polynomial identities and combinatorial methods

Polynomial Identitiesand CombinatorialMethods

edited by

Antonio GiambrunoUniversity of PalermoPalermo, Italy

Amitai RegevThe Weizmann Institute of ScienceRehovot, Israel

Mikhail ZaicevMoscow State UniversityMoscow, Russia

MARCEL DEKKER, INC. NEW YORK BASEL>>D E K K E R

Copyright 2003 by Marcel Dekker, Inc. All Rights Reserved.

Although great care has been taken to provide accurate and current information, neither the author(s)nor the publisher, nor anyone else associated with this publication, shall be liable for any loss, damage,or liability directly or indirectly caused or alleged to be caused by this book. The material containedherein is not intended to provide specific advice or recommendations for any specific situation.

Trademark notice: Product or corporate names may be trademarks or registered trademarks and areused only for identification and explanation without intent to infringe.

Library of Congress Cataloging-in-Publication DataA catalog record for this book is available from the Library of Congress.

ISBN: 0-8247-4051-3

This book is printed on acid-free paper.

HeadquartersMarcel Dekker, Inc.270 Madison Avenue, New York, NY 10016, U.S.A.tel: 212-696-9000; fax: 212-685-4540

Distribution and Customer ServiceMarcel Dekker, Inc.Cimarron Road, Monticello, New York 12701, U.S.A.tel: 800-228-1160; fax: 845-796-1772

Eastern Hemisphere DistributionMarcel Dekker ACHutgasse 4, Postfach 812, CH-4001 Basel, Switzerlandtel: 41-61-260-6300; fax: 41-61-260-6333

World Wide Webhttp://www.dekker.com

The publisher offers discounts on this book when ordered in bulk quantities. For more information,write to Special Sales/Professional Marketing at the headquarters address above.


Neither this book nor any part may be reproduced or transmitted in any form or by any means,electronic or mechanical, including photocopying, microfilming, and recording, or by any informationstorage and retrieval system, without permission in writing from the publisher.

Current printing (last digit):

1 0 9 8 7 6 5 4 3 2 1

PRINTED IN THE UNITED STATES OF AMERICA


PURE AND APPLIED MATHEMATICS

A Program of Monographs, Textbooks, and Lecture Notes

EXECUTIVE EDITORS

Earl J. Taft Zuhair NashedRutgers University University of Central Florida

New Brunswick, New Jersey Orlando, Florida

EDITORIAL BOARD

M. S. Baouendi Anil NerodeUniversity of California, Cornell University

San DiegoDonald Passman

Jane Cronin University of Wisconsin,Rutgers University Madison

Jack K. Hale Fred S. RobertsGeorgia Institute of Technology Rutgers University

S. Kobayashi David L. RussellUniversity of California, Virginia Polytechnic Institute

Berkeley and State University

Marvin Marcus Walter SchemppUniversity of California, Universitat Siegen

Santa BarbaraMark Teply

W. S. Massey University of Wisconsin,Yale University Milwaukee


LECTURE NOTES IN PURE AND APPLIED MATHEMATICS

1. N. Jacobson, Exceptional Lie Algebras2. L.-A. Lindahl and F. Poulsen, Thin Sets in Harmonic Analysis3. /. Satake, Classification Theory of Semi-Simple Algebraic Groups4. F. Hirzebruch et a/., Different/able Manifolds and Quadratic Forms5. I. Chavel, Riemannian Symmetric Spaces of Rank One6. R. B. Burckel, Characterization of C(X) Among Its Subalgebras7. B. R. McDonald et a/., Ring Theory8. Y.-T. Siu, Techniques of Extension on Analytic Objects9. S. R. Caradus et a/., Calkin Algebras and Algebras of Operators on Banach Spaces

10. E. O. Roxin et a/., Differential Games and Control Theory11. M. Orzech and C. Small, The Brauer Group of Commutative Rings12. S. Thornier, Topology and Its Applications13. J. M. Lopez and K. A. Ross, Sidon Sets14. W. W. Comfort and S. Negrepontis, Continuous Pseudometrics15. K. McKennon and J. M. Robertson, Locally Convex Spaces16. M. Carmeli and S. Matin, Representations of the Rotation and Lorentz Groups17. G. B. Seligman, Rational Methods in Lie Algebras18. D. G. de Figueiredo, Functional Analysis19. L. Cesari et a/., Nonlinear Functional Analysis and Differential Equations20. J. J. Schaffer, Geometry of Spheres in Normed Spaces21. K. YanoandM. Kon, Anti-Invariant Submanifolds22. W. V. Vasconcelos, The Rings of Dimension Two23. R E. Chandler, Hausdorff Compactifications24. S. P. Franklin and B. V. S. Thomas, Topology25. S. K. Jain, Ring Theory26. B. R. McDonald and R. A. Morris, Ring Theory II27. R. B. Mura and A. Rhemtulla, Orderable Groups28. J. R. Graef, Stability of Dynamical Systems29. H.-C. Wang, Homogeneous Branch Algebras30. E. O. Roxin et a/., Differential Games and Control Theory II31. R D. Porter, Introduction to Fibre Bundles32. M. Altman, Contractors and Contractor Directions Theory and Applications33. J. S. Golan, Decomposition and Dimension in Module Categories34. G. Fairweather, Finite Element Galerkin Methods for Differential Equations35. J. D. Sally, Numbers of Generators of Ideals in Local Rings36. S. S. Miller, Complex Analysis37. R Gordon, Representation Theory of Algebras38. M. Goto and F. D. Grosshans, Semisimple Lie Algebras39. A. I. Arruda eta/., Mathematical Logic40. F. Van Oystaeyen, Ring Theory41. F. Van Oystaeyen and A. Verschoren, Reflectors and Localization42. M. Satyanarayana, Positively Ordered Semigroups43. D. L Russell, Mathematics of Finite-Dimensional Control Systems44. P.-T. Liu and E. Roxin, Differential Games and Control Theory III45. A. Geramita and J. Seberry, Orthogonal Designs46. J. Cigler, V. Losert, and P. Michor, Banach Modules and Functors on Categories of Banach

Spaces47. P.-T. Liu and J. G. Sutinen, Control Theory in Mathematical Economics48. C. Byrnes, Partial Differential Equations and Geometry49. G. Klambauer, Problems and Propositions in Analysis50. J. Knopfmacher, Analytic Arithmetic of Algebraic Function Fields51. F. Van Oystaeyen, Ring Theory52. B. Kadem, Binary Time Series53. J. Barros-Neto and R. A. Artino, Hypoelliptic Boundary-Value Problems54. R. L Stemberg et a/., Nonlinear Partial Differential Equations in Engineering and Applied Science55. B. R. McDonald, Ring Theory and Algebra III56. J. S. Golan, Structure Sheaves Over a Noncommutative Ring57. T. V. Narayana et a/.. Combinatorics, Representation Theory and Statistical Methods in Groups58. T. A. Burton, Modeling and Differential Equations in Biology59. K. H. Kim and F. W. Roush, Introduction to Mathematical Consensus Theory


60. J. Banas and K. Goebel, Measures of Noncompactness in Banach Spaces61. O. A. Nielson, Direct Integral Theory62. J. E. Smith et a/., Ordered Groups63. J. Cronin, Mathematics of Cell Electrophysiology64. J. W. Brewer, Power Series Over Commutative Rings65. P. K. Kamthan and M. Gupta, Sequence Spaces and Series66. T. G. McLaughlin, Regressive Sets and the Theory of Isols67. T. L Herdman et a/., Integral and Functional Differential Equations68. R. Draper, Commutative Algebra69. W. G. McKay and J. Patera, Tables of Dimensions, Indices, and Branching Rules for Repre-

sentations of Simple Lie Algebras70. R. L. Devaney and 2. H. Nitecki, Classical Mechanics and Dynamical Systems71. J. Van Gee/, Places and Valuations in Noncommutative Ring Theory72. C. Faith, Injective Modules and Injective Quotient Rings73. A. Fiacco, Mathematical Programming with Data Perturbations I74. P. Schultz et a/., Algebraic Structures and Applications75. L Bican et a/., Rings, Modules, and Preradicals76. D. C. Kay and M. Breen, Convexity and Related Combinatorial Geometry77. P. Fletcher and W. F. Lindgren, Quasi-Uniform Spaces78. C.-C. Yang, Factorization Theory of Meromorphic Functions79. O. Taussky, Ternary Quadratic Forms and Norms80. S. P. Singh and J. H. Burry, Nonlinear Analysis and Applications81. K. B. Hannsgen et a/., Volterra and Functional Differential Equations82. N. L. Johnson et a/., Finite Geometries83. G. /. Zapata, Functional Analysis, Holomorphy, and Approximation Theory84. S. Greco and G. Valla, Commutative Algebra85. A. V. Fiacco, Mathematical Programming with Data Perturbations II86. J.-B. Hiriart-Urrutyetal., Optimization87. A. Figa Talamanca and M. A. Picarde/lo, Harmonic Analysis on Free Groups88. M. Harada, Factor Categories with Applications to Direct Decomposition of Modules89. V. I. Istratescu, Strict Convexity and Complex Strict Convexity90. V. Lakshmikantham, Trends in Theory and Practice of Nonlinear Differential Equations91. H. L. Manocha and J. B. Srivastava, Algebra and Its Applications92. D. V. Chudnovsky and G. V. Chudnovsky, Classical and Quantum Models and Arithmetic

Problems93. J. W. Longley, Least Squares Computations Using Orthogonalization Methods94. L. P. de Alcantara, Mathematical Logic and Formal Systems95. C. E. Au/l, Rings of Continuous Functions96. R. Chuaqui, Analysis, Geometry, and Probability97. L. Fuchs and L. Salce, Modules Overvaluation Domains98. P. Fischer and W. R. Smith, Chaos, Fractals, and Dynamics99. W. B. Powell and C. Tsinakis, Ordered Algebraic Structures

100. G. M. Rassias and T. M. Rassias, Differential Geometry, Calculus of Variations, and TheirApplications

101. R.-E. Hoffmann and K. H. Hofmann, Continuous Lattices and Their Applications102. J. H. Lightbourne III and S. M. Rankin III, Physical Mathematics and Nonlinear Partial Differential

Equations103. C. A. Baker and L. M. Batten, Finite Geometries104. J. W. Brewer et a/., Linear Systems Over Commutative Rings105. C. McCrory and T. Shifrin, Geometry and Topology106. D. W. Kueke et a/.. Mathematical Logic and Theoretical Computer Science107. B.-L Lin and S. Simons, Nonlinear and Convex Analysis108. S. J. Lee, Operator Methods for Optimal Control Problems109. V. Lakshmikantham, Nonlinear Analysis and Applications110. S. F. McCormick, Multigrid Methods111. M. C. Tangora, Computers in Algebra112. D. V. Chudnovsky and G. V. Chudnovsky, Search Theory113. D, V. Chudnovsky and R. D. Jenks, Computer Algebra114. M. C. Tangora, Computers in Geometry and Topology115. P. Nelson et a/., Transport Theory, Invariant Imbedding, and Integral Equations116. P. Clement et a/., Semigroup Theory and Applications117. J. Vinuesa, Orthogonal Polynomials and Their Applications118. C. M. Dafermos et a/., Differential Equations119. . O. Roxin, Modern Optimal Control120. J. C. Diaz, Mathematics for Large Scale Computing


121. P. S. Milojevft Nonlinear Functional Analysis122. C. Sadosky, Analysis and Partial Differential Equations123. R. M. Shortt, General Topology and Applications124. R. Wong, Asymptotic and Computational Analysis125. D. V. Chudnovsky and R. D. Jenks, Computers in Mathematics126. W. D. Wallis et al., Combinatorial Designs and Applications127. S. Elaydi, Differential Equations128. G. Chen etal.. Distributed Parameter Control Systems129. W. N. Everitt, Inequalities130. H. G. Kaper and M. Garbey, Asymptotic Analysis and the Numerical Solution of Partial Differ-

ential Equations131. O. Anno etal., Mathematical Population Dynamics132. S. Coen, Geometry and Complex Variables133. J. A. Goldstein et al., Differential Equations with Applications in Biology, Physics, and Engineering134. S. J. Andima et al., General Topology and Applications135. P Clement et al., Semigroup Theory and Evolution Equations136. K. Jarosz, Function Spaces137. J. M. Bayod et al., p-adic Functional Analysis138. G. A. Anastassiou, Approximation Theory139. R. S. Rees, Graphs, Matrices, and Designs140. G. Abrams et al., Methods in Module Theory141. G. L. Mullen and P. J.-S. Shiue, Finite Fields, Coding Theory, and Advances in Communications

and Computing142. M. C. JoshiandA. V. Balakrishnan, Mathematical Theory of Control143. G. Komatsu and Y. Sakane, Complex Geometry144. /. J. Bakelman, Geometric Analysis and Nonlinear Partial Differential Equations145. T. Mabuchiand S. Mukai, Einstein Metrics and Yang-Mills Connections146. L Fuchs and R. Gobel, Abelian Groups147. A. D. Pollington and W. Moran, Number Theory with an Emphasis on the Markoff Spectrum148. G. Dore et a/., Differential Equations in Banach Spaces149. T. West, Continuum Theory and Dynamical Systems150. K. D. Bierstedt et a/., Functional Analysis151. K. G. Fischer etal., Computational Algebra152. K. D. Elworthyetal., Differential Equations, Dynamical Systems, and Control Science153. P.-J. Cahen, et a/., Commutative Ring Theory154. S. C. Cooper and W. J. Thron, Continued Fractions and Orthogonal Functions155. P. Clement and G. Lumer, Evolution Equations, Control Theory, and Biomathematics156. M. Gyllenberg and L. Persson, Analysis, Algebra, and Computers in Mathematical Research157. W. O. Bray et a/., Fourier Analysis158. J. Bergen and S. Montgomery, Advances in Hopf Algebras159. A. R. Magid, Rings, Extensions, and Cohomology160. N. H. Pavel, Optimal Control of Differential Equations161. M. Ikawa, Spectral and Scattering Theory162. X. Liu and D. Siegel, Comparison Methods and Stability Theory163. J.-P. Zolesio, Boundary Control and Variation164. M. Kfizeketal., Finite Element Methods165. G. Da Prato and L Tubaro, Control of Partial Differential Equations166. E. Ballico, Projective Geometry with Applications167. M. Costabeletal., Boundary Value Problems and Integral Equations in Nonsmooth Domains168. G. Ferreyra, G. R. Goldstein, and F. Neubrander, Evolution Equations169. S. Huggett, Twistor Theory170. H. Cooket a/., Continua171. D. F. Anderson and D. E. Dobbs, Zero-Dimensional Commutative Rings172. K. Jarosz, Function Spaces173. V. Ancona et al., Complex Analysis and Geometry174. E. Casas, Control of Partial Differential Equations and Applications175. N. Kalton et al.. Interaction Between Functional Analysis, Harmonic Analysis, and Probability176. Z. Deng et al., Differential Equations and Control Theory177. P. Marcelliniet al. Partial Differential Equations and Applications178. A. Kartsatos, Theory and Applications of Nonlinear Operators of Accretive and Monotone Type179. M. Maruyama, Moduli of Vector Bundles180. A. Ursini and P. Agliano, Logic and Algebra181. X. H. Cao et al., Rings, Groups, and Algebras182. D. Arnold and R. M. Rangaswamy, Abelian Groups and Modules183. S. R. Chakravarthy and A. S. Alfa, Matrix-Analytic Methods in Stochastic Models


184. J. E. Andersen et at., Geometry and Physics185. P.-J. Cahen et al., Commutative Ring Theory186. J. A. Goldstein et al.. Stochastic Processes and Functional Analysis187. A. Sorbi, Complexity, Logic, and Recursion Theory188. G. Da Prato and J.-P. Zolesio, Partial Differential Equation Methods in Control and Shape

Analysis189. D. D. Anderson, Factorization in Integral Domains190. N. L. Johnson, Mostly Finite Geometries191. D. Hinton and P. W. Schaefer, Spectral Theory and Computational Methods of Sturm-Liouville

Problems192. W. H. Schikhofet a/., p-adic Functional Analysis193. S. Sertdz, Algebraic Geometry194. G. Can'sti and E. Mitidieri, Reaction Diffusion Systems195. A. V. Fiacco, Mathematical Programming with Data Perturbations196. M. Krizek et al., Finite Element Methods: Superconvergence, Post-Processing, and A Posteriori

Estimates197. S. Caenepeel and A. Verschoren, Rings, Hopf Algebras, and Brauer Groups198. V. Drensky et a/., Methods in Ring Theory199. W. B. Jones and A. Sri Ranga, Orthogonal Functions, Moment Theory, and Continued Fractions200. P. E. Newstead, Algebraic Geometry201. D. Dikranjan and L Salce, Abelian Groups, Module Theory, and Topology202. Z. Chen et a/., Advances in Computational Mathematics203. X. Caicedo and C. H. Montenegro, Models, Algebras, and Proofs204. C. Y. Yildirim and S. A. Stepanov, Number Theory and Its Applications205. D. E. Dobbs et a/., Advances in Commutative Ring Theory206. F. Van Oystaeyen, Commutative Algebra and Algebraic Geometry207. J. Kakol et a/., p-adic Functional Analysis208. M. Boulagouaz and J.-P. Tignol, Algebra and Number Theory209. S. Caenepeel and F. Van Oystaeyen, Hopf Algebras and Quantum Groups210. F. Van Oystaeyen and M. Saorin, Interactions Between Ring Theory and Representations of

Algebras211. R. Costa et al., Nonassociative Algebra and Its Applications212. T.-X. He, Wavelet Analysis and Multiresolution Methods213. H. Hudzik and L Skrzypczak, Function Spaces: The Fifth Conference214. J. Kajiwara et al., Finite or Infinite Dimensional Complex Analysis215. G. Lumerand L. Weis, Evolution Equations and Their Applications in Physical and Life Sciences216. J. Cagnol et al.. Shape Optimization and Optimal Design217. J. Herzog and G. Restuccia, Geometric and Combinatorial Aspects of Commutative Algebra218. G. Chen et al., Control of Nonlinear Distributed Parameter Systems219. F. AH Mehmeti et al.. Partial Differential Equations on Multistructures220. D. D. Anderson and I. J. Papick, Ideal Theoretic Methods in Commutative Algebra221. A. Granja et al., Ring Theory and Algebraic Geometry222. A. K. Katsaras et al., p-adic Functional Analysis223. R. Salvi, The Navier-Stokes Equations224. F. U. Coe/ho and H. A. Merk/en, Representations of Algebras225. S. Aizicovici and N. H. Pavel, Differential Equations and Control Theory226. G. Lyubeznik, Local Cohomology and Its Applications227. G. Da Prato and L. Tubaro, Stochastic Partial Differential Equations and Applications228. W. A. Camiellietal., Paraconsistency229. A. Benkirane and A. Touzani, Partial Differential Equations230. A. Illanes et al., Continuum Theory231. M. Fontana et al., Commutative Ring Theory and Applications232. D. Mond and M. J. Saia, Real and Complex Singularities233. V. Ancona and J. Vail/ant, Hyperbolic Differential Operators234. A. Giambruno et al., Polynomial Identities and Combinatorial Methods235. G. R. Goldstein etal., Evolution Equations

Additional Volumes in Preparation


Preface

This volume contains the proceedings of the conference on Polynomial Identities andCombinatorial Methods, held on the island of Pantelleria, Italy. It was the fourth in aseries of meetings in the last decade concerning the theory of associative andnonassociative algebras satisfying polynomial identities (Pi-algebras). The first of thesemeetings was a small workshop in Palermo, Italy, in 1992. The second was theconference entitled Methods in Ring Theory, held in 1997 in Trento, Italy. Theproceedings of that conference were published in the Marcel Dekker series Lecture Notesin Pure and Applied Mathematics, Volume 198, and it is now a standard reference forspecialists working in the area of polynomial identities.

Considerable progress could be observed in the theory of algebras withpolynomial identities during the years following the Trento conference. Some of themost important achievements in this area were due to a combination of algebraictechniques with analytical and combinatorial methods in the study of various numericalcharacteristics of Pi-algebras and their identities. Leading specialists in this area met inRehovot, Israel, in May 2000, in the Workshop on Growth Phenomena in Associative andLie Pi-algebras. Unfortunately, the results presented in Rehovot were not collected inone volume. To make up for this deficiency we hereby offer the proceedings of thePantelleria conference, presenting the up-to-date status and tendencies in this area.

The conference featured the latest results in the theory of polynomial identitiesand a presentation of different methods and techniques pertaining to different areas, suchas algebraic combinatorics, invariant theory, and representation theory, both of thesymmetric and the classical groups, and of Lie algebras and superalgebras.

During the conference one-hour invited lectures were given by Y. Bahturin, A.Belov, O. Di Vincenzo, M. Domokos, V. Drensky, E. Formanek, A. Giambruno, A.Guterman, P. Koshlukov, S. Mishchenko, V. Petrogradsky, C. Procesi, A. Regev, L. H.Rowen, I. Shestakov, and M. Zaicev. In addition, several other invited talks of shorterlengths were presented. This volume includes the papers of most of the principalspeakers and some other invited contributions related to the conference.

Even though the contents of this volume cover a broad range of themes, fromring theory to combinatorics to invariant theory, they still have a common thread in thetheory of polynomial identities. The book will be useful to all researchers working withpolynomial identities, varieties of associative algebras, Lie and Liebnitz algebras, andtheir generalizations. It will also be of interest to specialists in free algebras, growthfunctions of algebras, and, more generally, to mathematicians who apply numerical andanalytical methods in algebra.


The editors wish to express their appreciation to the following agencies andinstitutions that have contributed financial support: the Gruppo Nazionale per le StruttureAlgebriche Geometriche ed Applicazioni of the Istituto Nazionale di Alta Matematica,the national research project Algebre con Identita Polinomiali of the MURST, and theUniversity of Palermo.

Antonio GiambrunoAmitai Regev

Mikhail Zaicev


Contents

PrefaceContributors

1. Linearization Method of Computing Z2-Codimensions of Identities of theGrassmann AlgebraN. Anisimov

2. Cocommutative Hopf Algebras Acting on Quantum Polynomials andTheir InvariantsVyacheslav A. Artamonov

3. Combinatorial Properties of Free Algebras of Schreier VarietiesVyacheslav A. Artamonov, Alexander A. Mikhalev, andAlexander V. Mikhalev

4. Graded Algebras and Graded IdentitiesYu. A. Bahturin and M. V. Zaicev

5. Computational Approach to Polynomial Identities of Matrices: A SurveyFrancesco Benanti, James Demmel, Vesselin Drensky, and Plamen Koev

6. Poincare Series of Generic MatricesAllan Berele

1. Combinatorial Methods for the Computation of Trace CocharactersLuisa Carini

8. Polynomial Identities for Graded AlgebrasOnofrio Mario Di Vincenzo

9. Matrix Invariants and the Failure of Weyl's TheoremM. Domokos

10. Free Nilpotent-by-Abelian Leibniz AlgebrasVesselin Drensky and Giulia Maria Piacentini Cattaneo


11. Pebbles and Expansions in the Polynomial RingAdriano M. Garsia

12. Group Actions, Codimensions, and Exponential BehaviourA. Giambnmo

13. Monotone Matrix Maps Preserve Non-maximal RankA. Guterman

14. Explicit Decompositions of the Group Algebras FSn and FAnA. Henke and A. Regev

15. Graded and Ordinary Polynomial Identities in Matrix and Related AlgebrasPlamen Koshlukov

16. Varieties of Linear Algebras with Almost Polynomial GrowthS. P. Mishchenko

17. Algebras with Involution, Superalgebras, and Proper SubvarietiesVincenzo Nardozza

18. Gradings and Graded Identities of the Algebra of n x n UpperTriangular MatricesAngela Valenti


Contributors

N. Anisimov, Chair of Higher Algebra, Faculty of Mechanics and Mathematics, MoscowState University, 119899, Moscow, RussiaE-mail: [email protected]

Vyacheslav A. Artamonov, Department of Algebra, Faculty of Mechanics and Mathe-matics, Moscow State University, Leninsky Gory, 119992, GSP-2, Moscow, RussiaE-mail: [email protected]

Yu. A. Bahturin, Department of Mathematics and Statistics, Memorial University ofNewfoundland, St. John's, NF, A1A 5K9, Canada, and Department of Algebra, Facultyof Mathematics and Mechanics, Moscow State University, Moscow, 119992, RussiaE-mail: [email protected] ca

Francesca Benanti, Dipartimento di Matematica ed Applicazioni, Universita di Palermo,via Archirafi 34, 90123 Palermo, ItalyE-mail: [email protected]

Allan Berele, Department of Mathematics, DePaul University, Chicago, IL 60614 USAE-mail: [email protected]

Luisa Carini, Dipartimento di Matematica ed Applicazioni, University of Palermo, ViaArchirafi 34, 90123 Palermo, ItalyE-mail: [email protected]

James Demmel, Department of Mathematics, Computer Science Division, University ofCalifornia at Berkeley, Berkeley, CA 94720, USAE-mail: [email protected]

Onofrio Mario Di Vincenzo, Dipartimento Interuniversitario di Matematica, Universitadegli Studi di Bari, via Orabona 4, 70125 Bari, ItalyE-mail: [email protected]


Matyas Domokos. Renyi Institute of Mathematics, Hungarian Academy of Sciences,P.O. Box 127, 1364 Budapest, HungaryE-mail: [email protected] address (until February 2004):Department of Mathematics, University of Edinburgh, James Clerk Maxwell Building,King's Buildings, Mayfield Road, Edinburgh EH9 3JZ, ScotlandE-mail: [email protected]

Vesselin Drensky, Institute of Mathematics and Informatics, Bulgarian Academy ofSciences, Akad. G. Bonchev Str., Block 8,1113 Sofia, BulgariaE-mail: [email protected]

Adriano M. Garsia, Department of Mathematics, University of California, San Diego,La Jolla, CA 92093-0112 USAE-mail: [email protected]

Antonio Giambruno, Dipartimento di Matematica ed Applicazioni, Universita diPalermo, Via Archirafi 34, 90123 Palermo, ItalyE-mail: [email protected]

A. Guterman, Department of Higher Algebra, Faculty of Mathematics and Mechanics,Moscow State University, Moscow, 119899, RussiaE-mail: [email protected]

A. Henke, Department of Mathematics and Computer Science, University of Leicester,University Road, Leicester LE1 7RH, EnglandE-mail: [email protected]

Plamen Koev, Department of Mathematics, University of California at Berkeley,Berkeley, CA 94720, USAE-mail: [email protected]

Plamen Koshlukov, IMECC, UNICAMP, Cx.P. 6065, 13083-970 Campinas, SP, BrazilE-mail: [email protected]

Alexander A. Mikhalev, Department of Mechanics and Mathematics, Moscow StateUniversity, 119899, Moscow, RussiaE-mail: [email protected]

Sergey P. Mishchenko, Department of Algebra and Geometric Computations, Faculty ofMathematics and Mechanics, Ul'yanovsk State University, Ul'yanovsk, 432700, RussiaE-mail: [email protected]

Vincenzo Nardozza, Dipartimento di Matematica, Universita della Basilicata, C.daMacchia Romana, 85100 Potenza, ItalyE-mail: [email protected]


Ciulia Maria Piacentini Cattaneo, Department of Mathematics, University of Rome"Tor Vergata", Via della Ricerca Scientifica, 00133 Rome, ItalyE-mail: [email protected]

Amitai Regev, Department of Mathematics, The Weizmann Institute of Science,Rehovot 76100, IsraelE-mail: [email protected]

Angela Valenti, Dipartimento di Matematica ed Applicazioni, Universita di Palermo,Via Archirafi 34, 90123 Palermo, ItalyE-mail: [email protected]

M. V. Zaicev, Department of Algebra, Faculty of Mathematics and Mechanics, MoscowState University, 119992, RussiaE-mail: [email protected]


Linearization method of computing Z2 -codimensions of identities of the Grass-mann algebra

N. ANISIMOV Chair of Higher Algebra, Faculty of Mechanics and Math-ematics, Moscow State University, 119899, Moscow, Russia.E-mail: [email protected]

1 INTRODUCTION

Let A be an associative algebra over a field F of characteristic 0, and letG be a finite group of automorphisms and anti-automorphisms of A. Alsolet X = {.TI, x,2, } be a countable set of indeterminates. The free algebrawith the set of free generators < X\G >= {xf\i N,# G G} is called thealgebra of G-polynomials. The space

is called the space of multilinear G-polynomials in the variables xi,...,xn.The polynomial f(xi,...,xn,G) E Vn(x,G) is called an n-linear G-

identity if for arbitrary elements 0 ,1 , . . . . an of A, /(GI, . . . , an, G) = 0; herewe use the notation a9 g(a) for all a G A, g G. All n-linear G-identitiesform the ideal Idn(A, G) of n-linear G-identities. The sequence

Cn(A, G) = dirnp

is called the sequence of G-codimensions of the ideal of the G-identities ofthe algebra A. If the group G is generated by an element

basic notions of PI theory were introduced by A. Regev in [1] and thengeneralized by A. Giambruno and A. Regev in [2].

Properties of such sequences were widely investigated only in the tradi-tional case G = {id}: the general case is very incomplete. Moreover, thereare only a few algebras whose sequence of codimensions of identities havebeen computed exactly. One of these algebras is the infinite-dimensionalGrassmann algebra A. cn(A) = 2""~1 (see [3]). Note that the Grassmann al-gebra is of fundamental importance in PI theory, for example, it generatesa minimal variety of exponential growth. Questions that arise naturallyare: to compute the involutivc and the Z2-codimensions of the identitiesof the Grassmann algebra, and to describe the ideals of Z2-identities ofA. These questions were partially answered in [4]. For example, the se-quence of involutive codimensions of the Grassmann algebra was computedfor an arbitrary involution. But the general case of Z2-action on A wasinvestigated in [4] only for automorphisms of order two with linear actionon generators. In this article we point out the deep connection betweenarbitrary Z2-cdimensions of identities and Z2-cdimensions of identitiesfor linear automorphisms which were computed in [4]. The statements ofthe theorems of Section 2 contain an additional assumption on the au-tomorphism. By computing the structure of some automorphisms of theGrassmann algebra we show in Section 3 that the additional assumption isnatural for such automorphisms.

We now list some definitions and properties used later. Let A be theinfinite-dimensional Grassmann algebra with generators ei, 62, - . and defin-ing relations e,ej + Cj-e, = 0. Then the set of all ordered monomials{ejj .. . e,;jA; > 1.1 < i\ < .. . < i^} form a basis D\ of A. There is anatural Z2-grading on A, A = AO AI, where AO and AI are spanned bythe basic monomials of even and of odd length correspondingly. The Z2~graded elements of A commute by the following rule: if g A,; and h Aj,then gh = ( l)ljhg. The length a of the monomial a 6 D\ is the numberof generators in a: je,;, ... Cik = A;.

Let (p be an automorphism of order two on the algebra A. If it's actionon the generators is defined bv the formula

then the mapping defined on the generators by the formula

3

and homomorphically continued on A is also an automorphism of order twoon the Grassmann algebra. The automorphism tpi is a linear operator in


the space L span^{ei, 2, . . .}. In further considerations we will assumewithout loss of generality that the generators of the Grassmann algebraform the eigenbasis for the operator (pi in the space L. The subspacescorresponding to the eigenvalues 1 and 1 are denoted by L\ and L-\.

2 Zs-IDENTITIES OF THE GRASSMANN ALGEBRA

First we note that for an arbitrary automorphism (p of the Grassmann alge-bra A with dim LI = dimL_i = oo the c^-codimensions of the identities ofA were computed in [4]. But cn(A, k, at least one of the chosen monomials a^,..., a,ik isof a non-unit length. Since (p is a graded automorphism we may assume,without loss of generality, that a^ = e^aj- is of non-unit length. Then


Since aJ:J + X1 1=2 av I = "? 1 < "? therefore by induction

Since

Now suppose n > I. In this case we prove the equality cn(A, an) = 9j

and may be computed by the formula

(4)

In [4] it was proved that (r/) Idn(A, (^) if and only if

E (//("V)

Transform the value of the basic monomial xj,^-} . .. xj^\ f tne spaceVn(x,), computed on the monomials a\,... .an. Introduce the notation

hi = 0:

Let ^(jj) = ... = 9a(i,,) = I? and the other 3,; are equal to 0. Wealso use a natural lexicographic partial order on binary sequences: h =(hi, . . . , hn) < (51, . . . , gn) = g if /?.,; < 5,; for all i = 1, . . . , n. Now wetransform

~

V-

From condition (1) of the theorem and from Lemma 2.1 we obtain thatfor any natural number m. > I,

7 = 1

Since

Since (p is a graded automorphism, for any i, I < i < n, a\ is agraded element and one may apply Equation (4) to a^,^ , . . /ln) , whichtransforms the last equality as follows:

k=0 h. . . ( 7 )

Prove that for any k, 0 < k < I, and for any binary sequence h =

j'VK^o, (8)

using the system of Equations (5) on the coefficients aa^ of the polynomial/. Consider the following linear combination of some equations of thissystem:

For m fixed the number of sequences / with exactly j (j < m) unitson the first r entries is equal to C3TC~^ (all the units of / may appearonly in the first k. entries since / < /?,!), where CQ = ( ^ ) is the binomialcoefficient, C = 1 and Cf = 0 for a < (3; and also

i = l 7 = 1 ?'=r+l

Hence we may continue Equation (11) as follows:k m k kj

= E Y,(-Vm+ici

considered for the set of pairs (/, J) which differ in an essential way fromthose used in the proof of Theorem 2.1; condition (1) for the automorphism(p also changes. We list the corresponding statements with the necessarychanges in the proofs.

LEMMA 2.2 Let if : A > A be a graded automorphism of the Grassmannalgebra and let k be a natural number. If for any k generators e^,... ,eik

then for any basic monomials a^,..., a,;fc 6k

Proof. Consider arbitrary basic monomials a 6 D\ . We prove byinduction on the total length ^i'=i aij I f the chosen monomials. The baseof induction is the condition of the lemma. Suppose Z)i=i aij = rn > kand that the statement is true for all sets of basic monomials of totallength less than m. At least one of the monomials a^, . . . , a,^ is of non-unit length. Since

Since a^ + ^,;_2 aij\ = m, I < m therefore by inductionk

The element '^{e^} + e^ is graded and hence commutes or anticommut.eswith the monomial / . Finally we obtain

) - (-1) a'JW =

since we may apply the inductive assumption to the set of monomialse?:i.a7;2, . . . .a,-t. D

THEOREM 2.2 Let A 6e a graded automorphism of order two onthe Grassmann algebra A, dim LI = I < oo and assume that for any I + 1generators e^ , . . . , e,;i+1 ,

I+i

JTien

l l-C_! + 2" J] C^_! < cn(A ; ^ ) = cn(A, W) < 2" J] C^, n > /.

J=0 j=0

Proof. For n < / the statement of the theorem may be proved by the chainof inequalities (2).

Let n > 1. We prove that in this case the equality cn(A, i)holds, by proving (as in Theorem 2.1) the isomorphism

Define the isomorphism r : Vn(x,ip) > Vn(x,(p{) by r(x'f) = xf andprove that if (rf) 6 Idn,(A. ^/) then / Idn(A, yj). The inverse inclusionwas proved in [41. Let / 6 V n ( x , ( p ) be an arbitrary polynomial (see (3)).


Choose n arbitrary basic monomials ai, . . . ,an. In further considerationswe use the definitions of the sequences / = /(ai, . . . , an), J = J(ai, . . . , an)and the elements /} (cr) and gj1 (gi, . . . ,gn) listed before Equation (5).

The proof of Theorem 3.1 of [4] contains the statement that (rf) EIdn(A, (pi) if and only if

E (fin\)9(J?(9i, ,gn))

Substitute this expression in / (o - i , . . . . an):

EQ=1 ,; = = 9>^

(^-1) f/),TT) /1 r \o-i . . . o4 ' ; . (15)Now we prove that for any k, 0 < k < I and for any binary sequence

h (hi.... , hn), 53r=i ^'i k, holds

g>h

We use the system (13) where aa^ are the coefficients of the polynomial/. Consider following linear combination of some equations of this system:

1 * (E /^ W Ng. The mapping[ ]J ' : A > A^^ denned by the rule [g}] = QJ (g3 is denned by formula (23))has two properties which we use later:

for any g. h A and any natural number j, [g + /i]J = [g]J + [/i]J;

let g, h e A, g = Y = \ _ 9 j - , h = E i ^ / t - Then for any naturalnumber, / > 2,

< Ng + Nh, (24)

and [gh]1 = 0 for / > Ng + Nh.Without loss of generality we consider the action of the automorphismon the generator e-2. Denote y = 11,2 AQ, a = v% A], then by (21)

(p(e2) = e-2 + e.\y + a. (25)


Write the elements t, c, y and o. as a sum of Z-graded elements of A.

Nt Nc

y =

where the sum in ^+ is considered only for even j's and sum in ^,~k isconsidered only for odd fc's.

Since the automorphism (c). (26)Introduce one more notation. Let g 6 A in formula (23) be independent

of e\ (in particular, such are elements t,c,y,a). Then for any naturalnumber m

m 1

]m, (27)

because gm do not depend on e\ and [

Nt

t J(tm_ J+Sr J)- (29)

Now we apply anologous reasoning concerning Equation (25). The ele-ment (p(e-2) 62 changes its sign under the

LEMMA 3.1 If Equation (31) holds for all even m, < n then

where

Si = - E (ck + Skc)(tc s i . . . (t8l + Sst')(yj + S$-

-cktsi ...tsiyjj. (33)

Proof.Let

n-l

r=3Si = -

Then by definition, 5^+1 = E"r=3 [^(Gr-)]n+1 and from the set of equal-ities (31) it immediately follows that ^+1 = 50 + Si + ... + SN.

For shortness we prove (33) only in the case I = 0, the general case issimilar. First of all note that for any Z-graded element g^ E A\ ' whichdoes not depend on e\ holds:

0, m < k;[k.(34)

Then

-2S0 =n-l

(24),(

r=5

n-l r-2

r-2

E"k=3

n+1n-l r-2

r=5 fc=3

yn+l-r+ka i ore+l-t I _

r=5 fc=3

n-l

J=3


77-1

((Si +

that completes the proof of the equality (33) for I = 0. DSubstituting Equation (32) in (30) and using Lemma 3.1 we obtain for

m, = ?7,:-2an+i =

i + E S> - e^+

Nt

i=2

?:=2-j + E 5< -

j>2 /=!

E

Introduce the following notation

T; = i ^

fc+sj + .^+sj+T n+1

With these notations

-2nn+i =

-

ei^ + \ E+ cJ+i^"J' - lei E+ si("


Compute TI + Si for an arbitrary natural number l,Q < I < N, using(33):

cktsi...tsiyr+

...(tSl+S*l)(yr+Sry).

Applying (29) we obtain

Cktsi *s,2/r +

k+sl+.. .+sj+r=n+l

E[, r)7-=n+l

S*'1 ,^ + 5tsi) . . . (tsj + Sst')(yr + Sry}+

Ei>2

x (tsi + 5tai ) . . . (tai + Sst'}(yT + Sry) =(fc ,S 1 , . . . ,S j , r )

fc + s1+...+s,+t-=ij.+ l

E (c*(*-i +5t81) (tSl+S?)(yr + Sry)-cktSl . . . tsiyr] -

5*-a(tfll + Stsi) . . . (ts, + S?)(yr + Sry) + Tl+l.

We note here that TN+\ = 0. In fact, from formula (24) we have for anarbitrary fixed natural number M that

= b(^)]M- (36)

The element TJV+I may be represented as a sum of elements, each ofwhich is a sum of type (36), for / = N + 1 as a factor. This implies thatTN+l = 0 since tN+1 = 0.


Now substitute the expressions obtained for T} + Si in formula (35). Wehave

N ,-2on+i = -elS

1=0 (*,S1,...,s,

. . . (*s, + Sfl)(yr + S;) - cktsi . . . t s , y r . (37)LEMMA 3.2 For any even natural number m holds

... (tsi + Sf')(yr + Sry) - cktsi ... t8lyr (38)Proof. Recall that we prove the theorem by simultaneous induction on thetwo Equations (31) and (38). Check the base of induction for Equation(38). In fact for m = 2, Equation (38) is 0 = 0.

Let Equation (38) hold for all even m < n. We prove it for m = n. Firstwe prove that the sum of elements of the right hand side of (38) which donot, depend on the generator ei, is zero. Multiply both parts of (38) by e\and compute

(=0

(tsi + Sf')(yr + Sry) - cktsi . . . ts,yr). (39)Step 1. The item in the sum (39) which corresponds to / = 0 is equal to

, (40)

and we may use induction to represent the elements eiS"Note that for any g A represented in form (23) and for any fixed

natural number M holds

E Cfcicfc2

because c = ^k ck AI. Hence we may continue Equation (40) as follows

E" ^ sy+ l~k - - E" c* E ^ i ( E1=0 (k1,s1,...,sl,r)

(tai + Sstl}(yr + Sry) - ckltsi

E^r( E1=0 (fcl,*! ..... s;,r)

(fc,n , . . . ,s,,r)fc + s1+...+s i+r=n + l

Substituting this in (39) we have

S?) . . .

. . . (tai + S*')(yr + Sry) - cktSltS2 . . . tSlwhere

fij = N+l+i E elcfc*si tsj-^t3 X

i

l=q= E

-,

S?) . . . (tsi + S?)(yr + Sry)

Sst>+1) . . . (tSN+i + SstN+')(yr + Sry), l

The base of induction was proved in step 1. Suppose the equality (43)is true for all q < q. Prove it for q q + 1. Note that for q q the itemcorresponding to I = q in the right hand side of (43) is equal to

E

and to compute it we may once again use inductive representation for e\Syand property (41):

^ ^ cktsi ...tsx

N 1 E

( = 0

2/+T(fc.-'i s/. r)

A- + s1+... + s /+r=T7+l

9+1

5^+2)... (ts, + S*')(yr + Sry) -^fi,q+i.1=1

Substituting this in (43) we immediately obtain that Equation (43) alsoholds for q = q + 1.Step 3. For q = N + 1 Equation (43) is

N+l N+l

- EProve that for any ?;, 1 < i < N + 1 holds j=i fij = 0. We use

definitions (42) for the elements fa, and formulas (36), (28):


( k , r )

Since for fixed i > I holds

AT+1^-s -1

j=i

N

N+l

') = E **- N+lEN+l) - E

j-i-l j=ithen by linearity of [ ]m we obtain

N+l N+l

E / = E TO+T E

E(fc,r)

-1 n+1fcr

Hence j J] /,;j = 0 and by (44) the element (39) is zero.Step 4. The equality (39) = 0 means that Equation (38) holds (for m = n)in the part which do not depend on e\. Comparing (37) and (38) we seethat all the elements of (38) which depend on e\ are also contained in(37), with same coefficients, and (37) do not contain other elements whichdepend on e\. Hence for m = n (37) implies (38) for elements dependingon e\. D

Applying Lemma 3.2 to equality (37) we obtainN I

-2an+i = ^"1=0

that implies (31) for m, = n. Hence equality (31) is proved by induction foran arbitrary even m. Now the equality

1

completes the proof of the theorem. D


COROLLARY 3.1 Let (p : A > A be a graded automorphism of order twoon the Grassm.ann algebra A, such that dimL_i = 1- Then for an arbitrarynatural number i.j > 1 holds

-e = 0.

Proof. Let ip(ei) = -e\ + erf + c where t A0, c AI, t,N+l = 0.First let i.j > 2. By Theorem 3.1 we have if (erf = &i + e\Ui f(c,t)ui,tf>(erf = e-j + eiUj-f^tfuj where /(c,t) = ^c + c^^i ^M^, Ui,Uj A0.Then

Uj - f ( c , t ) u r f =

, i f ( c , t ) u j = 0,since the elements n,; and Uj are central, /(c. t) 6 AI and /2(c, t) = 0.

The only case left is i or j is equal to 1. Since tf is a graded automorphismwe may assume, without loss of generality, that i I. Then

AT

rf - erf = (-lei + erf + c}(elUj - -cuj - c(^ ^^-tk}urfk=i

N I N I

fc=l

N 1 ^

k=i k=iJV+1

1 N 1-^tk)u, = 0.

nCorollary 3.1 and Theorem 2.1 immediately imply

COROLLARY 3.2 Let A be a graded automorphism of order twoof the Grassmann algebra A, such that dimL_i = 1. Then

for all n > 2. DThe following theorem is one more corollary of Theorem 2.1.


THEOREM 3.2 Let (p : A * A be a graded automorphism of order twoof the Grassmann algebra A, such that dim LI = 1. In the general casethe action of the automorphism (f> on the generators of A is defined by theelements t, u\ AQ and c, Vi . AI (i = 2, 3, . . .) by the formulas

e i+ei* + c, (45)

(f(ei) - -e,; + eiUi + vit i - 2, 3, . . . , (46)where the elements Ui and Vi do not depend on e\ . Then

=lwhere N = N(t) is the minimal natural number such that tN+l = 0.

Proof. Let 7 be an automorphism of order two of A defined by (e?;) = &ifor all natural numbers i. Then 7 commutes with (p. In fact, since (p is agraded automorphism, 1 holds


Proof. Let 7 be the automorphism, defined by its action on the generators:7(e,;) = &i. As it was shown in the proof of Theorem 3.2, 7 commuteswith ) < (n + l)2n

for all n> 2. D

REFERENCES

[1] Regev A. Existence of identities in A B. Isr. J. Math. 1972, ^1,131-152.

[2] Giambruno A.; Regev A. Wreath products and P.I. algebras. J.PureAppl. Algebra 1985, 35, 133-149.

[3] Krakowski D.; Regev A. The polynomial identities of the Grassmannalgebra. Trans. Amer. Math. Soc. 1973, 181, 429-438.

[4] Anisimov N. Y. Zp-codimensions of ZP-identities of Grassmann alge-bra. Comm, Algebra 2001, 29(9), 4211-4230.


Cocommutative Hopf algebras acting onquantum polynomials and their invariants

VYACHESLAV A. ARTAMONOV Department of Algebra, Faculty ofMechanics and Mathematics, Moscow State University, Leninsky Gory,119992, GSP-2, Moscow, Russia.E-mail: [email protected]

1 INTRODUCTION

Let k be a field with a fixed matrix Q = (qij) Mat(?7,, k) whose entriesqij k* satisfy the relations qa = qij^ji = 1 for all i,j. Fix an integer rsuch that 0 < r < n. Denote by

A = k,Q[X\... ,X\Xr+l,...,Xn]the associative fc-algebra with a unit element generated by elements

i Xn, A j , . . . , Xrsubject to defining relations

XiXj = qijXjXi, 1 < i,j < n,X,X~l = X^Xi = 1, 1 < i < r.

The algebra A is an algebra of quantum polynomials. The elements q^ aremultiparameters. The algebra A can be viewed as a coordinate algebra ofa quantum affine space AQ if r = 0 [D] and a coordinate algebras of aquantum torus TQ if r = n [B]. We shall unify both cases and considerarbitrary r, 0 < r < n. Then A can be considered as a coordinate algebra


The aim of the present paper is to study actions of cocommutative Hopfalgebras on A. Throughout the paper we shall assume that the algebraA is a general algebra of quantum, polynomials that is all multiparametersQijt 1 < ?' < j : 5; 7'-, ai'e independent in the multiplicative group k* of thefield k.

Particular cases of this situation have been considered in [AC], [AW],[Al], [ACo]. In [AC] the authors study some automorphism groups and Liealgebras of derivation Der A under the assumption that r = 0. Derivationand automorphisms of A in the case r = n were considered in [OP]. Asystematic study of automorphism group and their invariants was carriedout in [AW] . An investigation of group of automorphisms of division ringsof fractions of A was made in [ACo], [Al]. Wre shall quote some of theseresults.

THEOREM 1 ([OP], [AW]). Suppose that 7 e Aut A and n > 3. Thenthere exist elements 71 , . . . ,7 k and an integer e = 1, such thatj(Xw) = "fwX^,, w = 1, . . . , n. In particular, if r < n, then e = 1 and

Aut A ~ k* x x fc* .r?. tim.es

Let G be a finite subgroup o/AutA. Then the subalgebra of invariants Ais left and right Noetherian and A is a finitely generated left and right AG-module. Suppose that I is a nonzero left ideal in A. Then I n A ^ 0.In particular if F is the division ring of fractions of A.. Then FG is thedivision ring of fractions o/A .

In the paper [ACo] we study automorphisms of the division ring F =k,q(X, Y) of a general quantum polynomial algebra A = k q [ X , Y ] , n = 2.If h is a rational function in one variable, not identically zero, then X i h(Y}X, Y H-* Y; and X .-+ X, Y ^ h(X}Y define automorphisms ofF and of k ( Y } ( ( X ; a ) ) . We shall also consider automorphisms {X, Y} i >{aX, i3X}, where a, ,3 k*, e = 1. We shall call these automorphismselementary.

THEOREM 2 ([ACo]). Let v> : kq[X,Y] -> F be a homomorphism of k-algebras. Then there exists a sequence of elementary automorphisms and aconjugation by a power series z is of the form,

; a ) ) , z3 k(Y).

such that the pair ij>(X). ijj(Y} can be replaced by a pair Xm, Ym withm = 1.


THEOREM 3 ([ACo]). Let f e kq(X,Y) \ k. Then the centralizer C(f)is a maximal sub field in kg (X, Y).THEOREM 4 ([Z]). Let F be a division ring of fractions of algebras ofquantum polynomials with n > 3 variables. If f G F\k, then the centralizerof f in F is a maximal sub field in F.

In noncommutative geometry [D] a quantum polynomial algebra A is acoordinate algebras of a quantum affine space AQ . It is considered togetherwith quantum Grassman algebra F. The algebra of functions on matricesof size n was introduced in the book [D] as a universal bialgebra coacting onthe pair of algebras A, F. According to these ideas in the present paper weconsider actions of cocommutative Hopf algebras H on the algebra A, n >3. From geometrical point of view these questions are related to a study ofcommutative quantum groups acting on the affine space AQ. It is necessaryto mention that cocommutative Hopf algebras considered in this paper arerelated to algebras of quantized differential operators on AQ (quantizedWeyl algebras) that were introduced by J. Alev, F. Dumas [ADI] and byE. Demidov [D]. In the present paper we also study automorphisms ofthe division ring F of skew quantum Laurent power series, generalizedderivations of A and of T actions of cocommutative Hopf algebras on A.The main results of the paper are Corollary 4 Theorem 9, Corollary 6,Theorem 11, Theorem 13.

NOTATION 1. Let Zr x (N U 0)n~r be the set of all multi-indicesu = (ui,... , un}, ? /! , . . . , un e Z, ur+i,... , un > 0. (1)

For a multi-index u from (1) we putx

u = xil x^ e A. (2)

A product Xu from (2) will be called a monomial.

2 DERIVATIONS AND AUTOMORPHISMS

The algebra A is a left and right Noetherian domain satisfying the Orecondition. Therefore it has a skew field of fractions F. As it is shown in[Al] that there exists an embedding of F into the skew ring

F=kQ((Xl,...,Xn)}

of quantum Laurent power series in the sense of Malcev-Neumann. Ele-ments of f are maps / : Zn > k and such that supp / = {m 6 Zn|/(m) ^


0} is Artinian with respect to the lexicographic order on Zn. An element/ T can be identified with a power series / = X^meZ" f(m}Xm- We saythat a monomial Xu from (2) occurs in an element / e JF if /(?/) ^ 0. Forany nonzero element / e JF denote by ||/|| the least element of supp/ inZ" with the lexicographic order.

An automorphism g (a derivation d) of F is continuous if g (respectively,9) is completely determined by the images g(Xj], 1 < i < n.

In this section we shall consider Lie algebras of (continuous) derivationand groups of (continuous) automorphisms of A (respectively, of .F).NOTATION 2. For an associative algebra A denote by Auk A the group ofautomorphisms of A. If A J- , then AutJF stands the group of continuousautomorphisms of T .DEFINITION 1. Let A be an associative algebra and g E AutA Alinear operator d on A is a g-derivation (or a skew g-derivation) if d(xy) =d(x)y + g(x)d(y) for all x, y e A. If u e A, then the linear operator &dg udefined as (a.dgu)(x} = ux g(x)u is called an inner g-derivation.

It is not hard to check that, every inner g-derivation is a g-derivation,and adg(aui + du?) = aa.dgui + {3a,dgU2 for all a,/3 k. If g is theidentical automorphism then any g-derivation is a derivation of A and aninner g-derivation is an ordinary inner derivation ad u.

NOTATION 3. Throughout the rest of the section we fix a continuous au-tomorphism g AutjF which has a finite order d> I such that

So j = I for any j 1, . . . , n, and ,; ^ 1 for some index i if d > 1. ByTheorem, 1 this is always the case if g G AutA and n > 3, n > r.

The following affirmation will be used throughout the paper.

PROPOSITION 1 ([MP]). Letp = ( p i , . . . ,pn), t = (ti, . . . ,tn) be multi-indices from Zr x (N U 0)n~r. Then

YPYt x x -


Proof. We have

XpXt = X^1 X

yPl yPn-1 ytAl " A n- l A l

tn-1 ypn+tn

It follows that

(adgXm)Xv = XmXv - g(Xv}Xm =

m yv fV\~

ym yvA X ~

". (3)

D

COROLLARY 1. Let v,m Z". Then the following are equivalent:1) elements v , m 6 Zn are independent;2) (a,dgXm)Xv = eXmXv, where 0 fc*.Proo/ By (3) (adgXm}Xv = 0 if and only if

Recall that 1, . . . ,n are roots of 1 of degree d. Raising the last equalityto the d-th power we conclude that

SI ' Sn

Since q^ j > i, are independent in fc*, we conclude that Vjmi Virrij = 0for all j > i and therefore for all j,i. It means that 1) holds. Conversely,1) implies 2). D

The next Theorem generalizes the corresponding result from [AC].


THEOREM 5. Let HI,. . . , un

LEMMA 3. Let m =~

for some M e Z in (5) and for some i. Then

Uj = XjXj + 6XjX^~ in (5) where Aj, 5 e k.Proof. Without loss of generality we can assume that A,; = 1. The left handside in (5) is a monomial

y-MA,: 1-eM

Hence for any monomial ^X7" occurring in MJ where T = (t\, . . . , tn] 6 Zand 6 e A;*, we have

s=l

_ f 1 - ql~Me,ej, T + eei = sej + Me,:,1 0, otherwise.

In the first case we have T = ee.j + (M e)e,;. In the second one

s=l

Then as in the proof of Lemma 2 we can conclude that

0, otherwise.

D

COROLLARY 3. Let m =we have

f1 in (5). Then for any index j

X,6M k.

We shall now prove Theorem 5 using the argument of [AC]. Supposethat Ui = Vi + v'^Xi), 1 < i < n, where suppf,; n Zej = 0. By Lemma 2and Corollary 2 we can replace each Uj by Uj (ad9 w)Xj and assume thatUi = v[, that is supp?/j C Ze^. By Corollary 3 there exists w' f suchthat supp w' C supp Uj and

(adgW^X] = u : i - X j X E j ,where Aj is from Corollary 3. Hence we can replace each ut by ut (adgw')Xt and assume that Uj = XjXj. Applying again Corollary 3 wededuce that ut XtXf for any index t = I, . . . , n. D


COROLLARY 4. Let d be either a continuous g-derivation of J- or a deriva-tion of A, where g is an automorphism of a finite order from, Notation 3.Then there exists an element a F (respectively, a, 6 A.) and scalarsAI, .. . , Xn 6 k such that d(Xi) \Xi + (ads w)(Xt) for all i I, ... , n.Proof. Put ?/.,; = d ( X i ) , I

Proof. The case s = I follows from Proposition 1, Corollary 1 and Propo-sition 2. Suppose that we have already proved that

ds(xv) =where v ^ 0 and V, v* k, v* Z, v* < sm + v. Proposition 1 entails

ds+l(Xv) = vd(Xsm+v]v [(adg w)Xsm+v

Apply Proposition 2. D

THEOREM 7. Let d be a g-derivation of A where g is the automorphismfrom. Notation 3. Suppose that there exists a nonzero polynomial f ( T ) ek[T] such that f(d] = 0. Then d(Xi) = \iXi, for any i = 1, . . . , n, whereAI, ... ,An 6 k.

Proof. Letcs^0 (9)

and Ui = d(Xi), i = 1, . . . , n. By Theorem 5 we get ut = \iXi+(adg w)Xifor some w e A. If w j= 0 then we get a contradiction with Proposition 3since f(u)Xv ^ 0 for some monomial Xv. DTHEOREM 8. Let char k = 0 andd a continuous derivation of J-. Assumethat there exists a that there exists a nonzero polynomial f ( T ) k[T] suchthat f ( d ) = 0. Then d = 0.Proof. By Corollary 4 we have d(Xi) = XiXi + (adw)Xi for some w Tand for all i = 1, . . . , n. Without loss of generality we can assume that theconstant term does not occur in w.

Put v = d ad w. Then

v(xm} = Xmx

m, xmek*. (10)

Let f ( T ) be from (9). Then for any monomial Xm, m Z", we have0 = f(d)Xm = cs(ad w + v)sXm + + c0Xm. (11)

Suppose that w ^ 0. Take m 6 Z" such that ||u>||,m are independent inthe additive Abelian group Zn. By Proposition 2 we have ||(adui)JXm|| =j||;|| + m, for any j > 0. If \\w\\ < 0, then from (11) we can deduce that

0

and therefore

s\\w\\ + m, > min (j'||^|| + m) > (s 1)||HI + fn.0 0. Since the constant term does not occur in w we canassume that ||u;|| > 0. It follows immediately from (11) and (10) thatf(Xm) 0 fr anY multi-index m Zn. It, means that there are onlyfinitely many values of Xm "?, 6 Zn .

If / Zn, then for any integer t by (10) we havetl

and hence txi is a root of /(T) for any integer t. There exist positiveintegers t\ < ti such that (^2 t \ ) X l = 0- Since char k = 0 it follows thatXi = 0 for any I Z". It follows that the constant term CQ = 0 and alsov(h] = 0 for all h e T . So d = ad w and

0 = f(d)Xm = cs(adw)sXm + + ct(adw)tXm,

in (11), where 0 < t < s and ct / 0. If we take m, 6 Zn such that ||io||, mare independent, then

ct(iuiwy-Xm = -ct+l(adw)t+lXm ----- cs(adw)sXm.

From Proposition 2 it follows that t < s. Then

t\\w\\+rn> min (;/|H| + m) > ( + 1)|H| + mt+l \\w\\, a contradiction. D

3 ACTIONS OF COCOMMUTATIVE HOPF ALGEBRAS

Suppose that a Hopf algebra H acts on an associative algebra A or A is a/e/it H -module algebra. This means that ^4 is a left ^-module and for anyh & H and a, b G ^ 4

provided


Suppose that the basic field k is algebraically closed, char A; = 0 and H isa cocommutative Hopf algebra. Then by [M, p. 76] the Hopf algebra His pointed and therefore by [M, Corollary 5.6.4] the algebra H is a smash-product H = Hi^kG, where H\ is an irreducible connected constituent of1 and G is a group acting in H\ by conjugations. Then G acts on A as anautomorphism group.

If char A; = 0, then by [M, Theorem 5.6.5] HI is the universal envelopeU(P(H}} of the Lie algebra P(H). Thus we obtain a Lie algebra homo-morphism P(H) DerA which is compatible with the action of G inP(H).

Let U be the universal enveloping algebra of DerA Then the groupG = Aut A acts on U by conjugations. So we can form a smash productHA = U$kG. There is the natural action of HA in A So we have provedTHEOREM 9. Let k be an algebraically closed field of characteristic zeroand H a cocommutative Hopf algebra acting on A. Then there exists ahomomorphism, of Hopf algebra H > HA compatible with actions of HA onA.

We shall now study in details the action of H\ on A. The center Z(A) =k. This means in particular that the base of Derint A consists of elementsadXu, where u 1r x (N U 0)n~r, u ^ 0. We shall consider the followingorder on the set of multi-indices Zr x (N U 0)"~r.

By Theorem 6 and PBW-theorem the base of the universal envelopingalgebra U = 7(Der A) consists of monomials

U l ) - - - ( & d X U m ) d l (12)where HI, . . . ,um, I Zr x (N U 0)n~r and ni > ... > um ^ 0. HereOl = d[l dl as in Notation 1.PROPOSITION 4. Let n > 3, m > 1, and

where u^ = (un, . . . Uin), 1 < i < m. Suppose that g 6 G and g(Xi) =&i.Xf according to Theorem 1, where oti, . . . ,an fc* and e = 1. Then

n W-v iiEV4-\-1Lm 4~\Vji\-U

eii+-+in/

7=1

U %


Here we assume that u> 1 ?//, = 0.Proof. By Theorem 6 for any index j = 1, . . . , n we have

X? = VjXv.

Hence dlXv = (v\l v'^)Xv. If t = (*i, . . . ,tn) G Zr x (NUO) n ~ r , then byProposition 1

(ad A"')*" = xtxv - xvxt =n iOv^\ I TT *il9/i )-( 11 fy (13)

D

THEOREM 10. Let k be a field of characteristic zero and n > 3. Then thefollowing classes coincide:1) the class of all left ideals in A generated by some set of monomials;2) the class of all two-sided ideals in A generated by some set of m,onomials;3) the class of H1\-subm,odules in A.Proof. According to Proposition 1 the conditions (1) and (2) are equivalent.By Proposition 4 any .//v-submodule is a left ideal in A. This means that wehave only to prove that any nonzero //A-submodule M in A is generated bysome set of monomials. Let / G M\0. Then there exists an automorphisma G Aut A such that

Q(A--) = {2Xl' ' = 1;\Xi, i > 1.

Let / = /o + + fm, where ft = X\gt where gt belongs to the subalgebrain A generated by X?, ... , Xn. Then for any d = 0, 1 , . . . , m- we have

ad(f) = /o + 2rf/i + ' '' + 2md/m G M. (14)

The system (14) can be viewed as a system of linear equations with un-knowns /o, /i,. .. , fm. The matrix of the system (14) is precisely the matrixof the Vandermonde determinant, which is nonzero. Multiplying the system(14) by the inverse matrix we obtain /j G M for any i = 0. 1, ... , m. Sim-ilar argument can be applied to each of the variables X%, ... , Xn showingthat each monomial occurring in / belongs in fact to M.

Conversely by Theorem 1 and Theorem 6 any right ideal generated bymonomials is an .H"A-submodule in A. D


4 INVARIANTS

In this section we shall deal with subalgebras A^ of invariants of cocommu-tative Hopf algebras H over an algebraically closed field k of characteristiczero acting on A under the assumption that r = 0 and n > 3. It follows fromTheorem 9 that without loss of generality we can always assume that H isa Hopf subalgebra in H\. Moreover by [M, Corollary 5.6.4] H = U(L)$kG,where L = P(H] is a Lie subalgebra in Der A and G = G(H) is a subgroupin Aut A.

Throughout this section as above we shall consider the lexicographicorder on the set of monomials and in (N U 0)".DEFINITION 2 ([M], p. 45). An extension of algebras A/ AH is calledSchelter-integral if for any element / A there exists a positive integer msuch that

0 (15)where q is a sum of elements of the form

aifni---adfndad+l, (16)

for some a\, . . . , ad+i G A^ such that n\ + + nd < m.PROPOSITION 5. Let the extension A/AH be Schelter-integral and s G(N U 0)n, s 7^ 0. Then there exist a positive integer t and elements

such that ts = u\ + + ud+i and monomials

occur in some polynomials fromProof. Take / = Xs. Then fm = aXms for any integer m > 0, wherea = a(m) e k*. Hence by (15) and (16) the monomial aXms occurs insome product of the form

where is an automorphism of A induced by the defining relations of A.It follows that /*, t m (n\ + + nd) occurs in some product of theform


Hence the questionable monomials occur in

Finally we have to observe that * and C*(a*) contain the same monomials.D

COROLLARY 5. Let the extension A/A^ be Schelter-integral. For anyindex i = 1, . . . , n there exists an integer W{ > 0 and an element gir G A.Hsuch that a monomial bX' ,b G k* , occurs in g,;.

Proof. Put t = m,;e?; G (N U 0)", where e, is from (8). By Proposition 5there exist elements u\, . . . Ud+i . (N U 0)n such that monomials

occur in some elements hi,... Jid+\ A.H . One can easily deduce thatUj = Uji&i G (N U 0)n, j = 1, ... ,77., and mn = uu + + Ud+i,i- Sincemi > 0 there exists Uji > 0 for some j and we can take g,; = hj. D

Fix an element v G L where L is a given Lie subalgebra in Der A. ByTheorem 6

(17)

where /3j G k and the constant term of u G A is zero. The followingstatement follows directly from Proposition 4.

PROPOSITION 6. Let f = Xl, where I = (k, . . . , /) G (N U 0)n. Then

j=l 3=1

PROPOSITION 7. Let the extension A/AH be Schelter-integral. Then L CDerint A.

Proof. Let /3j ^ 0 in (17). By Proposition 6 there exists an elementg G AH \ 0 which contains a monomial a.X*,t > 0. a G k*. Without lossof generality we can assume that t is minimal with respect to this property.Then 0 - v(g) = ("=i fadfig + ad(5), or

(18)


By Proposition 6 a monomial

fcatXt (19)occurs at the left hand side of (18). Suppose that

Then by (13)

-u= > 9=/e(Nuo)n ze(Nuo)n

(20)It follows from (19) and (20) that there exists l,z (N U 0)n such thatl + z = t&i where &i is from (8). Hence I = liSi, z = ZiCi, where /,,z,; NUOand t = li + Zi. Since t is minimal we conclude z,; = t and li = 0. Thismeans that the constant term of u is nonzero, a contradiction. Hence/?! = ... = (3n = 0 and v = ad u. DCOROLLARY 6. Let the extension A./AH be Schelter-integral and L ^ 0.Then A.H and L are commutative.

Proof. By Proposition 7 we have L C Derint A and A^ is contained in thecentralizer of each element u A such that adn e L. By the assumptionA^ ^ k. From [Al] we know that A^ is commutative. DTHEOREM 11. Let H be a Hopf subalgebra in H\ such that the extensionA/A is Schelter-integral. Suppose that A is a finite left (right) A. -moduleand some multiparameter g^, 1 < i < j < n, is algebraically independentover the sub field of rationals Q in k. Then L = 0 and H = kG, where G isa subgroup o/AutA.

Proof. If L 7^ 0 then A.H is commutative by Corollary 6. Hence by [P, The-orem 3.20] the algebra A is a Pi-algebra, satisfying some standard identity

Sm(yi, ,ym}= (-1) Vi yam = 0. (21)In order to prove the theorem we need to show that A does not satisfy (21).For any integer TO, > 0 we put yt = XiXJ1 , t = 1, . . . , m. Then

_

2/crl ' ' ' yam,


Therefore

Sm(yii- -.yrn) =

The monomials

in qij have different powers because of the uniqueness of ra-adic decom-position of any integer. Now the independence of QJJ over the subfield ofrationals Q in k entails 5 m (y i> . . . , ym) ^ 0. D

Each derivation d of A can be expanded to F, namely

Similarly if n > 3, then each automorphism of A from Theorem 1 can beexpanded to F. So the action of H\ = U(Dei A)[jA:(Aut A) can be alsoexpanded to F.

PROPOSITION 8. The action of H^, n > 3, can be expanded to T'.Proof. Let d = Y^Pjdj + at^n ^ Dor A. Since A C T the derivation adu isexpanded to J='. Now if / = \iXUl + 6 F, then supp/ C (N U 0)n isArtinian with respect to the lexicographic order. Then by Proposition 4,djf = XiUijX""1 + is an element of F, since it has the same support.Similar argument works for any automorphism because of Theorem 1. D

Let as above A: be an algebraically closed field of characteristic zero.Fix a Hopf subalgebra H in Ht\. Then H [/(L)jJfcG where L is a Liesubalgebra in DerA and G is a subgroup in Aut A.

PROPOSITION 9. Let FH be the set of all invariants for each element ofH. Then FH is-a skew subfield in F.Proof. If u. v FH and d 0 and elementsa . Q , . . . . am_i FL such that

X + om-lXrn-1 + + aiXi + OQ = 0.


Then there exists an element aj,j < m, with the smallest monomial rXd(viewed as an element of .F) such that rXdX^ cancels either with X orwith some C,Xd' ' X{ ', where j ^ f < m. In the first case rXd = X~j andin the second (C,Xd'}-l(rXd} = xf~j. But (CiXd'}-l(rXd} is the smallestterm of a~,la,j FL viewed as an element of f . So in any case there existsa nonzero element a e FL with the smallest term X*, t > 0.

Suppose that L ^ 0 and d = ^"==1 f3jdj + ad u L. As we have shownabove for any index i = 1, . . . ,n there exists an element a 6 FL with thesmallest term X\,t> 1. By Proposition 4

n

0 = d(a] = (J^ Pjdj)Xt + (ad u)X\ + = /M* + , (22)

where is a series with || ||, ||(adu)X*|| > ||X*||. Here ||/|| is the multi-index corresponding to the smallest monomial in /. So (22) implies /% = 0for any i.

Now d = a,du and FL is contained in the centralizer of u. Apply resultsfrom [Al], [ACo], [Z] ' DTHEOREM 13. Let H be a Hopf subalgebra in H^ such that F has a finiteleft (right) finite dimension over FH . Suppose k has characteristic zeroand some multiparameter qij,l < i < j < n, is algebraically independentover the subfield of rationals Q in k. Then L = 0 and H = kG, where G isa subgroup o/AutA.

Proof. Observe that FH C FL . So if L ^ 0 then FH is commutative. Thenwe can apply the argument of Theorem 11. D

Finally we have to mention that subalgebras of invariants of kG in A andin F were studied in details in [AW] (the case of at least three variables)and in [AD2] (the case of two variables).

REFERENCES

[Al] Artamonov V.A., Automorphisms of division ring of quantum rationalfunctions, Math. Sbornik - 191(2000), N 12, 3-26.

[A2] Artamonov V. A. Valuations on quantum fields, Commun. Algebra,29(2001), N 9, 3889-3904.

[A5] Artamonov V. A. Actions of Hopf algebras on quantum polynomials.In the book: Representation of algebras. Coelho/Merklen Ed. MarcelDekker Inc., NY., 2001, 11-20.


[ACo] Artamonov V.A., Cohn P.M., The skew field of rational functionson the quantum plane, J. Math. Sci, 93(1999), N 6, 824-829.

[AC] Alev J., Chamarie M., Derivations et automorphismes de quelquesalgebres quantiques, Comm. Algebra, 20(2992), N 6, 1787-1802.

[ADI] Alev J., Dumas F., Sur le corps des fractions de certaines alg'ebresquantiques, J. Algebra, 170(1994), N 1, 229-265.

[AD2] Alev J., Dumas F.. Invariants du corps de Weyl sous Faction degroupes finis, Comm. Algebra 25(1997), N 5, 1655-16723.

[AW] Artamonov V.A., Wisbauer R. Homological properties of quantumpolynomials, Algebras and representation theory, 4(2001), N 3, 219-247.

[B] Brookes C. J. B. Crossed products and finitely presented groups., J.Group Theory, 3(2000), 433-444.

[D] Demidov E. E., Quantum group, Moscow: Factorial, 1998, 127P.[GL1] Goodearl K.R., Letzter E.S., Prime factor of the coordinate ring of

quantum matrices, Proc. Amer. Math. Soc. 121(1994, N 4, 1017- 1025.[GL2] Goodearl K.R., Letzter E.S., Prime and primitive spectra of multi-

parameter quantum affine spaces, "Trends in ring theory" (Miskolc),1996, 39-58. CMS Conf. Proc. 22, AMS Providence RI, 1998.

[GL3] Goodearl K.R.. Letzter E.S., Quantum n-space as a quotient of clas-sical n-space, Trans. Amer. Math. Soc. 352(2000), N 12, 5855-5876.

[H] Herstein I. N. On the Lie and Jordan rings of a simple associative ring,Amer. J. Math., 77(1955), 279-285.

[M] Montgomery S., Hopf algebras and their actions on rings, RegionalConf. Ser. Math. Amer. Math. Soc., Providence RI, 1993.

[MP] McConnell J.C., Pettit J.J., Crossed products and multiplicative ana-logues of Weyl algebras, J. London Math. Soc. 38 (1988), N 1, 47-55

[Pj Procesi C., Rings with polynomial identities. Marcel Dekker, NewYork, 1973.

[O] Freddy van Oystaeyen. Algebraic geometry for associative algebras.Marcel Dekker Inc. NY, 2000, 302 pp.

[OP] Osborn J. P., Passman D.. Derivations of skew polynomial rings, J.Algebra, 176(1995), N 2, 417-448.


[Z] Zelenova S., Commutative subalgebras in a ring of quantum polynomi-als and a skew field of quantum Laurent series, Math. Sb., 192(2001),N 3, 53-64.


Combinatorial properties of free algebrasof Schreier varieties1

VYACHESLAV A. ARTAMONOV Department of Mechanics and Math-ematics, Moscow State University, Moscow, Russia.E-mail: [email protected]

ALEXANDER A. MIKHALEV Department of Mechanics and Mathe-matics, Moscow State University, Moscow, Russia.E-mail: [email protected]

ALEXANDER V. MIKHALEV Department of Mechanics and Mathe-matics, Moscow State University, Moscow, Russia.E-mail: [email protected]

ABSTRACT In this survey article we consider combinatorial properties of free al-gebras of Schreier varieties. Our main goal is to observe results concerning automorphicorbits of elements of free algebras of main types of Schreier varieties, on Schreier tech-nique, on applications of free differential calculus, on primitive, test, and generalizedprimitive elements of free algebras. In this text we do not touch standard bases of idealsof free algebras, elementary theories of free algebras, actions of the symmetric group anddimensions of homogeneous components, since these topics deserve some special fascicles.

xThis work was partially supported by RFBR, grants 00-15-96128, 02-01-00218, 02-01-00219, and by INTAS, grant 00-566.


1 MAIN TYPES OF SCHREIER VARIETIES

Let, K be a field. A variety fTJt of algebras over K is defined as a class ofalgebras closed under taking subalgebras, homomorphic images and directproducts. Let X be a set. An algebra Fgn(X) in OT is called the freealgebra with the set X of free generators if Fyn(X) is generated by X, andif / is an arbitrary map from X to any algebra A of SOT then / can beextended to a homomorphisiii Fyn(X) > A (it is clear that such extensionis unique). The algebra F^(X) is determined by X and fU? uniquely up toan isomorphism. For any nontrivial variety $Jl and any set X there existsthe free algebra F$n(X). If A G fOT. then there exists a set X (for example,X = A) such that A is a homomorphic image of Fy^(X). In particular, Ais isornorphic to a factor algebra of FOT(X). The cardinality of X is calledthe rank of F

algebras are Schreier. A. A. Mikhalev in [94] and A. S. Shtern in [169]showed that the variety of Lie superalgebras is Schreier. A. A. Mikhalevproved this result for the variety of color Lie psuperalgebras in [96].

It is quite natural now to formulate the following

PROBLEM 1.1 To give the complete classification of Schreier varieties ofalgebras.

U. U. Umirbaev [180, 182] obtained new examples of Schreier varietiesof algebras and gave necessary and sufficient conditions for a variety ofalgebras to be Schreier. As for subalgebras of free algebras of varieties offi-linear algebras, see articles [10, 16].

Let K be a field, X a set, and let F = F(X] be the free algebra ofa homogeneous variety of algebras with the set X of free generators overthe field K. For u . F ( X ) , by f.(u) = f - x ( u ) we denote the degree ofu. Consider now an arbitrary mapping yu: X > N (a generalized degreefunction), where N is the set of positive integers. Let T ( X ) be the freegroupoid of non-associative monomials in the alphabet X, S(X) the freesemigroup of associative words in X, and ~: T(X) > S(X) the bracketremoving homomorphism. We set [JL(X\ xn) = Y^i=i l^(xi) fr xi G X- If/j,(x) 1 for all x 6 X , then fi is just the usual degree , JJL = i. If a -F,a = ^aja,;, 0 7^ a,; K, o,,; are basic monomials, dj ^ as with j ^ s,then we set /i(a) max,;{ft(a'?;)}. By a we denote the leading part of a: = j>(aJ-)=M(a)aJaJ-A subset M of F is called independent if M is a set of free generatorsof the subalgebra of F generated by M. A subset M = {aj} of nonzeroelements of F is called reduced if for any i the leading part a of the elementa,; does not belong to the subalgebra of F generated by the set {a \ j ^ i}.

Let S = {sa | a e /} be a subset of F. A mapping u: S + S' C Ais an elementary transformation of 5" if either ui is a non-singular lineartransformation of S1, or W(SQ) = sa for all a /, a ^ /3, and ^(s^) sp +f({sa \ a ^ /?}), where / is an element of a free algebra of the same varietyof algebras. It is clear that elementary transformations of free generatingsets induce automorphisms of the algebra f; such automorphisms are calledelementary.

Let A be a free algebra of a homogeneous Schreier variety of algebras.One can transform any finite set of elements of the algebra A to a reducedset by using a finite number of elementary transformations and possiblycancelling zero elements. Every reduced subset of the algebra A is anindependent subset (this is what is called the Nielsen property). Moreover,by using Kurosh's method, one can construct a reduced set of generatorsfor any subalgebra of the free algebra A. Indeed, let B be a subalgebra of


A, M = Ujlo ^J' where Mj consists of G- homogeneous elements of degreej, if nonempty. We set MO = 0. Then, by induction, we suppose thatMO, . . . , M j have been defined. Let B, be the subalgebra of B, generatedby U}=o Mj ani be a maximal such subset. It is clear thatM generates the subalgebra B. Moreover, M is a reduced set.

Hence any subalgebra of A is free in the same variety of algebras (thisis what is called the Schreier property). In [85], J. Lewin proved thatfor a homogeneous variety of algebras, the Nielsen and Schreier propertiesare equivalent. By using this equivalence, it is easy to see that the au-tomorphism groups of the corresponding free algebras (of finite rank) aregenerated by elementary automorphisms (for free Lie algebras, this resultwas obtained by P. M. Cohn in [31], and for free non-associative algebrasby J. Lewin in [85]). A similar result for any homogeneous Schreier varietyof Jl-algebras was proved in [10].

Let 97? be a homogeneous variety of linear algebras over a field K. LetC be an algebra of the variety 97?, (y) the free algebra of rank one of thevariety 97? . We consider the free product B = (y) * C 97?. By B\ wedenote the subspace of elements of B with the degree one with respect toy. Then BI is a free one-generated C-bimodule in 97?. Consider C ffi BIwith the multiplication given by

(ai + mi) (02 + 7T?,2) O-lfl.2 + a\m2 + 777-10,2,

where a 1,0,2

for all a, J3 K, a, b e F(A"). The algebra F(X) is the free non-associativealgebra (i.e. the free algebra of the variety OTo). Let WQ = r(X), A =F(X). Then the algebra U(A) is the free associative algebra with the setS0 = {rw, lw | w 6 WQ] of free generators (for the details see [54, 180]).

Let OTI be the variety of all commutative algebras, I the two-sided idealof the free non-associative algebra F(X) generated by the set {ab baa, b G F ( X ) } . Then the factor algebra A = F(X}/I is the free algebra ofthe variety SUti with the set X of free generators (i.e. the free commutativenon-associative algebra).

Suppose that the set T(X) is totally ordered in such a way that fora, b 6 T(X) if(a) > l(b), then a > b (f.(u) denotes the degree of a monomialu T(X)). We construct the set W\ of all commutative regular monomialsinductively in the following way. First of all, X C W\. Furthermore,w 6 W\ if w = uv, u and v are commutative regular monomials, andu < v. Then W\ is a linear basis of A = F ( X ) / I ([156]). The universalmultiplicative enveloping algebra U(A) is the free associative algebra withthe set Si = {rw w e Wi} of free generators, [54, 180].

Let OT2 be the variety of all anti-commutative algebras, J the two-sidedideal of the free non-associative algebra F+(X) without unit element gen-erated by the set {aa \ a 6 F(X)}. Then the factor algebra A = F+(X)/Jis the free algebra of the variety OT2 with the set X of free generators(i.e. the free anti-commutative non-associative algebra). We construct theset W2 of all anti-commutative regular monomials inductively: X C W2\w 6 W% if w uv, u and v are anti-commutative regular monomials, andu < v. Then W2 is a linear basis of A = F(X}/J ([156]). The universalmultiplicative enveloping algebra U(A) is the free associative algebra withthe set 52 = {rw \ w 6 W?} of free generators, [54, 180].

Let G be an Abelian semigroup, K a field of characteristic differentfrom two; :GxG > K* a skew symmetric bilinear form (so-called acommutation factor, or a bicharacter), that is,

e(g,h)(h,g) = 1,e(9i +92,h) = ( g i , h ) e ( g 2 , h ) , e(g,hi + h2) = ( g , h i ) ( g , h 2 )

for all g, gi,g2, h, /M, h,2 e G;

G^ = {g 1.Let n be a primitive root of 1, en = e27ri/n, G = (/)n (g}n the direct sum


of cyclic groups of order n generated by a and b, respectively, e(f, /) = 1,(/)

Then L(X) is the free color Lie superalgebra with the set X of free gener-ators. In the case where char .ft' = p > 2, let LP(X] be the subalgebra of[K(X)]P generated by X. Then LP(X) is the free color Lie p-superalgebraon X. For more information on free Lie superalgebras we refer to themonographs [14, 124].THEOREM 1.2 ([31],[77],[85],[94]-[96],[155, 156, 169, 187]) Let X be a fi-nite set, X = { X I , . . . , X H } , K a field, charK 7^ 2, F = F(X) the freealgebra without the unity element on the set X of free generators of oneof the following varieties of algebras over a field K: the variety of all al-gebras; the variety of Lie algebras; varieties of color Lie superalgebras; thevariety of Lie p-algebras; varieties of color Lie p-superalgebras; varieties ofcommutative and anticomm.utative algebras.

1. Any finite subset of F can be transformed into a reduced subset bya finite sequence of elementary transformations (with cancellation ofpossible zeros);

2. Any reduced subset of the algebra F is an independent subset;3. The leading part of a polynomial on a reduced subset is a polynomial

on leading parts of elements of this subset;4- Any subalgebra of F is free;5. A subset M of F is independent if and only if it is linearly independent

modulo the square of the subalgebra of F generated by M (for freecolor Lie p-superalgebras we add to the square the ideal all p-powersof elements of M);

6. If \X\ < oo, then the automorphism group of the algebra F is gener-ated by elementary automorphisms.

PROBLEM 1.3 (P. M. Cohn, [31]) Is it true that the automorphism, groupof a free associative algebra of finite rank is generated by elementary auto-morphisms? The sam,e question for a polynomial algebra.

Note that, this problem has a positive solution for the polynomial algebrain two variables (H. W. E. Jung [63], W. van der Kulk [76]), and also for thefree associative algebra of rank 2 (A. J. Czerniakiewicz [36], L. G. Makar-Limanov [92]).PROBLEM 1.4 (M. Nagata, [130]) Let (p be an automorphism of the poly-nomial algebra K[x,y,z] given by

(f>(x) = x + z(x2-yz),(y) = y + 2x(%2 - yz) + z(x2 - yz}2,(p(z) = z.


7s it true that this automorphism, is not a composition of elementary auto-morphisms of K[x,y,z ?

Recently I. P. Shestakov and U. U. Umirbaev [153, 154] have proved thatif char if 0, then the Nagata automorphism is wild (it means that it isnot a composition of elementary automorphisms of K[x, y, z\).

Let X = {x1; . . . ,xn}. It follows from Theorem 1.2 that the free alge-bras F ( X ) are Hopfian (it means that if the algebra F(X) is generatedby elements HI ..... ?/, then these elements form a free generating set of

An algebra A over a field K with an anticommutative bilinear operation[x.y] and a trilinear operation A(x,y,z] is called an Akivis algebra if

[[x,y],z] + [[y,z],x] + [[z,x],y]= A(x,y,z) + A(y,z,x) + A(z,x,y)- A(y,x,z) - A(x,z,y) - A(z,y,x).

These algebras were introduced by M. A. Akivis as tangent algebras oflocal analytic groups. I. P. Shestakov and U. U. Umirbaev [152] provedthat the variety of Akivis algebras is Schreier, automorphisms of finitelygenerated free Akivis algebras are tame, the occurrence problem for freeAkivis algebras is decidable, finitely generated subalgebras of free Akivisalgebras are residually finite, the word problem is decidable for the varietyof Akivis algebras.

2 SCHREIER TECHNIQUEO. Schreier [150] introduced very useful technique to obtain free generatorsfor a subgroup of a free group from special set of cosets representatives.Later this technique was extended by many authors for different types offree algebras.

Subalgebras of the free associative algebra are not necessary free. Forexample, if we consider elements x2 and x3 in the free associative algebraK(x) = K[x], then x2x3 = x3x2. Hence the subalgebra generated by x2and x3 is not free. At the same time, we have P. M. Conh's result thatover a free associative algebra submodules of free left modules are free.P. M. Conh obtained this result using a generalization of the Euclideanalgorithm in the free associative algebra.

Let K be a field. The family {,; | i /} of elements of the free associativealgebra K(X) is called left /'-dependent if there exist elements bj. (E K(X),almost all zero, such that


or if some G,J is equal to zero.Otherwise the family {a,} is called left ^-independent.An element o K(X) is said to be left ^-dependent on a family {a,;} if

either a,; = 0, or there exist &,; e K(X), almost all zero, such that

for all i. Otherwise a is said to be left ^-independent on a family {a,}.In the free associative algebra over a field K the following analog of the

Euclidean algorithm takes place:

THEOREM 2.1 (Weak algorithm [28, 29]) Let QI, . . . ,an e K{X) and(a'l) < < f-(a>n)- Then if the family {ai, . . . ,an} is left ^-dependent,then there exists i, 1 < i < n, such, that the element 04 is left dependent onthe family {a\, . . . , a,;_i},

Note that the right side version of this algorithm also takes place inK(X).

Since the weak algorithm takes place in K(X), the algebra K(X) is a freeideal ring (a fir), i.e. K(X] is left fir and right fir, that is left (right) idealsof K(X] are free left (right, respectively) K(X) -modules, and moreover, ofunique rank. Therefore, over a left fir every submodule of a free left moduleis free, [30, 33].

Note that a very elegant proof of this fact can be given using P. M. Cohn'stheorem that the free product of fir's is again fir. In [86] J. Lewin with theuse of Schreier technique gave a direct proof of the fact, that K(X) is a fir.

Applying the weak algorithm P. M. Cohn [31] obtained the followingsufficient condition for a subalgebra of a free associative algebra to be free.Let B be a subalgebra of K(X}. We consider K(X) as a left B-modulerelative to the usual multiplication. A system of elements S {s,; i e /}of K(X) is said to be S-independent if for almost all zero elements a,

THEOREM 2.2 ([31]) If B is any subalgebra of K(X) such that K(X] isfree as a left B-module, with a B-independent basis, then B is a free as-sociative algebra over K. In particular, if B is a subalgebra o f K ( X } suchthat K(X) satisfies the weak algorithm as a B-module, then B is a freesubalgebra of K(X).


For more information on free ideal rings we refer to P. M. Conn's mono-graph [33].

Suppose that, F is a free group of rank n. and let H C F be a subgroupof a finite index \F : H\. Then H is a free group of the rank m, where m isdetermined by well known Schreier formula [150]: m 1 = (n 1) \F : H\.

J. Lewin [86] obtained the following analog of Schreier formula for freeassociative algebras. For a left ideal M of K(X) we denote by rank (M)the rank of the free left K(.X"}-module M.THEOREM 2.3 ([86]) Let \X\ = n and let I be a left ideal of the freeassociative algebra K(X] such that dimff(K(X} //) = k < oo. Then I is afree left K(X]-module and rank (/) = k(n 1) + 1.

For free Lie p-algebras G. P. Kukin [74] obtained the following analog ofSchreier formula.

THEOREM 2.4 ([74]) Let K be a field, p = charK > 2, \X\ = N < oo.Suppose that H is a subalgebra of the free Lie p-algebra LP(X],dim LP(X)/H = t < oo. Then rank (H} = pt(N-l) + l.

A. A. Mikhalev [94j-[96] obtained Schreier type formulas for free Liesuper algebras and p-superalgebras.

THEOREM 2.5 ([94]) Let char AT ^ 2,3, and let L = L(X) = L+L- bethe free color Lie superalgebra, rank (L) = \X\ = N < oo, and let H be a G-homogeneous subalgebra of L. H = H+Q

polynomial identities and combinatorial methods

Documents

russiamarcel dekker

headquartersmarcel dekker

customer servicemarcel

rutgers university madisonjack

massey university of

new york baseld e

new jersey orlando

current information