Algebra A.G. Kurosh 1908-1971 Algebra Proceedings of the International Algebraic Conference on the Occasion of the 90th Birthday of A. G. Kurosh Moscow, Russia, May 25-30, 1998 Editor Yuri Bahturin W DE G Walter de Gruyter · Berlin · New York 2000 Editor Yuri Bahturin Dept. of Mathematics and Statistics University of Newfoundland St. John's, NF, Canada 7997 Mathematics Subject Classification: 03-03, 03B70, 08-XX, 13-XX, 14-XX, 15-XX, 16-XX, 17-XX, 18-XX, 20-XX, 22-XX, 62-XX, 68-XX Keywords: Groups, rings, modules, Lie algebras, algebraic groups, representations, algebraic varieties, universal algebras, computer algebra © Printed on acid-free paper which falls within the guidelines of the ANSI to ensure permanence and durability. Library of Congress — Cataloging-in-Publication Data International Algebraic Conference on the Occasion of the 90th Birthday of A. G. Kurosh (1998 : Moscow, Russia) Algebra: proceedings of the International Algebraic Confe- rence on the Occasion of the 90th Birthday of A. G. Kurosh, Moscow, Russia, May 25-30, 1998 / editor, Yuri Bahturin. p. cm. Includes bibliographical references. ISBN 3 11 016399 3 (alk. paper) I. Algebra - Congresses. I. Kurosh, A. G. (Aleksandr Gen- nadievich) II. Bakhturin, IU. A. QA150 .152 1998 512-dc21 99-058019 Die Deutsche Bibliothek — C1P-Einheitsaufnähme Algebra : proceedings of the International Algebraic Confe- rence on the Occasion of the 90th Birthday of A. G. Kurosh, Moscow, Russia, May 25-30, 1998 / Ed. Yuri Bahturin. - Berlin ; New York : de Gruyter, 2000 ISBN 3-11-016399-3 © Copyright 2000 by Walter de Gruyter GmbH & Co. KG, D-10785 Berlin All rights reserved, including those of translation into foreign languages. No part of this book may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopy, recording or any information storage and retrieval system, without permis- sion in writing from the publisher. Printed in Germany. Typeset using the authors' T X files: L Zimmermann, Freiburg. E Printing: WB-Druck GmbH & Co., Rieden/Allgäu. Binding: Lüderitz & Bauer-GmbH, Berlin. Cover design: Thomas Bonnie, Hamburg. Preface The International Algebraic Conference, dedicated to the memory of one of the forerunners of modern algebra Alexander Gennadievich Kurosh, was held at Mos- cow Lomonosov State University from May 25 to May 30, 1998. More than 230 participants from at least 25 countries around the world attended the event. Most of them delivered lectures or shorter talks, about 160 in total. The Organizing Committee (Head: Alexei Ivanovich Kostrikin) had selected 25 plenary lectures and a number of principal sectional talks. Six Section Committees formed the remaining portion of the program. The names of the sections give an idea of the diversity of interests of the participants. Section 1. Group Theory. Section 2. Theory of Rings and Modules, Homological Algebra and fC-theory. Section 3. Lie Groups and Lie Algebras, Invariant Theory, Algebraic Groups. Section 4. Algebraic Geometry, Algebraic Number Theory, Commutative Algebra. Section 5. Algebraic Systems. Section 6. Computer Algebra and Algorithmic Problems. The main organizing force of the conference was formed by the faculty, staff and the graduate students of the Department (= Chair) of Higher Algebra. The administration of Moscow State University was also of great assistance, especially with the accommodation of the participants. Some expenses were covered by grants of the Russian Foundation for the Basic Research (Grant 98-01-10041), IN- TAS (Grant 98-MO48) and Pro Mathematica Fond. These grants assisted with the travel (actually, made it possible at all!) of the mathematicians from a num- ber of economically troubled countries and the publishing of the materials of the Conference, including this volume. The organizers greatly appreciate the support of Moscow University and the foundations mentioned. The atmosphere of the Conference was very friendly and businesslike. All the talks contained new research information and, in quite a few cases, the solutions to known problems, or simply outstanding achievements. Some of the talks resulted in the papers that are now suggested to the attention of the reader of the present volume. All the papers have been refereed, with invaluable help of the Department of Al- gebra of Moscow State University. The technical job of preparing the manuscripts for this volume was done by Ilya Bahturin. The Editor Table of Contents Schedule of Talks xi A. I. Kostrikin A word about Kurosh 1 A. A. Abramov, V. O. Belash and L. F. Yukhno Two methods for solving the linear problem of elimination 9 s I. Agoston, V. Dlab and E. Lukacs Strictly stratified algebras 17 N. Yu. Anisimov and E. E. Demidov Randomness: algebraic, statistical and complexity theory aspects 27 V. A. Artamonov General quantum polynomials 35 A. V. Astrelin, O. D. Golubitsky and E. V. Pankratiev Gröbner bases and involutive bases 49 Y. Bahturin, A. Giambruno and M. Zaicev Codimension growth and graded identities 57 R. M. Bryant and A. N. Krasil'nikov On certain non-finitely based varieties of groups 77 F. Campana ^-connectedness of compact Kahler manifolds, II. Solvable quotients of Kahler groups 85 A. A. Davydov Finite groups with the same character tables, Drinfel'd algebras and Galois algebras 99 R. I. Grigorchuk, W. N. Herfort and P. A. Zalesskii The profinite completion of certain torsion p-groups 113 M. Grinenko Birational correspondences of a double cone 125 H. Heineken Groups with neighborhood conditions for certain lattices 135 viii Table of Contents N. K. lyudu Subalgebra membership problem and subalgebra standard basis 137 D. A. Jordan Graded algebras generated by Eulerian derivatives 145 Y. Kachi and E. Sato Characterizations of projective spaces and hyperquadrics 155 U. Knauer Non additive Morita theory. A survey 167 A. I. Kostrikin and A. S. Dzhumadil'daev Modular Lie algebras: new trends 181 V. S. Kulikov On generic coverings of the plane 205 V. N. Latyshev General version of standard bases in linear structures 215 A. I. Lichtman Division rings generated by group rings and enveloping algebras 227 D. B. Nikolova and B. I. Plotkin Some notes on universal algebraic geometry 237 D. S. Passman and W. V. Temple Groups with all irreducible modules of finite degree 263 V. M. Petrogradsky Schreier's formulae for free Lie algebras, their applications and asymptotics 281 A. G. Pinus Conditional terms and their applications 291 Yu. G. Prokhorov Blow-ups of canonical singularities 301 A. Regev and A. Gouterman On the growth of identities 319 L. A. Shemetkov Radical and residual classes of finite groups 331 Table of Contents ix /. P. Shestakov Speciality and deformations of algebras 345 T. Vovkivsky Infinite torsion groups arising as generalizations of the second Grigorchuk groups 357 V. I. Yanchevskii Brauer groups of curves and reciprocity law in Brauer groups of their function fields 379 V. K. Zakharov and A. V. Mikhalev Mathematical systems 397