the OC 2001.

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[17] I.M. Duursma, M. Greferath, S.N. Litsyn, and S.E. Schmidt, A Z8 -linear lift of the binary Golay code and a nonlinear binary (96, 237 , 24)-code, IEEE Trans. Inform. Theory 47 (2001), no. 4, 1596–1598. [18] P. Gaborit, Mass formulas for self-dual codes over Z4 and Fq + uFq rings, IEEE Trans. Inform. Theory 42 (1996), no. 4, 1222–1228. [19] M. Greferath, A. Nechaev, and R. Wisbauer, Finite quasi-Frobenius modules and linear codes, J. Algebra Appl. Vol. 3, No. 3 (2004) 247-272. [20] M. Greferath and M.E. O’Sullivan, On bounds for codes over Frobenius rings under homogeneous weights, Discrete Math. 289 (2004), no. 1-3, 11–24. [21] M. Greferath and Ute Vellbinger, Efficient decoding of Zpk -linear codes, IEEE Trans. Inform. Theory 44 (1998), no. 3, 1288–1291. [22] A.R. Hammons, P.V. Kumar, A.R. Calderbank, N. J. A. Sloane, and P. Sol´e, The Z4 -linearity of Kerdock, Preparata, Goethals and related codes, IEEE Trans. Inform. Theory 40 (1994), 301–319. [23] M. Harada, P. Sol´e, and P. Gaborit, Self-dual codes over Z4 and unimodular lattices: a survey, Algebras and combinatorics (Hong Kong, 1997), Springer, Singapore, 1999, pp. 255–275. [24] Tor Helleseth and P. Vijay Kumar, The algebraic decoding of the Z4 -linear Goethals codes, IEEE Trans. Inform. Theory 41 (1995), no. 6, part 2, 2040–2048. [25] J. Carmelo Interlando, R. Palazzo, Jr., and M. Elia, On the decoding of Reed-Solomon and BCH codes over integer residue rings, IEEE Trans. Inform. Theory 43 (1997), no. 3, 1013–1021. [26] A.M. Kerdock, A class of low-rate nonlinear binary codes, Information and Control 20 (1972), 182–187; ibid. 21 (1972), 395. [27] P. Langevin and P. Sol´e, Duadic Z4 -codes, Finite Fields Appl. 6 (2000), no. 4, 309– 326. [28] F.J. MacWilliams, Combinatorial properties of elementary abelian groups, Ph.D. thesis, Radcliffe College, Cambridge MA, 1962. [29] T. Muegge, Existenzschranken f¨ ur Blockcodes u ¨ber endlichen Frobeniusringen, Diplomarbeit, 2002. [30] A.A. Nechaev, Kerdock codes in a cyclic form, Discrete Math. Appl. 1 (1991), 365– 384. [31] G.H. Norton and A. S˘ al˘ agean, On the key equation over a commutative ring, Des. Codes Cryptogr. 20 (2000), no. 2, 125–141. [32] F.P. Preparata, A class of optimum nonlinear double-error correcting codes, Information and Control 13 (1968), 466–473. [33] C. Rong, Tor Helleseth, and J. Lahtonen, On algebraic decoding of the Z4 -linear Calderbank-McGuire code, IEEE Trans. Inform. Theory 45 (1999), no. 5, 1423–1434. [34] C.E. Shannon, A mathematical theory of communication, Bell System Tech. J. 27 (1948), 379–423, 623–656. [35] K. Shiromoto, Singleton bounds for codes over finite rings, J. Algebraic Combin. 12 (2000), no. 1, 95–99. [36] K. Shiromoto and L. Storme, A Griesmer bound for linear codes over finite quasiFrobenius rings, Discrete Appl. Math. 128 (2003), no. 1, 263–274, International Workshop on Coding and Cryptography (WCC 2001) (Paris). [37] J.A. Wood, Duality for modules over finite rings and applications to coding theory, Amer. J. Math. 121 (1999), no. 3, 555–575.

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Semiperfect rings with T -nilpotent prime radical Nadiya M. Gubareni Institute of Econometrics & Computer Science, Technical University of Cz¸estochowa, D¸abrowskiego 69, 42-200 Cz¸estochowa, Poland [email protected] Vladimir V. Kirichenko Department of Mechanics and Mathematics, Kyiv Taras Shevchenko University, Volodymyrs’ka 64, 01033, Kyiv, Ukraine [email protected] Abstract. We consider semiperfect rings with T -nilpotent prime radical whose quotient rings by the prime radical are right Noetherian. We prove that the prime quiver of a serial ring A with a Noetherial diagonal is a disconnected union of cycles and chains. If A is a ring such that all finitely generated A-modules are serial, then A is serial. If 1 = e1 + ... + en is a decomposition of the identity of A into a sum of pairwise orthogonal local idempotents then all finitely generated A-modules are serial if and only if all finitely generated eAemodules are serial for all e = ei1 + ei2 + ei3 , where ei1 , ei2 , ei3 are different idempotents from 1 = e1 + ... + en .

1. Introduction All rings considered in this paper are assumed to be associative with identity 1 6= 0, and all modules are unitary right modules, unless otherwise specified. A ring is called decomposable if it decomposes into a direct product of two rings, otherwise the ring is indecomposable. The main notions of this paper are that of a prime quiver of a semiperfect ring and the notion of a prime block. We say that a semiperfect ring A is a prime block if its prime quiver P Q(A) is connected and it is known that 2000 Mathematics Subject Classification. 16P40, 16G10. Key words and phrases. semiperfect ring, piecewise Noetherian ring, quiver, prime quiver, prime block. 217

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a semiperfect ring A with T -nilpotent prime radical is a prime block if and only if A is indecomposable (see Theorem 4.6). An important class of semiperfect rings is formed by the serial rings. Artinian serial rings, also called “generalized uniserial rings,” were introduced by Nakayama [21, 22], who showed that all modules over them are serial. The structural description of right Noetherian serial rings was given in [9, 10], where, in particular, the following theorem was proved: A right Noetherian ring A is serial if and only if it decomposes into a finite direct product of a two-sided Noetherian serial ring and rings, which are Morita equivalent to quotient rings of rings of the following form: µ ¶ Hm (O) Mm,n (D) H(O, m, n) = , 0 Tn (D) where O is a local principal left and principal right ideal domain, D is its classical division ring of fractions, Tn (D) is the ring of upper triangular n × n matrices over D, Mm,n (D) is the space of all (m × n)-matrices over D and   O O ... O M O . . . O   Hm (O) =  . . . . . . . . . . . . , M M ... O where M is the Jacobson radical of O. An analogous theorem was proved by S. Singh in [24]: Let R be an indecomposable, nonprime, non-Artinian ring. Then R is a serial right Noetherian ring if and only if µ ¶ S M R= 0 T such that (a) S is a prime, serial Noetherian ring, which is not Artinian, and T is an indecomposable Artinian serial ring admitting a simple projective module. (b) M is an (S, T )-bimodule such that S M is a divisible torsion free module with rank S M = rank(T /B), B = annT M and T B is a direct summand of T T . (c) MT is a direct sum of finitely many isomorphic serial modules and rank(MT ) = rank(SS ). Note that a two-sided Noetherian ring A is serial if and only if it decomposes into a finite direct product of a two-sided Artinian serial ring and rings which are Morita equivalent to rings of the form Hm (O). The full description of serial two-sided Noetherian rings was also obtained by R.B. Warfield [26].

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The properties and the structure of Artinian serial rings were considered in many papers (see, for example, [4], [5], [15]–[20]). Let N be the set of positive integers. Denote by R(A) the Jacobson radical of a ring A. Proposition 1.1. If A is a serial two-sided Noetherian ring. Then \ Rn (A) = 0. n∈N

Proof. This follows from the fact that in Hm (O) one has that \ Rn (Hm (O)) = 0. n∈N

¤ Proposition 1.2. [7, p. 300] A right (left) serial ring A with nilpotent Jacobson radical R is right (left) Artinian. Theorem 1.3. Let A be a serial ring with the Jacobson radical R(A). Then A is two-sided Artinian if and only if R(A) is nilpotent. Proof. The proof follows from Proposition 1.2 and the following well-known Hopkins proposition: the radical R(A) of a right Artinian ring A is nilpotent. ¤ Theorem 1.4. Let A be a serial ring T with Jacobson radical R. Then A is two-sided Noetherian if and only if n∈N Rn = 0. Proof. T The “only if” part follows from Proposition L L ns1.1. Let n∈N Rn = 0 and let AA = P1n1 ... Ps be a decomposition of AA into a direct sum of principal right A-modules. Observe that for every m ∈ N such that Pi Rm 6= 0 the inclusion Pi Rm+1 ⊂ Pi Rm is proper. Indeed, if Pi Rm = Pi Rm+1 then Pi Rm = Pi Rm+1 = Pi Rm+2 = ...

T Thus, Pi Rm ⊂ n∈N Pi Rn and we have that Pi Rm = 0, a contradiction. Obviously, Pi Rm /Pi Rm+1 is a simple module and if x ∈ Pi Rm \Pi Rm+1 , then xA = Pi Rm and Pi is a Noetherian module for i = 1, ..., s. Consequently, A is right Noetherian. Analogously, A is left Noetherian. ¤ 2. Finitely generated serial modules The aim of this section is to prove the following Reduction Theorem. Theorem 2.1. (Reduction Theorem) Let A be a ring such that all finitely generated A-modules are serial. Then A is serial and if given 1 = e1 +...+en , a decomposition of the identity of A into a sum of pairwise orthogonal local idempotents, then all finitely generated A-modules are serial if and only if

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all finitely generated eAe-modules are serial for all e = ei1 + ei2 + ei3 , where ei1 , ei2 , ei3 are different idempotents from 1 = e1 + ... + en . We need the following assertions. Theorem 2.2. (Drozd-Warfield Theorem) (see, for example, [7, p. 323]). For a ring A the following conditions are equivalent: (1) A is serial; (2) any finitely presented right A-module is serial; (3) any finitely presented left A-module is serial. Proposition 2.3. (See [7, Proposition 12.3.1]) For a semiperfect ring A the following conditions are equivalent: (1) A is a right (left) serial ring; (2) for any two nonzero homomorphisms of right (left) principal Amodules fi : Pi −→ P i = 1, 2 one of the two equations: f1 = f2 x or f2 = f1 y is solvable; (3) for any two nonzero homomorphisms of left (right) principal Amodules fi : P −→ Pi i = 1, 2 one of the two equations f1 = xf2 or f2 = yf1 is solvable with x ∈ HomA (P1 , P2 ), y ∈ HomA (P1 , P2 ). Corollary 2.4. (See [7, Corollary 12.3.2]) Let A be a semiperfect ring and let 1 = e1 + ... + en be a decomposition of 1 ∈ A into a sum of local pairwise orthogonal idempotents. The ring A is right serial if and only if for each idempotent e, which is a sum of not more than three different local idempotents from {e1 , e2 , ..., en }, the ring eAe is right serial. Theorem 2.5. (Ivanov’s theorem, see [25, p. 90]). For a ring A, the following conditions are equivalent: (1) All finitely generated right A-modules are serial. (2) A is a serial ring, and all uniform right A-modules are uniserial. (3) All 2-generated right A-modules are serial. (4) A is a right semiuniform ring, and all indecomposable injective right A-modules are uniserial. (5) A is a right serial ring, and for every isomorphism f : M −→ N between arbitrary submodules M and N of any primitive cyclic A-modules xA and yA, respectively, either f or f −1 can be extended to a homomorphism between xA and yA. Let P be a finitely generated projective A-module that decomposes into a direct sum of n indecomposable modules. The endomorphism ring B = EndA P of the module P is called a minor of order n of A. By [7, Theorem 10.3.8] a ring is semiperfect if and only if any minor of the first order of this ring is semiperfect (or, equivalently, is local). Let C(A) and C(B) are the categories of all A-modules and all B-modules correspondingly.

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Denote by C(P ) the full subcategory of C(A) formed by all A-modules M such that there exists an exact sequence P (I) −→ P (J) −→ M −→ 0, where P (I) is the direct sum of Pi (i ∈ I) and all Pi are isomorphic to P . We consider the functors F : C(A) −→ C(B) and G : C(B) −→ C(A), where F (M ) = HomA (P, M ) and G(N ) = N ⊗B P . Theorem 2.6. ([2]). For any B-module N, G(N ) ∈ C(P ). The functors G and F determine an equivalence of the categories C(B) and C(P ). For a semiperfect ring A, the nontrivial minors of A coincide with the rings that are Morita-equivalent to the rings eAe (e2 = e and e 6= 0, 1). Definition 2.7. We will say that a ring A is an Sr -ring (Sl -ring) if all finitely generated right (left) A-modules are serial. Example. Let Q be the field of rational numbers and let p be a prime integer, m Zp = { ∈ Q|(n, p) = 1}. n The following Herstein-Small ring µ ¶ Zp Q A= O Q is an Sr -ring, but A is not an Sl -ring (see, [3]). Proposition 2.8. Any Sr -ring and any Sl -ring is serial. The proof follows from the Drozd-Warfield theorem. Lemma 2.9. (see [7, p. 241]) Suppose that the identity of a semiperfect ring A can be decomposed into a sum of pairwise orthogonal local idempotents in two different ways: 1 = e1 + . . . + en = f1 + . . . + fm . Then m = n and there exists a permutation τ of {1, . . . , n} and an invertible element a ∈ A such that fτ (i) = aei a−1 for i = 1, . . . , n. Proposition 2.10. Any minor of an Sr -ring (Sl -ring) is an Sr -ring (Sl ring). Proof. Let A be an Sr -ring. Then by Proposition 2.8 A is serial. We can assume that A is reduced, i.e., A/R(A) is a finite direct product of division rings. Let 1 = e1 + . . . + en be a decomposition of the identity of A into a sum of pairwise orthogonal local idempotents and {ei1 , . . . eim } is a subset of {e1 , . . . , en }. By Lemma 2.9 any nontrivial minor B is Morita-equivalent to a ring B = eAe, where e = ei1 + . . . + eim . Let P = eA. Obviously, any finitely generated module over a serial ring A is serial if and only if any finitely generated indecomposable A-module is uniserial.

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Let N be an indecomposable finitely generated B-module. By the Adjoint Isomorphism (see, for example, [7, p. 101]) we have HomA (G(N ), G(N )) ' HomA (N ⊗B P, G(N )) ' ' HomB (N, HomA (P, G(N )) ' HomB (N, N ). Consequently, G(N ) = N ⊗B P is an indecomposable finitely generated A-module and F G(N ) ' G(N )e = N . Let 1 = e + f . Then G(N ) = N ⊕ G(N )f is a direct sum of Abelian groups. Suppose that N is not a uniserial B-module. Then in N there exist two B-submodules M1 and M2 such that M1 ∩M2 6= M1 and M1 ∩M2 6= M2 . ˜ 1 = M1 A ⊂ G(N ) and M ˜ 2 = M2 A ⊂ G(N ). Obviously, M ˜ 1 6⊂ M ˜2 Write M ˜ 2 6⊂ M ˜ 1 . This contradiction proves that N is uniserial. and M ¤ Proof of Theorem 2.1. The “only if” part follows from Proposition 2.10. Show that if all rings of the form eAe are Sr -rings for all e = ei1 +ei2 +ei3 , where {ei1 , ei2 , ei3 } ⊂ {e1 , . . . , en }, then A is an Sr -ring. We shall prove this assertion by induction on n. The case n = 3 is clear. By Proposition 2.8 and Corollary 2.4 A is serial. Thus, by Ivanov’s Theorem it is enough to prove that any indecomposable injective A-module Q is uniserial. If Q is not uniserial then there exists a nonuniserial submodule M = q1 ei A + q2 ej A ⊂ Q. We may assume that i, j ≤ n − 1. Let f = en , e = 1 − f , A1 = eAe, A2 = f Af , X = eAf , Y = f Ae. Obviously, M A1 is a decomposable A1 -module. Indeed, if M A1 is indecomposable, then by the induction hypothesis M A1 is a uniserial A1 -module. Suppose that q1 ei A1 ⊂ q1 ej A1 . So, q1 ei A ⊂ q1 ej A and M is uniserial. Let M A1 = z1 A1 ⊕ z2 A1 . If z1 X ∩ z2 X = 0, then M is a decomposable submodule of the indecomposable injective module Q. This is impossible. Set T = z1 X ∩ z2 X. Obviously, T Y = 0 and the A-module T is an A2 module. By the induction hypothesis, T is uniserial. Let e = ei + ej + en for i 6= j, and e = ei +en for i = j. Put B = eAe. Obviously, T e = T and M B is an essential extension of T . By Ivanov’s Theorem any indecomposable Bmodule is uniserial. Consequently, M B is a uniserial B-module, and we can assume that q1 ei B ⊂ q2 ej B (possibly i = j). As above, q1 ei A ⊂ q2 ej A and we have a contradiction. The Reduction Theorem is completely proved. ¤ 3. The prime quiver of a serial ring with a right Noetherian diagonal The definition of the prime quiver P Q(A) for a semiperfect ring A is given in [7]. Since any serial ring A is semiperfect we have the definition of the prime quiver for a serial ring as well.

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In [13] the two-sided Peirce decomposition for the prime radical P r(A) of a semiperfect ring A was obtained in the form: 

 I1 A12 . . . A1t A21 I2 . . . A2t    (1) P r(A) =  .. .. ..  , ..  . . . .  At1 At2 . . . It where A¯ = A¯1 × · · · × A¯t is a decomposition of the diagonal A¯ = A/P r(A) into a direct product of a finite number of prime rings, ¯ 1 = f¯1 +· · ·+ f¯t is the ¯ ¯ corresponding decomposition of the identity 1 ∈ A into a sum of pairwise orthogonal central idempotents, Aij = fi Afj and Ii = P r(Ai ). In this paper we give the description of prime quivers of serial rings with right Noetherian diagonal. If A is a serial ring with a right Noetherian diagonal, then by Theorem 1.3 this diagonal is a two-sided Noetherian ring. Therefore it is possible simply to speak about a Noetherian diagonal. Theorem 3.1. Let A be a serial ring with a Noetherian diagonal. Then the prime quiver P Q(A) is a disconnected union of cycles and chains. Proof. Since P Q(A) = P Q(A/J 2 ) and P r(A/J 2 ) = J /J 2 , where J = P r(A), we can suppose that A is a serial ring with J 2 = 0. Consider the diagonal A¯ = A/J = A1 × · · · × At , where all rings A1 , . . . , At are indecomposable, and the corresponding decomposition 1 = f1 + · · · + ft of 1 ∈ A into a sum of pairwise orthogonal idempotents, central modulo P r(A). Note that, since J 2 = 0, fi J fj is a submodule of fi A, and fi J fj = fi Afj for j 6= i. Take a decomposition fi = ei1 + · · · + eimi of the idempotent fi into a sum of pairwise orthogonal local idempotents. Assume that a vertex i is a source of at least two arrows, the ends of which are s 6= i and t. It means that fi J fs and fi J ft are both nonzero. By Theorem 1.3 we may suppose that each Ai = fi Afi /fi J fi is either a division ring or a ring of the form Hm (O), where O is a discrete valuation ring. If the idempotent fi is local, which is always the case if Ai is a division ring, then the uniserial module fi J contains two submodules fi J fs and fi J ft , neither of which is contained in the other, which is impossible. Assume that fi Afi /fi J fi ' Hmi (Oi ), where Oi is a discrete valuation ring. Denote by R the Jacobson radical of fi Afj and take a decomposition fi = e1 + · · · + em of fi into a direct sum of local idempotents. Choose k so that ek Afs 6= 0. If ek J ft 6= 0, then ek A contains two submodules, ek Afs and ek J ft , neither of which is a submodule of the other, which is impossible. Suppose that ek J ft = 0 for all t 6= s, in particular, ek J ek = 0. If ek Rek Afs 6= ek Afs , then there are two submodules, ek Rek + ek Rek Afs and ek Afs , of the uniserial module ek A, neither of which contains the other. Hence, ek Rek Afs = ek Afs for every k such that ek Afs 6= 0. But ek Rek ⊂ ek Rel Rek + J for every l 6= k, hence

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ek Rek = ek Rel Rek and el Rek Afs 6= 0 for every l. Therefore, el J ft = 0 for every l, thus fi J ft = 0, a contradiction. In exactly the same way we can prove that two arrows cannot end at the same vertex. There are two types of finite connected graphs with these properties: a cycle and a chain. The theorem is proved. ¤ Examples. 1. Let Zp denote the ring of p-adic integers for some prime p in the field of p-adic numbers Qp , and Zp∞ = Qp /Zp . Define a ring R by (R, +) = Zp ⊕ Zp∞ , and for (λ, x), (µ, y) ∈ R, (λ, x)(µ, y) = (λµ, λy + µx). This ring was considered in [23] as an example of a ring, which is an injective cogenerator without chain conditions. It was shown that this ring is a local ring with maximal ideal (p, 0)R, which is its Jacobson radical and so it is not T -nilpotent. The prime radical P r(R) of A equals 0 ⊕ Zp∞ and so it is nilpotent. The diagonal R/P r(R) ' Zp is a Noetherian ring. The prime quiver of this ring is a one-point cycle (a loop). 2. Let O be a discrete valuation ring with division ring of fractions D. Consider the ring A with elements of the following form:   α β1 . . . βt−1 βt  0 α . . . βt−2 βt−1     .. .. . . .. ..  , . .  . . .   0 0 . . . α β1  0 0 ... 0 α where α ∈ O, βi ∈ D, i = 1, . . . , t − 1, βt ∈ D or βt ∈ D/O. Then A is a serial ring with Noetherian diagonal and nilpotent prime radical. The prime quiver of the ring A is a loop. This example can be considered as a generalization of example 1. 3. Suppose that O is a discrete valuation ring with the division ring of fractions D. Then Hm (O) is a serial Noetherian prime ring. Let D be the common division ring of fractions of the discrete valuation rings O1 ,. . . ,Oq . Suppose that ∆1 ,. . . ,∆p is a set of rings, each of which equals either Ok for some k = 1, . . . , q or D. Let H((∆1 , n1 ), . . . , (∆p , np )) denote the ring of block matrices (Mij ), i, j = 1, . . . , p, where • Mij is an ni × nj matrix with entries from D if i < j; • Mij = 0 if i > j; • Mii ∈ Tni (D) if ∆i = D; • Mii ∈ Hni (Ok ) if ∆i = Ok . Then H((∆1 , n1 ), . . . , (∆p , np )) is a serial ring with Noetherian diagonal. 4. Let A = H(O, m, n) be the particular case of the previous example for p = 2, ∆1 = O, n1 = m, ∆2 = D, n2 = n. It is a right hereditary serial

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ring. The prime quiver of this ring is a chain: ½ 1 2 n n+1 • −→ • −→ . . . −→ • −→ •

225

¾ .

The quiver of this ring is a union of a cycle: ½ ¾ 1 2 m−1 m 1 • −→ • −→ . . . −→ • −→ • −→ • and a chain: ½ m+1 m+2 m+n−1 m+n • −→ • −→ . . . −→ • −→ •

¾ .

4. Prime blocks of semiperfect rings Definition 4.1. [12] A semiperfect ring A is called a prime block if P Q(A) is a connected quiver. Obviously, any prime block is indecomposable, but the converse is not true in general. We recall the definition of T -nilpotency introduced by H. Bass. Definition 4.2. [1] A right (resp. left ) ideal I is called left (resp. right) T -nilpotent if for any sequence a1 , a2 , . . . , an , . . . . of elements ai ∈ I there exists a positive integer k such that ak ak−1 . . . a1 = 0 (resp. a1 . . . ak−1 ak = 0). An ideal I is called T -nilpotent if it is right and left T -nilpotent. From the decomposition (1) it is easy to see that if P r(A) is T -nilpotent, then for any idempotent e2 = e ∈ A the prime radical P r(eAe) of eAe is also T -nilpotent. The following theorem, which may be considered as some generalization of Nakayama’s Lemma for arbitrary right modules, is proved in [7] and [8]. Theorem 4.3. For any right ideal I in a ring A the following conditions are equivalent: (1) I is left T -nilpotent; (2) M I 6= M for any nonzero right A-module M . In an equivalent form, this theorem may be formulated as follows: Lemma 4.4. Let I be a right ideal in a ring A, which is left T -nilpotent. Then for any submodule N of a right A-module M the equality N +M I = M implies N = M . Theorem 4.5. [7, p. 292] The following conditions are equivalent for a ring A with T -nilpotent prime radical P r(A): (1) A is indecomposable; (2) the quotient ring A/P r2 (A) is indecomposable.

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Theorem 4.6. [7, p. 293] The prime quiver of a semiperfect ring A with T -nilpotent prime radical P r(A) is connected if and only if A is an indecomposable ring. Let A be an indecomposable serial ring with Noetherian diagonal and T nilpotent prime radical. Then by Theorem 2.1 and Theorem 4.6 it follows that the prime quiver of A is a chain or a cycle. In the case when the prime quiver is a chain the following theorem holds: Theorem 4.7. Let A be a serial ring with Noetherian diagonal and T nilpotent prime radical, the prime quiver of which is a chain. Then (P r(A))t = 0, where t is a number of vertices of PQ(A). Proof. Let A be an indecomposable serial ring with Noetherian diagonal and T -nilpotent prime radical, the prime quiver of which P Q(A) is a chain of the form ½ ¾ 1 2 t−1 t . • −→ • −→ . . . −→ • −→ • We can write the corresponding two-sided Peirce decomposition of the prime radical I = P r(A) of A:   I1 A12 . . . A1t A21 I2 . . . A2t    I = P r(A) =  .. .. ..  . .  . . . .  At1 At2 . . . It P and I = I 2 + t−1 i=1 Ai,i+1 . Let G be the set of matrices of the form:   0 a12 0 . . . 0 0 0 a23 . . . 0     .. .. ..  , .. g =  ... . . . .    0 0 0 . . . at−1,t  0 0 0 ... 0 where ai,i+1 ∈ Ai,i+1 , i = 1, 2, . . . , t − 1. Then I = GA + I 2 . Since I is T -nilpotent, Lemma 4.4 implies that I = GA, i.e., X P r(A) = GA = { gi ai }, gi ∈G, ai ∈A

from which we obtain that: A12 A21 = I1 , A12 A23 = A13 , . . . A12 A2t = A1t ; A23 A31 = A21 , A23 A32 = I2 , . . . , A23 A3t = A2t ; ............... At−1,t At1 = At−1,1 , At−1,t At2 = At−1,2 , At−1,t At3 = At−1,3 , . . . At1 = 0, At2 = 0, At3 = 0, . . . , It = 0.

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From these equalities we have that Akr = 0 for k > r and I1 = 0, . . . , It = 0. Obviously, (P r(A))t = 0. Moreover, taking into account that in this case the quotient ring A = A/I is isomorphic to the direct product of rings A11 , . . . , Att , we obtain that these rings are isomorphic to either division rings or rings of the form Hs (O). ¤ Remark. A ring A satisfying the conditions of Theorem 4.7 decomposes, as an abelian group, into a direct sum of the form A = A0 ⊕ P r(A), and A0 = A11 × A22 × · · · × Att , where Aii = fi Afi , i = 1, 2, . . . , t. 5. Piecewise Noetherian rings In this section we consider some generalization of (right, left) Noetherian rings, which is connected to the following theorem: Theorem 5.1. [7, Theorem 3.6.1]. Let A be an arbitrary ring with an idempotent e2 = e ∈ A. Suppose that f = 1 − e, eAf = X, f Ae = Y , and µ ¶ eAe X A = Y f Af is the corresponding two-sided Peirce decomposition of A. Then the ring A is right Noetherian (Artinian) if and only if the rings eAe and f Af are right Noetherian (Artinian), X is a finitely generated f Af -module, and Y is a finitely generated eAe-module. Note that in any right Noetherian ring A there exists a finite number of pairwise orthogonal primitive idempotents e1 , e2 , . . . , en such that 1 ∈ A is represented in the form: 1 = e1 + e2 + · · · + en . By Theorem 5.1 eAe is a right Noetherian ring for any primitive idempotent e ∈ A. Definition 5.2. A ring A is called a right piecewise Noetherian ring if it satisfies the following conditions: i) A has a finite set of pairwise orthogonal primitive idempotents e1 , e2 , . . . , en such that 1 = e1 + e2 + · · · + en ; ii) ei Aei is a right Noetherian ring for any primitive idempotent ei , i = 1, . . . , n. Analogously we can introduce left piecewise Noetherian rings. A ring is called a piecewise Noetherian ring if it is both right and left piecewise Noetherian. From above we get the following statement. Proposition 5.3. A right Noetherian ring is right piecewise Noetherian. Note that any serial ring is semiperfect, so the condition (i) always holds for such rings.

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Proposition 5.4. A serial right piecewise Noetherian ring is left piecewise Noetherian. Proof. The proof follows from [7, Proposition 9.4.1], which states that local serial right Noetherian rings are either discrete valuation rings or Artinian uniserial rings, and in both cases they are two-sided Noetherian. ¤ Lemma 5.5. Let e be an idempotent of a ring A. For any nilpotent ideal I ˜ = I. of eAe there exists a nilpotent ideal I˜ of A such that eIe Proof. Let e1 = e, e2 = 1 − e, Aij = ei Aej , (i, j = 1, 2). We set I˜ = I + IA12 + A21 I + A21 IA12 . Obviously, I˜ is a nilpotent two-sided ideal of ˜ = I. A and eIe ¤ Proposition 5.6. [7, Proposition 7.3.9]. Let P r(A) be the prime radical of a ring A, e2 = e ∈ A and e 6= 0. Then eP r(A)e coincides with the prime radical of eAe. Note that from the remark after Theorem 4.7 it follows that a serial ring with Noetherian diagonal and T -nilpotent prime radical, whose prime quiver is a chain, is always a serial piecewise Noetherian ring. Theorem 5.7. A serial piecewise Noetherian semiprime ring is Noetherian. Proof. By [14, Corollary 3.5] a serial semiprime ring is a finite direct product of prime rings. Therefore it is sufficient to prove that a serial prime piecewise Noetherian ring A is Noetherian. Let 1 = e1 + · · · + es be a decomposition of 1 ∈ A into a sum of pairwise orthogonal local idempotents. Assume that there exist idempotents ei and ej such that ei Aei = Aii is a discrete valuation ring, and ej Aej = Ajj is a division ring. Then the ring ¶ µ Aii Aij Λ= Aji Ajj µ ¶ 0 Aij is not prime, because it contains the two-sided nilpotent ideal . Aji 0 By Lemma 5.5 the ring A is not prime as well. If for all idempotents e1 , . . . , es the rings e1 Ae1 , . . . , es Aes are division rings, then the ring A is Artinian and therefore it is Noetherian. So we can suppose that all rings e1 Ae1 , . . . , es Aes are discrete valuation rings. By Theorem 5.1, if A is not Noetherian there exists an index j such that Aij is an infinitely generated right Ajj -module. Therefore Aij Rj = Aij , where Rj is the Jacobson radical of Ajj . Since the ring A is prime then Aji Aij 6= 0, therefore Aji Aij = Rjm . The equality Rjm = Aji Aij = Aji Aij Rj = Rjm+1 gives a contradiction. The theorem is proved. ¤ Theorem 5.8. If A is a serial piecewise Noetherian ring, then (1) The prime radical P r(A) is nilpotent;

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(2) The diagonal of A is Noetherian. Proof. By the hypothesis, eAe is a uniserial Noetherian ring for any local idempotent eAe. Then eAe is either a discrete valuation ring or an Artinian uniserial ring. In the first case the prime radical P r(eAe) = eP r(A)e = 0. In the second case the prime radical P r(eAe) = eP r(A)e is a nilpotent ideal of eAe. Therefore by [12, Theorem 1.3] the prime radical P r(A) is nilpotent. Obviously, a factor ring of a serial piecewise Noetherian ring is also serial and piecewise Noetherian. Since the diagonal of any ring is a semiprime ring, our statement follows from Theorem 5.7. ¤ In conclusion of this section we note some theorems connected with the notion of a right (left) perfect ring. Perfect rings are a generalization of Artinian rings and were introduced by H. Bass in [1]. Recall that a ring A is called right (resp. left) perfect if any right (left) A-module has a projective cover. A right and left perfect ring is called perfect. It is well known that any right (left) perfect ring is semiperfect. We shall use the important theorem proved by H. Bass in [1], which may be formulated in the following form (see also [6, Theorem 22.26]): Theorem 5.9. A semiperfect ring A is right perfect if and only if the Jacobson radical R of A is left T -nilpotent. From the definition of the right quiver for semiperfect rings it follows that the right A-module R/R2 is finitely generated. So taking into account [23, Lemma 11] we have the following corollary: Corollary 5.10. If A is left perfect and the right quiver Q(A) is defined, then A is right Artinian. Remark. Let A be a semiperfect and semi-distributive ring, then the quiver Q(A) is always defined, and if A is right (left) perfect, it is left (right) Artinian. Theorem 5.11. A right perfect serial ring is two-sided Artinian. Proof. Let A be a right perfect serial ring with the Jacobson radical R. We can suppose that A is a reduced ring. Since any serial ring is semiperfect, by Theorem 5.9 R is left T -nilpotent and therefore R is a nil-ideal. If R = 0, the proposition is trivial. Suppose R 6= 0. Due to Theorem 4.3, we have the proper inclusion R2 ⊂ R. By [24, Lemma 2.1] A/R2 is an Artinian ¯ = R/R2 6= 0. Therefore, by [10, (serial) ring with the Jacobson radical R ¯ such that R ¯ = p¯A¯ = A¯ ¯p. Theorem 2.8], there exists a nonzero element p¯ ∈ R Denote by p an element whose image under the natural epimorphism A to A¯ is p¯. Then we have the equalities pA + R2 = R2 + Ap = R. Hence, by Lemma 4.4, pA = R = Ap. Therefore R2 = pApA = p2 A and by induction, Rn = pn A for any integer n > 0. Since R is left T -nilpotent, it is a nilideal and thus there exists an integer m > 0 such that pm = 0. Therefore

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Rm = pm A = 0, i.e. R is nilpotent, whence, by [24, Lemma 2.1], A is Artinian. The proposition is proved. ¤ Theorem 5.12. A serial ring A with Artinian diagonal and T -nilpotent prime radical is two-sided Artinian. Proof. Note that the prime radical of rings with Artinian diagonal coincides with the Jacobson radical. Hence the rings considered in the theorem are right and left perfect. The proof of the theorem follows now from Theorem 5.11. ¤ References [1] H. Bass, Finitistic dimension and homological generalization of semiprimary rings. Trans. Amer. Math. Soc. 1960, v. 95, pp. 466–488. [2] Yu. A. Drozd , Minors and reduction theorems. Coll. Math. Soc. J. Bolyai, 1971, v. 6, pp. 173–176. [3] Yu. A. Drozd, On generalized uniserial rings, Mat. Zametki, 18, 1975, N 5, pp. 707– 710. [4] D. Eisenbud and P. Griffith, Serial rings, J.Algebra, 1971, 17, N 3, pp. 268–287. [5] D. Eisenbud and P. Griffith, The structure of serial rings, Pacif. J. Math., 1971, 36, pp. 109–121. [6] C. Faith, Algebra: Rings, Modules and Categories II - Springer-Verlag, BerlinHeidelberg-New York, 1976. [7] M. Hazewinkel, N. Gubareni, and V.V. Kirichenko, Algebras, Rings and Modules, Kluwer Acad. Publishers, 2004. [8] F. Kasch, Modules and Rings. Academic Press, New York, 1982. [9] V.V. Kirichenko, Generalized uniserial rings. Preprint IM-75-1, Kiev, 1975. [10] V.V. Kirichenko, Generalized uniserial rings. Math. Sb., vol. 99 (141), N◦ 4, 1976, p. 559–581 (English translation in: Math. USSR Sbornik, vol. 28, N◦ 4, 1976, pp. 501– 520). [11] V.V. Kirichenko, Right Noetherian rings over which all finitely generated modules are semi-chain modules. Dokl. Akad. Nauk Ukrain. SSR Ser. A, 1976, N◦ 1, pp. 9–12. [12] V.V. Kirichenko, S. Valio, and Yu. V. Yaremenko, Semi-perfect rings and their quivers. In: Infinite Groups and Related Algebraic Topics. Institute of Mathemetics NAS Ukraine, 1993, pp. 438–456. [13] V.V. Kirichenko, Hereditary and semi-hereditary serial rings. Zap. Nauch. Sem. LOMI AN USSR, vol. 114, 1982, pp. 137–147. [14] V.V. Kirichenko and M.A. Khibina, Semi-perfect semi-distributive rings. In: Infinite Groups and Related Algebraic Topics. Institute of Mathemetics NAS Ukraine, 1993, pp. 457–480. [15] H. Kupisch, Beitr¨ age zur Theorie nichthalbeinfacher Ringe mit Minimalbedingung. Grelles Journal , v. 201, 1959, pp. 100–112. ¨ [16] H. Kupisch, Uber eine Klasse von Ringen mit Minimalbedingung, Arch. Math., 1966, 17, pp.20–35. [17] H. Kupisch, Einreihige Algebren u ¨ber einem perfekten K¨ orper, J. Algebra, 1975, 33, N1, pp. 68–74. [18] I. Murase, On the structure of generalized uniserial rings, I, Sci. Papers College Gen. Ed., Univ. Tokyo, 1963, 13, pp. 1–22. [19] I. Murase, On the structure of generalized uniserial rings, II, Sci. Papers College Gen. Ed., Univ. Tokyo, 1963, 13, pp. 131–158.

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[20] I. Murase, On the structure of generalized uniserial rings, III, Sci. Papers College Gen. Ed., Univ. Tokyo, 1964, 14, pp. 11–25. [21] T. Nakayama, Note on uniserial and generalized uniserial rings. Proc. Imp. Acad. Tokyo, v. 16 (1940), pp. 285–289. [22] T. Nakayama, On Frobeniusean Algebras. II. Ann. Math., v. 42, N◦ 1 (1941), pp. 1–21. [23] B.L. Osofsky, A generalization of quasi-Frobenius rings. J. Algebra, v. 4 (1966), pp. 373–387. [24] S. Singh, Serial right Noetherian rings. Can. J. Math. v. 36 (1984), pp. 22–37. [25] A. Tuganbaev, Distributive Modules and Related Topics, Gordon and Breach Science Publishers, 1999. [26] R.B. Warfield, Serial rings and finitely presented modules. J. Algebra, v. 37 (1975), pp. 187–222.

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The structure of the baric algebras Henrique Guzzo, Jr. Instituto de Matem´atica e Estat´ıstica, Universidade de S˜ao Paulo, Caixa Postal 66281, S˜ao Paulo, SP, 05315-970, Brasil [email protected] Abstract. In this work we present some results on the structure of baric algebras. Among other topics, we define simple and semi-simple baric algebras and the bar-radical. Furthermore, we determine the barradical for Bernstein, Jordan, and alternative algebras. We define a nongenerator element and show that the bar-radical is the set of all nongenerator elements. Besides, we show that for some baric algebras, the zero socle and the socle coincide.

1. Baric algebras Baric algebras play a central role in the theory of genetic algebras. They were introduced by I.M.H. Etherington [6], aiming for an algebraic treatment of population genetics. But the whole class of baric algebras is too large, some conditions (usually with a background in genetics) must be imposed in order to obtain a workable mathematical object. With this in mind, several classes of baric algebras have been defined: train, Bernstein, special triangular, etc. As a sample of the work in the field of genetic algebra, see [14] and [15]. In this section we will give the basic definitions on the theory of the baric algebras, some of them appear in [1], [2], [9], [14] and [15]. Let F be a field, A an algebra over F , not necessarily associative, commutative or finite dimensional and ω : A → F be a nonzero homomorphism. The ordered pair (A, ω) will be called a baric algebra over F and ω its weight function. If B ⊆ A, we denote by bar(B) the set of all x ∈ B such that ω(x) = 0, that is, B ∩ ker ω. If I ⊆ bar(A) is a two-sided ideal of A, then I is called a b-ideal of A. If e ∈ A is such that e2 = e and ω(e) = 1 then e is called idempotent of weight 1. A b-homomorphism from (A, ω) to (A0 , ω 0 ) is a homomorphism of F -algebras ϕ : A → A0 such that ω 0 ◦ ϕ = ω. We will denote (A, ω) ∼ =b (A0 , ω 0 ) or A ∼ =b A0 , when there exists a b-isomorphism 0 0 from (A, ω) to (A , ω ). 233

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Let (A, ω) be a baric algebra and B be a subalgebra of A. When B 6⊆ bar(A), then B is called b-subalgebra of (A, ω) and we will denote B ≤ A or B < A when B 6= A. In this case (B, ω 0 ) is a baric algebra, where ω 0 = ω|B : B → F . If B is a b-subalgebra of (A, ω) and bar(B) is a two-sided ideal of bar(A), then B is called a normal b-subalgebra of (A, ω) and we will denote B E A or B C A if B 6= A. 2. The Krull-Schmidt theorem In [1] and [2], the concept of indecomposable baric algebras was introduced and a Krull-Schmidt theorem for algebras satisfying ascending and descending chain conditions was proved. Suppose (A, ω) is a baric algebra with idempotent e of weight 1, so A = F e ⊕ N where N = ker ω. The additive group (N, +) can be endowed with a structure of abelian M -group, see [13, Chap. V, Def. 1]. The set M is formed by all right and left multiplications Ra and La , where a belongs to A∪F and ψ : N × M → N defined by ψ(n, T ) = T (n). In this case, the M -subgroups of (N, +) are the two-sided ideals of the algebra A, contained in N . These are exactly the two-sided ideals of N that are invariant under Le and Re (in short, invariant). The M -endomorphisms of (N, +) are the mappings ϕ satisfying ϕ(n + n0 ) = ϕ(n) + ϕ(n0 ), ϕ(αn) = αϕ(n), ϕ(na) = ϕ(n)a, and ϕ(an) = aϕ(n) where n, n0 ∈ N, α ∈ F and a ∈ A. This set of conditions is equivalent to ϕ being F -linear, ϕ(nn0 ) = ϕ(n)n0 = nϕ(n0 ), and ϕ(en) = eϕ(n), ϕ(ne) = ϕ(n)e. According to [13, Chap. V, § 12] an abelian M -group N is decomposable if there are two nontrivial M -subgroups N1 and N2 of N such that N = N1 ⊕N2 . In our context, this concept is translated to the following definition: Definition 1. A baric algebra (A, ω) with an idempotent of weight 1 is decomposable if there are nontrivial two-sided ideals N1 and N2 of A, both contained in N = kerω, such that N = N1 ⊕ N2 . Otherwise, it is indecomposable. Clearly all two-dimensional algebras are indecomposable. Suppose (A1 , ω1 ) and (A2 , ω2 ) baric algebras with idempotents e1 and e2 respectively, so A1 = F e1 ⊕ N1 and A2 = F e2 ⊕ N2 . In the vector space F ⊕ N1 ⊕ N2 , we define the multiplication: (α, n1 , n2 )(α0 , n01 , n02 ) = (αα0 , n1 n01 + αe1 n01 + α0 n1 e1 , n2 n02 + αe2 n02 + α0 n2 e2 ). Clearly N1 and N2 are two-sided ideals of F ⊕ N1 ⊕ N2 . If A1 and A2 are at least two-dimensional, F ⊕ N1 ⊕ N2 is decomposable. This algebra is called the join of (A1 , ω1 ) and (A2 , ω2 ) and is denoted by (A1 ∨ A2 , ω1 ∨ ω2 ) or simply by A1 ∨ A2 , where (ω1 ∨ ω2 )(α, n1 , n2 ) = α. The idempotent (1, 0, 0) is the join of e1 and e2 and is denoted by e1 ∨ e2 .

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It is not difficult to prove that this construction is independent of the idempotents e1 and e2 . Since ((A1 , ω1 ) ∨ (A2 , ω2 )) ∨ (A3 , ω3 ) ∼ =b (A1 , ω1 ) ∨ ((A2 , ω2 ) ∨ (A3 , ω3 )) allows us to define recursively the join (A1 , ω1 ) ∨ · · · ∨ (An , ωn ). Definition 2. A baric algebra (A, ω), with N = kerω, satisfies d.c.c.(a.c.c.) if the M -group (N, +) satisfies d.c.c. (a.c.c.). Theorem 1. (Krull-Schmidt) Suppose (A, ω) satisfies both d.c.c. and a.c.c. and let (A1 , ω1 ), . . . , (An , ωn ), (B1 , γ1 ), . . . , (Bm , γm ) be indecomposable baric algebras such that (A, ω) ∼ =b (A1 ∨ · · · ∨ An , ω1 ∨ · · · ∨ ωn ) (A, ω) ∼ =b (B1 ∨ · · · ∨ Bm , γ1 ∨ · · · ∨ γm ). Then n = m and for some permutation i 7→ i0 of indices, we have (Ai , ωi ) ∼ =b 0 0 (Bi , γi ) for all i = 1, . . . , n. Example 1. The gametic algebras for polyploidy have been studied by several authors, see [8] and [15], for instance. The gametic algebras for simple Mendelian inheritance with n + 1 alleles, denoted G(n + 1, 2), are clearly decomposable for n ≥ 2: G(n + 1, 2) = G(2, 2) ∨ . . . ∨ G(2, 2) (n summands). The general case, including polyploidy, is described by G(n + 1, 2m), of which we list now some properties. It has a canonical basis (in Gonshor’s sense) consisting of all formal monomials X0i0 X1i1 . . . Xnin with i0 + i1 + · · · + in = m. The multiplication, clearly commutative, is given by: (X0i0 X1i1 . . . Xnin )(X0j0 X1j1 . . . Xnjn ) = ½ ¡2m¢−1 ¡i +j ¢ i +j −m i +j 0 0 0 0 X11 1 . . . Xnin +jn m m X0 = 0

if i0 + j0 ≥ m . if i0 + j0 < m

The weight function ω is 1 on X0m and 0 elsewhere in the basis. The monomial X0m is an idempotent of weight 1. In [1] we show that for every n ≥ 1 and m ≥ 2, G(n + 1, 2m) is indecomposable. 3. Normal and composition series for baric algebras As an application of the theory of finite length modules we introduce in [9] the concepts of the noetherian, artinian, and composition series for baric algebras. From [9], a baric algebra (A, ω) is defined to be b-Artinian (b-Noetherian) if for every sequence A1 , A2 , . . . , of normal b-subalgebras of A with A1 ⊇ A2 ⊇ . . . (A1 ⊆ A2 ⊆ . . . ), there exists k ≥ 1 such that Ak = At , for all t ≥ k. If (A, ω) is a baric algebra with an idempotent of weight 1, then (A, ω) is b-Artinian (b-Noetherian) if and only if (A, ω) satisfies d.c.c (a.c.c.).

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Definition 3. A normal series of a baric algebra (A, ω) is a descending chain A1 ⊇ A2 ⊇ . . . ⊇ An+1 of normal baric subalgebras of A, such that A = A1 and bar(An+1 ) = 0. Proposition 1. Let (A, ω) be a baric algebra and A1 , A2 , . . . , An+1 be normal baric subalgebras of A. Then the following conditions are equivalent: (1) A1 ⊇ A2 ⊇ . . . ⊇ An+1 is a normal series of A; (2) There exists e ∈ A idempotent of weight 1, such that e ∈ Aj for all j = 1, . . . , n + 1 and bar(A) = bar(A1 ) ⊇ bar(A2 ) ⊇ . . . ⊇ bar(An+1 ) = 0; (3) There exists e ∈ A idempotent of weight 1, such that A = A1 ⊇ A2 ⊇ . . . ⊇ An+1 = F e. Definition 4. Let (A, ω) be a baric algebra. The normal series A = A1 ⊇ A2 ⊇ . . . ⊇ An+1 = F e1 and A = B1 ⊇ B2 ⊇ . . . ⊇ Bm+1 = F e2 are equivalent when n = m, e1 = e2 and there exists a permutation i 7→ i0 such that Ai /bar(Ai+1 ) ∼ =b Bi0 /bar(Bi0 +1 ). Definition 5. Let A = A1 ⊇ A2 ⊇ . . . ⊇ An+1 = F e

(1)

A = B1 ⊇ B2 ⊇ . . . ⊇ Bm+1 = F e

(2)

be two normal series of a baric algebra (A, ω). The normal series (2) is a refinement of (1) if the sequence of Ai is a subsequence of the Bj . Theorem 2. (Schreier) Let A = A1 ⊇ A2 ⊇ . . . ⊇ An+1 = F e

(1)

A = B1 ⊇ B2 ⊇ . . . ⊇ Bm+1 = F e

(2)

be the normal series of a baric algebra (A, ω). Then there exists a refinement of (1) and a refinement of (2), which are equivalent. Definition 6. (1) A series A = A1 ⊇ A2 ⊇ . . . ⊇ An+1 = F e of a baric algebra (A, ω) is a proper series if Ai 6= Ai+1 . In this case we will denote A = A1 ⊃ A2 ⊃ . . . ⊃ An+1 = F e; (2) A normal proper series of a baric algebra is a composition series if it has no proper refinement. Theorem 3. (Jordan-H¨ older) Any two composition series for a baric algebra with the same idempotent are equivalent. Theorem 4. Let (A, ω) be a baric algebra with an idempotent e ∈ A of weight 1 and A 6= F e. Then A has a composition series if and only if A is Artinian and Noetherian.

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Example 2. Let G = (V (G), E(G)) be a finite graph, where V (G) is the set of vertices of G and E(G) is the set of edges of G. The edges α will be denoted by α = ab = ba, where a and b are the vertices linked by α. Let F be a field of characteristic not 2 and N (G) be the vector space that is the direct sum of U and Z, where U is the F -vector space freely generated by V (G) and Z is the F -vector space freely generated by E(G). We introduce in N (G) the following commutative multiplication (on the basis of U and Z): a ab = b; b ab = a; other products are zero. Let τ : N (G) → N (G) be the linear operator given by τ (u) = 21 u and τ (z) = 0, where u ∈ U and z ∈ Z. Then by [3], A(G) = F ⊕ N (G) with the following multiplication: (α, x)(α0 , x0 ) = (αα0 , xx0 + τ (αx0 + α0 x)) and weight function ω : A(G) → F given by ω(α, x) = α, is an exceptional Bernstein algebra. Moreover the Peirce decomposition of A(G) relative to the idempotent e = (1, 0) is A(G) = F e ⊕ U ⊕ Z. The graph G is connected if for all a, b ∈ V (G), there exists aa1 , a1 a2 , . . . , an b ∈ E(G). The graph G is simple if the number of edges joining any two points is at most 1. Let G be a connected and simple graph and |V (G)| > 1 or |V (G)| = |E(G)| = 1 and I be a nonzero baric ideal of A(G). Then by [3, Theorem 1], I = U ⊕ Z 0 , where Z 0 is a subspace of Z. If E(G) = {α1 , . . . , αn }, then A(G) ⊃ F e ⊕ U ⊕ hα2 , . . . , αn i ⊃ . . . ⊃ F e ⊕ U ⊃ F e is a composition series of A(G). Example 3. The gametic algebra G(2, 2) has dimension 2. Its commutative duplicate Z(2, 2) has a basis {c0 , c1 , c2 } such that: 1 1 c20 = c0 , c0 c1 = c1 , c0 c2 = c1 c2 = c22 = 0, c21 = c2 . 2 4 By [1], the nonzero baric ideals of Z(2, 2) are F c2 and bar(Z(2, 2)) = F c1 + F c2 , then Z(2, 2) ⊃ F c0 ⊕ F c2 ⊃ F c0 , is a composition series of Z(2, 2). Take now the commutative duplicate of Z(2, 2), which we will denote by D(Z(2, 2)), called the copular algebra. This algebra has a basis {c0 , . . . , c5 } with the following commutative multiplication: ½ 2 1 c0 = c0 , c0 c1 = 21 c1 , c0 c2 = 41 c3 , c21 = 41 c2 , c1 c2 = 81 c4 , c22 = 16 c5 . other products are zero. So, A ⊃ F c0 ⊕hc2 , . . . , c5 i ⊃ F c0 ⊕hc3 , c4 , c5 i ⊃ F c0 ⊕hc4 , c5 i ⊃ F c0 ⊕hc5 i ⊃ F c0 , is a composition series of D(Z(2, 2)).

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4. The bar-radical We have introduced in [10] the concept of the bar-radical for baric algebras. Definition 7. Let (A, ω) be a baric algebra and B ⊆ A. (1) B is a maximal (maximal normal) b-subalgebra of (A, ω) if B < A (B C A) and there is no C ≤ A (C E A), such that B ⊂ C ⊂ A. (2) B is a maximal b-ideal of (A, ω), if B is a b-ideal of (A, ω), B 6= bar(A) and there does not exist a b-ideal J of (A, ω), such that B ⊂ J ⊂ bar(A). In [10] we define b-simple baric algebra in the following way. A baric algebra (A, ω) is b-simple if for all normal b-subalgebras B of (A, ω) either bar(B) = 0 or bar(B) = bar(A). Definition 8. The bar-radical of (A, ω) is zero by definition if (A, ω) is bsimple, otherwise the bar-radical of (A, ω) is defined to be the intersection of all bar(B), where B is a maximal normal b-subalgebra of (A, ω). We will denote by rad(A) the bar-radical of a baric algebra (A, ω). By Theorem 4, we have ¡ ¢ Theorem 5. Let (A, ω) be a baric algebra. If A/bar A2 is Artinian and Noetherian, then the following conditions hold: ¡ 2¢ (1) A/bar ¡ 2 ¢A is finite dimensional; (2) rad A ⊆ rad(A) ⊆ A2 . Theorem 6. Let (A1 , ω1 ), . . . , (An , ωn ) be baric algebras having an idempotent of weight 1. Then rad(A1 ∨ . . . ∨ An ) = rad(A1 ) ⊕ · · · ⊕ rad(An ). Definition 9. Let (A, ω) be a baric algebra with an idempotent of weight 1. If there exist b-simple baric algebras (A1 , ω1 ), . . . , (An , ωn ) each of them having an idempotent of weight 1 such that (A, ω) ∼ =b (A1 ∨ . . . ∨ An , ω1 ∨ . . . ∨ ωn ), then (A, ω) is called b-semisimple. By Theorem 6 we have that if (A, ω) is a b-semisimple baric algebra, then rad(A) = 0. Theorem 7. Let (A, ω) be a b-Artinian baric algebra with an idempotent of weight 1. If rad(A) = 0, then (A, ω) is b-semisimple. As a consequence of this theorem, we have that if (A, ω) is an Artinian baric algebra with idempotent of weight 1 such that rad(A) = 0 and (A, ω) is indecomposable, then (A, ω) is b-simple. Example 4. If G is a graph, then rad(A(G)) = U .

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Example 5. The gametic algebra that describes k independent loci, each with two alleles, is the tensor product of k copies of G(2, 2), see [15, Chapter 6, Section B]. If {c0 , c1 } is a canonical basis for G(2, 2), so that c20 = c0 , c0 c1 = 21 c1 , c21 = 0, then {ci1 ⊗ . . . ⊗ cik : 0 ≤ ij ≤ 1, j = 1, . . . , k} is a canonical basis for Nk G(2, 2). A monomial ci1 ⊗ . . . ⊗ cik will be denoted by cσ , where σ = (i1 , . . . , ik ) ∈ {0, 1}k . The product cσ cσ0 is zero when σ and σ 0 have at least one 1 in the same position. In the other cases, cσ cσ0 = 21i cσ00 , where i is the number of 1’s in σ plus the number of 1’s in σ 0 , and σ 00 is obtained from σ and σ 0 by choosing the maximum element in each one of the k positions. N Denote by Vi the subspace of k G(2, 2) generated by all cσ where σ has exactly i 1’s. Then ¡ ¢ Nk G(2, 2) = V0 ⊕ V1 ⊕ . . . ⊕ Vk and dim Vi = dim Vk−i = ki . Moreover, V0 is generated by c0 ⊗. . .⊗c0 , and Vk by c1 ⊗. . .⊗c1 . The weight function is defined ´by ω(c0 ⊗ . . . ⊗ c0 ) = 1 and ω(cσ ) = 0 if cσ 6= c0 ⊗ . . . ⊗ c0 , ³N k so bar G(2, 2) = V1 ⊕ . . . ⊕ Vk . ³N ´ k In [10], we show that rad G(2, 2) = V2 ⊕ . . . ⊕ Vk . 5. Nongenerator element Let (A, ω) be a baric algebra. If S ⊆ A, then we denote (see [11]): (1) ∆ (S) = {B | B is a subalgebra of A (not necessarily baric), S ⊆ B and bar(B) is an ideal of A}; (2) A(S) = ∩B∈∆(S) B. Definition 10. Let (A, ω) be a baric algebra. The element x ∈ A is a nongenerator if for all S ⊆ A such that A(S ∪ {x}) = A one has A(S) = A. Theorem 8. Let (A, ω) be a b-Noetherian baric algebra. Then x ∈ rad(A) if and only if x is a nongenerator element. Let A be an algebra. Soc (A) is called the socle of A and is defined to be the sum of all minimal ideals of A. Zsoc (A) is called the zero socle of A and is defined to be the sum of all minimal ideals of A with zero product. Of course Zsoc (A) ⊆ Soc (A). Proposition 2. Let (A, ω) be a baric algebra, then Zsoc (A) ⊆ bar(A). Proposition 3. Let (A, ω) be a baric algebra, I be a minimal ideal of A and e ∈ A be an idempotent of weight 1. If there exists b ∈ A such that eb 6= 0 and for all a ∈ bar(A) there exists k ≥ 1 such that ak = 0 or k a = 0 or a[k] = 0, then I is a baric ideal of A.

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Theorem 9. Let (A, ω) be a baric algebra and e ∈ A be an idempotent of weight 1. Suppose that: (1) There exists b ∈ A such that eb 6= 0; (2) bar(A) is solvable; (3) If I is a baric ideal of A, then I 2 is a baric ideal of A. Then Zsoc (A) = Soc (A). 6. Some results about the bar-radical An alternative algebra A over F is an algebra in which x2 y = x(xy),

yx2 = (yx)x

for all x, y ∈ A. The left and right equations are known as, respectively, the left and right alternative laws. They are equivalent, in terms of associators (x, y, z) = (xy)z − x(yz), to (x, x, y) = (y, x, x) = 0

for all x, y ∈ A.

In [1] and [2], we show that the following hold. Theorem 10. If (A, ω) is a finite dimensional associative baric algebra, then (A, ω) contains an idempotent of weight 1. If (A, ω) is an alternative baric algebra, then for all x ∈ A the subalgebra of A generated by x is associative, so we have the following corollary. Corollary 1. If (A, ω) is a finite dimensional alternative baric algebra, then (A, ω) contains an idempotent of weight 1. Theorem 11. If (A, ω) is an alternative baric algebra, then rad(A) = R(A) ∩ (bar(A))2 , where R(A) is the nil radical of A (the maximal nil ideal of A). Let a be any element of an algebra A over F. The right multiplication Ra and the left multiplication La of A determined by a are defined, respectively, by Ra : x → xa, La : x → ax for all x ∈ A, which are linear operators on A. The set R(A) (and L(A)) of all right (and left) multiplications of A is a subspace of the associative algebra of all linear operators on A. We denote by M(A), the enveloping algebra of R(A)∪L(A); that is, the associative subalgebra of the associative algebra of all linear operators on A generated by right and left multiplications of A. If (A, ω) is a baric algebra, then M(A) is a baric algebra too. Theorem 12. If (A, ω) is a baric algebra such that A2 = A and its multiplication algebra M(A) is b-semisimple, then (A, ω) is b-semisimple.

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An nth -order Bernstein algebra is a commutative baric algebra (A, ω) n satisfying: x[n+2] = (ω(x))2 x[n+1] , for all x ∈ A and n is the smallest positive integer with such property, where x[1] = x, x[r+1] = x[r] x[r] , for r ≥ 1. If n = 1 then (A, ω) is called Bernstein algebra. An algebra A is power-associative if xi xj = xi+j for every i, j ≥ 1 and x ∈ A. A commutative algebra A is a Jordan algebra if (x2 y)x = x2 (yx) for all x, y ∈ A. In [12], we prove the following results. Theorem 13. If (A, ω) is a Bernstein algebra or a Jordan and nth -order Bernstein algebra, then rad(A) = (bar(A))2 . Theorem 14. If (A, ω) is a power-associative and nth -order Bernstein algebra, then rad(A) ⊆ (bar(A))2 . In [7], we show that Theorem 15. Let (A, ω) be a finite dimensional Jordan baric algebra. Then (1) rad(A) ⊆ R(A); (2) If rad(A) ⊆ (bar(A))3 , then rad(A) = R(A). References [1] R. Costa and H. Guzzo, Jr., Indecomposable baric algebras, Linear Algebra and its Appl. 183 (1993), 223–236. [2] R. Costa and H. Guzzo, Jr., Indecomposable baric algebras, II, Linear Algebra Appl. 196 (1994), 233–242. [3] R. Costa and H. Guzzo, Jr., A class of exceptional Bernstein algebras associated to graphs. Comm. Algebra 25 (1997), no. 7, 2129–2139. [4] M.A. Couto and H. Guzzo, Jr., The radical in alternative baric algebras. Arch. Math. (Basel) 75 (2000), no. 3, 178–187. [5] M.A. Couto and H. Guzzo, Jr., The multiplication algebra of a b-semisimple baric algebra. Comm. Algebra 29 (2001), no. 4, 1729–1740. [6] I.M.H. Etherington, Genetic algebras, Proc. Roy. Soc. Edinb. 59 (1939), 242–258. [7] J.C.M. Ferreira, and H. Guzzo, Jr., The bar-radical of Jordan baric algebras, Algebras Groups and Geometries 21, 387–398 (2004). [8] H.Gonshor, Special train algebras arising in Genetics, Proc. Edinb. Math. Soc. 2 (12), 41–53 (1960). [9] H. Guzzo, Jr., On normal and composition series for baric algebras, Nova Journal of Mathematics, Game Theory, and Algebra 4 (1), 25–38 (1995). [10] H. Guzzo, Jr., The bar-radical of baric algebras. Arch. Math. (Basel) 67 (1996), no. 2, 106–118. [11] H. Guzzo, Jr., Some properties of bar-radical of baric algebras. Comm. Algebra, 30 (2002), no. 10, 4827–4835. [12] H. Guzzo, Jr. and P. Vicente, The bar-radical in power-associative nth-order Bernstein algebras. Arch. Math. (Basel), 79 (2002), 27–33. [13] N. Jacobson, Lectures in Abstract Algebra, vol.I, Van Nostrand, 1966. [14] Yu. I Lyubich, Mathematical Structures in Population Genetics (Biomathematics 22, Springer, Berlin-Heidelberg-New York 1992).

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[15] A. W¨ orz, Algebras in Genetics. Lecture Notes in Biomathematics 36, Springer, BerlinHeidelberg-New York (1980).

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On torsion units of integral group rings of groups of small order Christian H¨ ofert and Wolfgang Kimmerle IGT, Fachbereich Mathematik, Universit¨at Stuttgart, D - 70550 Stuttgart, Germany [email protected] [email protected]

1. Introduction Let G be a finite group. Denote its integral group ring by ZG and let V (ZG) be the subgroup consisting of units with augmentation 1. H. Zassenhaus stated with respect to torsion subgroups of units of V (ZG) three conjectures [28]. ZC–1 Let u be a unit of finite order of V (ZG). Then u is conjugate within QG to an element of G.1 ZC–2 Let H be a subgroup of V (ZG) with the same order as G. Then H is conjugate within QG to G. ZC–3 Let H be a finite subgroup of V (ZG). Then H is conjugate within QG to a subgroup of G. These conjectures had a big influence on the development of the structure theory of integral group rings even if it turned out that ZC–2 and ZC–3 do not hold for all finite groups, for a recent survey we refer to [17]. The socalled first Zassenhaus conjecture ZC–1 , originally stated in [34], is still wide open. We say that the conjecture ZC–i (i = 1, 2 or 3) holds for a group G, if it is valid for V (ZG). The object of this note is a sketch of the proof of the following. Proposition 1.1. ZC–1 is valid for all groups of order ≤ 71. The proof of the proposition uses most of the known results. In this sense this note presents also a survey on the first Zassenhaus conjecture. ZC–1 might hold for all finite groups even if one has to say that the evidence (see the results mentioned in section 2) is small. Thus it makes sense to 1Elements of G are called the trivial units of ZG. 243

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investigate with the known results and methods possible candidates for a counterexample. The first obvious question is to find a group of smallest order for which ZC–1 is open. We remark in this context that in the meantime one knows that for ZC–2 there exist counterexamples of order 96 and 144 [3], [10], [11, Kap.III], whereas the first discovered counterexample due to Roggenkamp and Scott was of order 14400 [26], [30]. 2. Some known results Probably the most powerful result on ZC–1 known is the following due to A. Weiss. Theorem 2.1. [33] If G is nilpotent, then ZC–3 holds for G. Therefore in order to establish Proposition 1.1 we have not to consider groups of 16, 27, 32, 64. ZC–1 is open for metabelian even for metacyclic groups. Nevertheless in many special situations positive results for ZC–1 have been established for such groups [23], [24]. These results are of course very useful for groups of small order. The most far reaching theorem with respect to metacyclic groups appears to be the following. Theorem 2.2. [31] Let G ∼ = hai o X, where hai is a cyclic group of order n and X is an arbitrary abelian group of order m with (m, n) = 1. Then ZC–3 holds for G. The next three results also may be applied to several groups of small order. Theorem 2.3. [29] Let G ∼ = A o X, where A is an elementary abelian p−group and X is an arbitrary abelian group. If the action of X is faithful and irreducible on A, then ZC–1 holds for G. Theorem 2.4. [22] Let G ∼ = AoX, where A is an arbitrary abelian p−group and X is of prime order q. If |X| = q < p for all prime divisors p of |A| and if q is also prime, then ZC–1 is true for G. Theorem 2.5. [16] Let G be a Frobenius-group of order pm q n , with p and q prime. Then ZC–1 holds for G. We record two quick consequences of these theorems. Proposition 2.6. If the order of G is not divisible by a square of any prime, then ZC–3 holds for G. Also the next proposition is an immediate consequence of the above theorems. Proposition 2.7. If G is a group of order p2 q, where p and q are prime numbers, then ZC–1 holds for G.

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Recently M. Hertweck obtained the following result. It is not used in the proof of Proposition 1.1. It would, however, avoid some few calculations. Theorem 2.8. [12, Theorem 1.2] Let G be a finite group with normal Sylow p - subgroup S. Assume that S has an abelian complement. Then the first Zassenhaus conjecture holds for G. 3. Direct products The ZC–1-question may be considered for arbitrary G - adapted 2 subrings R of C. (ZC–1)R Let R be an integral domain. We say that G satisfies (ZC − −1)R if each torsion unit of RG with augmentation 1 is conjugated within Q(R)G to an element g ∈ G, where Q(R) denotes the quotient field of R. P Let Ci be a conjugacy class of the finite group G and x = g∈G z(g)g ∈ ZG. Then the partial augmentation νi of x with respect to Ci is just X z(g). νi = g∈Ci

If u is a torsion unit of V (ZG), then it is conjugate within QG to a trivial unit if and only if for each power of u all partial augmentations except one vanish [22, Theorem 2.7], [19]. There is an analogon to this unique-trace property for torsion units in RG. Here, for an element u ∈ RG the partial augmentations are defined in a natural way not within Z, but within R. Proposition 3.1. Let u ∈ V (RG) be a torsion unit of order k. Then u is conjugate within Q(R)G to an element g ∈ G if and only if for each divisor d of k there is a unique conjugacy class Cd of G with partial augmentation different from zero. The proof is similar to that of [19, Theorem 2] given by I.S. Luthar and I.B.S. Passi. Proposition 3.2. Let p be prime, ζp a primitive pth −root of unity and G an arbitrary finite group. If (ZC − −1)Z[ζp ] is valid then ZC–1 holds for G × E, where E is an elementary abelian p−group. The main idea of the proof is to treat with different projection of Z(G×Cp ) onto Z[ζ]G mapping a generator of Cp on the powers of a pth −root of unity. 2An integral domain R is called G - adapted if no prime divisor of |G| is invertible in

R.

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A similar result can be given if one replaces the prime p by n ∈ {4, 8} and E by a multiple of C4 , respectively C8 . We remark that it seems to be unknown whether ZC–1 is valid for a direct product G × A with A abelian if it holds for G. From Proposition 3.2 we get the following special case. Corollary 3.3. If ZC–1 holds for a group G, then ZC–1 is true for G × E, where E is an elementary abelian 2-group. As a summary of several known results we obtain now Corollary 3.4. If G is a finite subgroup of the orthogonal group O(3, R), then ZC–1 is valid for G. Proof. This is clear when G is abelian by Higman’s thesis [13]. If G is a dihedral group then the result follows from [25], [20]. ZC–1 for the tetrahedral group is proved in [1], for the octahedral group in [8], and for the icosahedral group in [19]. All other finite subgroups are a direct product of C2 with one of the previous groups. Thus Corollary 3.3 completes the result. ¤ Remark. Corollary 3.4 indicates that ZC–1 might hold at least for groups with irreducible representations of small degree. 4. Groups of small order By the results of section 2 the only possible group orders for a group G of order less or equal 72 for which ZC–1 is still unknown are 24, 36, 40, 48, 54, 56, 60, and 72. An examination - using GAP - of the groups of such orders by the results of section 2 and that ZC–1 is valid for the symmetric group S4 and the binary octahedral group [8], [7] rsp. yields a list of 30 groups up to order 60 that satisfy none of the theorems (and 18 groups of order 72). The remaining 30 groups up to order 60 for ZC–1 are now checked with ordinary characters by the Luthar - Passi method. For a more detailed explanation of this method we refer to [5] or [32]. There are some results that simplify the Luthar-Passi-Method, e.g., the following one due to Dokuchaev and Juriaans [7]. Proposition 4.1. Let N £ G. Suppose ZC–3 holds for the factor G/N . Then any finite subgroup H ≤ V (ZG) with (|H|, |N |) = 1 is conjugate to a subgroup of G. Within the Luthar - Passi method it is of course also used that the partial augmentations of nontrivial torsion elements of V (ZG) on central group elements have to be zero, which is an immediate consequence of Berman’s theorem [2], see also [27, Ch.5, p. 102].

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It turns out that the method of Luthar and Passi is quite a powerful tool for groups of small order. Together with Proposition 4.1 one gets enough information about the partial augmentations of the torsion units in V (ZG) to show the following result: Proposition 4.2. Let |G| < 72. Then ZC–1 holds for G, except G is isomorphic to one of the following groups: • G1 ∼ = GL(2, 3) • G2 ∼ = A4 o C4 In case of G1 torsion units of order 2, in case of G2 torsion units of order 2 and 4 remain. 5. The two remaining groups In this section we show that ZC–1 holds also for the groups G1 and G2 . The arguments use information on the partial augmentations obtained by the Luthar - Passi method. Let u ∈ V (ZG) be a torsion unit, N £ G a normal subgroup, and π : G −→ G/N the reduction map. π also induces a map on the partial augmentations by adding up those partial augmentations whose corresponding conjugacy classes fuse in G/N . Let G ∈ {G1 , G2 } and 1 6= N £ G. Because of Proposition 4.2 π(u) is conjugate to an element of G/N in Q(G/N ) and therefore exactly one partial augmentation of π(u) is different from zero. (1) G ∼ = GL(2, 3) The head of the character table of G is given by (obtained by GAP): i |CG (Ci )|

o(ri )

1 2 3 4 5 6 7 8 48 4 6 8 48 8 6 8 1a 2a 3a 4a 2b 8a 6a 8b

Let u ∈ V (ZG) be a torsion unit of order 2. Let νi be its partial augmentation with respect to Ci . Then it is clear that ν1 = ν3 = ν5 = ν7 = 0. The Luthar-Passi-method yields the following possible sets of partial augmentations that are in contradiction to ZC–1:

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(a) (b) (c) (d) (e) (f) (g)

C. H¨ ofert and W. Kimmerle

ν2 ν4 ν6 ν8 1 1 -1 0 -1 1 0 1 1 1 0 -1 -1 1 1 0 2 0 -1 0 2 0 0 -1 -1 0 1 1

χ2 (1,0) (1,0) (1,0) (1,0) (0,1) (0,1) (0,1)

χ3 (2,0) (2,0) (2,0) (2,0) (1,1) (1,1) (1,1)

χ4 (2,0) (2,0) (0,2) (0,2) (2,0) (0,2) (1,1)

χ5 (0,2) (0,2) (2,0) (2,0) (0,2) (2,0) (1,1)

χ6 (0,3) (2,1) (0,3) (2,1) (0,3) (0,3) (3,0)

χ7 (2,1) (0,3) (2,1) (0,3) (3,0) (3,0) (0,3)

χ8 (2,2) (2,2) (2,2) (2,2) (2,2) (2,2) (2,2)

where a tuple (a, b) on the righthand side of the table means µ0 (χi , u) = a and µ1 (χi , u) = b if µl (χi , u) is the multiplicity of ζ l as an eigenvalue of Di (u). Here ζ denotes a primitive 2nd -root of unity and Di is an ordinary representation affording χi . The reduction map π modulo the center of G – the factor group is isomorphic to S4 , the symmetric group of degree four – gives the following two possible cases: χ2 χ3 χ6 χ7 π(u) ∼ (1, 2) ∈ S4 (0,1) (1,1) (1,2) (2,1) π(u) ∼ (1, 2)(3, 4) ∈ S4 (1,0) (2,0) (1,2) (1,2) In each of the cases (a)-(g) we get a contradiction either to the eigenvalues of χ6 or to those of χ7 . Remarks. a) Recently M. Hertweck proved that if u is a torsion unit of V (ZG) – G denotes now again an arbitrary finite group – of order pm then all partial augmentations of classes whose representatives are of order pn with n > m vanish. Together with [22, Theorem 2.7] this shows that if G has a unique conjugacy class of noncentral elements of order p then ZC–1 holds for all elements of order p. This gives another way for proving ZC–1 for involutions P of V (ZGL(2, 3)). b) If u = g∈G z(g)g, then X z(g) ε(k) (u) = g∈G;o(g)=k

is called the k - generalized trace of u. A. A. Bovdi conjectured [4], see also [15], that for a torsion unit u ∈ v(ZG) of order pn ε(pn ) (u) = 1 and ε(pj ) (u) = 0 , if j < n. Note, if ZC–1 holds for p - elements of a finite group G then Bovdi’s conjecture follows. The result of Hertweck mentioned in a) shows that Bovdi’s conjecture holds for elements of order p, therefore in particular for groups with elementary abelian Sylow subgroups. For further results on Bovdi’s conjecture we refer to [15].

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(2) G ∼ = A4 o C4 Using GAP the character table of G is i |CG (Ci )|

o(ri ) ri2 ri3 ri5 χ1 χ2 χ3 χ4 χ5 χ6 χ7 χ8 χ9 χ10

1 48 1a 1a 1a 1a 1 1 1 1 2 2 3 3 3 3

2 8 4a 2a 4b 4a 1 -1 A -A . . 1 -1 A -A

3 48 2a 1a 2a 2a 1 1 -1 -1 -2 2 3 3 -3 -3

4 5 6 7 6 16 8 8 3a 2b 4b 4c 3a 1a 2a 2c 1a 2b 4a 4d 3a 2b 4b 4c 1 1 1 1 1 1 -1 -1 1 1 -A A 1 1 A -A -1 2 . . -1 2 . . . -1 1 -1 . -1 -1 1 . -1 -A -A . -1 A A

8 6 6a 3a 2a 6a 1 1 -1 -1 1 -1 . . . .

9 16 2c 1a 2c 2c 1 1 -1 -1 -2 2 -1 -1 1 1

10 8 4d 2c 4c 4d 1 -1 -A A . . -1 1 A -A

with A = i; ri is a representative of the conjugacy class Ci The same method as in (1) may be applied for a torsion unit u ∈ V (ZG) of order 2, respectively of order 4. The involved normal subgroups in the case of u2 = 1 are N = Z(G) ∼ = C2 and N ∼ = C2 × C2 . This suffices to show that ZC–1 holds for involutions in V (ZG). However in case of torsion elements of order 4 the reduction with respect to any normal subgroup does not lead to a final result. There are still remaining sets of partial augmentations, namely ν2 ν6 ν7 ν10 χ9 χ10 (a) 0 1 1 -1 (0,0,0,3) (0,3,0,0) (b) 1 0 -1 1 (0,3,0,0) (0,0,0,3) where the notation is similar to those in (1). To show that there is no torsion unit u ∈ V (ZG) with the properties (a) or (b) we work with a concrete faithful representation ϕ of the group G3. From now on u is as in (a) or (b). Let G∼ = A4 oC4 ∼ = [(C2 ×C2 )oC3 ]oC4 = [(ht1 i×ht2 i)ohri]ohsi. 3The idea for this solution was communicated to the first author by Martin Hertweck.

We thank him for this hint.

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The representation ϕ : G → C3×3 , defined via Ã ! Ã ! 1 · · −1 · · · −1 · · −1 · ϕ(t1 ) = , ϕ(t2 ) = · · −1 · · 1 Ã

! Ã ! · 1 · · i · · · 1 i · · ϕ(r) = and ϕ(s) = 1 · · · · i is faithful and irreducible. Therefore either the corresponding representation of χ9 or χ10 may be identified with ϕ. From the eigenvalues of u with respect to Wedderburn components of CG corresponding to these irreducible characters it follows that   i · · U := ϕ(u) = ±  · i · . · · i We show that this is impossible. P

Clearly U =

u(g)ϕ(g) is a Z - linear combination of elements of ϕ(G).

g∈G

Put H = C3 o C4 . The coset decomposition of G with respect to the normal S subgroup C2 × C2 leads to ϕ(G) = ϕ(C2 × C2 )ϕ(h). Because of ϕ(s2 ) = h∈H

−E we get that U must be an element of * + [ [ D·b hDiZ b, = b∈B

Z

b∈B

where D = ϕ(C2 × C2 ) and B = {ϕ(id), ϕ(r), ϕ(r2 ), ϕ(s), ϕ(r · s), ϕ(r2 · s)}. Therefore               

D=

1 · · · 1 · · · 1

  1    ·   ·

B=

· 1 ·

   ,  



· ·  ·  , · 1 1

1 · · · −1 · · · −1 1 · ·



· ·  1  , 1 · ·

   ,  

· · 1

−1 · · · −1 · · · 1 

1 ·  ·  , i · ·

i · ·

   ,  



−1 · · · 1 · · · −1

· i  ·  , · i ·

· · i



           

· ·  i  , · · i

· i ·

and 

 i  ·   .   ·

hDiZ consists only of diagonal matrices with integer entries. To get the entry (U )1,1 = i the equation S U = dϕ(r · s) + q must be satisfied with 0 6= d ∈ hDiZ and q ∈ hDiZ b. Because the matrix d is of the b∈B\{ϕ(r·s)}

form 

 w+x−y−z · ·  with w, x, y, z ∈ Z, · w−x−y+z · d= · · w−x+y−z

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we get the following equations: (U )1,1 = i = (dϕ(r · s))1,1 + (q)1,1 = (d)1,1 (ϕ(r · s))1,1 + (q)1,1 = (w + x − y − z)i + (q)1,1 (U )2,3 = 0 = (dϕ(r · s))2,3 + (q)2,3 = (d)2,2 (ϕ(r · s))2,3 + (q)2,3 = (w − x − y + z)i + (q)2,3 (U )3,2 = 0 = (dϕ(r · s))3,2 + (q)3,2 = (d)3,3 (ϕ(r · s))3,2 + (q)3,2 = (w − x + y − z)i + (q)3,2 . With (q)2,3 , (q)3,2 ∈ Z the last two equations imply 2(w − x) = 0 ⇒ w = x 2(y − z) = 0 ⇒ y = z and therefore (d)1,1 = 2(w + y) and (d)1,1 (ϕ(r · s))1,1 is an even multiple of i. Because (q)1,1 ∈ Z this gives the desired contradiction. Therefore U can’t be of the given form and a torsion unit u of V (ZG) with one of the properties (a) or (b) does not exist. Acknowledgment Dr. Kimmerle was supported by the Deutsche Forschungsgemeinschaft. References [1] P.J. Allen and C. Hobby, A characterization of units in ZA4 , J. of Algebra 66 (1980), 534–543. [2] S.D. Berman, On certain properties of integral group rings (Russian), Dokl. Akad. Nauk SSSR (N.S.) 91 (1953), 7–9. [3] P.F. Blanchard, Exceptional group ring automorphisms for groups of order 96, Comm. Algebra 29 (2001), no. 9, 4823 –4830. [4] A.A. Bovdi, The unit group of integral group rings (Russian), Uzhgorod Univ. 21 (1987), Dep.- Ukr. Ninti 24.09.87, N2712-UK 87. [5] V. Bovdi, C. H¨ ofert, and W. Kimmerle, On the first Zassenhaus conjecture for integral group rings, Publ. Math. Debrecen 65 (2004), no. 3-4, 291–303. [6] J.A. Cohn and D. Livingstone, On the structure of group algebras I, Canad. J. Math. 17 (1965), 583–593 . [7] M.A. Dokuchaev and S.O. Juriaans, Finite subgroups in integral group rings, Canad. J. Math. 48 (1996), no. 6, 1170–1179. [8] N.A. Fernandes, Torsion units in integral group ring of S4 , Boletim Soc. Brasil Mat.(1) 18 (1987), 1–10. [9] The GAP Group, GAP – Groups, Algorithms, and Programming, Version 4.2, Aachen, St. Andrews, (http://www.gap-system.org), 1999. [10] M. Hertweck, Eine L¨ osung des Isomorphieproblems f¨ ur ganzzahlige Gruppenringe von endlichen Gruppen, Ph.D. thesis, Stuttgart, 1998, Shaker-Verlag, ISBN 3-82656055–6058. [11] M. Hertweck, Contributions to the integral representation theory of groups, Elektr. Hochschulschrift, http://elib.uni-stuttgart.de/opus, Habilitationsschrift Universit¨ at Stuttgart, 2004. [12] M. Hertweck, On the torsion units of some integral group rings, Preprint 2004.

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[13] G. Higman, Units in group rings, D.Phil. thesis, Oxford Univ., 1940. [14] C. H¨ ofert, Die erste Vermutung von Zassenhaus f¨ ur Gruppen kleiner Ordnung, Diplomarbeit, Universit¨ at Stuttgart, 2004. [15] S.O. Juriaans, Torsion units in integral group rings, Canad. Math. Bull. Vol. 38(3) (1995), 317–324. [16] S.O. Juriaans and C.P. Milies, Units of integral group rings of Frobenius groups, J. Group Theory 3 (2000), 277–284. [17] W. Kimmerle, Group rings of finite simple groups, Resenhas IME-USP, 5(4) (2002), 261–278. [18] LAGUNA – Lie AlGebras and UNits of group Algebras, Version 3.0, developed by V. Bovdi, A.B. Konovalov, A.R. Rossmanith and Cs. Schneider (http://www.ukrgap.exponenta.ru/laguna.htm), 2003. [19] I.S. Luthar and I.B.S. Passi, Zassenhaus conjecture for A5 , J. Nat. Acad. Math. India 99 (1989), 1–5. [20] I.S. Luthar and Poonam Sehgal, Torsion units in integral group rings of some metacyclic groups, Res. Bull. Panjab Univ. Sci. 48 (1998), no. 1-4, 137–153. [21] I.S. Luthar and Poonam Trama, Zassenhaus conjecture for S5 , Comm. Alg. 19 (1991), no. 8, 2353–2362. [22] Z.S.Marciniak, J.Ritter, S.K. Sehgal, and A. Weiss, Torsion units in integral group rings of some metabelian groups, II, J. Number Theory 25 (1987), 340–352. [23] C.P. Milies, J. Ritter, and S.K. Sehgal, On a conjecture of Zassenhaus on torsion units in integral group rings II, Proc. AMS 97 (1986), 201–206. [24] C.P. Milies and S.K. Sehgal, Torsion units in integral group rings of metacyclic groups, J.Number Theory 19 (1984), 103–114. [25] T. Mitsuda, On the torsion units of integral dihedral group rings, Comm. Algebra 14 (1986), no. 9, 1707–1728. [26] K.W. Roggenkamp, Observations to a conjecture of H. Zassenhaus, Groups St. Andrews 1989 Conf. Proc. LMS Lect. Note Ser. 160 (1991), no. 2, 427–444. [27] R. Sandling, Graham Higman’s thesis “Units in Group Rings” in Integral Representations and Applications ed. by K. W. Roggenkamp, Springer Lecture Notes 882, 93–116, 1981. [28] S.K. Sehgal, Units in integral group rings, Pitman Monographs and Surveys in Pure and Applied Mathematics, Longman Scientific & Technical, Essex, 1993. [29] S.K. Sehgal and A. Weiss, Torsion units in integral group rings of some metabelian groups, J. Algebra, 103 (1986), 490–499. [30] L.L. Scott, On a conjecture of Zassenhaus, and beyond, Algebra, Proc. Int. Conf. Memory A. I. Malcev, Novosibirsk 1989, Contemp. Math. 131 (1992), no. 1, 325– 343. [31] A. Valenti, Torsion units in integral group rings, Proc. Amer. Math. Soc., 120 (1994), no. 1, 1–4. [32] R. Wagner, Zassenhausvermutung f¨ ur P SL(2, p), Diplomarbeit, Universit¨ at Stuttgart, 1995. [33] A. Weiss, Torsion units in integral group rings, J. Reine Angew. Math. 415 (1991), 175–187. [34] H. Zassenhaus, On the torsion units of finite group rings, Estudos de Matematica em homenagem ao Prof. A. Almeida Costa, Inst. de alta Cultura, Lisboa, 1974, 119–126.

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On a conjecture of Zassenhaus for metacyclic groups Stanley O. Juriaans Instituto de Matem´atica e Estat´ıstica, Universidade de S˜ao Paulo, Caixa Postal 66281, S˜ao Paulo, SP, 05315-970, Brasil [email protected] Sudarshan K. Sehgal Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Alberta, T6G 2G1, Canada [email protected] In honor of Prof. Francisco C´esar Polcino Milies on his sixtieth birthday. Abstract. Zassenhaus Conjecture is proven for some metacyclic groups. In particular the conjecture holds if G = hai× hxi with hxi a p-group acting faithfully on every Sylow p0 −subgroup of G. The case in which hai is of prime power order was proved by Polcino Milies-Sehgal.

1. Introduction Let ZZG be the integral group ring of a finite group G. Denote by UZZG the group of units and by U1 ZZG those of augmentation one. A well-known conjecture of Zassenhaus states: (ZC1): If u ∈ U1 ZZG is a torsion unit then u is rationally conjugate to a group element. That is, there exist elements g ∈ G, β ∈ U QG l such that, β −1 uβ = g. This conjecture has been verified for several families of groups including the following: Theorem 0 (Weiss [11]). If G is a finite nilpotent group then (ZC1) holds for G.

Key words and phrases. Group rings, torsion units, metabelian group. AMS Subject Classification: Primary 20C05. Secondary 16S34, 16U60. 253

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Theorem 00 (Polcino Milies-Ritter-Sehgal [6]). Let G be a metacyclic group of the form hai× hbi with a, b of coprime order. Then (ZC1) holds for G. In this paper we extend the last theorem, which was originally proved in a series of three papers, to allow the possibility of some common factors in the orders of a and b. First we fix some notation: Let G = A× X where A = B ×K. Let π be a set of rational primes, K, X cyclic π-groups and B a cyclic π 0 -group. We suppose that X acts faithfully on all Sylow subgroups of B. We shall denote by F0 the family of groups with these properties and by F1 a family of finite groups, closed with respect to forming subgroups and quotients, satisfying (ZC1). Our main result is: Theorem 1.1. Let G ∈ F0 . Suppose that K× X belongs to a family F1 , closed with respect to forming subgroups and quotients. If (ZC1) holds for all elements of F1 then (ZC1) holds for G. By applying Weiss’ theorem we have: Corollary 1.2. If G ∈ F0 and K, X are p-groups then (ZC1) holds for G. In Section 2 we derive some representations of G. In Section 3 we give some reductions and Section 4 is dedicated to possible counter examples. In Section 5 we give a complete proof of Theorem 00 , which is contained in [5], [6] and [7]. We also give a short proof of a result of Luthar and Trama [3]. In the last section we complete the proof of the theorem. We remark that our result includes the following result of [4]: Let G be a split extension of a cyclic group by a group of prime order. Then (ZC1) holds for G. 2. Representations of G In this section we determine some representations of G ∈ F0 . Let Φ ∈ Irr(G) be an irreducible complex representation of G. By [10, pg. 62] Φ is induced from ASΨ , where Ψ is an irreducible representation of A and SΨ is the stabilizer of Ψ in X namely SΨ = {x ∈ X : Ψ(x−1 ax) = Ψ(a), ∀a ∈ A}. Also the degree T of Φ is equal to [X : SΨ ]. From [8, 8.4.3] we see easily that ker(Ψ) = A Ker(Φ). Hence Φ is faithful on A if and only if Ψ is faithful on A. Fix a Φ faithful on A and let SΨ be as above. If x ∈ SΨ then Ψ(x−1 ax) = Ψ(a) for all a ∈ A. Since Ψ is faithful on A it follows that x−1 ax = a for every a ∈ A and thus x = 1. Hence we have proved that Φ is induced from A. Lemma 2.1. Let A = B × K be cyclic, G = A× X with X acting faithfully on B. If Φ ∈ Irr(G) is faithful on A then it is induced from A and has degree |X|.

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Corollary 2.2. Let G ∈ F0 and p a prime Tdividing |A|. Suppose that p2 divides |A| and Φp ∈ Irr(G) is such that A Ker(Φp ) has order p. Then Φp is induced from A and has degree |X|. T Proof. Set N = A Ker(Φp ). We shall see later on, Corollary 3.2, that G/N ∈ F0 . But Φp is defined on G/N and is faithful on A/N . The result follows from the Lemma. ¤ Let G ∈ F0 and let Φ0 ∈ Irr(G) be faithful on A. Set X = hxi, A = hai and Ψ ∈ Irr(A) such that Φ0 = ΨG . For the action of x on A we write ax = aj and let Ψ(a) = ξ, a complex root of unity, o(ξ) = o(a). Setting s = |X| we see easily that: 

ξ

 0  Φ0 (a) =  0 .  .. 0

0

0

···

ξj

0 .. .

···

0 ···

.

0 .. . ···

0 .. . .. .

0 .. 0

0

ξ





   , Φ0 (x) =   

 0  0   .. . 1

j s−1

0

1

0

···

0

1 .. .

··· .. . .. .

0 .. . ···

0 ···

0

 0 ..  . ..  .   1 0

So Φ0 : G −→ Ms (l Q(ξ)). Up to rational equivalence this is the only Ψ ∈ Irr(G) that is faithful on A, i.e., if Φ is an irreducible representation of G that is faithful on A then there exists a field automorphism σ ∈ Aut(l Q(ξ)) such that Φ = σ ◦ Φ0 . Note that if a1 , a2 ∈ A are such that Φ0 (a1 ), Φ0 (a2 ) are conjugate then they are already conjugate in G and this depends only on the fact that A is cyclic. To see this we only have to compare eigenvalues and remember that ξ and a have the same order. We shall frequently use this observation in the proof of our results. Let Φp be as in Corollary 2.2. It follows from this corollary that, up to rational equivalence, Φp is given by:     Φp (a) =    

ξp

0

0

···

0

ξ pj

···

0 .. .

0 .. .

0 .. .

0

···

0 ···

0 .. . .. .

0 .. . 0

0

ξ

pj s−1





   , Φp (x) =   

 0  0   .. . 1

0

1

0

···

0

1 .. .

··· .. . .. .

0 .. . ···

0 ···

0

 0 ..  . ..  .   1 0

So we see that Φp = Φ0 ◦ Ψp where Ψp : G −→ G is the homomorphism defined by Ψp (ak xl ) = apk xl . Note that Ker(Φp ) = Ker(Ψp ). The Corollary is still true if K = 1 and X acts faithfully on B. In this case Φ0 represents a more or less as above but the image of X consists of permutation matrices. In the next Lemma A = B × K and B, X are groups of coprime order. We write A = hai.

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Lemma 2.3. Let G = A× X with X acting faithfully on B. Suppose that u ∈ 1 + 4(G, A) and o(u) has exactly the same prime divisors as |A|. Then there exists a0 ∈ A such that Φ0 (u) and Φ0 (a0 ) are conjugate. Proof. Since X acts faithfully on B it must be abelian. As remarked above Φ0 maps elements of X to permutation matrices. Denote by u1 , · · · , ur the generators of the Sylow subgroups of hui. By [1, Theorem 2.9] every ui is rationally conjugate to an element of A. Each matrix Φ0 (ui ) has an eigenvalue of the form ξ si . It follows that Φ0 (u) has an eigenvalue of the form ξ t and we may suppose that it is the product of these ξ si . Write X = {1 = x0 , x1 , · · · , xs } and axi = aki . We want to show that ξ tki are all eigenvalues of Φ0 (u). But this is easily seen because Φ0 (xi ) is a permutation matrix and hence is fixed by the field automorphism σi : ξ −→ ξ ki . Therefore, applying σi to the entries of Φ(u) is the same as conjugating by Φ0 (xi ). To P see this write u = fk (a)xk and notice that Φ0 (aki ) = σi (Φ0 (a)). Then k P −1 Φ0 (x−1 fk (Φ0 (aki ))Φ0 (xk ) = σi (Φ0 (u)). i )Φ0 (u)Φ0 (xi ) = Φ0 (xi uxi ) = k

Since X acts faithfully on B and o(u) has the same prime divisors as |A| it follows that these eigenvalues are all distinct. In fact suppose that ξ tki = ξ tkj . Since ξ and a have equal orders it follows that atki = atkj . Write T k at = a1 a2 with a1 ∈ B and ha2 i B = 1. Then we have that ak1i = a1j and thus xi (xj )−1 centralizes a1 . Note now that o(a1 ) has exactly the same prime divisors as |B| and since the latter and |X| are relatively prime it follows that x = xi (xj )−1 centralizes B. To see this let b ∈ B be of prime order p, say, and notice that b is centralized by x. Write bx = bk for the action of x on b. It follows that k = 1 mod p. Since (o(x), |B|) = 1 it follows that x centralizes the Sylow p-subgroup containing b and hence the statement is proved. Therefore, xi = xj and the eigenvalues ξ tki are all distinct. It follows that Φ0 (u), Φ0 (at ) are conjugate. ¤ 3. Some reductions In this section we establish some reductions. Notations of former sections will be used freely and ∼ stand for conjugation of units or matrices. Lemma 3.1. Let p be a prime, hai a cyclic p-group, and X a p0 -group acting faithfully on hai. Then X acts faithfully on every nontrivial homomorphic image of hai. Proof. Let x ∈ X be given and set ax = aj . If hai is a nontrivial homorphic image of hai then ax = aj and hence if x acts trivially on a we have that j = 1 mod p. Since X is a p0 -group if follows that j = 1 mod o(a) and thus by assumption x = 1. ¤ Corollary 3.2. Let G ∈ F0 and N ⊂ B a proper subgroup. Then G/N ∈ F0 .

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Proof. By the last Lemma X acts faithfully on every Sylow subgroup of B/N and hence the result is proved. ¤ Lemma 3.3. Let G ∈ F0 , g ∈ G, g = byx with b ∈ B, y ∈ K, and x ∈ X. If x 6= 1, then g is a π-element. Proof. We use induction on the number of prime divisors of |B|. If B is of prime power order then, by [9, 37.16], bx is a π-element. Since K is normal in G it follows that g is a π-element. In the general case write b = b1 b2 with hb1 i a Sylow subgroup of hbi. By induction b2 x is a π-element and hence, by [8, 9.1.3] it is conjugate to an element of K× X. It follows that a conjugate h of g is of the form h = b3 y1 x1 with X 3 x1 6= 1, y1 ∈ K and b3 ∈ B has prime power order. It follows by the first case that h and hence g is a π-element. ¤ Lemma 3.4. Let G ∈ F0 be a counter example of least order to (ZC1) and let u ∈ U1 ZZG be a torsion unit that is not conjugate, in QG, l to a group element. If K× X ∈ F1 then u ∈ 1 + 4(G, A). Proof. Denote by W = 4(G) 4 (G0 ) the Whitcomb ideal. Since G/A and A are abelian we use Whitcomb’s Argument and [9, 31.3] to find g ∈ G such that u − g ∈ W and o(u) = o(g). Write g = byx, as in the last Lemma. If x 6= 1 then, by the last Lemma, u is a π-element and thus, by [9, 37.13], it is rationally conjugate to an element of ZZ(K× X). Since (ZC1) holds for the latter group we have a contradiction. So x = 1 and since W ⊂ 4(G, A) the result is proved. ¤ Lemma 3.5. Let G and u be as in Lemma 3.4 and K× X ∈ F1 . Then every prime dividing |A| divides o(u). Proof. Suppose that p is a prime dividing |A| and p does not divide o(u). Let N be a subgroup of G of order p. Note that N is normal in G. Now either G/N is isomorphic to K× X or G/N ∈ F0 . If the first case happens then we know that (ZC1) holds for this group. In the second case (ZC1) holds by the choice of G. Since N and u have coprime order we apply [1, Theorem 2.2] to obtain that u is conjugate, in QG, l to a group element, a contradiction. ¤ Lemma 3.6. Let G be a finite nilpotent-by-nilpotent group and A a normal subgroup of G. If Φ ∈ Irr(G) is faithful on A then it is injective on finite subgroups of 1 + 4(G, A). Proof. Let u ∈ 1 + 4(G, A) be such that Φ(u) = 1. We may suppose that u is of prime power order. Since G is nilpotent-by-nilpotent it follows, by [1, Theorem 2.9], that u is rationally conjugate to a group element A 3 a, say. So Φ(a) = 1 and u = 1. ¤

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4. Counter examples In this section we prove some results about characters and units. Unless explicitly stated, groups G in this section do not necessarily belong to F0 . Also ∼ will stand for conjugation of matrices or units. We begin by making the following assumptions: Assumptions: (1) G is a finite nilpotent-by-nilpotent-group belonging to some family F and it is a counter example to (ZC1). (2) A is a normal abelian subgroup of G. (3) Φ0 is a fixed irreducible representation of G that is faithful on A. It satisfies the following: If α ∈ U1 ZZG is a torsion unit that is rationally conjugate to a group element g ∈ A and if Φ0 (α) ∼ Φ0 (h), h ∈ G; then g ∼ h in G. (4) u ∈ 1 + 4(G, A) is a torsion unit of least order that is not rationally conjugate to a group element; |A| and o(u) have exactly the same prime divisors. (5) Φ0 (u) ∼ Φ0 (a0 ) for some a0 ∈ A. (6) Φ0 is faithful on finite subgroups of 1 + 4(G, A). In all that follows, G, Φ0 , a0 , and u will be as above. Lemma 4.1. Let Φ ∈ Irr(G) be so that there exists a prime p dividing o(u), but p does not divide o(Φ(u)). Then Φ(u) ∼ Φ(a0 ). Proof. Since G is nilpotent-by-nilpotent it follows, by [1, Theorem 2.9], that u is not of prime power order. Write u = u1 u2 where u1 is a p-element and u2 is a p0 -element. By our hypothesis Φ(u) is a p0 -element and hence Φ(u1 ) = 1. Since u2 is a p0 -element we can find infinitely many integers k ∈ IN such that k k k k up2 = u2 and u1p = 1. So we have up = u2 and thus Φ0 (u2 ) ∼ Φ0 (ap0 ). By k

k

(3) and (4) it follows that up ∼ ap0 and consequently Φ(u2 ) ∼ Φ(ap0 ). Now write a0 = a1 a2 where a1 is a p-element and a2 is a p0 -element. By k (6), o(ui ) = o(ai ) and thus ap0 = a2 . It follows that Φ(u2 ) ∼ Φ(a2 ). m m m Let m = o(u2 ). By (5), Φ0 (um 1 ) = Φ0 (u ) ∼ Φ0 (a0 ) = Φ0 (a1 ). By (3) m m and (4) it follows that u1 ∼ a1 and since p does not divide m and Φ(u1 ) = 1 we have that Φ(a1 ) = 1. Finally it follows that Φ(u) = Φ(u1 u2 ) = Φ(u2 ) ∼ Φ(a2 ) = Φ(a0 ). ¤ k

Corollary 4.2. Suppose that A is cyclic and p is a prime dividing the order of u but p2 does T not divide the order of u. If Φ ∈ Irr(G) is such that p divides |N = A Ker(Φ)| then Φ(u) ∼ Φ(a0 ). Proof. We use the notation of the proof of the last Lemma. Since p2 does not divide o(u) it follows that u1 and a1 have order p. Since A is cyclic and p divides |N | we must have that Φ(a1 ) = 1 and thus Φ(u) = Φ(u2 ). It

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follows that p does not divide the order of Φ(u). The result follows by the Lemma. ¤ P Let u = u(g)g ∈ ZZG,Pg0 ∈ G and denote by C the conjugacy class of g0 . Define u ˜(g0 ) = tC (u) = u(g) and Λ = [ZZG, ZZG]. It is easy to see that g∈C

u ∈ Λ if and only if u ˜(g) = 0 for every g ∈ G. It is also known that (ZC1) is equivalent to the following: For every u ∈ U1 ZZG of finite order there exists a unique conjugacy class C such that tC (u) 6= 0 (see [9, 41.5]). As in the previous sections we shall denote by W the Whitcomb ideal 4(G) 4 (G0 ). Suppose that A = hai and G = A× X with X an abelian group. Define Ψp : G −→ G by Ψp (ak xl ) = apk xl , xl ∈ X. We shall denote by s the order of X. This notation will be used in what follows. Theorem 4.3. Let p be a prime dividing |A|, π a set of rational primes and Ψp : G −→ G the homomorphism defined above. Suppose that Φ ∈ Irr(G) satisfies Φ = Φ0 ◦ Ψp . Suppose that A is not a π-group and that one of the following holds: (1) Elements of the form ak xl , x` 6= 1, are π-elements. (2) p does not divide the order of X. Then Φ(u) ∼ Φ(a0 ). P P Proof. Write u = fj (a)xj , ZZ(A) 3 fj (a) = αij ai . Then Ψp (u) = i 0≤j<s P P fj (ap )xj = f0 (ap ) + fj (ap )xj and, by [9, 7.1], up = 0≤j<s 1≤j<s P (fj (a)xj )p mod Λ + pZ ZG. Further, we have that up = f0 (ap ) + 0≤j<s P αij (ai xj )p mod Λ + pZ ZG. i,j6=0

Note that, by (1) and (2), elements of the form (ai xj )p either do not have the same prime divisors as A or are not in A. So if C = Cg0 is the conjugacy class of an element g0 ∈ A such that o(g0 ) and |A| have precisely the same prime divisors then tC ((aiT xj )p ) = 0 if xj 6= 1. It is easy to see that A Ker(Φ) has order p. Thus if p2 does not divide the order of u then we are are done by Corollary 4.2. So we may suppose that o(u) is divisible by p2 . By Whitcomb’s Argument and [9, 31.3] we can find gu ∈ G such that u − gu ∈ W and o(u) = o(gu ). Since Ψp is a homomorphism we see that Ψp (u) − Ψp (gu ) ∈ W and hence, by [9, 31.3], their orders are equal. But it should be noticed that W ⊂ 4(G, A) and thus gu ∈ A. It follows that Ψp (gu ) = gup and consequently Ψp (u) has o(u) p order o(u) p . So we have proved that u and Ψp (u) have order equal to p and hence, by assumption (4), they are rationally conjugate to group elements. We show that they are in fact conjugate to each other. Note first that since p2 divides the order of u, the orders of up and Ψp (u) have precisely the same

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prime divisors as |A|, which is not a π-group. Let C = Cg0 be as above. Then, by what was proved above, it follows that tC (up ) = tC (f0 (ap )) mod p and tC (Ψp (u)) = tC (f0 (ap )) and hence tC (up ) = tC (Ψp (u)) mod p. On the other hand there exist unique conjugacy classes C1 = Cg1 , C2 = Cg2 , with g1 , g2 having precisely the same prime divisors as |A|, such that tC1 (up ) 6= 0 and tC2 (Ψp (u)) 6= 0. Taking C = C1 we conclude that up and Ψp (u) are conjugate in QG. l Finally we have Φ(u) = Φ0 ◦ Ψp (u) ∼ Φ0 (up ) ∼ Φ0 (ap0 ) = Φ0 ◦ Ψp (a0 ) = Φ(a0 ). Hence the proof is complete. ¤ Remark. Note that all we needed is that the orders of the elements (ai xj )p that belong to A do not have the same prime divisors as |A|.

5. Some applications In this section we give a proof of a result of Luthar and Trama [3] which proves Theorem 00 . Theorem 5.1. (Luthar-Trama [3]) Let G = hai× X with X any group such that (|X|, o(a)) = 1. Let u ∈ 1 + 4(G, hai). Then u is rationally conjugate to a group element. Proof. Note that such groups form a family F that is closed with respect to forming quotients and subgroups. Let G be a counter example of least order in F. Let u ∈ U1 ZZG be an element of least order that is not conjugate in QG l to a group element and set N = CX (a). Note that N is normal in G and, by [1, Corollary 2.8], has order prime to that of u. If N 6= 1 then, by the choice of G and [1, Theorem 2.2], u would be rationally conjugate to a group element. Hence N = 1 and thus X is abelian and acts faithfully on hai. Let q be a prime divisor of o(a). If q does not divide o(u) we take N to be a Sylow q-subgroup of G and use [1, Theorem 2.2] and the choice of G to get a contradiction. Hence o(a) and o(u) have exactly the same prime divisors. Since u ∈ 1 + 4(G, hai) and the orders of hai and X are relatively prime we obtain, by Lemma 2.3, that Φ0 (u) ∼ Φ0 (a0 ) for some a0 ∈ hai. Take π to be the set of prime divisors of |X| and let Φp be as in section 2. It follows by Theorem 4.3 that Φp (u) ∼ Φp (a0 ) for every prime divisor of n. Let Φ ∈ T Irr(G) beTgiven. Suppose first that Φ is not faithful on hai. Then N = (A KerΦ) KerΦp 6= 1 for some prime divisor p of the order of A and, by Corollary 4.2, we may suppose that p2 divides the order of A. Consider the factor group G = G/N and denote by u the image of u in ZZG. Note that Φp and Φ are both defined on G and Φp is faithful on A/N . Also Φp (u) = Φp (u) ∼ Φp (a0 ) = Φp (a0 ), where a0 is the image of a0 in G. It follows, by the choice of G and the fact that A/N is cyclic, that u ∼ a0 and hence Φ(u) = Φ(u) ∼ Φ(a0 ) = Φ(a0 ).

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On the other hand, if Φ is faithful on A then Φ = σ ◦ Φ0 , where σ is a field automorphism. Since Φ0 (u) ∼ Φ0 (a0 ) it follows that Φ(u) ∼ Φ(a0 ). So we have seen that Φ(u) ∼ Φ(a0 ) for every Φ ∈ Irr(G). By [9, 37.5] u ∼ a0 in QG, l a final contradiction. There are no counter examples in F and the theorem is proved. ¤ Theorem 5.2. Let G = hai× X, X abelian, with o(a) and |X| relatively prime. Then (ZC1) holds for G. Proof. Denote by F the family of all such groups and let G be a counter example of least order in this family. Let also u ∈ U1 ZZG be an element of least order that is not rationally conjugate to a group element. By Whitcomb’s Argument and [9, 31.3] there exists G 3 gu = ak xl such that u − gu ∈ W , W being the Whitcomb ideal, and o(u) = o(gu ). By the choice of G and [9, 37.13] we may suppose that o(u) is divisible by every prime divisor of o(a). Hence, by [6, Lemma 2.3], xl is central in G and thus, by the choice of u, we may suppose that o(u) has the same prime divisors as n and u ∈ 1 + 4(G, hai). The result follows by Theorem 5.1. ¤ The case in which hai is a p-group was proved in [2] and A. Valenti proved a stronger version of this theorem (see [9, 42.9]). 6. Proof of the theorem We shall use freely the notations and results of the previous sections. The group G will always be in F0 . In particular, G and u will be as in section 3 and Φ0 is as in section 2. By Lemma 3.6, Φ0 is faithful on finite subgroups of 1 + 4(G, A). Proof of Proof of Theorem 1.1. Let G be a counter example of least order and let u ∈ U1 ZZG be an element of least order that is not rationally conjugate to a group element. We want to apply the results of section 4. We have to check that conditions (3) and (5) are satisfied: If α ∼ g ∈ A and Φ0 (α) ∼ Φ0 (h), h ∈ A then Φ0 (g) ∼ Φ0 (h) and, since A is cyclic, g ∼ h. By Lemma 2.3 Φ0 (u) ∼ Φ0 (a0 ) for some a0 ∈ A. By Theorem 4.3 we have that Φp (u) ∼ Φp (a0 ) for every prime divisor of |A|. Let Φ ∈ Irr(G) be given. We proceed as in the proof of Theorem 5.1. T T Suppose first that Φ is not faithful on hai. Then N = (A KerΦ) KerΦp 6= 1 for some prime divisor p of the order of A and, by Corollary 4.2, we may suppose that p2 divides the order of A. Consider the factor group G = G/N and denote by u the image of u in ZZG. Note that Φp and Φ are both defined on G and Φp is faithful on A/N . By the choice of G and the fact that A/N is cyclic it follows that u ∼ a0 and hence Φ(u) ∼ Φ(a0 ). If Φ happens to be faithful on A then Φ = σ ◦ Φ0 , where σ is a field automorphism. The fact that Φ0 (u) ∼ Φ0 (a0 ) implies that Φ(u) ∼ Φ(a0 ).

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We have proved that Φ(u) ∼ Φ(a0 ) for every Φ ∈ Irr(G) and hence, by [9, 37.5], u ∼ a0 in QG, l a final contradiction. Thus there are no counter examples and the theorem is proved. ¤ Acknowledgments Research supported by FAPESP, Brazil and NSERC, Canada. S.O. Juriaans would like to thank the University of Alberta, especially the department of Mathematics, for its warm hospitality in 1995 when this article was written. This work has not been previously submitted for publication. References [1] M.A. Dokuchaev and S.O. Juriaans, Finite subgroups in integral group rings, Canad. J. Math. 48 (1996), 1170–1179. [2] S.O. Juriaans, Torsion units in integral group rings, Commun. Algebra 22 (12) (1994), 4905–4913. [3] I.S. Luthar and P. Trama, Zassenhaus conjecture for certain integral group rings, Journal of Indian Math. Soc. 55 (1990), 199–212. [4] I.S Luthar and P. Trama, Torsion units in integral group rings of some metacyclic groups, Res. Bull. Panjab Univ. Sci. 48 (1998), 137–153. [5] C. Polcino Milies and S.K. Sehgal, Torsion units in integral group rings of metacyclic groups, J. Number Theory 19 (1984), 103–114. [6] C. Polcino Milies, J. Ritter, and S.K. Sehgal, On a conjecture of Zassenhaus on torsion units in integral group rings II, Proc. Amer. Math. Soc. 97 (2) (1986), 206–210. [7] J. Ritter and S.K. Sehgal, On a conjecture of Zassenhaus on torsion units in integral group rings, Math. Ann. 264 (1983), 257–270. [8] D.J.S. Robinson, A Course in the Theory of Groups, Springer-Verlag, New York, Heidelberg, Berlin, 1980. [9] S.K. Sehgal, Units of Integral Group Rings, Longman’s, Essex, 1993. [10] J.-P. Serre, Linear Representations of Finite Groups, Springer-Verlag N.Y. Heidelberg Berlin 1971. [11] A. Weiss, Torsion units in integral group rings, J. Reine Angew. Math. 415 (1991), 175–187.

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Nilpotent blocks revisited Burkhard K¨ ulshammer Mathematisches Institut, Friedrich-Schiller-Universit¨at, 07740 Jena, Germany [email protected]

The main purpose of this note is to give a self-contained proof of Puig’s structure theorem for nilpotent blocks. Puig’s original proof appeared in [8]. Textbook accounts were given by K¨ ulshammer in [6] and by Th´evenaz in [9]. The present proof makes use of simplifications discovered by K¨ ulshammer, Okuyama, and Watanabe [7]. The exposition is based upon lectures given by the author in Budapest in 2000. 1. Separable extensions Let F be a field, and let A be an (associative unitary) F -algebra. An A-bimodule M will always be required to satisfy αm = mα

for α ∈ F, m ∈ M.

Then the centralizer M A := {m ∈ M : am = ma for a ∈ A} is a vector space over F . Definition 1.1. An extension of A is an F -algebra B, together with a unitary homomorphism of F -algebras β : A → B. Often A will be a subalgebra of B, and β : A → B will be the inclusion map. For an arbitrary extension of F -algebras β : A → B, every (left) B-module N restricts to an A-module such that an := β(a)n

for a ∈ A, n ∈ N.

In a similar way, every B-bimodule M restricts to an A-bimodule; in this case we clearly have an inclusion of vector spaces M B ⊆ M A . Lemma 1.2. Let β : A → B be an extension of F -algebras, and let µ : B ⊗A B −→ B, x ⊗ y 7−→ xy, be the morphism of B-bimodules induced by the multiplication in B. Then the following assertions are equivalent: 263

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(1) µ splits as a morphism of B-bimodules; (2) There exists an element w ∈ (B ⊗A B)B such that µ(w) = 1B . Proof. (1) ⇒ (2). Let ν : B → B ⊗A B be a morphism of B-bimodules such that µ ◦ ν = idB . Since 1B ∈ Z(B) = B B , we have w := ν(1B ) ∈ (B ⊗A B)B and µ(w) = µ(ν(1B )) = idB (1B ) = 1B . (2) ⇒ (1). Let w ∈ (B ⊗A B)B such that µ(w) = 1B . Then the map ν : B −→ B ⊗A B, x 7−→ xw = wx is a morphism of B-bimodules, and µ(ν(x)) = xµ(ν(1B )) = xµ(w) = x1B = x for x ∈ B. ¤ If the conditions above are satisfied then β : A → B is called a separable extension of F -algebras. Proposition 1.3 (Higman). Let β : A → B be a separable extension of F -algebras. Then the following holds: (i) A short exact sequence of B-modules f

g

0 −→ L−→M −→N −→ 0 splits if its restriction to A splits. (ii) A B-module P is projective (injective) if its restriction to A is. Proof. P (i) Let w ∈ (B⊗A B)B such that µ(w) = 1B , and write w = kj=1 xj ⊗yj with x1 , y1 , . . . , xk , yk ∈ B. Moreover, let h ∈ HomA (N, M ) such that g ◦ h = idN . Now HomF (N, M ) is a B-bimodule with (xφy)(n) = xφ(yn)

for x, y ∈ B, φ ∈ HomF (N, M ), n ∈ N.

Moreover, H : B ⊗A B −→ HomF (N, M ), x ⊗ y 7−→ xhy, is a morphism of B-bimodules. Thus 0

h := H(w) =

k X

xj hyj ∈ HomF (N, M )B = HomB (N, M ),

j=1

and g(h0 (n))

k X = g( xj h(yj n)) = j=1

=

k X

xj g(h(yj n)) =

j=1

k X

xj yj n = µ(w)n = 1B n = n

j=1

for n ∈ N . (ii) is an easy consequence of (i). ¤

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A morphism between two extensions β : A → B and γ : A → C of A is a homomorphism of F -algebras φ : B → C that is also a morphism of A-bimodules. Theorem 1.4 (Wedderburn-Malcev). Let β : A → B be a separable extension of F -algebras, and let γ : A → C be an arbitrary extension of F -algebras. Moreover, let I be a nilpotent ideal of C, and let ρ : B → C/I be a unitary morphism of extensions. Suppose that ρ lifts to a morphism of A-bimodules τ0 : B → C. Then ρ lifts to a unitary morphism of extensions τ : B → C. Proof. Let ν : C → C/I, c 7→ c + I denote the canonical map. Then ν ◦ γ : A → C → C/I is also an extension of A, and we wish to show that there exists a morphism of extensions τ : B → C such that ν ◦ τ = ρ. This means that τ (x) + I = ρ(x) = τ0 (x) + I

for x ∈ B.

P Let w ∈ (B ⊗A B)B such that µ(w) = 1B , and write w = kj=1 xj ⊗ yj with x1 , y1 , . . . , xk , yk ∈ B. Since I is nilpotent, it suffices to construct morphisms of A-bimodules τ0 , τ1 , τ2 , . . . : B → C such that n

n

(i) τn+1 (x) + I 2 = τn (x) + I 2 , n n (ii) τn (xy) + I 2 = τn (x)τn (y) + I 2 for x, y ∈ B and n = 0, 1, 2, . . . . The existence of τ0 follows from our hypothesis. Suppose that τ0 , . . . , τn have already been constructed successfully. Then n

n

τn (1B )2 + I 2 = τn (1B ) + I 2 by (ii), and

τn (1B ) + I = τn−1 (1B ) + I = . . . = τ0 (1B ) + I = ρ(1B ) = 1C/I = 1C + I n

by (i). This implies that τn (1B ) + I 2 following morphisms of A-bimodules:

n

= 1C + I 2 . We consider the

n

θ : B ⊗A B −→ I 2 , x ⊗ y 7−→ τn (xy) − τn (x)τn (y), n

λ : B ⊗A B ⊗A B −→ I 2 , x ⊗ y ⊗ z 7−→ θ(x ⊗ y)τn (z), n

η : B −→ I 2 , x 7−→ λ(x ⊗ w) =

k X

θ(x ⊗ xj )τn (yj ).

j=1

Easy computations show that τn (x)θ(y ⊗ z) + θ(x ⊗ yz) = θ(xy ⊗ z) + θ(x ⊗ y)τn (z)

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and n+1

τn (x)η(y) − η(xy) + η(x)τn (y) + I 2 k X = [τn (x)θ(y ⊗ xj )τn (yj ) − θ(xy ⊗ xj )τn (yj ) j=1

+θ(x ⊗ xj )τn (yj )τn (y)] + I 2 =

n+1

k X [θ(x ⊗ y)τn (xj )τn (yj ) − θ(x ⊗ yxj )τn (yj )+ j=1

= θ(x ⊗ y)

k X

+θ(x ⊗ xj )τn (yj y)] + I 2

n+1

τn (xj yj ) − λ(x ⊗ yw) + λ(x ⊗ wy) + I 2

n+1

j=1

n+1

n+1

= θ(x ⊗ y)τn (1) + I 2 = θ(x ⊗ y) + I 2 n+1 = τn (xy) − τn (x)τn (y) + I 2 for x, y, z ∈ B. Hence

τn+1 : B −→ C, x 7−→ τn (x) + η(x) is a morphism of A-bimodules satisfying (i) and (ii).

¤

We now turn to group algebras. Let H be a subgroup of a finite group G, and let F H and F G denote the corresponding group algebras. The following result is not needed for the proof of the main result of this paper, but it motivates some of the later arguments. Proposition 1.5 (Maschke). The inclusion map F H → F G is a separable extension of F -algebras if and only if the index |G : H| is not divisible by the characteristic p of F . Proof. Suppose first that F H → F G is a separable extension. The map X X g : F [G/H] −→ F, αxH xH 7−→ αxH xH∈G/H

xH∈G/H

is a morphism of F G-modules. It induces a short exact sequence of F Gmodules E : 0 −→ Ker(g) −→ F [G/H] −→ F −→ 0. The morphism of F H-modules h : F → F [G/H], α 7→ αH, shows that the restriction of E to F H splits. Hence, by Proposition 1.3, E splits as well. Hence there is a trivial F G-submodule U of F [G/H] such that F [G/H] = U ⊕ Ker(g). But the only trivial F G-submodule of F [G/H] is P F xH∈G/H xH. We conclude that X 0 6= g( xH) = |G : H|1F . xH∈G/H

Thus |G : H| is not divisible by p = char(F ).

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Now suppose, conversely, that |G : H| is not divisible by p = char(F ), and let T be a transversal for G/H. Then X w := |G : H|−1 t ⊗ t−1 ∈ F G ⊗F H F G t∈T

is independent of T . This implies that X gw = |G : H|−1 gt ⊗ t−1 g −1 g = wg

for g ∈ G.

t∈T

Hence w ∈ (F G ⊗F H F G)F G and µ(w) = |G : H|−1

P t∈T

tt−1 = 1. For any

subgroup H of G, we have a unitary subalgebra (F G)H = (F G)F H = {x ∈ F G : hx = xh for h ∈ H} of F G, and for any subgroup I of H, we have a relative trace map X I H TrH x 7−→ hxh−1 . I : (F G) −→ (F G) , hI∈H/I H Standard properties imply that the image (F G)H I of TrI is an ideal of (F G)H . ¤

Now let b be a primitive idempotent in Z(F G). Then the F -algebra B := F Gb = bF G is called a block of F G. We denote by P a defect group of B. By definition, P is a subgroup of G, minimal with respect to the condition b ∈ (F G)G P . It is well known that P is a p-subgroup of G where p = char(F ). The following result is related to Proposition 1.5 and not really necessary for the remainder of the paper. Proposition 1.6. In the situation above, the map F P → B, x 7→ xb, is a separable extension of F -algebras. P Proof. Let x ∈ (F G) TrG P (x) = b, and let T be a transversal for P such that−1 G/P . Then w := t∈T txb ⊗ bt ∈ B ⊗F P B is independent of T . Then, for g ∈ G, we have X gw = gtxb ⊗ bt−1 g −1 g = wg. t∈T

Hence w ∈ (B ⊗F P B)B , and µ(w) = b.

P t∈T

G txbt−1 = TrG P (xb) = TrP (x)b = ¤

In this section, we closely followed the exposition in [7].

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2. Brauer pairs As the name indicates, Brauer pairs were first used by R. Brauer [2]. The concept was extended by Alperin and Brou´e [1]. A somewhat different but equivalent approach was given by Brou´e and Puig [3], and this is the approach we will be using here. In the following, let F be a field of characteristic p > 0, and let G be a finite group. A Brauer pair (P, e) of F G consists of a p-subgroup P of G and a primitive idempotent e in Z(F CG (P )). We recall the Brauer homomorphism X X BrP : (F G)P −→ F CG (P ), αg g 7−→ αg g, g∈G

g∈CG (P )

a homomorphism of F -algebras defined for every p-subgroup P of G. We also recall the following standard facts: X \ Ker(BrP ) = (F G)PQ and (F G)P1 = Ker(BrQ ). Q
1
Theorem 2.1 (Broue–Puig). Let (P, e) be a Brauer pair of F G, and let Q be a subgroup of P . Then there exists a unique primitive idempotent f in Z(F CG (Q)) such that BrQ (i)f = BrQ (i) for every primitive idempotent i in (F G)P with BrP (i)e 6= 0. Proof. Suppose that f1 and f2 are primitive idempotents in Z(F CG (Q)) satisfying the condition above. There certainly exists a primitive idempotent i in (F G)P such that BrP (i)e 6= 0. Then 0 6= BrQ (i) = BrQ (i)f2 = BrQ (i)f1 f2 , so f1 f2 6= 0 and f1 = f2 . Thus the uniqueness is proved. In order to show existence we argue by induction on |P : Q|. Supose first that P = Q and that i is a primitive idempotent in (F G)P such that BrP (i)e 6= 0. Since BrP (i) is a primitive idempotent in F CG (P ) this implies that BrP (i)e = BrP (i), so the condition is satisfied with f := e. Now suppose that Q < P , so that Q < NP (Q) =: R. By induction, there is a primitive idempotent g in Z(F CG (R)) such that BrR (i)g = BrR (i) for every primitive idempotent i in (F G)P with BrP (i)e 6= 0. The Brauer homomorphism BrR maps Z(F CG (Q))R into Z(F CG (R)). Let f be a primitive idempotent in Z(F CG (Q))R such that BrR (f )g = g, and let f 0 be a primitive idempotent in Z(F CG (Q)) such that f f 0 = f 0 . Moreover, let T be 0 the stabilizer of f 0 in R, f = TrR T (f ). P under theR conjugation action. Then But f ∈ / Ker(BrR ) = S
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whenever Q < S ≤ R.

Nilpotent blocks revisited

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So let S be a subgroup of R with Q < S. Since BrR (i) 6= 0 there exists a primitive idempotent j in (F G)R such that ij = j = ji and BrR (j) 6= 0. Then BrR (j)g = BrR (ji)g = = BrR (j)BrR (i)g = BrR (j)BrR (i) = BrR (ji) = BrR (j) 6= 0. By induction, there is a primitive idempotent h in Z(F CG (S)) such that BrS (j)h = BrS (j) 6= 0. Then 0 6= BrS (ji)h = BrS (j)BrS (i)h; in particular, we have BrS (i)h 6= 0. By induction, there also exists a primitive idempotent h0 in Z(F CG (S)) such that BrS (i)h0 = BrS (i) 6= 0. Then 0 6= BrS (i)h0 h; in particular, we have h0 h 6= 0. Thus h0 = h and BrS (i)h = BrS (i). Now 0 6= BrR (j)g = BrR (j)BrR (f )g = BrR (BrQ (j)f )g. But BrQ (j) is a primitive idempotent in BrQ ((F G)R ) = F CG (Q)R . Thus BrQ (j)f = BrQ (j). Hence BrS (j)BrS (f )h = BrS (j)h 6= 0; in particular, we have BrS (f )h 6= 0. Since BrS (f ) ∈ Z(F CG (S)) this implies BrS (f )h = h. Hence BrS (BrQ (i)(1 − f )) = BrS (i)(1 − BrS (f )) = BrS (i)h(1 − BrS (f )) = 0, and our first approximation is proved. We conclude that \ BrQ (i)(1 − f ) ∈ Ker(BrS : (F CG (Q))S −→ Q<S≤R P −→ F CG (S)) = (F CG (Q))R Q = BrQ ((F G)Q ).

(This last equality is a standard property of the Brauer homomorphism.) Hence BrQ (i)(1 − f ) = (1 − f )BrQ (i) ∈ BrQ (i(F G)PQ i). But i(F G)PQ i is a proper ideal of the local F -algebra i(F G)P i; thus it is nilpotent. Since BrQ (i)(1 − f ) is an idempotent we obtain BrQ (i)(1 − f ) = 0. ¤ In the situation of Theorem 2.1 we write (Q, f ) ≤ (P, e). The following result is an easy consequence. Corollary 2.2. Let (P, e), (Q, f ), and (R, g) be Brauer pairs of F G. Then the following holds: (i) (Q, f ) ≤ (P, e) if and only if Q ≤ P and BrP (i)e 6= 0 6= BrQ (i)f for some primitive idempotent i in (F G)P . (ii) If (R, g) ≤ (Q, f ) and (Q, f ) ≤ (P, e) then (R, g) ≤ (P, e). It follows that ≤ is a partial order on the set of Brauer pairs of F G. Moreover, G acts by conjugation on this set, and the action of G is compatible with the partial order ≤. Now let b be a primitive idempotent in Z(F G) and let B = F Gb = bF G be the corresponding block. A Brauer pair (P, e) of F G is called a Brauer B-pair if (1, b) ≤ (P, e). This means that BrP (b)e 6= 0.

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Every Brauer pair (P, e) of F G is a Brauer B-pair for a unique block B of F G, and then every Brauer pair (Q, f ) of F G with (Q, f ) ≤ (P, e) is also a Brauer B-pair. Theorem 2.3 (Sylow). Let B be a block of F G with defect group P . Then there exists a Brauer B-pair (P, e). Moreover, if (Q, f ) is an arbitrary Brauer B-pair then (Q, f ) ≤ (P, e)x for some x ∈ G. Proof. Let b be the primitive idempotent in Z(F G) such P that B = F Gb. P . We write b = Then TrG (a) = b for some a ∈ (F G) P i∈I i with pairwise orthogonal primitive idempotents i in (F G)P . Then X X G G P P b = bTrG (a) = Tr (ba) = Tr (ia) ∈ TrG P P P P ((F G) i(F G) ). i∈I

i∈I

P P Each summand TrG P ((F G) i(F G) ) is an ideal in Z(F G). Rosenberg’s P P Lemma implies that b is contained in TrG P ((F G) i(F G) ) for some i ∈ I. If i ∈ (F G)PQ for some proper subgroup Q of P then b ∈ (F G)G Q contradicting the choice of P . Thus i ∈ / (F G)PQ for every proper subgroup Q of P . P Rosenberg’s Lemma implies that i ∈ / Q

Thus there is x ∈ G such that Q ⊆ P x and BrQ ((F G)Q ix (F G)Q f ) 6= 0. Hence 0 6= BrQ (ix )f , and (Q, f ) ≤ (P, e)x . ¤ 3. Source algebras Let F be an algebraically closed field of characteristic p > 0, and let G be a finite group. Moreover, let b be a primitive idempotent in Z(F G), and let P be a defect group of the corresponding block B = F Gb. By Theorem 2.3 and its proof, there exists a primitive idempotent i in (F G)P and a Brauer B-pair (P, e) such that P P b ∈ TrG P ((F G) i(F G) ) ⊆ F GiF G and

BrP (i)e 6= 0.

It follows that i = ib ∈ B and B = F GbF G = F GiF G = BiB. This implies that the F -algebras B and iBi are Morita equivalent. We call i a source idempotent and iBi = iF Gi a source algebra of B. It is not difficult to show that i is essentially unique, and that every primitive idempotent i0 in (F G)P b with BrP (i0 ) 6= 0 is a source idempotent of B. However, these facts are not needed in the following.

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Proposition 3.1. In the situation above, the map F P −→ iBi, x 7−→ xi = ix is a separable extension of F -algebras. P P Proof. We write b = kj=1 TrG P (xj iyj ) with x1 , y1 , . . . , xk , yk ∈ (F G) . Each summand TrG / P (xj iyj ) is contained in the local F -algebra Z(B). Since b ∈ J(Z(B)) we must have c := TrG (x iy ) ∈ / J(Z(B)) for some j ∈ {1, . . . , k}. j j P Thus c has an inverse c−1 ∈ Z(B). Then −1 −1 b = cc−1 = TrG = TrG P (xj iyj )c P (xj iyj c ).

This shows that b = TrG suitable x, y ∈ (F G)P . We choose a P (xiy) for P transversal T for G/P and set v := t∈T txi ⊗ iyt−1 ∈ F G ⊗F P F G. Then v is independent of T . Thus, for g ∈ G, we have X gv = gtxi ⊗ iyt−1 g −1 g = vg t∈T

for g ∈ G. This means that v ∈ (F G ⊗F P F G)F G and X µ(v) = txiyt−1 = TrG P (xiy) = b. t∈T

We consider iF Gi ⊗F P iF Gi as a subset of F G ⊗F P F G and set w := ivi ∈ (iF Gi ⊗F P iF Gi)iF Gi . Then µ(w) = iµ(v)i = ibi = i, and the result follows. ¤ In the following, we consider F G as an F P -bimodule. The indecomposable direct summands of F G have the form F [P gP ] where P gP is a double coset. Now we note that iF Gi is a direct summand of the F P -bimodule F G. We write iF Gi = M1 ⊕ · · · ⊕ Mr with indecomposable F P -bimodules M1 , . . . , Mr . For j = 1, . . . , r, we choose an element gj ∈ G and an isomorphism of F P -bimodules φj : F [P gj P ] → Mj . Then xj := φj (gj ) ∈ φj (P gj P ) =: Xj and Pj := P ∩ gj P gj−1 ≤ P . Let ej be the unique primitive idempotent in Z(F CG (Pj )) such that (Pj , ej ) ≤ (P, e). Proposition 3.2. In the situation above, we have (Pj , ej )gj ≤ (P, e) for j = 1, . . . , r. Proof. For u ∈ Pj , we have uxj gj−1 u−1 = uφj (gj )gj−1 u−1 gj gj−1 = φj (u · gj · gj−1 u−1 gj )gj−1 = xj gj−1 . Thus xj gj−1 ∈ (F G)Pj . Now X := X1 ∪ · · · ∪ Xr is an F -basis of iF Gi that is invariant under left and right multiplication with elements in P . We can extend X to an F -basis Y of F G with the same property. Then Y gj−1 is an F -basis of F G that is invariant under left and right multiplication with elements in Pj . Since xj gj−1 ∈ Y gj−1 ∩ (F G)Pj we have BrPj (xj gj−1 ) 6= 0.

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Since xj gj−1 = i·xj gj−1 ·gj igj−1 there are primitive idempotents ij in i(F G)Pj i −1

and i0j in i(F G)gj

P j gj

i such that

0 6= BrPj (ij · xj gj−1 · gj i0j gj−1 ) = BrPj (ij )BrPj (xj gj−1 )BrPj (gj i0j gj−1 ). Thus the primitive idempotents BrPj (ij ) and BrPj (gj i0j gj−1 ) are contained in the same block of F CG (Pj ). Now (Pj , ej ) ≤ (P, e) implies that BrPj (ij )ej

= BrPj (ij i)ej = BrPj (ij )BrPj (i)ej = BrPj (ij )BrPj (i) = BrPj (ij i) = BrPj (ij ).

Hence BrPj (ij ) and BrPj (gj i0j gj−1 ) are contained in F CG (Pj )ej . Hence 0 6= BrPj (gj i0j gj−1 )ej = BrPj (gj i0j gj−1 )BrPj (gj igj−1 )ej ; in particular, we have BrPj (gj igj−1 )ej 6= 0. Since also Brgj P g−1 (gj igj−1 )gj egj−1 6= 0 j

we get (Pj , ej ) ≤

gj (P, e)gj−1 .

¤

The block B of F G is called nilpotent if the following condition is satisfied: (∗) Whenever (Q, f ) is a Brauer B-pair such that (Q, f ) ≤ (P, e) and whenever g ∈ G is such that (Q, f )g ≤ (P, e) then there are elements c ∈ CG (Q) and u ∈ P such that g = cu. From now on we suppose that our block B of F G is nilpotent. Then, by Proposition 3.2, we may assume that gj ∈ CG (Pj ) for j = 1, . . . , r. The map F P −→ F P ⊗F iF Gi, u 7−→ u ⊗ ui (u ∈ P ) is an extension of F -algebras. Moreover, the map σ : F P ⊗F iF Gi −→ iF Gi, u ⊗ x 7−→ x (u ∈ P, x ∈ iF Gi) is a morphism of extensions of F P . Moreover, I := Ker(σ) = J(F P ) ⊗F iF Gi is a nilpotent ideal of F P ⊗F iF Gi. Theorem 3.3 (Puig). There exists an extension of F -algebras α : F P −→ Mat(n, F ) (i.e., a representation of F P ) and an isomorphism of extensions of FP, iF Gi −→ F P ⊗F Mat(n, F ). Proof. The morphism of extensions σ : F P ⊗F iF Gi → iF Gi induces an isomorphism of extensions σ ¯ : (F P ⊗F iF Gi)/I → iF Gi. Let ρ := σ ¯ −1 . We can consider iF Gi and F P ⊗F iF Gi as permutation modules over F [P × P ]. We claim that, for j = 1, . . . , r, the stabilizers of xj ∈ iF Gi and of 1 ⊗ xj ∈ F P ⊗F iF Gi coincide. Indeed, let u, v ∈ P such that uxj v = xj . Then

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ugj v = gj , so u = gj v −1 gj−1 ∈ P ∩ gj P gj−1 = Pj . Since gj ∈ CG (Pj ) this implies u = gj−1 ugj = v −1 and u(1 ⊗ xj )v = uv ⊗ uxj v = 1 ⊗ xj , and our claim is proved. It follows that there is a morphism of F P -bimodules τ0 : iF Gi −→ F P ⊗F iF Gi such that τ0 (xj ) = 1 ⊗ xj for j = 1, . . . , r. Then σ ¯ (τ0 (xj ) + I) = σ(τ0 (xj )) = xj , i.e., τ0 (xj ) + I = ρ(xj ) for j = 1, . . . , r. Thus τ0 lifts ρ. Since F P → iF Gi is a separable extension, Theorem 1.4 implies that there is a morphism of extensions τ : iF Gi → F P ⊗F iF Gi lifting ρ. Hence σ(τ (x)) = σ ¯ (τ (x) + I) = σ ¯ (ρ(x)) = x for x ∈ iF Gi; in particular, we have F P ⊗F iF Gi = τ (iF Gi) ⊕ I. Let M be a simple iF Gi-module. Then F P ⊗F M is an F P ⊗F iF Gimodule. We can restrict F P ⊗F M to iF Gi (via τ ) and then further to F P (via u 7→ ui). Then F P acts on F P ⊗F M in such a way that u(v ⊗ m) = uv ⊗ ium for u, v ∈ P , m ∈ M . Thus F P ⊗F M is a projective F P -module. Since F P → iF Gi is a separable extension, Proposition 1.3 implies that F P ⊗F M is a projective iF Gi-module as well. On the other hand, F P ⊗F M has a composition series of length |P |, with all composition factors isomorphic (both over F P ⊗F iF Gi and over iF Gi). This implies that M is the only simple iF Gi-module, up to isomorphism, and F P ⊗F M is its projective cover. In particular, F P ⊗F M is a faithful iF Gi-module. But F P ⊗F J(iF Gi) annihilates F P ⊗F M . Thus the map τ

φ : iF Gi−→F P ⊗F iF Gi −→ F P ⊗F (iF Gi/J(iF Gi)) is injective. Comparing dimensions we see that φ is even an isomorphism, an isomorphism of extensions of F P . The result follows since iF Gi/J(iF Gi) ∼ = Mat(n, F ) for some n. ¤ Corollary 3.4. Let B be a nilpotent block of F G with defect group P . Then B is isomorphic (as an F -algebra) to Mat(m, F P ) for some m. Proof. We know that B is Morita equivalent to its source algebra iBi, and by Theorem 3.3 iBi is Morita equivalent to F P . Since F P is a local F -algebra, the result follows. ¤ References [1] J. Alperin and M. Brou´e, Local methods in block theory, Ann. Math. (2) 110 (1979), 143–157. [2] R. Brauer, On the structure of blocks of characters of finite groups, Proc. Second Internat. Conf. Theory of Groups (Austral. Nat. Univ., Canberra, 1973), pp. 103– 130, Springer-Verlag, Berlin 1974. [3] M. Brou´e and L. Puig, Characters and local structure in G-algebras, J. Algebra 63 (1980), 306–317.

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[4] K. Hirata and K. Sugano, On semisimple and separable extensions over noncommutative rings, J. Math. Soc. Japan 18 (1966), 360–373. [5] L. Kadison, New Examples of Frobenius Extensions, Amer. Math. Soc., Providence 1999. [6] B. K¨ ulshammer, Lectures on Block Theory, Cambridge University Press, Cambridge 1991. [7] B. K¨ ulshammer, T. Okuyama and A. Watanabe, A lifting theorem with applications to blocks and source algebras, J. Algebra 232 (2000), 299–309. [8] L. Puig, Nilpotent blocks and their source algebras, Invent. math. 93 (1988), 77–116. [9] J. Th´evenaz, G-Algebras and Modular Representation Theory, Clarendon Press, Oxford 1995.

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Decomposition of central units of integral group rings Yuanlin Li Department of Mathematics, Brock University, St. Catharines, Ontario, L2S 3A1, Canada [email protected] Michael M. Parmenter Department of Mathematics and Statistics, Memorial University of Newfoundland, St. John’s, NL, A1C 5S7, Canada [email protected]

The following theorem concerning integral group rings ZG was proved by Polcino Milies and Sehgal in [4] (see also Jespers, Juriaans, de Miranda, and Rogerio [3]). Theorem. Let Φ(G) denote the F C subgroup of a group G and let T = T Φ(G) be the torsion subgroup of Φ(G). Then every central unit u of ZG can be written in the form u = wg with w ∈ ZT and g ∈ Φ(G). Recently Dokuchaev, Polcino Milies, and Sehgal [1] characterized precisely those groups G such that ZG has only trivial central units, using the above theorem to reduce the general case to a case with more manageable groups (the case where G is finite had been settled earlier by Ritter and Sehgal [6]). Clearly the decomposition u = wg is not unique - in fact, u = (wt−1 )(tg) for any t ∈ T . In this paper we investigate the following question. Question. Is it possible to arrange the decomposition u = wg in such a way that w and g are both central units in ZG? We will say that a group G has the Central Decomposition Property (CDP) if the above question has a positive answer. Our initial belief was that many groups would not have CDP, but this has proved difficult to check. However, we do present one construction that yields examples of groups without CDP in the next section. After that we verify that certain classes of groups do have CDP. For information and notation concerning units of integral group rings the reader should consult [5] or [7]. 275

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1. Example In this section we give an example of a central unit u in an integral group ring such that any decomposition u = vg, where v ∈ ZT and g ∈ G, has the property that g ∈ / Z(G). First recall that a group G has the normalizer property in ZG if NU (G) = Z(U) · G (here U denotes the group of units of ZG). For many years it was an open question as to whether all integral group rings had this property (for information see [7]), but an example showing this is not the case was constructed by Hertweck [2]. Example 1.1. Let T be a finite group that does not have the normalizer property, and choose w ∈ NU (T ) with w ∈ / Z(U) · T . Let G be the semidirect product of T and an infinite cyclic group < x > −1 where tx = tw for all t ∈ T . We will show that u = wx is a central unit in ZG and moreover, that u is the example we are looking for. First note that since w is invariant under conjugation by x to show that u is central it is enough to show that u commutes with elements of T . But −1 if t ∈ T, tu = twx = (tw )x = (tw )w since w ∈ NU (T ), and so tu = t and u is central. To see that u is an example of the type we want, assume u = vg with v ∈ ZT and g ∈ G. Then w = vgx−1 and, since w and v are both in ZT, gx−1 ∈ T . But w ∈ / Z(U) · T so we must have v ∈ / Z(U) and thus g ∈ / Z(G) as desired (note that U denotes U(ZT ) throughout this example). ¤ We do not know any other method of constructing such examples. 2. Results The simplest case to consider when looking for groups with CDP is G =< T, g > where T / G is finite and g is of infinite order, and we will assume that without further comment in the rest of this section. Also, in view of the previous example, we will assume that T satisfies the normalizer property in ZT . If u is a central unit in ZG, we have u = vg i where v ∈ U(ZT ) by the Polcino Milies and Sehgal Theorem. Also v ∈ NU (T ) (since u is central and T /G), so the normalizer property tells us that v = zt1 , where z ∈ Z(U(ZT )) and t1 ∈ T . Hence u = zt1 g i . Here are a few basic facts. Lemma 2.1. Using the notation just described, we have the following. (i) z and t1 commute. (ii) t1 and g i commute. (iii) z and g i commute. (iv) zt1 and g commute.

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Proof. (i) z is central in ZT . (ii) Since u is central, and using (i), we have zt1 g i = t1 zg i = zg i t1 . So t1 g i = g i t1 . (iii) zt1 and t1 both commute with g i . (iv) zt1 g i and g i both commute with g. ¤ We begin by noting a few easily proved families of groups with CDP. Proposition 2.2. G has CDP in each of the following cases. (i) T abelian. (ii) g p ∈ Z(G) for some prime p. (iii) Z(T ) = {1}. (i) We have u = zt1 g i as described earlier. Since zt1 commutes with g by Lemma 2.1(iv) and also with all elements of T, zt1 is central in ZG. Hence g i is in Z(G) and we are done. (ii) Again we have u = zt1 g i . If p | i , then g i ∈ Z(G). If p 6 | i then α1 p + α2 i = 1 for some integers α1 , α2 and g = (g p )α1 (g i )α2 . Since z commutes with g i by Lemma 2.1 (iii) and with g p , z commutes with g. So z is central and t1 g i ∈ Z(G). (iii) Again u = zt1 g i . We know zt1 commutes with g by Lemma 2.1 (iv), so gzg −1 gt1 g −1 t−1 = z. Since conjugation by g per1 mutes conjugacy class sums in ZT, gzg −1 is central in ZT and so gt1 g −1 t−1 1 ∈ Z(T ). Hence g commutes with both zt1 and t1 , hence with z. Thus z is a central unit in ZG, and t1 g i ∈ Z(G). ¤

Proof.

Here is our main result. Theorem 2.3. Assume that T is a p-group and Z(T ) is cyclic. Then G has CDP. Proof. Again we have u = zt1 g i as described earlier. We may also assume u has augmentation 1. Say Z(T ) =< x > and |Z(T )| = pl . Since < x > is characteristic in T and T / G, xg = xk for some integer k where p 6 |k. Also, as in the proof of −1 −1 i Proposition 2.2(iii), gzg −1 gt1 g −1 t−1 1 = z so gt1 g t1 = x for some i (note −1 gzg and z are central units in ZT ). First consider the case that i, k, and pl are such that there exists j0 , 0 ≤ j0 ≤ pl − 1 such that kj0 + i ≡ j0 (mod pl ). Then we have (zx−j0 )g = −1 i−j0 z g x−kj0 = z(gt1 g −1 t−1 = zx−i xi−j0 = zx−j0 . So zx−j0 is a central 1 ) x −j j unit in ZG and u = (zx 0 )(x 0 t1 g i ) gives the result. So we can assume no such j0 exists. Next observe that if k 6≡ 1 (mod p) and j1 6≡ j2 (mod pl ) then (k −1)j1 6≡ (k −1)j2 (mod pl ) and so kj1 +i−j1 6≡

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kj2 + i − j2 (mod pl ). But then there exists j0 such that kj0 + i − j0 ≡ 0 (mod pl ), which is a contradiction. Therefore we can assume k ≡ 1 (mod p). Since z is a central unit in ZT we have l z = a0 + a1 x + · · · + apl −1 xp −1 + α where α is a Z-linear combination of conjugacy class sums of size > 1. Note that a0 + a1 + · · · + apl −1 ≡ 1 (mod p). Also we have l

z = gzg −1 gt1 g −1 t−1 = (a0 + a1 xk + · · · + apl −1 xk(p −1) + αg )xi 1 l = a0 xi + a1 xk+i + · · · + apl −1 xk(p −1)+i + αg xi . Since neither the support of α nor the support of αg xi contains any powers of x we must have l

l

a0 + a1 x + · · · + apl −1 xp −1 = a0 xi + a1 xk+i + · · · + apl −1 xk(p −1)+i . So for any j, 0 ≤ j ≤ pl − 1, aj = akj+i = ak2 j+ki+i = · · · . Choose nj minimal such that k nj j + k nj −1 i + · · · + ki + i ≡ j (mod pl ). Hence (kj + i − j)(k nj −1 + k nj −2 + · · · + 1) ≡ 0 (mod pl ). Since k ≡ 1 (mod p) and we are assuming kj + i − j 6≡ 0 (mod pl ), this says that p|nj . But this holds for all j, 0 ≤ j ≤ pl − 1, and thus p|a0 + a1 + · · · + apl −1 , contradicting a0 + a1 + · · · + apl −1 ≡ 1 (mod p). So this case cannot occur and we are done. ¤ Open question Let G =< T, g > where T /G is finite, g is of infinite order, and T satisfies the normalizer property in ZT . Does G have CDP? Acknowledgments This research supported in part by NSERC. References [1] M. Dokuchaev, C. Polcino Milies, and S.K. Sehgal, Integral group rings with trivial central units II, Comm. Algebra (to appear). [2] M. Hertweck, A counterexample to the isomorphism problem for integral group rings, Ann. of Math 154 (2001), 115–138. [3] E. Jespers, S.O. Juriaans, J.M. de Miranda, and J.R. Rogerio, On the normalizer problem, J. Algebra 247 (2002), 24–36. [4] C. Polcino Milies and S.K. Sehgal, Central units of integral group rings, Comm. Algebra 27 (1999), 6233–6241. [5] C. Polcino Milies and S.K. Sehgal, An Introduction to Group Rings, Algebras and Applications, Kluwer Academic Publishers, Dordrecht, 2002. [6] J. Ritter and S.K. Sehgal, Integral group rings with trivial central units, Proc. Amer. Math. Soc. 108 (1990), 327–329. [7] S.K. Sehgal, Units in Integral Group Rings, Longman Scientific and Technical Press, Harlow, 1993.

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Generic units in ZC Zbigniew Marciniak Institute of Mathematics, Warsaw University, Banacha 2, 02–097 Warsaw, Poland [email protected] Sudarshan K. Sehgal Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, T6G 2G1, Canada [email protected]

1. Background Let ZG denote the integral group ring of a group G and let UZG denote its group of invertible elements. Let ε : ZG → Z be the augmentation homomorphism: ³X ´ X ε ag g = ag . g∈G

g∈G

For any unit u ∈ U(ZG) holds ε(u) = ±1. The subgroup of units with augmentation 1 will be denoted by U1 ZG and called the group of normalized units. We have UZG = ±U1 ZG. Let C = Cn = hxi be the cyclic group of order n. We have QC =

⊕ X

Q(ξ d ),

d|n

where ξ = ξn is a primitive complex n-th root of unity. Under the above identification, we have the embedding ZC ⊂

⊕ X

Z[ξ d ] = M,

d|n

where M is the maximal order of the algebra QC. Let M× denote the group of invertible elements in M. Then ´ X ³ ϕ(d) The rank of UZC = the rank of M× = −1 . 2 2
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For details consult [6], [7], [8], [9]. In the present paper we are interested in the following: Question. When is an element f (x) =

X

ai xi ∈ ZC

i

a unit for (many values of) n? 2. Examples of units Clearly, the elements ±g, g ∈ G, are invertible in ZG. These are the trivial units. The other known units are motivated by the cyclotomic units of Z[ξ]. Namely, for (i, n) = 1 we have 1−ξ i 1−ξ ξi + · · ·

v = 1 + ξ + · · · + ξ i−1 = v −1 =

1−ξ 1−ξ i

=

1−ξ ik 1−ξ i

=1+

+ ξ i(k−1) ∈ Z[ξ]

for some k satisfying ik ≡ 1 (mod n). We can consider something similar in ZC: α = 1 + x + · · · + xi−1 7→ (i, · · · , v, · · · ) ∈ M. All entries except the first one are units in the corresponding components. The first one is equal to i hence α is not a unit in ZC. This obstruction was overcome by Bass - he suggested a modification of α in the following way: u = (1 + x + · · · + xi−1 )ϕ(n) + u 7→ (1, · · · , v, · · · ) ∈ M × .

1−iϕ(n) n

(1 + x + · · · + xn−1 ),

These u’s are called Bass cyclic units and B1 (C) is the usual notation for the group generated by them. ¯ ¯ Theorem 2.1. ¯U1 ZC : B1 (C)¯ < ∞. There are also other ways to handle the augmentation obstruction: 1 + xj + · · · + xj(i−1) 1 + x + · · · + xi−1 with (ij, n) = 1. It can be proved that uij ∈ UZC; this is a Hoechsmann unit. uij =

We denote by H(C) the group generated by these units. We also define ® Y(C) = Hhxd i : d|n . There is no need to modify the augmentation, if we consider an alternating sum 1 − x + x2 − · · · + (−1)k−1 xk−1 . This is a unit iff n is odd and (k, 2n) = 1. We call them alternating units.

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3. The new units Consider in U1 ZC5 the units v = x + x−1 − 1

or u = xv = 1 − x + x2 .

These units are very beautiful as they are invertible in all ZCn with (n, 6) = 1. This can be seen in a number of ways, e.g., u is an alternating unit. Definition 3.1. [5] A monic polynomial f (X) ∈ Z[X] defines a generic unit if there exists a natural number D such that f (x) is a unit in ZCn for all (n, D) = 1. We can classify all such polynomials. Theorem 3.2. [5] A monic polynomial f (X) defines a generic unit iff it is shifted cyclotomic and f (1) = ±1. A polynomial is shifted cyclotomic means all its roots are zero or roots of unity. Thus f (X) above is of the form Y Xk · φmi (X) i

where φmi is the (irreducible) mi -th cyclotomic polynomial with φmi (1) = ±1. The following lemma recalls the well-known properties of cyclotomic polynomials. Technical Lemma 3.3. Y (X d − 1)µ(m/d) . (i) φm (X) = d|m α1 −1 s −1 ···pα s

(ii) φpα1 ···pαs s (X) = φp1 ···ps (X p1 1

(iii) If (m, l) = 1 then φml (X) =

Y

).

φl (X d )µ(m/d) .

d|m

(iv) If (e, m) = 1 then φm (X e ) =

Y

φmd (X).

d|e

(v) φm (1) = 1 ⇔ m is not a prime power ⇔ ξm − 1 is invertible in Z[ξm ]. In the above µ denotes the Moebius function; for proofs see [4] and [1]. So when is φm (x) ∈ ZCn a unit? Proposition 3.4. The element φm (x) ∈ ZCn is a unit iff m/(m, n) is not a prime power. Proof. (⇒) Suppose φm (x)q(x) = 1. Then φm (X)q(X) = 1 + (X n − 1)p(X)

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for some p(X) ∈ Z[X]. Plugging in ξm for X we get n 1 = −(ξm − 1)p(ξm ). n − 1 is a unit in Z[ξ ]. Then m/(m, n), the order of ξ n , This means that ξm n m is not a prime power, as claimed. The converse is similar. ¤

Definition 3.5. Let G(Cn ) denote the the subgroup of UZCn generated by the set {φm (x) : m/(m, n) is not a prime power}. We call it the subgroup of generic units. Theorem 3.6. (i) G(Cn ) ∩ Cn = hx2 i. (ii) G(Cn ) is generated by units φpq (xd ), (pq, n) = 1, d ∈ Z.

where

p 6= q are primes,

To see (i), pick two distinct primes p, q ≡ −1(mod n). Then φpq (x) ∈ G(Cn ) and φp (xq ) 1 + xq + · · · + xq(p−1) = φp (x) 1 + x + · · · + xp−1 −1 1 + x + · · · + x−(p−1) = = (x−1 )p−1 = x2 . 1 + x + · · · + xp−1 The proof of (ii) reduces to a sequence of transformations involving the identities from the Technical Lemma. Let us recall the various subgroups of units: • the Bass cyclic group; • H(C) generated by all Hoechsmann units φpq (x) =

1 + xk + · · · + xk(`−1) with (k`, |C|) = 1; 1 + x + · · · + x`−1 • Y(C) generated by ∪{H(C 0 ) : C 0 ≤ C}. uk` =

The following inclusions hold for any finite cyclic group C: B1 (C) ⊂ Y(C) ⊂ U1 ZC. We also know that, generally speaking, the containments are proper. Theorem 3.7. G(Cn ) = Y(Cn ). The proof relies on the observation that φpq (x) =

φp (xq ) 1 + xq + · · · + xq(p−1) = = upq (x) φp (x) 1 + x + · · · + xp−1

is a Hoechsmann unit. The identities from the Technical Lemma then complete the proof.

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4. An independence criterion Let us introduce a log map L∗ . For that we need some notation. We denote by τk the endomorphism of Cn sending x to xk , 1 ≤ k ≤ n. Let En be the semigroup of all such endomorphisms. We have [ En = {τk : 1 ≤ k ≤ n} = {τk : 1 ≤ k ≤ n, (k, n) = d}. d|n

¯ The map τk 7→ τk/(k,n) identifies the subset corresponding to d¯n with the automorphism group An/d = {τ` : 1 ≤ ` < n/d, (`, n/d) = 1} ≈ Authxd i. Let ZEn be the contracted semigroup ring of En . Let En∗ be the factor semigroup En /hτ∗ i where τ∗ is the standard involution x 7→ x−1 . Finally, let ZEn∗ denote the contracted semigroup ring of En∗ ; clearly ZEn∗ = ZEn /(τ∗ − 1)ZEn . In the sequel we will use the identifications a a En ≈ τd An/d and En∗ ≈ τd A∗n/d , where A∗n/d = An/d /hτ∗ i. d|n

d|n

For each positive integer m define L(φm (x)) ∈ ZEn by the formula X ³m´ µ L(φm (x)) = τd . d d|m ¯ Notice that the divisors d¯m that are multiples of n give no input ¡into the¢ sum, as τn = 0 ∈ En and ring is contracted. Let L∗ φm (x) ¡ the semigroup ¢ denote the image of L φm (x) under the natural homomorphism ZEn → ZEn∗ . Theorem 4.1. We have a well-defined map L∗ : G(Cn ) −→ ZEn∗ such that L∗ (u · v) = L∗ (u) + L∗ (v) holds for all u, v ∈ G(Cn ). The kernel of this map coincides with the group Cn ∩ G(Cn ). Corollary 4.2. Let u1 , . . . , us be generic units. If the elements L∗ (ui ) are additively independent in ZEn∗ then the units ui are multiplicatively independent, i.e., they form a basis of a free abelian subgroup of UZCn . The proof of the theorem above is based on the Theorem 4.3 (Bass Independence Theorem). Let κi , 0 < i < n, be integers such that κn−i = κi for all i. If for all complex characters χ : Cn −→ C holds n−1 Y ¡ ¢κ e 1 − χ(xi ) i = 1 i=1

then κi = 0 for all i. Here e : C −→ C, e(z) = z for z 6= 0 and e(0) = 1.

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This in turn is an extension of Franz Independence Theorem, which depends on the properties of L-functions, see [8]. 5. An explicit basis of a finite index subgroup of UZCn We describe an algorithm that provides an explicit basis of a finite index subgroup in the group of units of ZCn . It generalizes a result from [3], yielding such an algorithm when n is a prime power. The general idea is to pick a few “stem” units φmj (x) and subsets Sj ⊂ En so that the union of partial orbits {φmj (xa ) : a ∈ Sj } forms a basis. To simplify the presentation, let n be odd. First we pick the stem units. Assume that n = pα1 1 · . . . · pαν ν with 2 < p1 < · · · < pν . Then An = Aut(Cn ) = Aut(Cpα1 ) × · · · × Aut(Cpαν ν ) 1

and each factor in this product is cyclic. Choose aj ∈ An so that An = ha1 i × · · · × has i is the above direct product decomposition. Now, for j = 1, . . . , s, pick pairs of different primes qj , qj0 so that aj = τqj = τqj0 in An . This is possible by the Dirichlet’s theorem on arithmetic progressions. Our stem units are φqj qj0 (x), 1 ≤ j ≤ s. ` Now we define the subsets Sj ⊂ En ≈ d|n τd An/d . [ We construct them as unions Sj = τd Sjd with Sjd ⊂ An/d . d|n

Let d|n be fixed. Let ρd : An −→ An/d , ρd (τk ) = τk|hxd i = τk(mod n/d) , bi = ρd (ai ), and Bi = hbi i ≤ An/d . Let t be the largest index for which Bt 6= 1 and let r = #Bt . Consider the subset T ⊂ Bt , covering exactly half of Bt , defined as T = {bit : i < r/2}. Then we set ¢ ¡ ¢ ¡ Sjd = (Bj \ {1}) × Bj+1 × · · · × Bt−1 × Bt ∩ B1 × · · · × Bt−1 × T . In particular, Std = T \ {1} and Sjd = ∅ for j > t. It also follows that Sjd = ∅ S when Bj = 1. We set Sj = {τd Sjd : d|n}. Theorem 5.1. For any cyclic group Cn the following set of generic units φqj qj0 (xa ),

a ∈ Sj ,

1≤j≤s

forms a basis of a free abelian, finite index subgroup in UZCn . Example 5.2. Consider the group C65 . Aut C65 = C4 × C12 = ha1 i × ha2 i,

a1 = τ53 ,

a2 = τ41 .

We pick two pairs of primes q1 = 53, q10 = 313; xq1 q2 = 41, q20 = 431; xq2

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0

= xq1 0 = xq2

= a1 (x), = a2 (x).

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The stem units are u1 = φ53·313 (x) and u2 = φ41·431 (x). Define the sets S1 = {ai11 ai22 : 1 ≤ ii ≤ 3, 0 ≤ i2 < 6} ∪ {τ13 a1 }, S2 = {ai22 : 1 ≤ i2 < 6} ∪ {τ5 ai22 : 1 ≤ i2 < 6}. A basis is {u1 (xa ) : a ∈ S1 } ∪ {u2 (xa ) : a ∈ S2 }. 6. Alternating-like units Recall that for odd integers k the element uk = 1 − x + x2 − · · · + xk is an alternating unit in ZCn , if n is odd and (k, 2n) = 1. It is known that the subgroup generated by all alternating units is of finite index in U1 ZC iff n = pα or n = 3pα , p is an odd prime and 2 or −2 is a primitive root mod n. Observe that 1 + x3 φ2 (x3 ) 1 − x + x2 = = 1+x φ2 (x) and similarly 1 + xk φ2 (xk ) uk = = . 1+x φ2 (x) (m)

More generally, we can consider uk = φm (xk )/φm (x) with (n, mk) = 1. This is a unit in G(Cn ). Let us fix m relatively prime to n and vary (k, n) = 1. We get a family of alternating-like units. Let A(m) be the subgroup generated by them. When is it of finite index? Let m = q1α1 · . . . · qsαs and let ai ∈ A∗n be the coset of τqi for i = 1, . . . , s. Theorem 6.1. (i) If A∗n is not cyclic then the group A(m) of alternating-like units is of infinite index in the group of units UZCn . (ii) If A∗n is cyclic then A(m) is of finite index in UZCn iff each ai generates A∗n . (iii) In particular, for every cyclic p-group Cpα there exists a prime q such that the alternating-like units of the form φq (xk )/φq (x) generate a finite index subgroup in UZCpα . 7. Polynomials defining generic units Let f (X) ∈ Z[X] be a polynomial defining a generic unit. We have classified f (X) under the assumption that f is monic. Can we remove this condition? Not so far - it seems to be not easy. However, we can do this for polynomials f of degree 2 or 3: Theorem 7.1.

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(i) The only quadratic polynomial that defines a nontrivial normalized unit in the group ring of a cyclic group is f (X) = X 2 − X + 1. (ii) There exist no nontrivial units of degree 3. Acknowledgments Research partially supported by NSERC Grant No. A5300 and Polish KBN Grant No. 1 P03A 005 26. References [1] H. Hasse, Number Theory, Springer Verlag, 1980. [2] K. Hoechsmann, Constructing units in commutative group rings, Manuscripta Math. 75 (1992), pp. 5–23. [3] K. Hoechsmann and J. Ritter, Constructible units in abelian p-group rings, J. Pure Applied Algebra 68 (1990), pp. 325–339. [4] S. Lang, Algebra, Addison-Wesley, Reading Mass., 1965. [5] Z. Marciniak and S.K. Sehgal, Generic units in abelian group rings, to appear in J. Group Theory. [6] C. Polcino Milies and S.K. Sehgal, An Introduction to Group Rings, Kluwer, Dordrecht, 2002. [7] S.K. Sehgal, Topics in Group Rings, Marcel Dekker, New York, 1978. [8] S.K. Sehgal, Units in Integral Group Rings, Longman, Essex, 1993. [9] S.K. Sehgal, Group Rings, in: Handbook of Algebra, Vol. 3, Elsevier Science B.V., 2003.

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On quasi-Frobenius semigroup algebras Boris V. Novikov Kharkov National University, Kharkov, Ukraine [email protected] Abstract. We define quasi-Frobenius semigroups and find necessary and sufficient conditions under which a semigroup algebra of a 0-cancellative semigroup is quasi-Frobenius.

I.S. Ponizovskii was first who considered quasi-Frobenius semigroup algebras over a field for finite semigroups. In [7] he described the commutative case completely. Here we study another class of semigroups (0-cancellative). As an example we consider so-called modifications of finite groups [3, 4, 5]. A semigroup S is called 0-cancellative if for all a, b, c ∈ S from ac = bc 6= 0 or ca = cb 6= 0 it follows a = b. In what follows S denotes a finite 0-cancellative semigroup. As the next assertion shows, these semigroups can be called elementary analogously to [7, 6]. Let e be an identity of S, H the subgroup of invertible elements of S (if S does not contain an identity, we set H = ∅), N = S \ H. Lemma 1. N is a nilpotent ideal of S. Proof. If ab = e then bab = b, whence (0-cancellativity) ba = e. Hence every right inverse element is also left inverse. This implies that N is an ideal. Let a ∈ S. Since S is finite, am = an where, e. g., m < n. It follows from 0-cancellativity that either a ∈ H or am = 0. Therefore N is a nil-semigroup, and since |N | < ∞, N is nilpotent [2]. ¤ If A is a semigroup or a ring we shall denote by lA (B) and rA (B) the left and right annihilators of a subset B ⊂ A, respectively. Analogously with Ring Theory, we call a semigroup S quasi-Frobenius if rS lS (R) = R for every right ideal R and lS rS (L) = L for every left ideal L of S. Lemma 2. If S is quasi-Frobenius then S contains an identity (so H 6= ∅). Proof. Suppose that S does not have an identity. Then S is nilpotent by Lemma 1. Let S n = 0, S n−1 6= 0. Then 0 = lS rS (0) = lS (S) ⊃ S n−1 , which is impossible. ¤ 287

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Let F be a field, F0 S the contracted semigroup algebra of S (i.e., a factor of F S obtained by gluing the zeroes of S and F S). For A ⊂ S we denote by F0 A the image of the vector space F A in F0 S. Lemma 3. If F0 S is quasi-Frobenius then S is quasi-Frobenius too. Proof. First show that rF0 S (F0 A) = F0 rS (A) lF0 S (F0 A) = F0 lS (A)

(1) (2)

for any subset A ⊂ S. Indeed, rF0 S (F0 A) = rF0 S (A) ⊃ F0 rS (A). X X Conversely, if αs s ∈ rF0 S (A) (αs ∈ F ) then αs as = 0 for a ∈ A. s∈S

s∈S

Different nonzero summands on the left side of this equality cannot equal one another because of 0-cancellativity. Hence X X αs s = αs s ∈ F0 rS (A). s∈S

s∈rS (A)

The equation (2) is proved analogously. Now for right ideal R ⊂ S we get from (1) and (2): F0 rS lS (R) = rF0 S [F0 lS (R)] = rF0 S lF0 S (F0 R) = F0 R. Further, F0 A = F0 B implies A ∪ 0 = B ∪ 0 (here 0 is the zero of S). Since 0 ∈ R ⊂ rS lS (R) we have rS lS (R) = R. Similarly lS rS (L) = L. ¤ Let S contains an identity (i.e., H 6= ∅), N n = 0 6= N n−1 . Denote M (S) = N n−1 . Evidently S \ 0 is a disjoint union of cosets of H. Since N is an ideal, HM (S) ⊂ M (S) and M (S) consists of cosets of H as well. Lemma 4. If S is quasi-Frobenius then M (S) = Ha∪0 for any a ∈ M (S)\0. Proof. Let a ∈ M (S). Since Ha ∪ 0 is a left ideal in S, Ha ∪ 0 = lS rS (Ha ∪ 0) = lS (N ) ⊃ M (S). So Ha ∪ 0 = M (S).

¤

Lemma 5. If S is quasi-Frobenius then for all a ∈ M (S) \ 0 and b ∈ S \ 0 there is a unique element x such that xb = a. Proof. Uniqueness of x follows from 0-cancellativity. The assertion is evident for b ∈ H. Let b ∈ N k \ N k+1 . If k = n − 1 the statement follows from Lemma 4. We use decreasing induction on k, supposing that our statement holds for bigger k’s .

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If N b = 0 then b ∈ rS (N ) = rS lS (M (S)) = M (S), i.e., k = n − 1, the case that has already been considered. If N b 6= 0 then cb 6= 0 for some c ∈ N . In this case cb ∈ N k+1 , so accordingly to the assumption of induction xcb = a for some x. ¤ Now we are able to prove the main result. Theorem 1. For a finite 0-cancellative semigroup S the following conditions are equivalent: (i) S is quasi-Frobenius; (ii) H 6= ∅ and M (S) is the least nonzero ideal; (iii) F0 S is Frobenius; (iv) F0 S is quasi-Frobenius. Proof. 1 =⇒ 2 was already proved (see Lemmas 2 and 4). 2 =⇒ 3. We use Theorem 61.3 from [1]. Fix some a ∈ M (S) \ 0 and define a linear function on F0 S: X f( αs s) = αa . s∈S\0

Every element from Kerf is of the form A =

X αs s. Let αs 6= 0. By Lemma s6=a

5 there is x ∈ S such that xs = a. Since a 6= 0, xt 6= a for any t 6= s. Hence xA 6∈ Kerf , i.e., Kerf does not contain left (and similarly right) ideals. By Theorem 61.3 [1], F0 S is Frobenius. 3 =⇒ 4 is evident. 4 =⇒ 1 was already proved (Lemma 3). ¤ *** A vast variety of examples of finite 0-cancellative semigroups is given by modifications of groups [3, 4, 5]. Recall that a modification G(∗) of a group G is a semigroup on the set G0 = G ∪ {0} with an operation ∗ such that x ∗ y is equal either to xy or to 0, while 0∗x=x∗0=0∗0=0 and the identity of G is the same for the semigroup G(∗). In other words, to obtain a modification, one must erase the contents of some inputs in the multiplication table of G and insert there zeros so that the new operation would be associative. It is clear that modifications are 0-cancellative. Denote as above by H the subgroup of all invertible elements in G(∗). By Lemma 1 its complement N = G(∗) \ H is a two-sided nilpotent ideal if G is finite. Let N n = 0 6= N n−1 = M . We shall describe quasi-Frobenius modifications for n ≤ 3. If n = 1 the semigroup G(∗) turns into the group G with an adjoint zero, so its algebra F0 G(∗) ∼ = F G is always quasi-Frobenius.

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Let n = 2. If G(∗) is quasi-Frobenius then N = M = Ha ∪ 0. Therefore when G has a subgroup H of the index 2 we can build a quasi-Frobenius modification G(∗) giving its multiplication by ½ xy if x ∈ H or y ∈ H, x∗y = 0 otherwise. Let n = 3. Fix a ∈ M \ 0; then M = Ha ∪ 0 = aH ∪ 0. Let b ∈ N \ N 2 . By Lemma 5 for every h ∈ H there is unique x ∈ G(∗) such that x ∗ b = ha. Therefore for all x ∈ G(∗), b ∈ N \ N 2 from xb ∈ Ha it follows x ∗ b 6= 0. So the operation ∗ must have the form ½ xy if x ∈ H or y ∈ H or xy ∈ Ha, x∗y = (3) 0, otherwise. Hence we have Proposition 1. Let G be a finite group, H its proper subgroup having the nontrivial normalizator NG (H) 6= H, a ∈ NG (H) \ H. Then a modification G(∗) is given by (3) is quasi-Frobenius. Proof. We need only to check associativity of ∗. This is equivalent to proving the statement (x ∗ y) ∗ z = 0 ⇐⇒ x ∗ (y ∗ z) = 0. If a, b ∈ N then a ∗ b ∈ M (G(∗)), so x ∗ y ∗ z = 0 when x, y, z ∈ N . Hence it is sufficient to establish associativity only in the case when one out of elements x, y, z is contained in H. Let, e. g., x ∈ H. We have to prove that xy ∗ z = x(y ∗ z).

(4)

However, xy and y belong or do not belong to H simultaneously; also xyz and yz belong or do not belong to aH simultaneously. Therefore xy ∗ z = 0 ⇐⇒ xy 6∈ H & z 6∈ H & xyz 6∈ H ⇐⇒ y 6∈ H & z 6∈ H & yz ∈ 6 H ⇐⇒ y ∗ z = 0 ⇐⇒ x(y ∗ z) = 0. ¤ We conclude this note by a remark. Theorem 1 and the results of [7] are too similar, nevertheless the considered classes of semigroups (commutative and 0-cancellative) are rather different. It would be of interest to find a larger class of semigroups that includes both above-mentioned classes and yields to some analogue of Theorem 1. Acknowledgment I am thankful to Prof. C.P. Milies for the opportunity to take part in Conference.

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References [1] Ch.W. Curtis and I. Reiner, Representation Theory of Finite Groups and Associative Algebras. J.Wiley & Sons, New York-London, 1962. [2] N. Jacobson, Structure of rings. Publ. AMS, R. I., 1956. [3] V.V. Kirichenko and B.V. Novikov, On the Brauer monoid for finite fields. In: Finite Fields and Applications, Springer, 2000, 313–318. [4] B.V. Novikov, On the Brauer monoid. Matem. Zametki, 57 (1995), N4, 633–636 (Russian). Translation in Math. Notes 57 (1995), no. 3-4, 440–442. [5] B.V. Novikov, On modification of the Galois group. Filomat (Yugoslavia), 9(1995), N3, 867–872. [6] J. Okni´ nski, Semigroup Algebras. Marcel Dekker, Inc., New York, 1991. [7] I.S. Ponizovskii, On Frobeniusness of a semigroup algebra of a finite commutative semigroup. Izv. Akad. Nauk SSSR, 32 (1968), 820–836 (Russian).

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Twisted loop algebras and Galois cohomology Arturo Pianzola Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Alberta, T6G 2G1, Canada [email protected] Dedicated to C´esar Polcino Milies on his sixtieth birthday.

A vida amigo, ´e arte do encontro Embora haja tanto desencontro pela vida. Vinicius de Moraes. 1. Loop algebras Throughout this note k will denote an algebraically closed field of characteristic 01. The unadorned symbol ⊗ will always stand for tensor product over our base field k. We fix once and for all a choice of compatible primitive n-th roots of unity e = ζ for all n, e ≥ 1. Our primary objects of study, (ζn ) in k × . Thus, ζen n loop algebras, are constructed out of a pair (A, σ) consisting of a k-algebra A together with an automorphism σ of A that is of finite order (henceforth denoted by m). Before defining loop algebras, let us introduce three rings R ⊂ Sm ⊂ Sb that will play an important role in their study. They are R = k[t±1 ], Sm = k[t±1/m ], and Sb = lim Sm . −→ This last ring has a very simple interpretation. As a space, Sb can (and will) be naturally identified with ⊕q∈Q ktq . The multiplication is then given by bilinear extension of tp tq = tp+q . Let then A be a k-algebra. At this point we do not put any assumption on the nature of A (in fact A may end up being something more general than an algebra as we shall see in the superconformal case later). Let σ ∈ Autk A 1This assumption, though not crucial, does facilitate the exposition considerably with-

out detracting from the spirit of the topic. For the more general point of view, the reader can turn to [ABP] and [P]. 293

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be of finite order m. We have the eigenspace decomposition i A = ⊕m−1 i=0 A¯i , A¯i := {a ∈ A | σ(a) = ζm a}

where − : Z → Zm := Z/mZ is the canonical map. We then define the loop algebra of (A, σ) by (1.1)

b L(A, σ) := ⊕i∈Z A¯i ⊗ ti/m ⊂ A ⊗ Sm ⊂ A ⊗ S.

The loop algebra has a natural k-algebra structure (infinite dimensional whenever A 6= 0). Of course, if A is one of the “usual” type of algebras (e.g., associate, Lie, Jordan,...), then so is L(A, σ). Example. Clearly L(A, Id) = A ⊗ R. We say that L(A, σ) is trivial if L(A, σ) ' A ⊗ R (isomorphism as k-algebras). Remark. It follows from the definition that L(A, σ) has a natural R-algebra structure. The contrast between k-isomorphisms and R-isomorphisms of loop algebras is very delicate, and it is usually handled using the centroid of A⊗R (see [ABP]). For the time being, let us point out that k-automorphism of A ⊗ Sb given by a ⊗ tq 7→ a ⊗ t−q maps L(A, σ) onto L(A, σ −1 ). Thus these two loop algebras are always isomorphic as k-algebras (but not necessarily so as R-algebras as we shall see). 2. The Loop algebra principle For motivational as well as historical reasons, let us first look at loop algebras of a finite dimensional simple Lie algebra g. We then have the exact sequence (2.2)

¯

1 → Gad → Aut g → F → 1.

Here Gad is the (abstract) group of k-points of the (algebraic) group Gad of “adjoint type” corresponding to g, and F is the finite group of automorphisms of the corresponding Coxeter-Dynkin diagram. This sequence splits: a section of ¯ is easily seen to exist by the presentation of g by generators and relations coming form the Cartan matrix. The central result of the theory is the following. Theorem 2.3. (Kac) With the above notation we have (i) L(g, σ) ' L(g, τ ) ⇔ σ ¯ ∼ τ¯±1 (where ∼ denotes conjugacy on the finite group F ); (ii) L(g1 , σ1 ) ' L(g2 , σ2 ) ⇒ g1 ' g2 ; (iii) Every affine Kac-Moody Lie algebra (derived modulo its center) is isomorphic to a loop algebra L(g, π) (with g and π ∈ F unique to the extent of (i) and (ii)). To put this into perspective, we turn to the classical theory. We can construct the Lie algebra g by starting with the free Lie algebra on generators ei , fi , hi : 1 ≤ i ≤ rank(g), and imposing the relations coming from the

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corresponding Cartan matrix. Though this abstract presentation of g is quite useful (in fact we just finish using it for constructing a section F ,→ Aut g) above, it is also important to know that g can be identified with a very concrete object: For example, if we start with the matrix of type B2 , then g is nothing but so5 . Kac-Moody algebras are defined by generators and relations as above, but using “generalized” Cartan matrices. Part (iii) of Kac’s Theorem says that, at least in the affine case (which is hands down the most interesting one besides the finite case), the algebras have very concrete realizations in terms of loop algebras. In this paper we will be interested in the philosophy captured by parts (i) and (ii) of the theorem (something that we will refer to as the “Loop algebra principle” below). Here are the constituents of these results. (P1) σ¯1 ∼ σ¯2 ±1 ⇒ L(g, σ1 ) ' L(g, σ2 ). (P2) L(g, σ1 ) ' L(g, σ2 ) ⇒ σ ¯1 ∼ σ ¯2±1 . (P3) L(g1 , σ1 ) ' L(g2 , σ2 ) ⇒ g1 ' g2 . The implication in (P1) is readily available in the literature [K1]. The proof is based on the so-called Gantmacher decomposition of the automorphism σ. The trick is that with respect to a clever choice of Cartan subalgebra, one can write σ = λπ with λ “diagonal” and π in F . One then argues that λ can be “erased,” namely L(g, λπ) ' L(g, π). On the other hand, (P2) and (P3) amount to the 16 types of affine KacMoody Lie algebras (Tables I, II, III as they are known) are indeed all nonisomorphic. Some awkward case-by-case specialization arguments can be put together to justify this claim. One can of course use conjugacy of Cartan subalgebras to prove this fact (but conjugacy was not available until much later on in the Kac-Moody game ([PK]). The main spirit of this paper is to explain how these results (and their analogs for arbitrary algebras) can be explained using Galois cohomology. Motivating by the foregoing we formulate the following. Loop algebra principle: The isomorphism class of L(A, σ) depends only on the outer part σ ¯ of σ. The reader will immediately be struck by the unprecise nature of this statement, but this is intentionally so. First of all, our A is completely arbitrary, so the principle presumes some notion of “inner” and “outer” part of an automorphism. There may be no natural way of doing this, in which case the principle can be turned around to indicate what the “correct” definition of inner automorphism is (namely one which makes the principle true!)2 2One can incorporate the existence of the group of inner automorphisms into the Prin-

ciple itself. See [P2].

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The principle can also be thought of in different degrees of strictness, like a weak version encompassed in spirit by (P1), and a hard one encompassing all three conditions. Here is an example. Theorem 2.4. [ABP] The hard Loop Algebra Principle holds for symmetrizable indecomposable Kac-Moody Lie algebras. Here the group of inner automorphisms is generated by eadx with x in a real root space and the diagonal automorphisms corresponding to Hom(Q, k × ). The “outer” group F is generated by the symmetries of the Coxeter-Dynkin diagram and also ω (the Cartan involution) if the underlying Cartan matrix is not of finite type. The ingredients of the proof are a generalized Gantmacher decomposition and Galois cohomology. 3. Some geometry Though it will seem at first as if we are going away from loop algebras, we are simply starting on a new path that will lead us back to them. Let us begin by looking at the following classical assertion. X.

A (topological) line bundle L over X looks locally like the line A1X over

A good example is the case when X = S1 is the circle, and L the M¨obius strip. Of course L looks nothing like a cylinder (which is the line over X) but locally they agree. So X can be covered by open subsets Ui ' R and over each Ui our L looks like A1Ui ∼ = R2 . For each point x ∈ Ui ∩ Uj the transition function ϕij = ϕi ◦ ϕ−1 j yields an × automorphism of the fiber R. Thus ϕij (x) ∈ R and we have a continuous map ϕij : Ui ∩ Uj → R× . Thus if OX is the structure sheaf of regular (= × ) its subsheaf of continuous in this case) functions on X and GmX (or OX units, then we can think of ϕij as an element of GmX (Ui ∩ Uj ). This allows us to attach to L an element [L] of the first Cˇech cohomology H 1 (X, GmX ) of X with values on the sheaf GmX . In fact, since compatible local data can be glued together (the continuous incarnation of descent theory), we have a natural correspondence. (3.1)

Isomorphism classes of line bundles over X ↔ H 1 (X, GmX )

With this in mind, we now go back to loop algebras. We now have the following (admittedly rather more outlandish looking) fact. (3.2)

L(A, σ) looks locally over X = Spec R like A ⊗ R.

Now X = Spec R is the geometric object that corresponds to R = k[t±1 ]. Recall the ring extension R ⊂ Sm = k[t±1/m ]. This yields a scheme morϕ phism U := Spec Sm → Spec R = X. Of course ϕ is not an open inversion

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(unless m = 1), but it is still a rather nice map (´etale and finite, in fact Galois) that can be thought of as an open cover of X (by one open set!) on a “topology” on X, called the ´etale topology, which is finer than the Zariski topology. We saw observed that line bundle becomes trivial when restricted to some open subsets of X. The idea here is the same: L(A, σ) as an object over Spec R becomes trivial, namely A ⊗ R, when “restricted” to Spec S. This is the content of the following straightforward lemma. Lemma 3.3. L(A, σ) ⊗R Sm 'Sm −alg A ⊗ Sm 'Sm −alg (A ⊗ R) ⊗R Sm . Just as in the classical situation as illustrated in (3.1) “objects that look locally like the fiber” are classified by the “cohomology of the sheaf of automorphisms of the fiber.” Accordingly we have that Isomorphisms classes of R-sheaf of algebras that are locally isomorphic to A ⊗ R ↔ H 1 (X, Aut AR ). This is part of the theory of principal homogeneous spaces (torsors for short in the case of schemes) developed by A. Grothendieck (after a crucial first step by J.-P. Serre in the case of varieties). It is not our intention to go into any details about this theory (see [SGA3], [Mil] and [DG]). The important point is that in spirit, this theory allows for algebras over R that look locally (in the ´etale or more generally the flat topology on Spec R) like A ⊗ R to be classified by some Cˇech-like nonabelian H 1 . This whole philosophy will for simplicity be referred to as the“Zen of torsors.”3 We have L(A, σ) Ã [L(A, σ)] ∈ H 1 (X, Aut AR ), L(A, σ) ∼R−alg L(A, τ ) ⇔ [L(A, σ)] = [L(A, τ )]. In this generality Aut AR is just the sheaf of groups over R given by Aut AR (R0 ) = AutR0 −alg A ⊗k R0 for any (associative unital commutative) R-algebra R0 . The torsor L corresponding to L = L(A, σ) is the R-sheaf of sets L : R0 7→ IsomR0 −alg (L ⊗R R0 , A ⊗ R0 ). It is clear what the (right) action of Aut AR is on the sheaf L. Of course L need not be isomorphic to Aut AR . (For example L(R) = ∅ for any of the twisted affine Kac-Moody algebras). But L “becomes” Aut AR when we restrict to Sm since L(Sm ) 6= ∅ by Lemma 3.3. Let us finish our discussion of torsors by pointing out that because Aut AR need not be abelian, our H 1 (X, Aut AR ) are not groups, but simply sets with a distinguished element (namely the isomorphism class of the trivial torsor corresponding to A ⊗ R). To reassure the reader that all this level of abstraction can be put to good use, let us begin by “explaining” why Kac’s Theorem is true. 3Variations on a theme of Grothendieck, namely the “yoga of descent.”

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The starting point is the sequence of k-algebraic groups (3.4)

¯

1 → Gad → Aut g → Fk → 1,

whose k-points yield (2.2) (so Fk is the constant k-group corresponding to F ). Passing to cohomology, we obtain (3.5)

ψ

H 1 (R, GR ) → H 1 (R, Aut gR ) → H 1 (R, FR )

where for convenience we have written G instead of Gad . Throughout, the subindex R reminds us of the fact that we are now working on the category of affine group schemes over R (or affine R-groups for short). Let us first look at H 1 (R, FR ). Grothendieck’s work (see [SGA1]) tells us that this H 1 measures Galois extensions of R with Galois group F , and can be computed by 1 H 1 (R, FR ) ' Hct (π1 (R), F ). This last is the “usual” continuous cohomology of the (profinite) algebraic fundamental group π1 (R) of R (based at the geometric point t = 1). The Comparison Theorem (or Hurwitz formula) tells us that π1 (R) has the same finite quotients as the (usual) fundamental group of C[t±1 ]. In fact π1 (R) = b This is a cyclic group (in the profinite sense). Because F is a constant Z. 1 is group, the action of π1 (R) on FR is trivial, so a 1-cocycle in our Hct b → F . Since ϕ is nothing but a (continuous) group homomorphism ϕ : Z then determined by ϕ(1) we may identify ϕ with an element of F . Finally, two elements of F (namely two 1-cocycles) are cohomologous if they are conjugate. Thus (3.6)

H 1 (R, FR ) ' Conjugacy classes of elements of F.

This is a very satisfactory position to be in, as (3.6) is precisely the righthand side of Kac’s Theorem (when one takes into account that π ∼ π −1 in F since F ' Z2 , Z3 , or S3 ). Next we have to look more closely to the torsor L(σ) attached to L(σ) = L(g, σ). Our extension S = Sm of R is Galois with Galois group Γ = Zm given by ¯ 1t1/m = ζm t1/m . This group Γ acts naturally on Aut gR (S) = AutS−alg g ⊗ S by conjugation (3.7)

¯if = (1 ⊗ ¯i) ◦ f ◦ (1 ⊗ (−¯i)).

Because L(σ) is trivialized by S, we have [L(σ)] ∈ H 1 (S/R, Aut gR ) ' H 1 (Γ, AutS g ⊗ S) where this last is the usual (nonabelian) cohomology corresponding to the action (3.7) above. The standard 1-cocycle of [L(σ)] is given by ¯ 1 7→ 1⊗σ −1 . It then follows that in (3.5) we have ψ : [L(σ)] 7→ σ ¯ −1 . Theorem 3.8. The map ψ : H 1 (R, Aut gR ) → H 1 (R, FR ) is bijective.

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Before turning to the proof, let us observe that this result not only proves but also explains why Kac’s result is true. Indeed, the “Zen of torsors” above and (3.6) yield that the bijectivity of ψ amounts to L(σ) ∼R−alg L(τ ) ⇔ σ ¯ ∼ τ¯. All that is left is to contrast this with L(σ) ∼k−alg L(τ ). Using the fact that the centroid of L(σ) is R, and that any k-isomorphism of L(σ) and L(τ ) induces an isomorphism of the respective centroids, it follows (see [ABP] for details) that (3.9)

L(σ) 'k L(τ ) ⇒ L(σ) 'R L(τ ) or L(σ) 'R L(τ −1 ).

Since in F every element is conjugate to its inverse, Kac’s Theorem follows.

It would seem that the injectivity of ψ amounts to the kernel H 1 (R, GR ) of ψ being trivial. But this need not be true. The problem lies in the fact that our H 1 are not groups but only sets with distinguished elements. Now there is an old trick of Serre, called “twisting,” that makes any element of H 1 (R, FR ) the distinguished element. The preimage of this element is now measured by means of the twisted group π GR . (One twists by cocycles, or torsors in general. As we have seen, however, in our case we may think of cocycles in Z 1 (R, FR ) ,→ H 1 (R, Aut gR ) as elements of F .) These twisted groups are themselves R-groups, which are forms of GR (they have no analogue over k). The injectivity of ψ amounts to all twisted cohomologies H 1 (R, π GR ) vanishing. 4. The case of Lie superalgebras Again we abide by convention and denote our finite dimensional simple Lie superalgebra by g rather than A. The first person to look at the loop (super)algebras corresponding to these algebras was V. Serganova ([S]). The results therein, and much more, can be obtained by cohomological methods. Theorem 4.1. [GP] The hard Loop Algebra Principle holds for finite dimensional Lie superalgebras. By definition, automorphisms of g preserve parity. This is an algebraic condition, so we can look at the algebraic group Aut g of automorphisms of g. The connected component G0 of Aut g (and not the simply connected group of the even part of g) is the correct group of inner automorphisms that makes Loop Algebra Principle hold). We have the exact sequence 1 → G0 → Aut g → F → 1

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where F is a finite constant group. The groups G0 and F are as follows: g G0 = Aut0 g Fk Split sl(m|n) (SLm × SLn × Gm )/(µm × µn ) Z2,k No psl(n|n), n > 2 (SLn × SLn × Gm )/(µn × µn ) Z2,k × Z2,k No psl(2|2) (SL2 × SL2 × SL2 )/(µ2 × µ2 ) Z2,k Yes sp(n) (SLn × Gm )/µn 1k Yes psq(n) PGLn Z4,k Yes osp(2l|2n) (SO2l × Sp2n )/µ2 Z2,k Yes osp(2l + 1|2n) SO2l+1 × Sp2n 1k Yes F (4) (Spin7 × SL2 )/µ2 1k Yes G(3) G2 × SL2 1k Yes D(α), α3 6= 1, (SL2 × SL2 × SL2 )/(µ2 × µ2 ) 1k Yes α 6= −(2)±1 D(α), (SL2 × SL2 × SL2 )/(µ2 × µ2 ) Z2,k Yes α ∈ {1, −2, −1/2} D(α), α3 = 1, (SL2 × SL2 × SL2 )/(µ2 × µ2 ) Z3,k Yes α 6= 1 W (n) NW (n) o GLn 1k Yes S(n) NS(n) o GLn 1k Yes S 0 (2l) NS 0 (2l) o SL2l 1k Yes H(2l) NH(2l) o ((SO2l × Gm )/µ2 ) Z2,k Yes H(2l + 1) NH(2l+1) o (SO2l+1 × Gm ) 1k Yes The canonical map (4.2)

H 1 (R, Aut g) → H 1 (R, FR )

is bijective and the loop algebras are therefore parameterized as follows L(g, σ) 'R L(g, τ ) ⇔ σ ¯ ∼ τ¯, while L(g, σ) 'k L(g, τ ) ⇔ σ ¯ ∼ τ¯±1 . Furthermore, the base algebra is an invariant L(g1 , σ1 ) ∼k L(g2 , σ2 ) ⇒ g1 ' g2 . As explained in §2, to establish (4.2) one must show that H 1 (R, π G) = 1 for all twisted versions of G. Unlike §2, the groups now in question are not in general reductive, and even their semisimple parts may be of neither adjoint nor simply connected type. The vanishing of H 1 is established in [GP] by choosing, for each type of superalgebra, a suitable exact sequences that reduces the semisimple case to the results established in [P]. To deal with twisted tori, a different approach is needed. We illustrate how this goes by looking at the case of the twisted multiplicative group π GmR . The Weil restriction of GmS by S/R. This is the R-group whose functor of points is

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given by

RS/R GmS (R0 ) = GmS (S ⊗R R0 ) = (S ⊗R R0 )× . The norm map N : S → R leads to the exact sequence of R-groups 1 → π GmR → RS/R GmS → GmR → 1. Passing to cohomology yields N

S × → R× → He´1t (R, π GmR ) → He´1t (R, RS/R GmS ). This last H 1 vanishes by Shapiro’s Lemma, while the norm map is surjective (since k is algebraically closed). Thus H 1 is trivial. As we shall presently see, the vanishing of He´1t (R, −) for the above twisted groups is part of a much more general and deeper fact. 5. Serre’s Conjecture I If K is a field then Spec K is reduced to one element. Thus, the only Zariski cover of Spec K is the trivial one. However, there are many beautiful covers of Spec K in the ´etale topology. In fact, if F is a Galois extension of K, then Spec F → Spec K is such a covering. Usual Galois cohomology thus fits nicely within the theory we have looked at. Serre’s Conjecture I is one of the most striking and deepest results in classical Galois cohomology. Theorem 5.3. Let G be a connected linear algebraic group over a field K of dimension 1. If char K 6= 0, assume further that G is reductive. Then H 1 (K, G) = 1. Note. If K is perfect the result is due to Steinberg. The general case was proved later by Borel and Springer. The concept of dimension 1 of K is rather technical (the Brauer group Br(K) is trivial for every finite separable field extension K of k). It suffices for us to know that, as a consequence of T’sen’s theorem, the field K = k(t) is of dimension 1. It should be clear by now that the Loop Algebra Principle is intimately related to the vanishing of certain H 1 (R, G). Unfortunately, R is not a field and our G, though reductive (as affine group schemes over Spec R in the sense of [SGA3]), do not in general come from linear algebraic groups (as exemplified by the twisted groups already encountered). It seems natural to ask for a ring analogue of Serre’s conjecture I. Which rings R then, have the property that H 1 (R, G) = 1 for all reductive (connected) R-group schemes? There is an immediate obstruction. If S/R is finite ´etale then RS/R GmS is a reductive R-group and H 1 (R, RS/R GmS ) = H 1 (S, GmS ) = Pic(S). It is thus inevitable to make the following assumption on R. TPEC Every finite connected ´etale cover S of R has trivial Picard group.

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Nicely enough, in dimension one this condition is essentially sufficient. Theorem 5.4. [P2] Let X = Spec D be a Dedekind scheme whose generic fiber is the spectrum of a field K of dimension 1. The following two conditions are equivalent: (i) X satisfies condition TPEC, (ii) H 1 (X, G) = 1 for all reductive group schemes G over X. From this we get the following version of the Loop Algebra Principle. Corollary 5.5. Let Aut A ⊂ GLk (A) be the linear algebraic group of automorphisms of a finite dimensional k-algebra A. Let Fk := Aut A/Aut0 A be the (finite constant) group of connected components of Aut A and − : Aut A(k) → F := Fk (k) the corresponding canonical (abstract) group homomorphism. Finally, let σ and τ be two elements of Aut A(k) of finite order. Then (i) L(A, σ) 'R-alg L(A, τ ) ⇔ σ ¯ ∼ τ¯ (ii) If the centroids of the k-algebras L(A, σ) and L(A, τ ) coincide with R (with R acting naturally by scalar multiplication; for example, if A is central and perfect), then L(A, σ) 'k-alg L(A, τ ) ⇔ σ ¯ ∼ τ¯±1 . 6. Lie superconformal algebras In this final section we illustrate the flexibility of the cohomological point of view we have taken. For basic concepts related to superconformal algebras see [K2], especially Chapter 2. Let A/C[δ] be a Lie superconformal algebra. We will assume that, as a C[δ]-module, A is free of finite rank (this covers all the N -superconformal cases, which is what interests us). With σ ∈ Aut A of order m, we construct the loop algebra as usual by using (1.1). The important thing is that the n-product of the affinization A ⊗ Sb are given by (6.6)

a ⊗ f(n) b ⊗ g =

X

a(n+j) b ⊗ δ (j) (f )g

j≥0

b is a Lie superconformal b δ] where δ = d/dt and δ (j) = j!1 δ j . Then A/C[ b is easily seen to be superalgebra where δb = δ ⊗ 1 + 1 ⊗ δ, and L(A, σ) ⊂ A a subalgebra (in the conformal sense). The quotient spaces b b =: LieA b δA b Lie(A, σ) := L(A, σ)/δL(A, σ) ⊂ A/ are Lie algebras with respect to the bracket induced by the 0-product. Both of these are naturally R-modules but not in general R-Lie algebras. b does not come from that This stems from the fact that the 0-product of A of A by R-linear extension.

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The functoriality that has served us so well heretofore is now lost. One b on Aut bA b (automorphisms can nevertheless define a continuous action of Z S b that are S-linear), b of A and still study the isomorphism classes of Lie(A, σ) b Aut bA). b The by means of the continuous nonabelian cohomology H 1 (Z, S absence of functoriality also brings a mysterious possibility into the picture. b = Aut A(S), b the groups Aut A and Unlike the algebra case where AutSbA b may a priori be completely unrelated. As we shall see below, this is AutSbA likely the case. We finish by looking at the development of the N = 2 and N = 4 superconformal algebras through the years. As we shall see, this is at the same time both highly amusing and instructive (see [SS] for details). N = 2: Like in the classical case, the analogue of the Ramond and NevueSchwartz algebra are originally thought to be different. Later on, these two algebras are viewed as part of an infinite discrete family of algebras. There is also an “exotic” algebra that no one looks at (or cares about). Pictorially, at this is the situation ··· ◦ ◦··· ◦ ◦···• A direct calculation shows that Aut A = C× o Z2 . If this was the algebra case, cohomology would at once dictate that only two distinct isomorphism classes of algebras can exist, hence that the above picture cannot be correct. But as we have cautioned above, functoriality may now be lost: There is no b from of that of Aut A. This a priori way of telling the nature of AutSb(A) however does not happen and b = Sb o Z2 Aut bA S

b Sb× o (so that Aut A seems to behave like the functor Gm oZ2,k ). Since H 1 (Z, Z2 ) ' Z2 the above picture is wrong. Indeed in [SS] the authors exhibit an explicit isomorphism between any of the two algebras of the series depicted by · · · ◦ · · · ◦ · · · above (in particular the Nevue-Schwartz and Ramond algebras are in this case isomorphic). So the correct picture, just as cohomology would have predicted, is in fact ◦ • N = 4: The tables are now reversed. The group Aut A = (SL2 (C) × SL2 (C))/±1 is connected (so one will expect all loop algebras to be trivial), yet the physicists assert the existence of an infinite discrete family of nonisomorphic algebras! Again, the correct answer would be given by computing a suitable 1 b Aut bA). b H (Z, Unfortunately, the author does not know4 the structure S b But there are two possible (or likely) answers. They of the group AutSbA. are as follows. 4Due to his own limitations. A. Retakh assures him this can be done.

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b = (SL2 (S) b × SL2 (S))/±1. b Possibility 1. AutSbA If so, all loop algebras are isomorphic and (again like in the case N = 2) the discrete family is a mirage. This scenario is unlikely given the physicists’ belief 5. Could cohomology account for a discrete infinite family of algebras? It does. Furthermore, the answer would be quite beautiful and in perfect agreement with the physicists’ picture. b = (SL2 (S) b × SL2 (C))/±1. Possibility 2. AutSbA 1 Now our H is basically the set of conjugacy classes of elements of finite order of SL2 (C). This infinite discrete set is exactly of the kind appearing in the parametrization given in [SS] (allowing passage from the compact to the complex case). References [ABP] B. Allison, S. Berman, and A. Pianzola, Covering algebras II: Isomorphisms of Loop algebras, J. Reine Angew. Math. 571 (2004) 39–71. [DG] M. Demazure and P. Gabriel, Groupes alg´ebriques. Tome I: G´eom´etrie alg´ebrique, ´ g´en´eralit´es, groupes commutatifs. (French) Masson & Cie, Editeur, Paris; NorthHolland Publishing Co., Amsterdam, 1970. [GP] D. Grantcharov and A. Pianzola, Automorphisms and twisted loop algebras of finite dimensional simple Lie superalgebras, Inter. Math. Res. Not. 73 (2004) 3937–3962. [K1] V. Kac, Simple irreducible graded Lie algebras of finite growth, Math USSR-Izvestija 2 (1968), 1271–1311. [K2] V. Kac, Vertex Algebras for Beginners, Second edition, University Lecture Series 10, American Mathematical Society, Providence, RI, 1998. ´ [Mil] J.S. Milne, Etale Cohomology, Princeton University Press, Princeton, 1980. [P] A. Pianzola, Affine Kac-Moody Lie algebras as torsors over the punctured line, Indag. Math. 13 (2002), 249–257. [P2] A. Pianzola, Vanishing of H 1 for Dedekind rings and applications to loop algebras, C. R. Acad. Sci. Paris (to appear). [PK] D. Peterson and V. Kac, Infinite flag varieties and conjugacy theorems, Proc. Nat. Acad. Sci. USA 80 (1983), 1778–1782. [SS] A. Schwimmer and N. Seiberg, Comments on the N = 2, 3, 4 superconformal algebras in two dimensions, Physics Letters B 184 (1987), 191–196. [S] V. Serganova, Automorphisms of simple Lie superalgebras, Math. USSR Izvestiya 24 (1985), 539–551. [SGA1] S´eminaire de G´eom´etrie Alg´ebrique du Bois Marie, 1960–1961, Revˆetements ´etales et groupe fondamental, dirig´e par A. Grothendieck, Lecture Notes Math. 224, Springer-Verlag, Berlin-New York, 1971. [SGA3] S´eminaire de G´eom´etrie Alg´ebrique du Bois Marie, 1963–1964, Sch´emas en groupes, dirig´e par M. Demazure et A. Grothendieck, Lecture Notes Math. 151– 153, Springer-Verlag, Berlin-New York, 1970. [SGA4] S´eminaire de G´eom´etrie Alg´ebrique du Bois Marie, 1963–1964, Th´eorie des topos et cohomologie ´etale des sch´emas, Lecture Notes Math. 269, 270, 305, SpringerVerlag, Berlin-New York, 1972–1973.

5Though admittedly, the N = 2 fiasco does not inspire much confidence.

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− Presentation of the group of units of ZD16 ´ Antonio Pita and Angel del R´ıo Departamento de Matem´aticas, Universidad de Murcia, Campus de Espinardo, 30100 Murcia, Spain [email protected] [email protected] Abstract. We obtain a presentation of the group of units of the integral − = a, b|a8 = 1 = b2 , ba = a3 b. group ring of the group D16

One of the Open Problems listed by Sehgal in [9] asks for obtaining presentations by generators and relations of the group of units ZG∗ of the integral group ring ZG for some ﬁnite groups G. This problem has been solved up to ﬁnite index for seven of the nine non abelian groups of order 16. (Recall that Higman described the structure of ZG∗ for G abelian [4].) Namely it is well known since Higman that Z(Q8 × C2 )∗ = Q8 × C2 [9] (see notation below); if G is D8 × C2 , C4 C4 , Q16 or H then ZG∗ contains a subgroup of ﬁnite index that is a direct product of free groups (see [5] and [6]). Furthermore, a concrete direct product of free groups F of ZG∗ with minimal index in ZG∗ has been computed in [8]; a presentation of ZG∗ has + . Thus at this moment there been obtained in [7] for the groups D and D16 are two groups of order 16 for which no presentation of ZG∗ by generators − . Here is the and relations is known, namely these groups are D16 and D16 notation: Cn Q4n D2n C4 C4 H D + D16 − D16

= = = = = = = =

Cyclic group of order n. Quaternion group of order 4n. Dihedral group of order 2n. a, b : a4 = b4 = 1, ba = a−1 b. . a, b : a4 = b4 = (ab)2 = [a2 , b] = 1. 2 2 4 2 a, b, c : a = 1 = b , c = 1, ba = c ab, c central. a, b : a8 = 1 = b2 , ba = a5 b. a, b : a8 = 1 = b2 , ba = a3 b

AMS classification index: Primary 16U60, Secondary 11R27, 16S34, 20C05. 305

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− In this paper we obtain a presentation of the group of units of ZD16 . A − matricial description of the group of units of ZD16 has been obtained in [5] − − is isomorphic to (ΓF3 )(±D16 ) who proved that the group of units√ of ZD16 where Γ is the subgroup of SL2 (Z[ −2]) formed by the matrices of the form √ √ 4m + 2m√ −2 4n + 4n √−2 (1) I+ 2e + e −2 4f + 2f −2

with m, m , n, n , e, e , f, f ∈ Z. Thus we only have to obtain a presentation of Γ. In order to achieve this goal, we use that Γ is Kleinian and hence we can apply Poincar´e’s Method to obtain a presentation of Γ from a fundamental polyhedron (see [1],[2] or [3]). Recall that a Kleinian group is a group that acts discontinuously on the 3-dimensional hyperbolic space by the Poincar´e obius Transformations. extension of the action of PSL2 (C) by M¨ A matricial description of the group of units of ZD16 has also been obtained in [5]. Unfortunately our methods do not apply to obtain a presen∗ because the matrix subgroups describing ZD ∗ in [5] are not tation of ZD16 16 − is of Kleinian type while D16 is not. Kleinian. In the terminology of [7], D16 Our main result is the following theorem. − − is isomorphic to (ΓF3 )(±D16 ), Theorem 1. The group of units of ZD16 where F3 is a free group of rank 3 and Γ is given by the following presentation

A0 , A1 , B0 , B1 , C, D, I, J, E0 , E1 , F0 , F1 , G0 , G1 , H0 , H1 , K0 , K1 , K2 , K3 , Γ= L0 , L1 , M0 , M1 , N0 , N1 , P0 , P1 , Q0 , Q1 , R0 , R1 , S, T

where:

F0−1 K2−1 JK0 E0−1 K2−1 IK0 N0−1 K2−1 M0 K0 P0−1 K2−1 R0 K0 [S, T ]

⎧ S, ⎪ ⎪ ⎪ ⎪ T, ⎪ ⎪ ⎪ ⎪ A, ⎪ ⎪ ⎨ i Bi , Xi = C, ⎪ ⎪ ⎪ ⎪ D, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ Li , ⎩ Qi ,

if if if if if if if if

Xi Xi Xi Xi Xi Xi Xi Xi

= = = = =

F1−1 K3−1 J −1 K1 E1−1 K3−1 I −1 K1 N1−1 K3−1 M1 K1 P1−1 K3−1 R1 K1 X0−1 X1 X1−1 X0

= = = = =

(2) 1

= Ai or Bi ; = Ei or Fi ; = Fi , Hi or Ni ; = Hi , Mi or J; = Ei , Gi or Pi ; = Gi , Ri or I; = Mi , Ri , Hi or Gi ; = Ni , Pi , Hi or Gi .

So the ﬁrst step is obtaining a fundamental polyhedron of the action of Γ on H3 . We achieve this goal by computing a variation of Ford’s fundamental polyhedron (see, e.g., [1]), that is, we compute a fundamental polyhedron of Γ formed by the intersection of a fundamental polyhedron of Γ∞ , the subgroup of Γ formed by the elements that stabilize ∞, and the exterior © 2006 by Taylor & Francis Group, LLC

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of the isometric half-spheres of the elements of Γ \ Γ∞ . We use Poincar´e’s model of H3 of the hyperbolic 3-dimensional space, that is, H3 = {(x, y, r) ∈ R3 : r > 0} and the action by isometries of SL2 (C) on H3 is as explained in [2]. Since √ 1 4 1 4 −2 , Γ∞ = 0 1 0 1 a fundamental polyhedron of Γ∞ is the inﬁnite strip F∞ = [−2, 2] × √ √ + [−2 2, 2 2] × R . a b Recall that the isometric circle of an element g = ∈ SL2 (C), c d with c = 0 is the circle given by the equation |cz + d| = 1, that is, the circle with centre −d/c and radius 1/|c|. The isometric half-sphere of g is the (euclidean) half sphere of H3 with the same centre and radius as the isometric circle of g. Remark 2. Notice that an isometric circle is in fact a circumference rather than a circle. We use this notation to keep the most extended terminology. By the inside part of an isometric circle we mean the open circle limited by it and the exterior part is the complement of the inside part. The inside and the exterior part of an isometric half-sphere are deﬁned similarly. We need a technical lemma. Lemma 3. If x ∈ C is of one of the following forms: ⎫ √ a + b −2 ⎬ √ with a, b ∈ Z and i, j = ±2 ( 3i + 2a) + ( 3j + b)√ −2 ⎭ 1 + 2a + ( 12 + b) −2

(3)

then x is not inside the isometric circle of an element of Γ. Proof. Consider an element g√∈ Γ of the form (1), and its isometric sphere +2f 1√ √ −2 and radius r = . Let S0 denote S, with centre c = − 1+4f 2e+e −2 |2e+e −2| the interior part of S. √ Assume ﬁrst that x = a + b −2 with a, b ∈ Z. Then √ √ √ e −2)| |c − x| = r| − (1 + 4f + 2f −2) − (a + b −2)(2e +√ = r| − 1 − 4f − 2ae + 2be − (2f + ae + 2be) −2|. √ If x ∈ S0 and y = −1 − 4f − 2ae + 2be − (2f + ae + 2be) −2 then |y| < 1 and hence y = 0, which is not possible. √ Assume now that x = ( 3i + 2a) + ( 3j + b) −2 with a, b ∈ Z and i, j = ±2. Then |c − x| = 3r |y| where √ y = −3 − 12f − 2ei − 12ae + 2je + 6be + (−6f − ie − 6ae − 2je − 6be) −2 © 2006 by Taylor & Francis Group, LLC

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and 1 represents the elements of √ thus x ∈ S0 if and only if |y| < 3. Figure √ Z[ −2] of norm at most 3. Write y = y1 + y2 −2 with y1 = −3 − 12f − 2ei − 12ae + 2je + 6be and y2 = −6f − ie − 6ae − 2je − 6be. Since y1 is odd and y2 is even, because i is even, a glance at Figure 1 shows that y = ±1. Thus y2 = 0 and hence 6 divides ie + 2je. This implies that if i = j then 0 ≡ e + 2e ≡ −e − 2e ≡ 2e − 2e mod 3 and if i = −j then 0 ≡ e − 2e ≡ −e + 2e ≡ 2e + 2e mod 3, a contradiction in either case.

3 2 1 321 1 2 3

1 2 3

Figure 1

√ Finally assume that x = 1 + 2a + ( 12 + b) −2 with a, b ∈ Z. Then |c − x| = r|y| where √ y = −1 − 4f − 2e − 4ae + e + 2be − (2f + e + 2ae + e + 2be) −2 x11 x12 and x ∈ S0 if and only if y = 0 if and only if x = c. Write g = x21 x22 √ 22 = x and hence with xij ∈ Z[ −2]. Then c = −x x21 √ −2x22 = x21 [(2 + 4a) + (1 + 2b) −2]. √ Since det √ x21 divides √ g = 1, x21 and x22 are coprime in Z[ −2] and therefore 2 in Z[ −2], that is, x21 is ±1 or ±2. Since x21 = 2e + e −2 = ±1, we deduce that √ √ 1 + 4f + 2f −2 = x22 = ±(2 + 4a + (1 + 2b) −2), √ Proposition 4. Let Γ be the subgroup of SL2 (Z[ −2]) formed by the matrices of the form (1) with m, m ,√n, n ,√e, e , f, f ∈ Z and let D be the intersection of F∞ = [−2, 2] × [−2 −2, 2 −2] × R+ with the exterior of the half-spheres with centre and radius in one of the cases of the following table. a contradiction.

Centre √ Radius f f’ √1 ±(2f − ( 12 + 2f ) −2) −1, 0 0, ±1 2 √ 1 1 ±( 2 + 2f + f √−2) −1, 0 0, ±1, ±2 2 1 1 √ ±(f − ( 4 + f ) −2) −2, −1, 0, 1 ±1 8 f √ 1 1 ±( 4 + f + 2 −2) ±1 ±1, ±3 4 © 2006 by Taylor & Francis Group, LLC

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Then D is a fundamental polyhedron of the group Γ. Proof. As mentioned above F∞ is a fundamental polyhedron of the stabilizer Γ∞ of ∞ and the intersection D1 of F∞ and the exterior of the isometric half-spheres of Γ \ Γ∞ is a fundamental polyhedron of Γ. We want to show that D = D1 . To prove D1 ⊆ D we show that the half-spheres mentioned in the proposition are isometric half-spheres of elements of Γ \ Γ∞ . The 12 half-spheres of radius √12 are the isometric half-spheres of the following elements √ √ −3 − 2 √−2 (−1)i 8 √−2 , Ai = −(−1)i −2 −3 + 2 −2 √ √ 1 + 2 √−2 −(−1)i√ 4 −2 , Bi = 1 − 2 −2 (−1)i −2 √ 1 0 −2 −3 4 √ , D= √ C= −2 1 −3 − −2

for i = 0, 1 and their inverses. The 20 spheres of radius half-spheres of the following elements

1 2

(4)

are the isometric

√ √ i 1 − 4 −2 −(−1) −3 + 4 −2 16 (−1)i√ 20 √ = , F , i −3 − 4 −2 −(−1)i 2 1 + 4 −2 (−1)i 2 √ √ i i 1 − 2 −2 −(−1) −3 + 2 −2 4 (−1)√ 8 √ Gi = , H , = i −3 − 2 −2 −(−1)i 2 1 + 2 −2 (−1)i 2 1 0 −3 4 I= , J= −2 1 2 −3

Ei =

for i = 0, 1 and their inverses. The 16 spheres of radius half-spheres of the following elements

√1 8

√ i8 1−2 √ −2 −(−1) √ , Ki = −(−1)i 2 −2 −7 + 2 −2 √ j8 1+2 √ −2 (−1)√ , Kj = −(−1)j 2 −2 −7 − 2 −2 √ √ −3 − 2 √−2 (−1)i 4 √−2 , Li = −(−1)i 2 −2 −3 + 2 −2 √ √ 5 + 2 √−2 −(−1)i√ 8 −2 Qi = 5 − 2 −2 (−1)i 2 −2

(5)

are the isometric

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for i = 0, 1, j = 2, 3 and their inverses. Finally the 16 spheres of radius 14 are the isometric half-spheres of the following elements √ i8 5 − 2 −2 −(−1) √ , Mi = −(−1)i 4 5 + 2 −2 √ i 24 5 − 6 −2 −(−1) √ , Ni = −(−1)i 4 5 + 6 −2 (7) √ −3 + 6 −2 (−1)i√ 20 , Pi = −3 − 6 −2 (−1)i 4 √ i4 −3 + 2 −2 (−1)√ Ri = −3 − 2 −2 (−1)i 4 for i = 0, 1 and their inverses. 4

2

3

2

1

1

2

2

3

0

-2 2 2

-1 0 -2

1 4

Figure 2

2

Figure 3: Fundamental Polyhedron of Γ

Figure 2 represents the bases of the √ 64 half-spheres mentioned and the four half-planes x = ±2 and y = ±2 −2. Let P be the set of intersections of two of these bases. Notice that each element of P belongs to 4 bases. Each side of D is included in one of the mentioned half-spheres or in one of the four half-planes. Figure 3 represents the bottom boundary of D. The intersections of 2 of the faces of D are geodesic lines of H3 with extremes in elements of P . Now it is easy to see that D is the convex hull of the intersections of pairs of faces of D. Abusing the notation we can refer to D as the convex hull of P in H3 . A tedious but straightforward computation shows that each element of P is of one of the forms of (3). By Lemma 3 every element of P is in the exterior part of any isometric sphere of Γ \ Γ∞ . Since D1 is convex, D1 contains the convex hull of D, that is D1 ⊇ D. Now that a fundamental polyhedron of Γ has been obtained in Proposition 4, the next step is looking for a set of generators of Γ. It is well known that © 2006 by Taylor & Francis Group, LLC

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the side-pairing transformations of Γ for D generate Γ [2]. A side-pairing transformation of Γ for D is an element g ∈ Γ such that sg = D ∩ g−1 (D) has dimension 2. The sets sg for g a side-pairing transformation are called sides of D. It is clear that sg = g(sg ) = sg−1 . The association s → s is called the pairing of D by Γ. T −1 F0−1

E0−1

E0

F0

K2

K0

N0−1 P0−1 A−1 1

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T

Figure 4: Side-Pairing Transformations. The side-pairing transformations of Γ∞ for F∞ are √ 1 4 1 4 −2 S= , T = 0 1 0 1

(8)

and their inverses and they are also side-pairing transformations of Γ for D. A straightforward computation shows that the elements of (4), (5), (6), and © 2006 by Taylor & Francis Group, LLC

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(7) and their inverses are also side-pairing transformations of Γ for D. To show that there are not more side-pairing transformations of Γ for D it is enough to realize that the sides sg of the elements of the forms (4), (5), (6), (7), and (8) and their inverses cover the boundary of D. Figure 4 represents the projection of the fundamental polyhedron D in C with the side-pairing transformations associated to each side. The pairing of D by Γ can also be read oﬀ from Figure 4 because sg = sg−1 . It is important to notice that the sides of S and T and their inverses are the vertical faces of D as it is emphasized in Figure 4. Notice that the sides and faces of D coincide. By Poincar´e’s Method, Γ has a presentation where the generators are the side-pairing transformation and the relations are the reﬂection and cycle relations. The reﬂection relations are of the form g2 = 1 where g is a sidepairing transformation such that sg = sg . The fundamental polyhedron D does not yield to any reﬂection relation as it is clear from Figure 4. The cycle relations are associated to cycles of D, which are sequences of the form (a1 , g1 , a2 , g2 , ..., an , gn ) such that: 1. g1 , ..., gn are side-pairing transformations of Γ for D with gi+1 = gi−1 for all i = 1, ..., n − 1. 2. a1 , ..., an are edges of D with sgi ∩ sgi+1 = ai+1 for all i = 1, ..., n − 1. In fact, sgi and sgi+1 are the only two sides of D containing the edge ai . 3. gi (ai ) = ai+1 for all i = 1, ..., n. Here an+1 stands for a1 . If c = (a1 , g1 , ..., an , gn ) is a cycle of D, then g = gn gn−1 . . . g2 g1 has ﬁnite order and the cycle relation associated to c is gm = 1 where m is the order of g. Looking for the cycles of D is now another tedious straightforward calculation. Figure 5 codiﬁes the results of this calculation.

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Figure 5: Cycles. Each picture in Figure 5 is a copy of Figure 4 where the edges of one cycle has been emphasized. The 33 relations in (2) are the cycle relations associated to the cycles represented by the 33 pictures of Figure 5. For example, the four edges of the ﬁrst picture are the four vertical edges of D and gives rise to the cycle

(sS ∩ sT , S, sS −1 ∩ sT , T, sS −1 ∩ sT −1 , S −1 , sS ∩ sT −1 , T ), © 2006 by Taylor & Francis Group, LLC

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which yields the relation [S, T ] = 1. The second picture represents the cycle (sA0 ∩ sF0 , A0 , sA−1 ∩ sF −1 , F1−1 , sA1 ∩ sF1 , A1 , sA−1 ∩ sF −1 , F0−1 ) 0

1

1

0

F0−1 A1 F1−1 A0

= 1, which is one of the relations of and induces the relation −1 −1 the form X0 X1 X1 X0 = 1 in (2). The reader can now complete the list of relations by reading the 31 relations of (2) from the remaining 31 pictures of Figure 5, by using also Figure 4. Acknoledgments Both authors have been partially supported by D.G.I. of Spain and Fundaci´ on S´eneca of Murcia. References [1] A.F. Beardon, The Geometry of Discrete Groups, Springer, 1983. [2] J. Elstrodt, F.Grunewald, and J. Mennicke, Groups Acting on Hyperbolic Space. Harmonic Analysis and Number Theory, Springer, 1998. [3] B. Fine, The Algebraic Structure of the Bianchi Groups, Marcel Dekker, 1989. [4] G. Higman, Units in group rings, D. Phil. Thesis, University of Oxford, Oxford, 1940. [5] E. Jespers and M. M. Parmenter, Units of group rings of groups of order 16, Glasgow Math. J. 35 (1993), 367–379. ´ del R´ıo, A structure theorem for the unit group of the integral [6] E. Jespers and A. group ring of some finite groups, J. Reine Angew. Math. 521 (2000), 99–117. ´ del R´ıo, and M. Ruiz, Group of units of integral group rings of Kleinian [7] A. Pita, A. type, (to appear in Trans. AMS). ´ del R´ıo and M. Ruiz, Computing large direct products of free groups in integral [8] A. group rings, Comm. Algebra 30 (2002) (4), 1751–1767. [9] S.K. Sehgal, Units in Integral Group Rings, Longman Scientific and Technical Essex, 1993.

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Engel theorem for Jordan superalgebras Ivan Shestakov Instituto de Matem´atica e Estat´ıstica, Universidade de S˜ao Paulo, Caixa Postal 66281, S˜ao Paulo, SP, 05315-970, Brasil Sobolev Institute of Mathematics, Novosibirsk, Russia [email protected] Konstantin Okunev Novosibirsk State University, Novosibirsk, Russia Dedicated to professor C´esar Polcino Milies on his 60th birthday. Abstract. It is proved that a finite dimensional Engelian Jordan superalgebra over an infinite field of characteristic 6= 2 is nilpotent.

1. Introduction An algebra J over a field F of characteristic 6= 2 is called Jordan if it satisfies the identities xy = yx, (x y)x = x2 (yx). 2

(1)

It is well known that a finite dimensional Jordan nil algebra is nilpotent [2], [6]. In particular, every solvable finite dimensional Jordan algebra is nilpotent. This result is not true any more for Jordan superalgebras: an example of a finite dimensional solvable Jordan superalgebra that is not nilpotent is given in [4]. In fact, Jordan superalgebras behave more like Lie algebras; an odd part of a Jordan superalgebra has a product “of Lie type.” But for Lie algebras, the ordinary notion of nilness is useless since every Lie algebra is nil of index 2. A proper substitution for this is the notion of an Engelian element: an element a of an algebra A is called Engelian if the operator of right multiplication Ra is nilpotent. The classical Engel Theorem states that a finite dimensional Lie algebra is nilpotent if and only if all its elements are Engelian [1]. This theorem admits also a generalization for finite dimensional Lie superalgebras [3]. 315

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Thus, it seems natural to ask whether an analogue of the Engel Theorem is also valid for Jordan superalgebras. In the present paper we give a positive answer to this question in case of an infinite ground field. 2. Engelian Jordan superalgebras Recall that a Z2 -graded algebra J = J0 +J1 over a field F of characteristic 6= 2, 3 is called a Jordan superalgebra if it satisfies the super-identities xi xj = (−1)ij xj xi , (super-commutativity), ((xl xi )xj )xk + (−1)ij+ik+jk ((xl xk )xj )xi + (−1)jk xl ((xi xk )xj ) = (xl xi )(xj xk ) + (−1)ik+jk (xl xk )(xi xj ) + (−1)ij (xl xj )(xi xk ), (2) where xi ∈ Ji , i = 0, 1. In case when char F = 3, one should also add identity (1) for even elements (see [5]). Denote the right side of (2) as F = F (xl , xi , xj , xk ). It follows easily from super-commutativity that F is super-symmetric on its graded variables. In particular, F (xl , xi , xj , xk ) = (−1)ij+jl+il F (xj , xi , xl , xk ). Therefore, the same is true for the left side of (2), and we have ((xl xi )xj )xk + (−1)ij+ik+jk ((xl xk )xj )xi + (−1)jk xl ((xi xk )xj ) = (xl (xi xj ))xk + (−1)ij+ik (xl (xj xk ))xi + (−1)jk (xl (xi xk ))xj . (3) Let us rewrite identities (2) and (3) in operator form, using operators Rx of right multiplication on x: Rxi Rxj Rxk + (−1)ij+ik+jk Rxk Rxj Rxi + (−1)jk R(xi xk )xj = Rxi Rxj xk + (−1)ik+jk Rxk Rxi xj + (−1)ij Rxj Rxi xk = Rxi xj Rxk + (−1)ij+ik Rxk xj Rxi + (−1)jk Rxi xk Rxj .

(4)

Observe that the even part J0 of a Jordan superalgebra J = J0 + J1 is an ordinary Jordan algebra and the odd part J1 is a Jordan bimodule over J0 . We will denote A = J0 , M = J1 . A Jordan superalgebra J = A + M is called Engelian if for any homogeneous element a ∈ A ∪ M the operator Ra is nilpotent. Below J = A + M is a finite dimensional Engelian Jordan superalgebra over an infinite field F of characteristic 6= 2. Denote by R(J) the subalgebra of the associative algebra EndF J generated by all the operators Ra , a ∈ J. This subalgebra is called the multiplication algebra of J. The algebra J is nilpotent if and only if R(J) is nilpotent [6]. For a subalgebra B of the algebra J denote by RJ (B) the subalgebra of End J generated by all the operators Rb , b ∈ B. Notice that the elements

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of RJ (B) act on J, and the multiplication algebra R(B) is a homomorphic image of RJ (B) (a restriction of RJ (B) to B). Observe also that the nilpotence of B in general does not yield that of RJ (B), but since B ⊆ J, the nilpotence of RJ (B) always implies that of B. The following proposition plays a crucial role in our proof. Proposition 2.1. Let B be a subsuperalgebra of the superalgebra J = A+M of the form B = I + F v, where B 2 ⊆ I and v ∈ A ∪ M . Assume that the algebra RJ (I) is nilpotent; then the algebra RJ (B) is nilpotent as well. Proof. Let us denote, for simplicity, B ∗ = RJ (B), I ∗ = RJ (I). Let Q = I ∗ + I ∗ B ∗ be the right ideal of B ∗ generated by I ∗ . It suffices to prove that B ∗ is nilpotent modulo Q, that is, (B ∗ )n ⊆ Q for some n. In fact, then an easy induction on m gives (B ∗ )nm ⊆ (I ∗ )m + (I ∗ )m B ∗ , and the nilpotency of I ∗ implies that of B ∗ . The proof differs essentially for the cases when the element v is even and when it is odd. 1. v ∈ A. Let us show that in this case (B ∗ )3 ⊆ Q. It suffices to consider the elements of the form Rv Ra Rb , where a, b ∈ I ∪ {v}. If a ∈ I then we have by (4) Rv Rb Ra ∈ −Ra Rb Rv + RJ (B 2 ) + RJ (B)RJ (B 2 ) ⊆ Q. If a = v, then by (4) again we have 2Rv Rb Rv = −Rv2 b + 2Rvb Rv + Rv2 Rb ∈ Q. This completes the case 1. 2. v ∈ M . Let w = Ra1 · · · Rak ∈ B ∗ . We will call the number d(w) = k the length of the word w. In what follows, for elements w, u ∈ B ∗ we will write w ≡Q u if w − u ∈ Q. Lemma 2.2. Let w = Ra1 . . . Rak ∈ B ∗ . If ai = a ∈ I for some odd i, 1 ≤ i ≤ k, then w ≡Q 0. Proof. If i = 1 then evidently w ∈ Q. Otherwise w = w0 Rx Ry Ra w1 for some x, y ∈ B, and we have by (4) w = w0 (±Ra Ry Rx ± R(ax)y ± Rax Ry ± Ray Rx ± Rxy Ra )w1 . Observe that (ax)y, ax, ay, xy ∈ I. Therefore w is a sum of elements of the same type that either have smaller length or have the factor Ra , a ∈ I, on position i − 2. Applying the same transformation to the summands, we will eventually express w as a sum of words of the type Ra0 w0 , where a0 ∈ I, w0 ∈ B ∗ . Since every such an element lies in Q, we get w ∈ Q as well. ¤

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Lemma 2.3. Let w = Rv Rs , s ∈ (M ∪ A) ∩ I. Then there exists a natural number n such that wn ∈ Q. Proof. Consider first the case when s ∈ M . The superalgebra J is finite dimensional and Engelian, hence there exists a natural number m such that Rx2m = 0 for all x ∈ M . Since F is infinite, every partial linearization of this equality holds on M as well [6]. In particular, for any x, y ∈ M we have X 0 = (Rx2m )∆m (y) = Rx1 Rx2 · · · Rx2m , x where the sum on the righthand side runs on all different summands with m of the factors Rxj equal to Rx and other m factors equal to Ry . Substitute there x = v, y = s. By Lemma 2.2, all the summands that contain the factor Rs on odd position, belong to Q. Hence we have (Rv Rs )m ≡Q 0, which proves the lemma in this case. Let now s ∈ A. By (4) and Lemma 2.2, we have 2Rv Rs Rv Rs + Rv Rs2 v = 2Rv Rs Rvs + Rv2 Rs2 ≡Q 0, hence 2(Rv Rs )2 ≡Q −Rv Rs0 , where s0 = s2 v ∈ I ∩ M . Observe that, by Lemma 2.2, for any u ∈ B ∗ of even length we have uQ ⊆ Q. Therefore, 2m (Rv Rs )2m = (2(Rv Rs )2 )m ≡Q (−Rv Rs0 )m ≡Q 0, proving the lemma.

¤

Lemma 2.4. Let W = {Rv Rb | b ∈ B}. Then W is nilpotent modulo Q, that is, there exists a natural number N such that W N ⊆ Q. Proof. Consider the subalgebra C = alghW ∪Qi of the algebra B ∗ generated by W and Q. By Lemma 2.2, W Q ⊆ Q, and since Q is a right ideal in B ∗ , we have Q C C. It suffices to prove that the quotient algebra C/Q is nilpotent. Observe that it is generated by the set W = {Rv Rs + Q | s ∈ I0 ∪ I1 ∪ {v}}. Denote ws = Rv Rs . It follows from (4) Rv Rs Rv Rt ± Rv Rt Rv Rs ± Rv R(st)v = Rv (±Rv Rst ± Rt Rsv ± Rs Rtv ), which implies by Lemma 2.2 that ws wt ± wt ws ≡Q ±w(st)v , and hence the set W is weakly closed. It follows from Lemma 2.2 (and the Engelianess of J) that the set W is nil. By [1, Theorem 2.2.1], the algebra alghW i = C/Q is nilpotent. ¤ Let us show now that (B ∗ )2N ⊆ Q. Consider w = Ra1 . . . Ra2N ∈ (B ∗ )2N , where ai ∈ B0 ∪B1 . If at least one of a1 , a3 , . . . , a2N −1 belongs to I then w ∈ Q by Lemma 2.2. Otherwise we can assume that a1 = a3 = · · · = a2N −1 = v and w ∈ W N ⊆ Q by Lemma 2.4. This proves Proposition 2.1. ¤ Theorem 2.5. A finite dimensional Engelian Jordan superalgebra J = A + M over an infinite field F of characteristic 6= 2 is nilpotent.

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Proof. Let B be a subsuperalgebra of J such that B ∗ is nilpotent (for example, B = 0). We will show that there exists a bigger subsuperalgebra B1 = B + F v that still satisfies the same condition. Since J is finite dimensional, this eventually will imply that R(J) = J ∗ is nilpotent and so J is nilpotent. Observe that the chain of subspaces J ⊇ JB ∗ ⊇ J(B ∗ )2 ⊇ · · · is strictly decreasing and ends by 0. Hence, there exists a ∈ A ∪ M such that a 6∈ B but aB ⊆ B. Now, we have several possibilities. 1. a ∈ A, a2 B ⊆ B. In this case, by [6, Proposition 3.2], we have ak B ⊆ B for any natural k. Since a is nilpotent, there exists such k that ak−1 6∈ B, ak ∈ B. Put v = ak−1 , then B1 = B + F v is a subsuperalgebra of J that satisfies the conditions of Proposition 2.1. Thus (B1 )∗ is nilpotent. 2. a ∈ A, and there exists b ∈ B ∩ A such that v = a2 b 6∈ B. In this case, as in the proof of [6, Theorem 4.3, case 2], one can show that v 2 ∈ B, vB ⊆ B, hence again B1 = B + F v is a subsuperalgebra of J such that (B1 )∗ is nilpotent. 3. a ∈ A, and there exists b ∈ B∩M such that v = a2 b 6∈ B. The element v is odd, hence v 2 = 0. Repeating the arguments from [6, Theorem 4.3, case 2], we see that vB ⊆ B, hence again B1 = B + F v satisfies Proposition 2.1. 4. a ∈ M . In this case the subspace B1 = B + F a is a subsuperalgebra that satisfies Proposition 2.1. Thus, in any case the subsuperalgebra B can be enlarged to B1 . This proves the theorem. ¤ Acknowledgment Dr. Shestakov was partially supported by CNPq, grant 304633/03-8. References [1] N. Jacobson, Lie Algebras, (Wiley, 1962) (reprinted in 1979 by Dover, New York). [2] N. Jacobson, Structure and Representations of Jordan Algebras, AMS, Providence, RI, 1968. [3] V.G. Kac, Lie superalgebras, Adv. Math. 26 (1977), 8–96. [4] I. Shestakov, Superalgebras and counter-examples, Sibirskii Matem. Zh., 32, N 6 (1991), 187–196; English transl.: Siberian Math.J. 32, N 6 (1991), 1052–1060. [5] I. Shestakov, Alternative and Jordan superalgebras, Siberian Adv. Math., 9, No. 2 (1999), 83–99. [6] K.A. Zhevlakov, A.M. Slin’ko, I.P. Shestakov, and A.I. Shirshov, Rings that are Nearly Associative, Moscow, Nauka, 1978 (in Russian); English translation by Academic Press: New York, 1982.

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A characterization of centre-by-finite subgroups of division algebras Mazi Shirvani Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Alberta, Canada [email protected] Dedicated to Cesar Polcino on his 60th birthday. Abstract. A group-theoretical characterization of the finitely generated centre-by-finite subgroups of (finite-dimensional) division algebras in all characteristics is given.

1. Introduction We determine all finitely generated centre-by-finite groups that can be embedded into finite-dimensional division algebras. The division algebras generated by such groups are called projective Schur division algebras (there are also more general projective Schur algebras) by Lorenz and Opolka, [6]. In [1], Aljadeff and Sonn prove that such division algebras have abelian splitting fields. For their purposes, the determination of the exact structure of these groups was not necessary. These groups, however, are a relatively useful ingredient in the study of projective representations of finite groups over arbitrary fields, so their structure is of some interest. In addition, such groups can also be regarded as central extensions of those finite groups, which possess a faithful projective representation of reduced degree equal to 1. Here the term reduced degree is used in an obvious extension of the term as used for ordinary representations of finite groups (e.g., [5], Section 5): If G is a finite group, then the reduced ordinary degrees of the irreducible representations of G over a field k of characteristic zero are the matrix degrees of the simple components of the group ring kG (the ordinary degrees are then the product of the reduced degrees and the Schur indices). In the same way, the reduced projective degrees of a finite group G may be defined as the matrix degrees of finite-dimensional simple k-algebras R, which are spanned over k by a group X of units, with the property that 321

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X is a central extension of G. Having reduced degree 1 then means that the algebra R is a division algebra. Unlike reduced ordinary degrees, nontrivial division algebras can occur in positive characteristic. The results depend on whether the characteristic is zero or not, and are stated separately. Recall that an Amitsur group is a finite subgroup of a division ring of characteristic zero (for a list, see 2.1 below). Theorem 1.1. Let k be a field of characteristic zero, and let X be a finitely generated centre-by-finite group. Let T be the torsion subgroup of X. If D = k[X] is a central division k-algebra, then T is an Amitsur group such that k[T ] is a division algebra. Conversely, if X is a finitely generated centre-by-finite group such that the torsion subgroup of X is an Amitsur group, then there exists a finitedimensional division algebra D of characteristic zero that is generated by X over the centre Z(D). The exact converse of the first half of the theorem can be false; see the comments at the end of Section 2. The statement of the next theorem refers to the subgroup Γp (T ) of Aut(T ), defined for cyclic groups T after Theorem 3.1 in Section 3. Theorem 1.2. Let k be a field of characteristic p > 0, and let X be a finitely generated centre-by-finite group. Let T be the torsion subgroup of X. If D = k[X] is a central division k-algebra, then T is cyclic of order not divisible by p, and X acts on T as a subgroup of Γp (T ). Conversely, if X is a finitely generated centre-by-finite group such that the torsion subgroup T of X is cyclic of order not divisible by p, and X acts on T as a subgroup of Γp (T ) ⊆ Aut(T ), then there exists a finite-dimensional division algebra D of characteristic p that is generated by X over the centre Z(D). The following notation and simple facts will be used throughout: Let X be a finitely generated centre-by-finite group (abbreviated henceforth to X ∈ ZF ). By Schur’s Theorem ([4], IV.2.3) the derived group X 0 is finite, and so the set of torsion elements in X forms a normal subgroup τ (X). Clearly, X/τ (X) is free abelian of finite rank. For any field k, the group ring kX can be obtained from kT by a series of skew Laurent polynomial extensions. To be precise, if X = X/T = hg1 i × . . . hgn i, then the group ring khT, g1 i is, in a natural way, isomorphic to the skew Laurent polynomial ring kT [x±1 1 ], where the action of x1 on kT is the same as conjugation by g1 . Similarly, khT, g1 , g2 i is isomorphic to the skew Laurent polynomial ring khT, g1 i[x±1 2 ], and so on. Such extensions are well known to be Noetherian.

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2. Proof of Theorem 1.1 The first half of Theorem 1.1 is trivial, since if D = k[X] a division algebra of characteristic zero, then T = τ (X) is an Amitsur group by definition, and k[T ] ⊆ D is a division algebra. We need some information before we can prove the other half. We include a list of the Amitsur groups from [7], 2.1.4 and 2.1.5, as the list, in this particular form, appears not to be well known (see Amitsur’s original paper [2] for presentations of these groups by generators and relations). We write ordn (i) for the multiplicative order of the integer i modulo the integer n (coprime to i, of course). Also, for n ≥ 8, we let Qn denote the (generalized) quaternion group of order n. Theorem 2.1. A finite group T is embeddable into a division ring of characteristic zero if and only if T is isomorphic to one of the following groups: (1) Z-groups: (1a) Cn , the finite cyclic group of order n. (1b) Cm o C4 , where m is odd and C4 acts by inversion on Cm . (1c) A group of the form T0 × T1 × . . . × Ts , where s ≥ 1, the orders |T0 |, |T1 |, . . . , |Ts | are coprime, T0 is cyclic, each of T1 , . . . , Ts is noncyclic of the form Cpa o(Cqb1 ×. . .×Cqrbr ), where p, q1 , . . . , qr 1 are distinct primes, and each Cpa oCqb is noncyclic and satisfies the following condition: If Cqα is the kernel of the action of Cqb on Cpa , then either q = 2, p ≡ −1 mod 4, and α = 1, or q = 2, p ≡ −1 mod 4, and 2α+1 6 | (p2 − 1), or q = 2, p ≡ 1 mod 4, and 2α+1 6 | (p − 1), or q > 2, and q α+1 6 | (p − 1). Moreover, for each noncyclic factor Cpa o Cqb of Ti , the integer qordqα (p) does not divide ord|T /Ti | (p). (2) Cm oQ2t , where m is odd, an element of Q2t of order 2t−1 centralizes Cm , and an element of Q2t of order 4 inverts Cm . (3) Q8 × M , where M is a Z-group of odd order m, and ordm (2) is odd. (4) SL(2, 3) × M , where M is a Z-group of odd order m, and ordm (2) is odd. (5) h2, 3, 4i, the binary octahedral group (of order 48). (6) SL(2, 5), the binary icosahedral group (of order 120). 2 We also need the following result. Lemma 2.2. [pfA20] Let T be an Amitsur group. Then the rational group ring QT has a unique component that is a division ring and contains an isomorphic image of T .

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Proof. The structure of the Amitsur groups listed above shows that we need only verify the result for SL(2, 5), h2, 3, 4i, SL(2, 3), and metacyclic Amitsur groups, since every other Amitsur group is a direct product of such groups. We write A(F ) for the quaternion algebra with centre F generated by ı,  subject to ı2 = −1, 2 = −1, and ı = −ı. This is clearly a division ring if F is a real field. In the case of h2, 3, 4i and SL(2, 3), the structure is extremely elementary and we omit the calculations. The division rings containing a copy of the group are the ones at the end of each decomposition. If ω is a primitive cube root of unity, then QSL(2, 3) ∼ = QAlt(4) ⊕ Q(ω)2×2 ⊕ A(Q) √ √ Qh2, 3, 4i ∼ = QSym(4) ⊕ Q(ω, 2)2×2 ⊕ A(Q( 2)). The structure of SL(2, 5) can be most easily obtained by using [3], Theorem 6.1 and Table II. In the notation of loc cit, the faithful characters of SL(2, 5) are the following: η1 , of (complex) degree 6, character field Q, and all local indices equal to 1 except for completions at {∞, 2} where the indices are 2. This gives the rational group algebra component A(Q)3×3 . θ1 , of degree 4, character field Q, and all local indices equal to 1 except for completions at {∞, 3} where the³ indices ´ are 2. This corresponds to −1,−3 the quaternion division algebra D = , and gives the rational group Q algebra component D2×2 . √ ζ1 and ζ2 , each of degree 4, character field Q( 5), and all local indices equal to 1 except for the completion at {∞} where the index is 2. These √two complex characters arise from the quaternion division algebra A(Q( 5)). The complete decomposition is √ 3×3 QSL(2, 5) ∼ ⊕ A(Q( 5)). = QAlt(5) ⊕ D2×2 ⊕ A(Q) The case of metacyclic Amitsur groups will be done in a number of steps. If k is any field of characteristic coprime to the order of the finite group G, P then we write eH = |H|−1 h∈H h ∈ R = kG, where H / G. Step 1. If H, K / G, then eH eK = eHK : Elementary. Step 2. If H, K / G, then eH R ∩ eK R = eHK R: Immediate from Step 1. ¡ Step 3. If H / G and e is a central idempotent in R, then H ⊆ ker G→ ¢ eR if and only if eH e = e: Also elementary. Q ∼ H1 × . . . × Hn . Then Step 4. Let H1 , . . . , Hn / G ³such that ni=1 Hi = ¢´ Qn ¡ dimk (eH1 R + . . . + eHn R) = |G| 1 − i=1 1 − 1/|Hi | : For, let Vi = eHi R. Then Vi1 ∩ . . . ∩ Vim = eHi1 ...Him R by Steps 2 and 3, and by definition dimk (eHi1 ...Him R) = |G|/|Hi1 , . . . , Him | = |G|/|Hi1 | . . . |Him |. Thus, by inclusion-exclusion,

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P P dimk (V1 + . . . Vn ) = ni=1 (|G|/|Hi |) − ni6=j (|G|/|Hi ||Hj |) + . . . , which is the given formula. Step 5. Let H1 , . . . , Hn be minimal normal subgroups of G of coprime orders. If e is a nonzero central idempotent of R that is orthogonal to eH1 , . . . , eHQn , then Q H1 . . . Hn ∩ ker(G → eR) = 1: Let K = ker(G → eR). Then K ∩ i Hi = i (K ∩ Hi ) since the orders are coprime. Being minimal, each K ∩ Hi = 1 or Hi . The latter is impossible because Hi ⊆ K, implies that e = eHi e = 0 in view of Step 3. Step 6. Let H1 , . . . ³ , Hn be minimal normal ´ L subgroups of G of coprime m orders, and write R = eH1 R + . . . + eHn R j=1 ej R, where the ej R are Q simple. Then i Hi embeds into each ej R, and dimk (eH1 R + . . . + eHn R) = ³ ¢´ Qn ¡ |G| 1 − i=1 1 − 1/|Hi | : This follows from Steps 4 and 5, on noting that Q ∼ i Hi = H1 × . . . × Hn . Step 7. Let G be a finite group with cyclic socle S, and write |G| = mm0 , where m and |S| are divisible by exactly the same primes. Then QG contains one or more simple components into which G projects faithfully, and the total Q-dimension of such components is m0 ϕ(m): The minimal normal Q subgroups H1 , . . . , Hn are cyclic of prime order, and by definition S = i Hi . If G does not project faithfully into a simple component eR = eQG, then some Hi ⊆ ker(G → eR), so eR ⊆ eHi R by Step 3. Therefore, the simple components into which G does not project faithfully all lie in eH1 R + . . . + eHn R, the total dimension of which is given by ³Step 6. The remaining ¢´ Q ¡ components (if any) have total dimension |G|−|G| 1 − ni=1 1 − 1/|Hi | = Q mm0 p|m (1 − 1/p) = m0 ϕ(m), as claimed. Step 8. If G is a metabelian Amitsur group, then QG contains a unique component into which G projects faithfully (and this component is, of course, a division ring): The metabelian Amitsur groups are the groups in parts (1)– (3) of Theorem 2.1. By inspection, any such G has a unique minimal normal subgroup for every prime dividing |G|, so the integer m0 of Step 7 is equal to 1. Thus, by Step 7 the faithful components of QG have total Q-dimension equal to ϕ(G). On the other hand, in the process of classifying Amitsur groups, one shows that the division algebra generated over Q by an Amitsur group G is uniquely determined by G. For metacyclic Amitsur group G, the division algebra has Q-dimension ϕ(G). This being the total Q-dimension of the faithful components of QG computed above, there can be only one such component, as required. ¤ We can now prove the second half of Theorem 1.1. Let X ∈ ZF be finitely generated, and assume that T = τ (X) is an Amitsur group. Let ∆ be the unique component of QT , which is a division ring and contains

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an isomorphic copy of T . The rational group ring QX contains QT , and X acts by conjugation on QT . The uniqueness of ∆ implies that ∆ is stabilized under conjugation by X. In particular, if e ∈ QT is the idempotent determining ∆, then e is central in QX. The ring R = QXe is easily seen to be obtained from ∆ by a series of skew Laurent polynomial extensions (in the notation at the end of Introduction, the variables in the successive extensions are g1 e, . . . , gn e). Theorem 1.4.3 of [7] now implies that R is an Ore domain, and therefore has a division ring of quotients D. Once we have shown that X embeds into D, we have D = k[X], where k = Z(D), which shows that D is finite-dimensional over k because X is centre-by-finite. To show that X embeds into D, suppose x ∈ X satisfies xe = e. As QX is obtained from QT by a series of polynomial extensions, it is evident that no element x 6∈ T can satisfy xe ∈ QT . Thus xe = e implies that x ∈ T , and so x = 1 because T embeds into ∆. In other words, X embeds into R. Note that the centre of D is a rational function field over the centre of ∆. The above construction also helps to explain why the obvious converse of the first part of Theorem 1.1 can fail to be true. To be precise, let X ∈ ZF , let T be an Amitsur group, and let k be a field of characteristic zero. Suppose we have a division algebra D = k[X]. The existence of the ring homomorphism θ : kX → D means that the group ring kT has a division algebra component ∆ into which T projects faithfully, and such that ∆θ ∼ = ∆. However, obviously examples exist in which the conjugation action of X on kT does not stabilize any division ring component of kT . In such an example, there exists an element x ∈ X such that ∆∆x = 0 in θ kT , whence (∆θ )x = 0, an obvious impossibility. In other words, for such examples no embedding of X into a division algebra D can exist. 3. Proof of Theorem 1.2 Let X ∈ ZF be finitely generated, and let D be a finite-dimensional division k-algebra, where k has characteristic p > 0. Then T = τ (X) is a finite subgroup of a division ring of positive characteristic, so T is cyclic by Wedderburn’s Theorem on finite division rings. The difficulties in this case stem from the fact that the group ring Fp T usually has more than one simple component into which T projects. I have been unable to find a reference for the the following result, so a proof is included. Theorem 3.1. Let n be a positive integer, and p a prime not dividing n. Let A = hai be a cyclic group of order n, and for each d|n let Ed be the set of primitive idempotents of the group ring Fp A such that e ∈ Ed if and only if ea has order exactly d. Then Aut(A) acts transitively on each Ed . Proof. If d < n, then the conclusion for Ed follows from that for En , since G = Aut(A) induces the full automorphism group of hai/had i for any divisor d of n. Write R = Fp A.

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Suppose first that n 6= p is an odd prime. Let t = ordq (p), and let c is the {e0 , e1 , . . . es } be all the primitive idempotents of R, where e0 = hai trivial idempotent. The components e1 R, . . . , es R are all isomorphic to Fpt . If an element x ∈ R is fixed by G = Aut(A), then x = α + βe0 for some α, β ∈ Fp since G acts transitively on {a, a2 , . . . , an−1 }. If, in addition, x is an idempotent, then x must be one of 0, 1, e0 , 1 − e0 , since being an idempotent is equivalent to the equations α2 = α and β 2 + 2αβ = β in Fp . Hence, if e is the sum of all the ei in the G-orbit of e1 , then e is an idempotent of R that is fixed by G, so it must be 1 − e0 . In other words, G acts transitively on the set {e1 , . . . es } = Eq , proving the result in this case. We also need to verify the result for n = 4. If p ≡ 1 mod 4, then we have a primitive fourth root of unity ζ ∈ Fp . The primitive idempotents of R are P ei = 4−1 3r=0 (ζ −i a)r for i = 0, . . . , 3, and E4 = {e1 , e3 } since aei = ζ i ei for all i. It is immediate that the inversion automorphism a 7→ a−1 of hai interchanges e1 and e3 , so G acts transitively on E4 . On the other hand, in (2) the case where p ≡ −1 mod 4, we have R ∼ = Fp ⊕ Fp2 (with a mapping to 1 and −1 in the first two components). Therefore, E4 consists of a single element in this case, so the transitivity of the action of G is trivial. The result when n = q m is a power of a prime q now follows by induction on m ≥ 2 (m ≥ 3 when q = 2). Let b = aq and set T = Fp hbi. Let P i e ∈ En have the form e = q−1 i=0 ti a , where the ti ∈ T . Let j ∈ [1, q − 1], m−1 m−2 and consider the automorphism θ of A defined by aθ = a1+jq = bjq a. P P q−1 i )θ = ijq m−2 ai . Then bθ = b, so eθ = q−1 t (a t b i=0 i i=0 i Note that T e ⊂ Re is also a field, and be has order exactly q m−1 since ae has order q m = n. Let f ∈ T be such that P the primitive Pidempotent q−1 i = i , so t = t f T e = T f . Then ef = e, so e = ee = q−1 (et )a (f t )a i i i i i=0 i=0 for all i. Note that the field T f is generated by bf . m−2 If eθ = e, then ti (1 − bijq ) = 0 in T for all i ∈ [1, q − 1]. Multiplying by f we obtain an equation in the field T f . Now ti 6= 0 for some i ≥ 1 m−2 (else e ∈ T ), so (bf )ijq = 1 in T f for the same i. Since i and j are both coprime to q, and bf has order a power of q, this leads to the contradiction m−2 (bf )q = 1. In other words, eθ 6= e for every j ∈ [1, q − 1], and we have obtained q distinct elements eθ of En by this procedure. Every such idempotent satisfies beθ = (be)θ = (bf )θ = bf . Conversely, it is obvious that for every idempotent f ∈ En/q of T , there exists an e ∈ En such that ef = e (in fact, there are at least q such e, by the above construction). The restriction of G = Aut(A) to hbi yields Aut(hbi). Hence, the inductive hypothesis implies that for any two f, f 0 ∈ En/q ⊆ T , there exists φ ∈ G such that f 0 = f φ . Putting this together with the construction of the previous paragraph, we have a G-orbit of En containing at least q|En/q | elements. An easy dimension count shows that |En | = q|En/q |, so there is in fact only one orbit.

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Finally, consider the case where n is a composite integer. The group G is the direct product of the full automorphism groups of the Sylow subgroups Q of A, each of which acts transitively on the corresponding set E|Q| , by the earlier argument. Together with the tensor product decomposition of Fp A into the components Fp Q, we obtain the result in the general case. ¤ Continuing with the notation of the above theorem, let e ∈ En and H = {γ ∈ Aut(A) : eγ = e}. The transitivity of the action of Aut(A) on En implies that the stabilizers of the other idempotents in En are the conjugates of H in the abelian group Aut(A). That is, all e ∈ En have the same stabilizer H in Aut(A). In other words, Aut(A) contains a subgroup Γp (A) such that • If γ ∈ Γp (A), then eγ = e for all e ∈ En . • If γ 6∈ Γp (A), then γ does not fix any e ∈ En . The subgroup Γp (A), which of course only depends on the integers n and p, is rather difficult to determine explicitly (although the proof of Theorem 3.1 provides some information about it). The proof of Theorem 1.2 is now immediate. Let D = k[X] be as in the first half of the theorem, and consider the ring homomorphism θ : Fp X → D. There exists a unique e ∈ E|T | such that eθ 6= 0. If x ∈ X were to act on T as an automorphism not in Γp (T ), then we would have the contradiction θ 0 = (ex )θ = (eθ )x = 1 in D. Thus, X must act as a subgroup of Γp (T ), as required. Conversely, let X ∈ ZF be finitely generated, and assume that T = τ (X) is cyclic and X/CX (T ) is a subgroup of Γp (T ) in the natural way. The final condition implies, by definition, that X centralizes every idempotent e ∈ E|T | ⊆ Fp T ⊆ Fp X. The proof that X embeds into a division ring over Fp now proceeds exactly as in the proof of Theorem 1.1 given in the latter part of Section 2. Acknowledgment Research partially supported by the Natural Sciences and Engineering Research Council of Canada. References [1] E. Aljadeff and J. Sonn, Projective Schur division algebras are abelian crossed products, J. Algebra 163 (1994), 795–805. [2] S.A. Amitsur, Finite subgroups of division rings, Trans. Amer. Math. Soc. 80 (1955) 361–386. [3] W. Feit, The computations of some Schur indices, Israel J. Math. 46 (1983), 274–300. [4] B. Huppert, Endliche Gruppen I, Springer, Berlin, 1983. [5] B. Huppert, Representation Theory at Mainz, in: Representation Theory of Finite Groups and Finite-Dimensional Algebras (Ed. G. O. Michler and C. M. Ringel) Progress in Mathematics 95 (1991), 1–36.

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[6] F. Lorenz and H. Opolka, Einfache Algebren und projektiv Darstellungen u ¨ber Zahlk¨ orpern, Math. Z. 162 (1978), 175–182. [7] M. Shirvani and B.A.F. Wehrfritz, Skew Linear Groups, Cambridge University Press, Cambridge, 1986.

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Isomorphic rational group algebras Ana Cristina Vieira Departamento de Matem´atica, Instituto de Ciˆencias Exatas, Universidade Federal de Minas Gerais, 30123-970, Belo Horizonte, Brasil [email protected] Guilherme Leal Instituto de Matem´atica, Universidade Federal do Rio de Janeiro, Caixa Postal 68530, 21910, Rio de Janeiro, Brasil [email protected] Abstract. Our main goal is to classify groups H such that QG ∼ = QH when G is a finite nilpotent of class 2 group with cyclic center.

1. Introduction The isomorphism problem asks whether a ring isomorphism RG ∼ = RH implies that G ∼ = H. This classical problem appears for the first time in regard to integral group rings in G. Higman’s Ph.D. thesis. In 1947, T. M. Thrall formulated it in the following terms: Given a group G and a field F , determine all groups H such that F G ∼ = F H. It was first considered by S. Perlis and G. Walker in 1950 when they settled that finite abelian groups are determined by their rational group algebras. In this paper, pursuing an answer for the isomorphism problem in Thrall sense, we classify the groups H with rational group algebra isomorphic to QG where G is a finite nilpotent class 2 group with cyclic center. Since all groups considered are finite and the classification is up to isomorphism we, most of the time, replace the condition QG ∼ = QH by QG = QH. With this set, we can state our main theorem. Theorem 1.1. Let G be a nilpotent class 2 finite p-group with cyclic center. If QG = QH, then H is also nilpotent of class 2, G0 = H 0 and the center of G/N is isomorphic to the center of H/N for any subgroup N of G0 . Conversely assume that G and H are p-groups of nilpotence index 2 of equal order with cyclic center such that G0 ∼ = H 0 and the center of G/NG is isomorphic to the center of H/NH for any subgroup NG of G0 and NH its correspondent in H 0 , then 331

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(1) If p is odd or p is even and G has no extra-special group as homomorphic image, we have QG ∼ = QH. (2) If G is a 2-group with an extra-special group of order 22n+1 as homomorphic image, this image must be unique and H also has an extra-special group of same order as homomorphic image. We have QG ∼ = QH, if and only if, the extra-special homomorphic images of G and H are isomorphic. 2. Beginning the proof Throughout this paper all groups will be finite. If N is a subgroup of a b the idempotent 1 P group G we call N h∈N h, such idempotent is central |N | c if and only if N is normal in G. If g ∈ G we write gb for hgi. Let Z(G) be the center of G, for an element g of a group G we denote its conjugacy class by Cg and the sum of the elements of this set by Cbg . We start with a technical but very useful lemma that appears in [4]. Lemma 2.1. Let G be a finite group and g ∈ G. If g −1 Cg ∩ Z(G) 6= {1}, then G contains a central element z of prime order so that Cbg = Cbg zb. Proof. By assumption there exist h ∈ G and 1 6= z ∈ Z(G) so that h−1 gh = zg. Hence for any positive integer n we get h−n ghn = z n g. It then follows easily that hziCg ⊆ Cg . So zCg = Cg and therefore Cbg = Cbg zb. Obviously replacing z by some power of z, if necessary, we may assume that z has prime order. ¤ We can split the rational group algebra c0 ⊕ QG · (1 − G c0 ). QG = QG · G c0 is isomorphic to Q[G/G0 ] and, since it is commutative The first part QG · G and contains all commutative ideals of QG, we call it the commutative part of QG. The second part is usually called the noncommutative part and is denoted by 4(G, G0 ). Proposition 2.2. Let G be a nilpotent class 2 p-group with cyclic center. If QG ∼ = QH, then H is also nilpotent class 2, the center of G is isomorphic to the center of H and G0 = H 0 . Proof. First we assume that QG = QH. Note that this implies that |G| = |H| and, since their commutative parts are the same, Q[G/G0 ] = Q[H/H 0 ], therefore |G0 | = |H 0 |. Let z be an element of order p in the center of G, since G is a nilpotent class 2 group with cyclic center from [4] we have that the central idempotent e = 1 − zb is primitive in QG.

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We first show that the component Q[G]e affords a faithful representation of H. Let tr(e) be the coefficient of 1 in e, called the trace of e. The dimension of the ideal Q[G]e is |G| · tr(e) = |G|(p − 1)/p. If the representation of H afforded by Q[G]e is not faithful then the prob = N b . It imjection H → He has a nontrivial kernel N such that eN plies that Q[G]e is a simple component of Q[H/N ] a contradiction since |H/N | ≤ |G|/p. So H has a faithful irreducible representation, therefore its center is cyclic. Now let x be an element of order p in the center of H such that xe = ze. We can compute dimensions using the traces and see that QG(1 − x b) = QG(1 − zb). So (1 − x b) = (1 − zb), i.e. x b = zb and we have x · zb = x · x b=x b = zb = z · zb and so x = x(b z + e) = xb z + xe = zb z + ze = z. By induction we get that G0 ⊂ H 0 and since |G0 | = |H 0 | we have G0 = H 0 so H 0 is central and H is a nilpotent class 2 group. To complete the proof note that if g ∈ G is a noncentral element of G and Cg its conjugacy class, Lemma 2.1 shows that Cbg (1 − zb) is zero, therefore the support of any central element in QG(1 − zb) = QH(1 − zb) is contained in the center of G(1 − zb). Since (1 − zb) affords faithful representations of G and H, the centers of G and H are isomorphic.¤ ¤ By considering the hypothesis of the last result, let H 0 = G0 ⊂ Z(G) be generated by a of order pn . The subgroups of G0 are n−1

hai, hap i, . . . , hap

n

= zi, {ap = 1},

c0 ) are and by [4] the central primitive idempotents of QG(1 − G pn−2 )a[ pn−1 , . . . , (1 − b (1 − zb), (1 − a[ a)abp . pn−2 )a[ pn−1 affords an irreducible representation The idempotent e = (1 − a[ n−1 n−1 of Ge ∼ = G/hap i so G/hap i has cyclic center and we can apply again the previous proposition to get the first part of our main theorem. Conversely let p be an odd prime and let’s assume that G and H are nilpotent class 2 p-groups with cyclic center such that G0 ∼ = H 0 and the center of G/NG is isomorphic to the center of H/NG for any subgroup NG of G0 and NH the subgroup of the same order in H 0 . By assumption we have G/G0 ∼ = H/H 0 so that the commutative parts of QG and QH are isomorphic. Moreover by [4] 4(G, G0 ) and 4(H, H 0 ) have the same primitive central idempotents because the groups are nilpotent class 2 and G0 ∼ = H 0 . Then the simple components of 4(G, G0 ) and 4(H, H 0 ) have the same dimension over Q (see Lemma 1.2 page 32 [5]) and, since the

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center of G/NG is isomorphic to the center of H/NG , we can see that the components MNG and MNH , which afford irreducible representations of G/NG and H/NH respectively, have isomorphic centers. Finally by [6] or Lemma 4.7 page 564 on [5] the Schur index of such components must be 1 and MNG ∼ = MNH . The problem that appears in the case p = 2 is that the Schur index of each noncommutative component can be 1 or 2. If the index is 2, Lemma 4.7 page 564 on [5], assures that this component is a matrix ring over the hamiltonian quaternion algebra with coefficients on the center of the component, so the components are isomorphic if and only if the Schur indexes are the same. [ 2i−2 )a 2i−1 the Now in a simple component with central idempotent (1 − a[ i−2

element a2 has order 2 and locally it is −1. √ If the order of the center i−1 2 of G/ < a > is bigger than 2 we will have −1 in the center of this component, in this case a possible hamiltonian quaternion algebra splits and the Schur index is 1. With that remark, to finish the proof of the main theorem, we just need to ¯0 ¯ such that Z G ¯=G deal with the case when G has a homomorphic image G ¯ and |Z G| = 2. We will treat this situation in the next section.

3. Extra-special groups Particular nilpotent class 2 groups are the extra-special p-groups. Any such p-group G is nonabelian and G0 = Z(G) = Φ(G) has order p, where Φ(G) denotes the Frattini subgroup of G. A consequence of the definition, the abelianization of G is elementary p-abelian. It is clear that a nonabelian group G of order p3 is extra-special and if p = 2 it is isomorphic to D = ha, b | a2 = b2 = 1, (ab)4 = 1i (Dihedral group) or Q = ha, b | a4 = 1, a2 = b2 , ab = a−1 i (Quaternion group). In case p is an odd prime, G is isomorphic to one of the following groups 2 (Theorem 5.1 page 203 [3]): N = hx, y | xp = y p = 1, xy = xp+1 i or M = hx, y, z | xp = y p = z p = 1, [x, z] = [y, z] = 1, z = [x, y]i. It is possible to establish the structure of an extra-special p-group. In fact, such a group G is isomorphic to a central product of r ≥ 1 nonabelian subgroups of order p3 . Moreover, we have |G| = p2r+1 and by considering central products (Theorem 5.2 page 204 [3]): (1) If p = 2, G ∼ = Dk Qr−k , for some k. For k ≥ 1 we have Dk Qr−k ∼ = DQr−1 , if k is odd and Dk Qr−k ∼ = Qr , if k is even. The groups DQr−1 and Qr are not isomorphic. (2) If p is odd, G ∼ = N k M r−k , for some k. For k ≥ 1 we have N k M r−k ∼ = r−1 k r−k r ∼ NM , if k is odd and N M = M , if k is even. The groups N M r−1 and M r are not isomorphic.

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We have:

QD ∼ = 4Q ⊕ M2 (Q), QQ ∼ = 4Q ⊕ H, ∼ QN = Q ⊕ (p + 1)Q[ζp ] ⊕ Mp (Q[ζp ]), QM ∼ = Q ⊕ (p + 1)Q[ζp ] ⊕ Mp (Q[ζp ]) where H is the classical quaternion algebra over the rational field and ζp is a primitive pth root of the unit. Remark 3.1. As we can observe from Proposition 2.2, if G is an extraspecial p-group and H a group such that QG ∼ = QH then H is an extraspecial group of the same order as G. Now, if G and H are extra-special p-groups of the same order, then G/G0 and H/H 0 are elementary p-abelian groups of the same order and so we have c0 ∼ c0 . We have to investigate the noncommutative component QG · G = QH · H 0 4(G, G ) in order to conclude whether QG ∼ = QH or not. Proposition 3.2. Let p be an odd prime and G be an extra-special p-group of order p2n+1 . Then 4(G, G0 ) ∼ = Mpn (Q[ζp ]), where ζp is a primitive pth root of the unit. Proof. The center of G has only two irreducible idempotents over Q and by [4], all central idempotents of 4(G, G0 ) of a nilpotent class 2 group are central; therefore, for an extra-special group we have that 4(G, G0 ) is simple. As in the last paragraph of the proof of Proposition 2.2, the support of any central element in this noncommutative component is in the center of G. So the center of 4(G, G0 ) is Q(ζp ). If p is an odd prime, the Schur index of QG is one, [6] or Lemma 4.7 page 564 of [5], so we have that 4(G, G0 ) is a matrix ring over Q[ζp ]. Since the dimension of 4(G, G0 ) over Q is p2n+1 − p2n , the result follows. ¤ Corollary 3.3. Let p be an odd prime and G an extra-special p-group. Then QG ∼ = QH, if and only if H is an extra-special group of the same order as G. Let us consider the case p = 2. We have that, up to isomorphism, there exist only two extra-special 2-groups. We prove that: Proposition 3.4. Let G be an extra-special 2-group of order 22n+1 , n ≥ 2. (1) If G ∼ = Q1 Q2 ...Qn (central product of quaternion groups) then (i) If n is odd then 4(G, G0 ) ∼ = M2n−1 (H). (ii) If n is even then 4(G, G0 ) ∼ = M2n (Q). (2) If G ∼ D Q ...Q (central product of one dihedral and n − 1 quater= 1 2 n nion groups) then: (i) If n is odd then 4(G, G0 ) ∼ = M2n (Q) (ii) If n is even then 4(G, G0 ) ∼ = M2n−1 (H).

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Proof. We can consider the center of G as {1, z}. When G ∼ = Q1 Q2 ...Qn we have each Qi is a quaternion group generated by ai and bi with a2i = b2i = [ai , bi ] = z but ai and bi commute with all other generators aj e bj , j 6= i. On the other hand, if G ∼ = D1 Q2 ...Qn the relations are a2i = b2i = [ai , bi ] = z, for i ≥ 2 and [ai , bj ] = [ai , aj ] = 1, for i 6= j while a1 and b1 are generators of order 2 of D1 such that [a1 , b1 ] = z and commute with ai , bi , i 6= 1. In any case we define the set A = {a1 b1 ai bi | i = 2, . . . , n} of n − 1 commuting elements of order 2 in G, therefore N = hAi is an elementary abelian 2-subgroup of G of rank n − 1. n Y (1 + a1 b1 aj bj ) 1 X b By considering the idempotent N = n−1 h= , we 2 2 h∈N

j=2

b 1−z has dimension 2n+1 over Q. have that QGN 2 Now we have to consider two cases. At first we consider the central product G∼ = Q1 Q2 . . . Qn . If n is odd, the elements σ = a1 a2 · · · an and τ = b1 b2 · · · bn commute with b and we have σ 2 = τ 2 = z and τ σ = zστ. each factor of N 0 Since z = −1 on the component QG( 1−z 2 ) and the dimension of 4(G, G ) n 0 ∼ over Q is 2 , we conclude 4(G, G ) = M2n−1 (H). In the other case, n is even and then λ = a1 a2 a3 · · · an is an element b . The trace of the of order 2 in G and commutes with the factors of N 1+λ b 1−z 1 idempotent e = 2 N 2 is 2n+1 and the dimension of the left ideal QGe over Q is 2n . In this case 4(G, G0 ) ∼ = M2n (Q). Finally let us consider the central product G∼ = D1 Q2 . . . Qn . If n is odd, we observe the element γ = b1 a2 a3 · · · an has order 2 and b . The trace of the idempotent e = 1+γ N b 1−z commutes with the factors of N 2 2 1 is 2n+1 and so the left ideal QGe has dimension 2n over Q. In this case 4(G, G0 ) ∼ = M2n (Q). Now if n is even then the elements α = b1 a2 a3 · · · an and β = b1 b2 b3 · · · bn b and we have α2 = β 2 = z and αβ = zβα. commute with each factor of N Again we use that z = −1 on the component QG( 1−z 2 ) and the dimension of 4(G, G0 ) over Q is 22n to conclude that 4(G, G0 ) ∼ ¤ = M2n−1 (H). Now we finish the proof of the main theorem with the following. Corollary 3.5. Let G be an extra-special 2-group of order 22n+1 , n ≥ 2. Then QG ∼ = QH, if and only if, G ∼ = H. Acknowledgment Dr. Leal is partially supported by CNPq, Brasil.

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References [1] D.B. Coleman, Finite groups with isomorphic group algebras, Trans. Amer. Math. Soc., 105 (1962), 1–8. [2] C.W. Curtis and I. Reiner, Methods in Representation Theory, Vol. 1, John Wiley, New York, 1987. [3] D. Gorenstein, Finite Groups, Harper and Row Publishers, New York, 1968. [4] E. Jespers, G. Leal, and A. Paques, Central idempotents in the rational group algebra of finite nilpotent groups, J. Algebra Appl., 2, 1, (2003), 57–62. [5] D.S. Passman, The Algebraic Structure of Group Rings, John Wiley, New York, 1977. [6] P. Roquette, Realisierung von Darstellungen endlicher nilpotenter Gruppen, Arch. Math. 9 (1958), 241–250.

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