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Combinatorial and Asymptotic Methods of Algebra Non A.I. Kostrikin I. R. Shafarevich (Eds.) Algebra VI Combinatorial and Asymptotic Methods of Algebra. Non-Associative Structures Heidelberg New York Barcelona Budapest Hong Kong London Milan Paris Tokyo Springer Consulting Editors of the Series: A. A. Agrachev, A. A. Gonchar, E. E Mishchenko, N.M. Ostianu, V.P Sakharova, A. B. Zhishchenko List of Editors, Authors and Translators Title of the Russian edition: Editor-in-Chief Itogi nauki i tekhniki, Sovremennye problemy matematiki, Fundamental’nye napravleniya, Vol. 57, Algebra 6 R. V. Gamkrelidze, Russian Academy of Sciences, Steklov Mathematical Institute, Publisher VINITI, Moscow 1990 ul. Vavilova 42,117966 Moscow, Institute for Scientific Information (VINITI), ul. Usievicha 20a, 125219 Moscow, Russia, e-mail: [email protected] Consulting Editors A. I. Kostrikin, Moscow University, MEHMAT, 119899 Moscow, GSP- 1, Russia Library of Congress Cataloging-in-Publication Data I. R. Shafarevich, Steklov Mathematical Institute, ul. Vavilova 42, Algebra 6. English. 117966 Moscow, Russia Algebra VI : combinatorial and asymptotic methods of algebra : nonassociative structures / A.I. Kostrlkfn. I.R. Shafarevich teds.). p. cm. -- (Encyclopaedla of mathematical sciences ; v. 57) Includes bibliographical references and index. Authors ISBN 3-540-54699-5 (Berlin : alk. paper). -- ISBN O-367-54699-5 (New York : alk. paper) E. N. Kuz’min, Mathematical Institute of the Siberian Division of the Russian 1. Nonassociative algebras. 2. Algebra, Honologlcal. Academy of Sciences, 4 University Ave., 630090 Novosibirsk, Russia 3. Combinatorial analysis. I. Kostrlktn. A. I. (Alekser Ivanovlch) II. Shafarevlch. I. R. (Igor’ Rostlslavovlch). 1923- . V. A. Ufnarovskij, Mathematical Institute of the Academy of Sciences III. Title. IV. Title: Algebra 6. V. Series. of Moldavia, 5 ul. Grosula, 277028 Kishinev, Moldavia OA252.A4413 1994 and Department of Mathematics, University of Lund, Box 118,22 100 Lund, 512--dc20 93-25596 Sweden, e-mail: [email protected] CIP 1. P. Shestakov, Mathematics Institute of the Russian Academy of Sciences, Universitetskij prospekt 4,630090 Novosibirsk, Russia Mathematics Subject Classification (199 1): IBPxx, 16Wxx, 17-xX, 17D10,17D25,18Gxx, 2OF10, Translator 20N05,2ONO7,55XX, 68Q20,68Rxx, 8 1R50 R. M. DimitriC, Department of Mathematics, University of California, Evans Hall, Berkeley, CA 94702, USA, e-mail: dimitric @ocf.Berkeley.edu ISBN 3-540-54699-5 Springer-Verlag Berlin Heidelberg New York ISBN O-387-54699-5 Springer-Verlag New York Berlin Heidelberg This work is subject to copyright. All rights ate reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in dam banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9.1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. 0 Springer-Vetlag Berlin Heidelberg 1995 Printed in Germany Typesetting: Springer TPX in-house system SPIN 10020719 41/3142 - 5 4 3 2 1 0 - Printed on acid-free paper Contents I. Combinatorial and Asymptotic Methods in Algebra V. A. Ufnarovskij 1 II. Non-Associative Structures E. N. Kuz’min, I. P. Shestakov 197 Author Index 281 Subject Index 283 \ I. Combinatorial and Asymptotic Methods in Algebra V. A. Ufnarovskij Translated from the Russian by R. M. Dimitrii: Contents Preface to the English Edition ....................... 5 Introduction ................................. 6 51. Basic Objects and Constructions .................. 8 1.1. Introduction ........................... 8 1.2. Ways of Defining Infinite-dimensional Algebras ....... 10 1.3. Combinatorial Approach .................... 12 1.4. Connections Between Different Ways of Defining by Generators and Relations ................... 14 1.5. Modules ............................. 16 1.6. The Tensor Product ...................... 18 1.7. Coalgebras. Hopf and Roos Algebras ............. 19 1.8. Elements of Homological Algebra ............... 22 52. Normal Words and a Grijbner Basis of an Ideal of a Free Algebra 27 2.1. Introduction ........................... 27 2.2. Basic Notation. Degree and Order ..... - ........ 27 2.3. Normal Words. Decomposition of a Free Algebra into an Ideal and Its Normal Complement ......... 28 2.4. Grobner Basis. Complete System of Relations ........ 29 2.5. Reduction. Composition. The Composition Lemma ..... 29 2.6. Examples ............................. 30 2.7. Grobner Basis in the Commutative Case ........... 32 2.8. Regular Words. Basis of a Free Lie Algebra ......... 34 2.9. A Composition Lemma for Lie Algebras ........... 37 V. A. Ufnarovskij I. Combinatorial and Asymptotic Methods in Algebra 3 2.10. The Triangle Lemma . 39 6.3. Comparison and Equivalency of Words. 2.11. Applications and Examples of Complete Systems of Relations 39 Periodic and Regular Words .................. 101 2.12. Comments . 42 6.4. Theorem on Height ....................... 104 6.5. Nilpotency. Sandwiches ..................... 106 $3. Graded Algebras. The Golod-Shafarevich Theorem. 6.6. Nilpotency and Growth in Groups .............. 110 Anick’s Resolution . 42 6.7. Comments ............................ 114 3.1. Introduction ..................... 42 3.2. Graded Algebras .................. 43 57. The Algebras of Polynomial Growth. The Monomial Algebras . 115 3.3. The Method of Generating Functions ....... 44 7.1. Introduction ........................... 115 3.4. The Free Product. Exact Sequences ....... 46 7.2. PI-algebras. Basic Definitions and Facts ........... 116 3.5. The Golod-Shafarevich Theorem ......... 47 7.3. Calculation of Hilbert Series and Series of Codimensions 3.6. Obstructions. The n-chains. The Graph of Chains . 49 in Varieties of Algebras ..................... 118 3.7. Calculating the Hilbert Series ........... 51 7.4. PI-algebras. Representability. T-primary and Non-matrix 3.8. Anick’s Resolution ................. 54 Varieties. The Equality Problem ............... 122 3.9. Calculating the Poincare Series .......... 59 7.5. Varieties of Semigroups ..................... 125 3.10. Associated Lattices of Subspaces and Algebras with One 7.6. The Automaton Monomial Algebras ............. 127 Relation. Algebras with Quadratic Relations . 63 7.7. Finite Global Dimension. Radical and PI-properties of 3.11.Comments . 67 the Monomial Algebras ..................... 129 7.8. Examples of Growth of Algebras. Properties of 54. Generic Algebras. Diophantine Equations ........... 67 the Gel’fand-Kirillov Dimension ................ 130 4.1. Introduction ......................... 67 7.9. Finite GrGbner Basis and Strictly Ordered Algebras .... 133 4.2. Order Properties for Series ................. 68 7.10. Lie Algebras of Polynomial Growth .............. 135 4.3. Algebras of Global Dimension Two ............ 70 7.11. Comments ............................ 136 4.4. Conditions of Finite-dimensionality ............ 73 4.5. Properties of the Set of Hilbert Series .......... 74 58. Problems of Rationality ....................... 137 4.6. Action of a Free Algebra. Diophantine Equations .... 76 8.1. Introduction ........................... 137 4.7. Comments .......................... 81 8.2. Rational Dependence. Formulation of the Fundamental Theorem ............................ 137 $5. Growth of Algebras and Graphs . 82 8.3. Examples of Non-rational Series ................ 139 5.1. Introduction ........................... 82 8.4. Contiguous Sets. Quadratic Algebras. Relation with 5.2. The Growth. Gel’fand-Kirillov Dimension. Superdimension 83 Hopf Algebras .......................... 141 5.3. The Exponential Growth .................... 86 8.5. Differential Algebras ...................... 144 5.4. The Growth of a Series ..................... 87 8.6. Comments ............................ 148 5.5. The Growth of the Universal Enveloping Algebra ...... 88 5.6. The Growth of Graphs ..................... 89 $9. Local Rings. CW-complexes ..................... 149 5.7. Graphs for Normal Words ................... 90 9.1. Introduction ........................... 149 5.8. Transformations of Graphs. Formulas for Normal Words. 9.2. Regularity. Complete Intersection. Rationality ....... 149 Calculation of the Hilbert Series ............... 92 9.3. Koszul Complex ................ i ........ 151 5.9. An Algorithm for Calculating the Growth of an Algebra . 94 9.4. Nilpotent Algebras and Rings with the Condition m3 = 0 153 5.10 ‘. Regular Languages. Automaton Algebras .......... 95 9.5. Hopf Algebras, Differential Algebras and CW-complexes . 155 5.11. Comments . 98 510. Other Combinatorial Questions ................... 156 56. Combinatorial Lemmas and Their Applications to Questions 10.1. Introduction ........................... 156 of Nilpotency. The Growth of Groups ............... 98 10.2. Hyperbolic Groups ....................... 156 6.1. Introduction .......................... 98 10.3. Quantum Groups and Quadratic Algebras .......... 158 6.2. The Avoidable Words ..................... 99 10.4. Comments ............................ 161 4 V. A. Ufnarovskij I. Combinatorial and Asymptotic Methods in Algebra 5 $11. Appendix ............................... 161 Preface to the English Edition 11.1. Computer Algebra ....................... 161 11.2. Commutative Grobner Basis ................. 162 Almost five years have past
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