Non-Associative Algebra and Its Applications (Lecture Notes in Pure and Applied Mathematics)

Non-Associative Algebra and Its Applications PURE AND APPLIED MATHEMATICS A Program of Monographs, Textbooks, and Lect...

Author: Lev Sabinin | Larissa Sbitneva | Ivan Shestakov

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Non-Associative Algebra and Its Applications

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

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

Zuhair Nashed University of Central Florida Orlando, Florida

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

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

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

Non-Associative Algebra and Its Applications Edited by

Lev Sabinin Morelos State University (UAEM) Morelos, Mexico

Larissa Sbitneva Morelos State University (UAEM) Morelos, Mexico

Ivan Shestakov Sao Paolo University Sao Paolo, Brazil

Boca Raton London New York

DK3005_Discl.fm Page 1 Tuesday, November 22, 2005 9:32 AM

Published in 2006 by CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2006 by Taylor & Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group No claim to original U.S. Government works Printed in the United States of America on acid-free paper 10 9 8 7 6 5 4 3 2 1 International Standard Book Number-10: 0-8247-2669-3 (Softcover) International Standard Book Number-13: 978-0-8247-2669-0 (Softcover) Library of Congress Card Number 2005053861 This book contains information obtained from authentic and highly regarded sources. Reprinted material is quoted with permission, and sources are indicated. A wide variety of references are listed. Reasonable efforts have been made to publish reliable data and information, but the author and the publisher cannot assume responsibility for the validity of all materials or for the consequences of their use. No part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, please access www.copyright.com (http://www.copyright.com/) or contact the Copyright Clearance Center, Inc. (CCC) 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. CCC is a not-for-profit organization that provides licenses and registration for a variety of users. For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. Library of Congress Cataloging-in-Publication Data Sabinin, Lev V. (Lev Vasil’evitch) Non-associative algebra and its applications / Lev Sabinin, Larissa Sbitneva, Ivan Shestakov. p. cm. -- (Lecture notes in pure and applied mathematics ; v. 246) Includes bibliographial references. ISBN 0-8247-2669-3 (alk. paper) 1. Nonassociative algebras. I. Sbitneva, Larissa. II. Shestakov, I.P. III. Title. IV. Series. QA252.S23 2005 512’.46--dc22

2005053861

Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com Taylor & Francis Group is the Academic Division of Informa plc.

and the CRC Press Web site at http://www.crcpress.com

Dedication

This book is dedicated to the memory of Professor Lev Vasilievich Sabinin

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Preface

This book reﬂects the last achievements in the intensively developing ﬁeld of nonassociative algebra. It contains the ﬁnal versions of talks presented at the V International Conference on Nonassociative Algebra and Its Applications (July 27–August 2, 2003, Oaxtepec, Mexico). The V International Conference on Nonassociative Algebra and Its Applications is a continuation of a 15-year-old sequence of international conferences devoted to nonassociative algebra and its applications: NONASS-1—Novosibirsk, Russia, 1988 NONASS-2—Sumcha, Uzbekistan, 1990 NONASS-3—Oviedo, Spain, 1993 NONASS-4—S˜ ao Paulo, Brazil, 1998 NONASS-5—Oaxtepec, M´exico, 2003 The book covers a wide range of topics focused on the following: 1. Lie algebras 2. Nonassociative rings and algebras 3. Quasigroups and loops (algebraic and smooth) and related systems 4. Applications of nonassociative algebra to geometry 5. Applications of nonassociative algebra to physics and natural sciences The theory of nonassociative algebras has seen new impetuous developments in recent years. From physics, geometry, and algebraic topology new nonassociative structures have appeared, such as superalgebras, coalgebras, pairs, and triple systems. These structures proved to be interesting from a purely algebraic point of view; they produced innovative ideas and methods that have helped to solve some old algebraic problems. The structure theory of these new structures is still in a formative stage and many unsolved problems remain, even in a ﬁnite dimensional case. On the other hand, the theories of the main classes of nonassociative algebras, namely, alternative, Jordan, and Malcev algebras, are far from completion, especially in the inﬁnite dimensional case. For example, the structure of free algebras, the description of prime degenerate algebras and of irreducible representations are still being developed, including questions on speciality for Jordan and Malcev algebras. Moreover, such classical nonassociative algebras as octonions or Albert algebra still attract the attention both of nonassociative algebraists and of specialists from other areas, and many natural questions on the structure, identities, and invariants of these algebras still being studied. The contributions to this volume represent both the “linear” and “non-linear” trends in nonassociative algebra research. Along with recent results in nonassociative rings and algebras, results in the theory of quasigroups and loops are presented. Special attention is given to applications of smooth quasigroups and loops to diﬀerential geometry and relativity. The theory of smooth quasigroups generalizing Lie groups theory has been established in the works of Dr. L.V. Sabinin and applied successfully to geometry and mathematical

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physics (L.V. Sabinin, A.I. Nesterov). In particular, the generalized theory of gravity has been developed in an algebraic way with promising applications to discrete space-time models. It has been disclosed that, in general, nonassociativity is an algebraic equivalent of nonlinearity. There are some applications of the above methods to Poisson mechanics, etc., being an alternative to the theory of diﬀerentiable manifolds. The book contains also a wide range of applications of nonassociative algebra to theoretical physics presented at the Conference. Finally, we include into the book English translation of two Russian publications that greatly stimulated research in nonassociative algebra in former Soviet Union. It is the well-known survey by A.I. Shirshov “Some problems in the theory of rings that are nearly associative” and “Dniester Notebook: Unsolved problems in the theory of rings and modules”. All the chapters included to the Proceedings have been refereed, and the editors thank the referees for their invaluable help. The V International Conference on Nonassociative Algebra and Its Applications was ﬁnancially supported by the Morelos State University (UAEM), Mexican National University (UNAM), by research projects (CONACyT), and “Quantasmagor” A.C. Lev Sabinin, Larissa Sbitneva, Ivan Shestakov The Editors

While this book was being prepared for publication, one of the editors, the Chairman of the Organizing Committee of the V NONASS Conference, Professor Lev Vasilievich Sabinin, passed away. This book is dedicated to his memory. Larissa Sbitneva and Ivan Shestakov

Lev Vasilievich Sabinin June 21, 1932–June 4, 2004

Life is good for only two things, discovering mathematics and teaching mathematics. Sim´eon Poisson (1781-1840) Lev Vasilievich Sabinin, a brilliant world renowned geometer died on June 6, 2004 in Cuernavaca, Morelos, Mexico, after a heart attack. Lev Vasilievich was born in Voronezh, Russia, on June 21, 1932. In 1950 he started his studies in mathematics at Moscow University. From 1955 to 1958 he was engaged in postgraduate studies in diﬀerential geometry and Lie groups under the supervision of Prof. P.K. Rashevsky, and earned his Ph.D. degree with the dissertation entitled “Mirror Symmetries of Riemannian Spaces” (1959). Out of these topics, diﬀerential geometry, and Lie groups grew future special ﬁeld of creativity. From 1958–1965 Lev Vasilievich held various positions in research and teaching at the Institute of Mathematics (Siberian Scientiﬁc Centre, Novosibirsk) and at Novosibirsk University. There his scientiﬁc interests were concentrated on the topics that advanced from the studies of geometry of Riemannian spaces—geometry of homogeneous spaces, Lie groups and Lie algebras, fundamentals of involutive geometry, and introduction of the concept of the trisymmetric space. At that time he had contacts with the prominent mathematician and academician, Anatoli Mal’tsev, who became his mentor in algebraic investigations. In 1965 Lev Vasilievich was invited to the position of Reader, and after some years became the head of the Department of Algebra and Geometry in the Friendship of Nations University, Moscow, teaching and supervising students and postgraduates from many countries, and carrying forward the work started in Novosibirsk, culminating with the D.Sc. dissertation, “Involutive Geometry of Lie Algebras” (Kazan University, Russia 1971). From 1972-1974 Lev Vasilievich became Professor of the Department of Mathematics of Ife University, Nigeria, where he established a scientiﬁc school in nonassociative algebra, active until today. Here he started a new ﬁeld of investigation of smooth quasigroups and loops with a pioneering paper, “On the Geometry of Loops” (1972), where a remarkable construction relating quasigroup and group theory, the Baer-Sabinin construction, was presented. After the return from Nigeria in 1974, Lev Vasilievich held the permanent professorship at Friendship of Nations University, Moscow, and since 1995 also several visiting and extraordinary posts in Mexico (Mathematical Institute, UNAM; Michoacan University, Morelia; Quintana Roo University, Chetumal). In 2001 he became full professor at Morelos University, Cuernavaca, Mexico, where he founded research groups in nonassociative algebra and diﬀerential geometry and quasigroups and mathematical physics. Lev Vasilievich is undoubtedly one of the most prominent Russian masters of geometry. For 50 years his research has made essential contributions to the geometry of various spaces (homogeneous, symmetric, Riemannian, and numerous subtypes), to the problems of Lie algebras, connected with geometries of these spaces, and to the diﬀerential geometry of

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various nonassociative systems (smooth quasigroups, loops, odules, etc.) with applications to the geometry of manifolds (odular structures on manifolds, local geodesic loops, etc.). The inﬁnitesimal aspects of these nonassociative systems reveal an interesting and rich algebraic structure of binary-ternary algebras with special identities, giving algebraic equivalents to several purely geometric concepts (as aﬃne connections, etc.). This concenter of problems seems to be of main importance. It forms about half of the papers by Lev Vasilievich, and may be summarized under terms such as nonlinear geometric algebra, odular geometry, and theory of smooth quasigroups and loops, which mark new substantial chapters of mathematics created by Lev Vasilievich. Besides its value for pure mathematics, this branch has also promising perspectives for applications in theoretical physics, in particular, a new approach to the discrete space-time in quantum gravity. Lev Vasilievich published more than 100 scientiﬁc papers and four monographs, “Analytic Quasigroups and Geometry” (Moscow, 1991), “Involutive Geometry of Lie Algebras” (Moscow, 1997), “Smooth Quasigroups and Loops” (Kluwer, Dordrecht, 1999), “Mirror Geometry of Lie Groups, Lie Algebras and Homogeneous Spaces” (Kluwer, Dordrecht, 2004). He has translated and edited several important volumes on geometry and mathematics in general. Lev Vasilievich was a member of the Siberian Mathematical Society, the Moscow Mathematical Society, the American Mathematical Society, and the Mexican Mathematical Society. He was also a member or chairman of numerous editorial boards, seminars, and councils. Of speciﬁc importance was his post as co-chairman of the famous Seminar on Vector and Tensor Analysis at Moscow University. He has been awarded numerous honorary medals, grants, awards by scientiﬁc institutions from Russia, the United States, the United Kingdom, and Mexico. During his entire career Lev Vasilievich shared his time and energy between research and teaching. He has designed and delivered a large variety of lecture courses, many (8) of them have been published. Under his supervision 16 students have successfully performed their research programs and earned Ph.D. and D.Sc. degrees in mathematics, in mathematics education and mathematical physics. We have tried to describe the achievements and scientiﬁc legacy of Lev Vasilievich but we have failed to give the full image of his charismatic personality. The moving power of mathematical invention is not only reasoning but also imagination and a love of beauty as well. A scientist, tutor, colleague, and mentor of exceptional intellectual creativity, mental inﬂuence and talent, Lev Vasilievich is sorely missed by his family, friends, and the scientiﬁc community. L. Bokut, V. Kharchenko, J. L˜ ohmus, G. Moreno, A. Nesterov, M. Rosenbaum, and I. Shestakov

Mathematical Research of Professor Lev Sabinin

Summary of Mathematical Research “Subsymmetric Riemannian Spaces and Mirror Symmetries” (Moscow University, 1955– 1958). “Geometry of Reductive Homogeneous Spaces and Trisymmetric Spaces” (Institute of Mathematics, Siberian Branch of the USSR Academy of Science, Novosibirsk University, 1958–1965). “Involutive Geometry of Lie Algebras and Homogeneous Spaces”. “Theory of Loops and Quasigroups, Analytic Loops” (Friendship of Nations University, 1965–1972). “Geometry of Loops. Applications of Loops Theory to Diﬀerential Geometry” (University of Ife, Nigeria, 1972–1974). “Smooth Quasigroups and Loops, Smooth Universal Algebra, Nonlinear Geometric Algebra” (Friendship of Nations University, Moscow, 1974–1995). “Foundations of Smooth Quasigroups with applications to Geometry and Mathematical Physics. Mirror Geometry of Lie Groups, Lie Algebras and Homogeneous Spaces” (Michoacan University, University of Quintana Roo, Morelos State University, Mexico, 1995–2004). From 1958–1972 he created and developed a new approach to Lie groups, Lie algebras and Homogeneous spaces: “Mirror Geometry of Lie Group, Lie Algebras and Homogeneous Spaces.” From 1971–1994 he created and developed a new branch of Mathematics, “Nonlinear Geometric Algebra” (based on universal algebra and diﬀerential geometry). Dr. Sabinin has published over 125 scientiﬁc papers, 8 lecture notes, and four monographs on algebra, geometry, diﬀerential equations, mathematical physics, and mathematical education in Russian and English. Description of Mathematical Research L.V. Sabinin oﬃcially started his research in mathematics during postgraduate studies at the Department of Diﬀerential Geometry of Moscow University (1953–1955), where he earned an M.S. degree in mathematics with a dissertation entitled “Reductive Spaces with Absolute Projective Connection,” published later (see [19]). He extended his research in diﬀerential geometry of homogeneous spaces during his doctoral studies (1955–1958) and introduced and studied subsymmetric spaces (spaces with mirrors), one remarkable generalization of the famous symmetric spaces of E. Cartan. It resulted in articles [1], [2] and the doctoral dissertation, “Mirror Symmetries of Riemannian Spaces,” earned at Moscow University in 1959 (see [3]). The motivation for this research was evident: to create, eventually, a universal geometric method for classiﬁcation and exploration of arbitrary homogeneous Riemannian spaces. This method, mirror geometry, was based on the special system of mirrors and was developed, in general, at the Siberian Scientiﬁc Centre, Novosibirsk (1959–1965). In particular, the concept of trisymmetric space was introduced (see [4]–[7]).

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During 1965–1972 (Friendship of Nations University) the research in the mirror (involutive) geometry was continued at the level of Lie algebras. The notions of involutive, isoinvolutive, and hyperinvolutive sums, as well as of principal involutive automorphisms were introduced and studied (papers [10]–[18], [20]). In this way the new powerful method of investigation of Lie algebras and Lie groups was founded. In particular, all trisymmetric spaces with compact simple basic groups were classiﬁed ([17], [18]). On the basis of these results he was awarded a doctorate in mathematics. His dissertation was entitled “Involutive Geometry of Lie Algebras” (Kazan University, Russia, 1971), see [22]. In 1972 Dr. Sabinin started research in a new ﬁeld of mathematics, in the theory of smooth quasigroups and loops. He discovered a remarkable relationship between the quasigroup theory and the group theory. This construction now is known as Baer-Sabinin construction (see [24]–[27]). Analyzing the notion of aﬃne connection, Dr. L. Sabinin introduced the concepts of odule, geodesic odule, geoodular space, and reformulated the geometry of aﬃne connection in the language of smooth loops and odules (nonassociative geometry). This theory allows us to generalize conventional diﬀerential-geometric constructions up to topological, or the discrete level (see [29], [30]). See, also, his fundamental presentation of nonassociative methods in diﬀerential geometry in [33]. Later, Dr. L. Sabinin (jointly with his student P. Miheev) constructed the theory of smooth Bol loops, a remarkable generalization of Lie groups and smooth Moufang loops theories. In particular, the notion of Bol algebra was introduced (see [38], [40], [44], [45], [49]). Further, he introduced and studied the important notion of geometric odule (see [56], [62], [93], [94]). At that time (1987–1988) Professor L. Sabinin started studies in the inﬁnitesimal theory of smooth loops. He introduced the proper inﬁnitesimal object, hyperalgebra, and developed the analog of Lie theory for smooth loops (see [59], [60]). In ﬁnal form he presented this theory in [63], [79], [86], [119]. He wrote a number of up-to-date surveys on the matter of smooth quasigroups, see [64], [66], [67], [72], [121], [126]. In 1990 he introduced the notion and started the studies of transsymmetric spaces (see [69]), one promising generalization of symmetric spaces in the context of smooth quasigroups theory. The complete theory of such spaces was elaborated later in his scientiﬁc school (see [76], [117], [126]). As well, a number of publications of Dr. Lev Sabinin treated remarkable classes of smooth quasigroups and loops: half Bol loops, hyporeductive loops, F-quasigroups, being of value in applications to diﬀerential geometry and mathematical physics (see [105], [111], [73], [77], [78], [88], [89]). Notable relationships between smooth quasigroups and homogeneous spaces were studied in [99], [100]. A serie of publications of Prof. L. Sabinin concerns mathematical physics (gravity, quantum theory, discrete space-time) and introduces some algebraic analog of Einstein equations (see [122], [123], [125], [127], [129]). Some problems of quantum theory in a nonassociative setting were considered in [107]. Among the many mathematical achievements of Professor Sabinin it is worth noting the inﬁnitesimal theory of smooth Bol loops and nonassociative geometry (geoodular spaces of L. Sabinin), the description of diﬀerential-geometric structures in the language of smooth quasigroups, giving, in particular, a new approach to the discrete space-time in gravity. Results of L.V. Sabinin in smooth quasigroups theory have been summarized in the monograph, “Smooth Quasigroups and Loops” (1999), Kluwer Academic Publishers. Dordrecht. The Netherlands [119].

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The results of L.V. Sabinin in mirror geometry have been summarized in the monograph, “Mirror Geometry of Lie Algebras, Lie Groups and Homogeneous Spaces,” 2004, Kluwer Academic Publishers, Dordrecht, The Netherlands [128]. Prof. L. Sabinin designed, delivered and published many original lecture courses in mathematics (see the list of publications). As well, participated in mathematical education at university level and published several papers on the subject. Under his supervision, 11 students earned Ph.D. degrees in mathematics, 4 students earned Ph.D. degrees in mathematics education, and one Ph.D. student earned a Dr.Sci. degree in mathematical physics. Also, he supervised many students from diﬀerent countries who earned M.S. degree (21 students) and B.S. degrees in mathematics (43 students). Books and Contributor Chapters Methods of Nonassociative Algebra in Diﬀerential Geometry. Supplement to Russian translation of S. Kobayashi and K. Nomizu, “Foundations of Diﬀerential Geometry,” Vol. 1, Nauka Press, Moscow, 1981, pp. 293–339. (Russian) MR 84b: 53002. Smooth Quasigroups and Geometry. Problems in Geometry. Vol. 20, 75-110. 1988, VINITI Press, Moscow. Co-author P.O. Miheev (Russian) MR 90b:53063. Quasigroups and Diﬀerential Geometry. Quasigroups and Loops: Theory and Applications. Collective monograph (O. Chein, H. Pﬂugfelder and J.D.H Smith, eds.) Heldermann Verlag, Berlin, 1990. Ch. XII, pp. 357–430. Co-author P.O. Miheev (English). MR 93g:20133. Analytic Quasigroups and Geometry. Monograph. Friendship of Nations University Press. Moscow, 1991 (Russian). MR 95d:53013. Involutive Geometry of Lie algebras. Monograph. Friendship of Nations University, Moscow, 1997 (Russian). Smooth Quasigroups and Loops. Monograph. Kluwer Academic Publishers, Dordrecht, The Netherlands, 1999. ISBN 0-7923-5920-8 (English). Mirror Geometry of Lie Groups, Lie Algebras and Homogeneous Spaces. Monograph. Kluwer Academic Publishers, Dordrecht, The Netherlands, 2004. ISBN 1-4020-2544-0 (HB) (English). Lecture Notes Lectures on Higher Mathematics. Part 1. Novosibirsk University Press. Novosibirsk. 1962. 63 pp. (Russian). Lectures on Higher Mathematics. Part 2. Novosibirsk University Press. Novosibirsk. 1963. 76 pp. (Russian). Fundamental Algebraic Structures. Lecture notes. Friendship of Nations University Press. Moscow. 1979. 85 pp. (Russian). Modules, Matrices, Forms. Lecture notes. Friendship of Nations University Press. Moscow. 1981. 71 pp. (Russian). Aﬃne Plane. Lecture notes. Friendship of Nations University Press. Moscow. 1982. 29 pp. (Russian). Hermitian Forms. Structure of Endomorphisms. Lecture notes. Friendship of Nations University Press. Moscow. 1984. 50 pp. (Russian). The Theory of Smooth Bol Loops. Lecture notes. Friendship of Nations University Press. Moscow. 1985. 81 pp. Co-author P.O. Miheev (Bilingual: in English and in Russian) MR 87j:22030. Zbl. 584:53001. Endomorphisms of Vector Spaces. Structural Theory. Lecture notes. Friendship of Nations University Press. Moscow. 1987. 37 pp. (Russian).

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Mathematical research of Lev Sabinin Translated books

S. Kobayashi, K. Nomizu. Foundations of Diﬀerential Geometry. Vol. 1 (1963), Vol. 2 (1969). Interscience publishers. (English). Russian translation in “Nauka” Press. Moscow. 1981. MR 84c:53001 S. Kobayashi. Transformation Groups in Diﬀerential Geometry. Springer–Verlag. 1972. (English). Russian translation in “Nauka” Press. Moscow. 1986. MR 87g:53001. O. Kowalski. Generalized Symmetric Spaces. Lecture Notes in Mathematics 805. SpringerVerlag. 1980. (English). Russian translation in “Mir” Press. Moscow. 1984. MR 85e:53060. Edited books Mathematics in Concepts, Deﬁnitions and Terms. Vol. 1 (1978) 352 pp. Vol. 2 (1982) 320 pp. Library for Teachers of Mathematics. Prosveshchenie Press. Moscow (Russian) Authors: O.V. Manturov, Yu.K. Solntsev, Yu.I. Sorkin, N.G. Fedin. MR 84a: 00002a, MR 84b: 00002b. Editor of the Proceedings of the V International Conference “Non Associative Algebra and its Applications” (contract with Marcel Dekker) 2005, Taylor and Francis Group, CRC Press, Boca Raton, FL. Translations of Books Edited by Professor Sabinin Marcel Berger (with co-authors). Probl`emes de g´eom´etrie comment´es et r´edi´ ges. CEDIC/FERNAND NATAN (French). Russian translation in “Mir” Press. Moscow. 1989. MR 90c:51001. V. Kharchenko, L. Sbitneva, and I. Shestakov

Publications [1] On the geometry of subsymmetric spaces, Scientiﬁc Reports of Higher School. Ser. Phys.–Math. Sci., 3, 1958, 46–49 (Russian) [2] On the structure of the groups of motions of homogeneous Riemannian spaces with axial symmetry, Scientiﬁc Reports of Higher School. Ser. Phys.–Math. Sci. 6, 1958, 127–138 (Russian) [3] Mirror symmetries of Riemannian spaces, Synopsis of Ph.D. in mathematics dissertation. Moscow University Press, 1959, 7 pp. (Russian). [4] The geometry of homogeneous Riemannian spaces and intrinsic geometry of symmetric spaces, Reports of Ac. of Sci. of USSR (Math.) vol. 129, no. 6, 1959, 1238–1241 (Russian). MR 28#1556 [5] On the explicit expression of forms of connection for quasisymmetric spaces, Reports of Ac. of Sci. of USSR (Math.) 132, no. 6, 1960, 1273–1276 (Russian). MR 22#9941 [6] Some algebraic identities in the theory of homogeneous spaces, Siberian Math. Journal 2, no. 2, 1961, 279–281 (Russian). MR 23#A2482 [7] On the geometry of tri-symmetric Riemannian spaces, Siberian Math. Journal 2, no. 2, 1961, 266–278 (Russian). MR 24#A2350 [8] Lectures on Higher Mathematics, Part 1, Novosibirsk University Press, Novosibirsk, 1962, 63 pp. (Russian).

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[9] Lectures on Higher Mathematics, Part 2, Novosibirsk University Press, Novosibirsk, 1963, 76 pp. (Russian). [10] On isoinvolutive decompositions of Lie algebras, Reports of Ac. of Sci. of USSR (Math.) 165, no. 5, 1965, 1003–1006 (Russian). MR 33#5788 [11] Isoinvolutive decompositions of Lie algebras, Soviet Math. Dokl., Amer. Math. Soc. 165, no. 5, 1965, 1554–1557 (English). MR 33#5788 [12] On principal invomorphisms of Lie algebras, Reports of Ac. of Sci. of USSR 175, no. 1, 1967, 31–33 (Russian). MR 36#224 [13] Principal invomorphisms of Lie algebras, Soviet Math. Dokl., Amer. Math. Soc. 8, 1967, 807–809 (English). MR 36#224 [14] On involutive sums of Lie algebras, Transactions of Sem. on Vector and Tensor Analysis, Moscow University 14, 1968, 94–113 (Russian). MR 40#7392 [15] Involutive duality in simple compact Lie algebras, Proceedings of Geometric seminar. Inst. of Sci. Information of Ac. of Sci. of the USSR 2, 1969, 277–298 (Russian). MR 41#8596 [16] Principal invomorphisms of compact Lie algebras, Transactions of Sem. on Vector and Tensor Analysis, Moscow University 15, 1970, 188–226 (Russian). MR 45#5279 [17] On the classiﬁcation of tri-symmetrical spaces, Reports of Ac. of Sci. of the USSR (Math.) 194, no. 3, 1970, 518–520 (Russian). [18] On the classiﬁcation of tri-symmetrical spaces, Soviet Math. Dokl., Amer. Math. Soc. 11 no. 5, 1970, 1245–1247 (English). [19] Reductive spaces with absolute projective connection, Collection of Math. papers. Friendship of Nations University, Moscow, 1970, 127–139 (Russian). MR 51#4112 [20] Homogeneous Riemannian spaces with (n − 1)-dimensional mirrors, Collection of Math. papers. Friendship of Nations University, Moscow, 1970, 116–126 (Russian). MR 51#4111 [21] On non-classical solutions of Pfaﬀ systems and the geometry of functionally-aﬃne connections, in book: Collection of Math. Papers. Friendship of Nations University, Moscow, 1970, 140–154 (Russian). MR 51#4078 [22] Involutive geometry of Lie algebras, Synopsis of Dr.Sci. in mathematics dissertation, Kazan, 1971, 17 pp. (Russian). [23] Tri-symmetric spaces with simple compact groups of motions, Transactions of Sem. on Vector and Tensor Analysis, Moscow University 16, 1972, 202-226 (Russian). MR 48#1110 [24] On the equivalence of categories of loops and homogeneous spaces, Reports of Ac. of Sci. of the USSR (Math.) 205, no. 3, 1972, 533–536 (Russian). MR 46(1973) # 9220 [25] On the equivalence of categories of loops and homogeneous spaces, Soviet Math. Dokl., Amer. Math. Soc. 13, no. 4, 1972, 970–974 (English). MR 46(1973) # 9220 [26] The geometry of loops, Mathematical Notes, Nauka Press 12, no. 5, 1972, 605–616 (Russian). MR 49(1975) # 5216. Zbl. 258:20006 [27] Loop geometries, Mathematical Notes, Plenum Publishing Company. 12, no. 5, 1972, 799–805 (English). MR 49(1975) # 5216 Zbl. 258:20006 [28] On the existence of half-Bol loops, Ann. sti. ale Univ. Al. I.Cuza. Jasi. Sect. In math 1a 22, no. 2, 1976, 147–148, Romania. Co-author B.L. Sharma. (English). MR 55(1977) # 3132 [29] Odules as a new approach to a geometry with a connection, Reports of Ac. of Sci. of the USSR (Math.) 233, no. 5, 1977, 800–803 (Russian). MR 57(1979)#1340

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[30] Odules as a new approach to a geometry with a connection, Soviet Math. Dokl., Amer. Math. Soc. 18, no. 2, 515–518, 1977, (English). MR 57(1979) # 1340 [31] Fundamental algebraic structures, Lecture notes. Friendship of Nations University Press, Moscow, 1979, 85 pp. (Russian). [32] Modules, matrices, forms, Lecture notes. Friendship of Nations University Press, Moscow, 1981, 71 pp. (Russian). [33] Methods of Nonassociative Algebra in Diﬀerantial Geometry, Supplement to Russian translation of S. Kobayashi and K. Nomizu “Foundations of Diﬀerential Geometry” vol. 1, Nauka Press, Moscow, 1981, pp. 293–339. (Russian). MR 84b: 53002 [34] Successful text-book for future mathematicians, Herald of Higher School, Moscow 10, 1981, 79–80 (Russian). [35] On a symmetric connection in the space of an analytic Moufang loop, Reports of Ac. of Sci. of the USSR (Math.) 262, no. 4, 1982, 807–809. (Russian). Co-author P.O. Miheev. MR 84e:53027 [36] On a symmetric connection in the space of an analytic Moufang loop, Soviet Math. Dokl., Amer. Math. Soc. 25, no. 1, 1982, 136–138 (English). Co-author P.O. Miheev. MR 84e:53027 [37] Aﬃne plane, Lecture notes. Friendship of Nations University press, Moscow, 1982, 29 pp. (Russian). [38] On analytic Bol loops, Webs and Quasigroups, Kalinin University press, 1982, 102–109 (Russian). Co-author P.O. Miheev. MR 84c:22007. Zbl. 499:20044 [39] On the geometry of almost symmetric spaces, in book: Applied problems of Diﬀerential Geometry, Moscow District Pedagogical Institute. Dep No 1648–82. Institute of Sci. Information, Ac. of Sci. of the USSR (VINITI Press) Moscow, 1982, 14–15 (Russian). [40] On the geometry of smooth Bol loops, Webs and Quasigroups, Kalinin University press, 1984, 144–154 (Russian). Co-author P.O. Miheev. Zbl. 568.53009, MR 88c:53001 [41] On the canonical reductants of spaces with constant curvature, Webs and Quasigroups, Kalinin University Press, 1984, 76–83 (Russian). Co-author S.S. Yantranova. MR 88c:53001 [42] Hermitian forms. Structure of endomorphisms, Lecture notes. Friendship of Nations University press, Moscow, 1984, 50 pp. (Russian). [43] Prominent Mathematics Educator. (On 90 birthday anniversary of I.K. Andronov.), National Education 9, 1984 (Russian). Co-authors O.V. Manturov, V.N. Shapkina. [44] On the diﬀerential geometry of Bol loops, Reports of Ac. of Sci. of the USSR (Math.) 281, no. 5, 1985, 1055-1057 (Russian). Co-author P.O. Miheev. MR 86k: 53022. Zbl. 587:53021 [45] On the diﬀerential geometry of Bol loops, Soviet Math. Dokl., Amer. Math. Soc. 31, no. 2, 1985, 389–391 (English). Co-author P.O. Miheev. MR 86k: 53022. Zbl. 587:53021 [46] On local analytic loops with the right alternative identity, Problems of the theory of Webs and Quasigroups, Kalinin University press, 1985, 72–75 (Russian). Co-author P.O. Miheev. Zbl. 572.20056. MR 88d: 22006 [47] On the structure of T,U,V-isospin in the theory of higher symmetry, in book: Some applications of diﬀerential geometry, Moscow District Pedagogical Institute. Dep No 4531–85. Institute of Sci. Information, Ac. of Sci. of the USSR (VINITI Press) Moscow, 1985, 21–30 (Russian).

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[48] On canonical base of algebra g2 , in book: Some applications of diﬀerential geometry, District Pedagogical Institute. Dep No 4531–85. Institute of Sci. Information, Ac. of Sci. of the USSR (VINITI Press) Moscow, 1985, 31–42 (Russian). [49] The Theory of Smooth Bol loops, Lecture notes. Friendship of Nations University press, Moscow, 1985, 81 pp. (Bilingual: in English and in Russian). Co-author P.O. Miheev. MR 87j:22030. Zbl. 584:53001 [50] In memory of Petr Konstantinovich Rashevskii (1907–1983). Obituary, Transactions of Sem. on Vect. and Tens. Analysis, Moscow University 22, 1985, 4–5 (Russian). Co-authors: O.V. Manturov, S.P. Novikov, V.V. Troﬁmov, A.T. Fomenko MR 90m: 01074 [51] Tangent aﬃne connections of loopuscular structures, Webs and Quasigroups, Kalinin University Press, 1986, 86–89 (Russian). MR 88m: 53043 [52] On the theory of special smooth loops, in book: Principle of inclusion and invariant tensors, Moscow District Pedagogical Institute. Dep. No 426–B86 Institute of Sci. Information, Ac. of Sci. of the USSR (VINITI Press) Moscow, 1986, 113–117 (Russian). [53] On local analytic loops and their corresponding hyperalgebras, in book: Proceedings of 9 conference of young researchers (Friendship of Nations University, 1986) Dep. No 6848–B86, Institute of Sci. Information, Ac. of Sci. of the USSR (VINITI Press) Moscow, 1986, 34–54 (Russian). Co-author P.O. Miheev. [54] On the postgraduate studies in the Institutes of Ministry of Education, in book: On teaching Geometry at Educational Institutes, Methodical instructions. Ministry of Education of the USSR Press, Moscow, 1986, 11–12 (Russian). [55] On the identity of quasi–linearity in diﬀerentiable linear geodiodular manifolds in book: Invariant tensors, Moscow District Pedagogical Institute. Dep. No 6553-B86, Institute of Sci. Information, Ac. of Sci. of the USSR (VINITI Press) Moscow, 1986, 9–18 (Russian). Co-authors O.A. Matveev, S.S. Yantranova. [56] Geometric odules, Webs and Quasigroups, Kalinin University press, 1987, 88–98 (Russian). Zbl. 637.53014. MR 88i:53041 [57] Problems in Geometry (review), New foreign books, ser A, Mir Press 2, 1987, 19–20 (Russian). [58] Endomorphisms of vector spaces. Structural theory, Lecture notes. Friendship of Nations University Press, Moscow, 1987, 37 pp. (Russian). [59] On the inﬁnitesimal theory of local analytic loops, Reports of Ac. of Sci. of the USSR (Math.) 297, no. 4, 1987, 801–804 (Russian). Co-author P.O. Miheev. MR 89g:22003. Zbl. 659:53018 [60] On the inﬁnitesimal theory of local analytic loops, Soviet Math. Dokl., Amer. Math. Soc. 36, no. 3, 1988, 545–548 (English). Co-author P.O. Miheev. MR 89g:22003. Zbl. 659:53018 [61] On Nonlinear Geometric Algebra, Webs and Quasigroups, Kalinin University press, 1988, 32–37 (Russian). MR 89e: 20121 [62] On the problem of universality of the Identity of Geometricity, Webs and Quasigroups, Kalinin University Press, 1988, 84–87 (Russian). Co-author A.M. Shelehov MR 89f: 22004 [63] Diﬀerential equations of smooth loops, Transactions of Sem. on Vector and Tensor Analysis, Moscow University 23, 1988, 133-146 (Russian). MR 91h: 53002 [64] Smooth Quasigroups and Geometry, Problems in Geometry, VINITI Press, Moscow, 20, 1988, 75–110 (Russian). Co-author P.O. Miheev. MR 90b:53063 [65] In memory of Anatolii Mikhailovich Vasiliev (1923–1987). Obituary, Transactions

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[67]

[68] [69] [70] [71]

[72]

[73] [74]

[75]

[76]

[77]

[78] [79] [80] [81]

Mathematical research of Lev Sabinin of Sem. on Vect. and Tens. Analysis, Moscow University, 23, 1988, 5–6 (Russian). Co-authors: O.V. Manturov, S.P. Novikov, V.V. Troﬁmov, A.T. Fomenko Diﬀerential geometry and Quasigroups, Transactions of Institute of Mathematics. Siberian branch of Ac. of Sci. of the USSR 14, 1989, 208–221 (Russian). Nauka Press. Novosibirsk. MR 91f: 53015 Algebraic structures of Nonlinear Geometric Algebra, Quasigroups and systems of quasigroups. Mathematical explorations 113, 1990, 83–88 (Russian). Stiinca Press. Kishinev. MR 91f:53016 On the inﬁnitesimal theory of smooth Hyporeductive Loops, Webs and Quasigroups, Kalinin University Press, 1990, 33–39 (Russian). MR 91e:53003 On the geometry of Transsymmetric spaces, Webs and Quasigroups, Kalinin University press, 1990, 64–68 (Russian). Co-author L.L. Sabinina. MR 91f: 53046 Reductive spaces and left F-quasigroups, Webs and Quasigroups Kalinin University Press, 1990, 89–94 (Russian). Co-author L.V. Sbitneva. MR 91h: 20093 Smooth hyporeductive loops, Variational methods in contemporary geometry. Collection of papers, Friendship of Nations University Press, Moscow, 1990, 50–69 (Russian). MR 92g: 22008 MR 92d:00023 Quasigroups and Diﬀerential Geometry, in book: Quasigroups and Loops: Theory and Applications. Collective monograph, (O. Chein, H. Pﬂugfelder and J.D.H. Smith, eds.) Heldermann Verlag, Berlin, 1990, Ch. XII, pp. 357–430 (English). Co-author P.O. Miheev. MR 93g:20133 On smooth hyporeductive loops, Reports of Ac. of Sci. of the USSR (Math.) 314, no. 3, 1990, 565–568 (Russian). MR 92d: 22002 Quasigroups, Geometry and Physics, Proceedings of International meeting “Quasigroups and nonassociative algebras in Physics” Transactions of Institute of Physics. Estonian Ac. of Sci. 66, 1990, 24–53 (English). Tartu, Estonia. MR 93d: 20125 Smooth hyporeductive loops, Proceedings of the International Conference “Nonlinear Geometric Algebra–89”. (Friendship of Nations University. Moscow. January 24–31, 1989), Webs and Quasigroups, Tver University press, 1991, 129–137 (English). MR 92f:53003 On geometry of transsymmetric spaces, Proceedings of the International Conference “Nonlinear Geometric Algebra–89.” (Friendship of Nations University. Moscow. January 24–31, 1989), Webs and Quasigroups, Tver University Press, 1991, 117–122 (English). Co-author L.L. Sabinina. MR 92f:53003 Reductive spaces and left F-quasigroups, Proceedings of the International Conference “Nonlinear Geometric Algebra–89” (Friendship of Nations University. Moscow. January 24–31, 1989), Webs and Quasigroups, Tver University Press, 1991, 123–128 (English). Co-author L.V. Sbitneva. MR 92f:53003 On smooth hyporeductive loops, Soviet Math. Dokl., Amer. Math. Soc. 42, no. 2, 1991, 524–526 (English). MR 92d:22002 Analytic Quasigroups and Geometry, Monograph. Friendship of Nations University Press, Moscow, 1991, 112 pp. (Russian). MR 95d:53013 On the theory of ϕ-spaces, Transactions of Sem. on Vector and Tensor Analysis, Moscow University, 24, 1991, 180-185 (Russian). Smooth Quasigroups and Loops. New results, in book: Contemporary Mathematics. Proceedings of the International Conference on Algebra (dedicated to the Memory of A.I. Malcev) (L. Bokut, Yu. Ershov, O. Kegel, A. Kostrikin, eds.), 131 (Part 1), 1992, 707–712 (English). American Math. Soc. MR 93i:22002

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[82] On ﬂat geoodular spaces, Webs and Quasigroups, Tver University Press, 1992, 4–9 (English). MR 94f:53023 [83] On the law of addition of velocities in Special Relativity, Advances in Mathematical Sciences, Moscow Math. Soc., 48, no. 5, 1993, 183–184 (Russian). Co-author P.O. Miheev. MR 95a:83011 [84] On the law of composition of velocities in Special Relativity theory, Russian Mathematical Survey, London Math. Soc., 48, no. 5, 1993, 183–184 (English). Co-author P.O. Miheev. MR 95a:83011 [85] On diﬀerential equations of Smooth Loops, Advances in Mathematical Sciences, Moscow Math. Soc. 49, no. 2, 1994, 165–166 (Russian). MR 95f:22009 [86] On diﬀerential equations of smooth loops, Russian Mathematical Survey, London Math. Soc. 49, no. 2, 1994, 172–173 (English). MR 95f:22009 [87] Almost Symmetric and Antisymmetric spaces of Aﬃne connection, Reports of Russian Ac. of Sci. (Math.), 337, no. 4, 1994, 454–455 (Russian). Co-author P.O. Miheev MR 95g:53016 [88] On Left Loops with the second Half Bol Property, Webs and Quasigroups, Tver University Press, 1994, 55–56, (English). Co-author Yu.A. Selivanov. MR 97e:53022 [89] Half Bol Loops, Webs and Quasigroups, Tver University Press, 1994, 50–54 (English). Co-author L.V. Sbitneva. MR 97e:53022 [90] Geoodular axiomatics of Aﬃne spaces, Note di Matematica, University ‘Degli Studi Di Lecce’, 14, no. 1, 1994, 109–113 (English). Italy. MR 98c:20121 [91] On gyrogroups of A. Ungar, Advances in Math. Sciences, Moscow Math. Soc. 50, no. 5, 1995, 251–252 (Russian). [92] On gyrogroups of A. Ungar, Russian Mathematical Survey, London Math. Soc., 50, no. 5, 1995, 1095–1096 (English). [93] Diﬀerential equations of Geometric Odule, Reports of Russian Ac. of Sci. (Math.) 344, no. 6, 1995, 745–748 (Russian). MR 96m:53033 [94] Diﬀerential equations of Geometric Odule, Doklady Mathematics, 52, no. 2, 1995, 268–270 (English). Interperiodica Publishing. MR 96m:53033 [95] On the theory of Left F-quasigroups, Algebras, Groups and Geometries, Hadronic Press. 12, 1995, 127–137 (English). Co-author L.L. Sabinina. MR 96d:20066 [96] Geodesic Loops and some classes of Aﬃnely connected manifolds (Survey on Odular geometry), Herald of Friendship of Nations University (Math.), 2, no. 1, 1995, 135–143 (English). Co-author O.A. Matveev. [97] On the structure of T, U, V isospins in the theory of Higher Symmetry, Herald of Friendship of Nations University (Math.), 2, no. 1, 1995, 130–134 (English). [98] The theory of Smooth Hyporeductive and Pseudoreductive Loops, Algebras, Groups and Geometries, Hadronic Press, 13, no. 1, 1996, 1–24 (English). MR 97c:22003 [99] Homogeneous spaces and Quasigroups, Communications of Higher School (Math.), Kazan University Press, no. 7, 1996, 77–84 (Russian). MR 97k:53053. Zbl. 874:53037 [100] Homogeneous spaces and Quasigroups, Russian Mathematics (Iz. VUZ Matematica), Plenum Publishers, USA, 40, no. 7, 1996, 74–81 (English). MR 97k:53053. Zbl. 874:53037 [101] On diassociativity of smooth monoalternative loops, Advances in Mathematical Sciences, Moscow Math. Soc., 51 no. 4, 1996, 747–749 (Russian). MR 98d:22003 [102] On diassociativity of smooth monoalternative loops, Russian Mathematical Survey, London Math. Soc., 51, no. 4, 1996 (English). MR 98d:22003

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[103] Left Quasigroups and Reductive spaces, Algebras, Groups and Geometries, Hadronic Press. USA, 13, 1996, 479–487 (English). Co-author L.V. Sbitneva. MR 97k:20119 [104] On the inﬁnitesimal theory of Reductive loops, Reports of Russian Ac. of Sci. (Math.), 353, no. 1, 1997, 26–28 (Russian). [105] An Inﬁnitesimal Theory of Reductive Loops, Doklady Mathematics, 55, no. 2, 1997, 185–187, Interperiodica Publishers (English). [106] Smooth Loops and Thomas precession, Hadronic Journal, 20, 1997, 219–237 (English) Hadronic Press. Co-author: A.I. Nesterov [107] Smooth Loops, generalized coherent states and geometric phases, International Journal of Theoretical Physics, 36, no. 9, 1997, 1981–1989 (English). Plenum Publishing Company. Co-author: A.I. Nesterov. MR 98i:81095 [108] Involutive Geometry of Lie algebras, Monograph. Friendship of Nations University, Moscow, 1997, 336pp. (Russian). [109] Undergraduate Geometry: new approach, in book: Proceedings of 6-th International Symposium in Mathematics Education “Elfriede Wenzelburger” (UNAM. Mexico. October 13–15. 1997), 1997, 161–164, Grupo Editorial Iberoam´erica, Mexico. [110] On C 1 -smooth commutative Moufang loops and distributive quasigroups, Webs and Quasigroups, Tver University Press, 1996/97, 86–88 (English). Co-authors L.L. Sabinina, R. Jimenez. [111] On the notion of Gyrogroup, Aequationes Mathematicae, Birkh¨ auser, Basel, Switzerland, 56, no. 1–2, 1998, 11–17 (English). Co-authors: L.L. Sabinina, L.V. Sbitneva. MR 99i:83004 [112] Quasigroups, Geometry and Nonlinear Geometric Algebra, Acta Applicandae Mathematicae, Kluwer Academic Publishers, 50, 1998, 45–66 (English). MR 99h:20106 [113] Smooth Loops I, Algebras, Groups and Geometries, Hadronic Press. 15, no. 2, 1998, 127–153 (English). [114] The problems of Mathematics Education at Universities with international contingent of students, in book: Proceedings of the International conference “Function spaces. Diﬀerential operators. Problems of Mathematical Education” dedicated to the 75th birthday of Corresponding Member of Russian Ac. Sci. Prof. L.D. Kudryavtsev, Russian Friendship University Press, Moscow , 1998, 3, 129–130 (English). [115] On Bol-Bruck loops, Webs and Quasigroups, Tver University Press, 1998–99, 106–108 (English). Co-authors L.L. Sabinina. [116] Generalized Bol loops, Herald of Friendship of Nations University (Math.), 6, no. 1, 1999, 182–190 (English). Co-author C. Castillo. [117] Perfect transsymmetric spaces, Publicaciones Mathematicae, Debrecen. Hungary. 54, no. 3–4, 1999, 303–311, (English). Co-authors: L.L. Sabinina, L.V. Sbitneva. [118] Methods of Non-associative algebra in Diﬀerential geometry, in book: Diﬀerential Geometry and Applications. Proceedings of the 7-th International Conference on Differential Geometry and Applications (Satellite Conference of ICM in Berlin. Brno. Czech Republic. August 10–14, 1998) Masaryk University. Brno Czech Republic, 1999, 419–427 (English). [119] Smooth Quasigroups and Loops, Monograph. Kluwer Academic Publishers, Dordrecht. The Netherlands, 1999, xvi+250. ISBN 0-7923-5920-8 (English). [120] Smooth quasigroups and loops. Recent achievements and open problems, Nonassociative algebra and its Applications. (Eds. R. Costa, A. Grishkov, H. Guzzo, L. Peresi) in book: Lecture Notes in Pure and Applied Mathematics, Marcel Dekker. USA, 211, 2000, 337–344 (English).

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[121] Smooth Quasigroups and loops: Forty-ﬁve years of incredible growth, Proceedings of the International Conference “Loops–99” (Prague. Czech Republic. July 27–August 1, 1999), Commentationes Mathematicae Universitatis Carolinae, 41, no. 2, 2000, 377–400 (English). [122] Non-associative geometry and discrete structure of space-time, Commentationes Mathematicae Universitatis Carolinae, 41, no. 2, 2000, 347–357 (English). Czech Republic. Co-author A.I. Nesterov. [123] Nonassociative geometry: Towards discrete structure of spacetime, Physical Review D 62, 2000, 081501 (R), 081501-1–081501-5. Rapid communications. American Physical Society. (English). Co-author A.I. Nesterov. [124] Analytic decompositions of smooth Bol-Bruck loops, Webs and Quasigroups, Tver University Press, 2000, 94–97, (English). Co-autor L.V. Sbitneva. [125] Nonassociative Geometry and Discrete Space-Time, International Journal of Theoretical Physics, Plenum Publishing Company. 40, no. 1, 2001, 351–358 (English). [126] From symmetric to transsymmetric spaces, Diﬀerential Geometry and Applications (DGA), Elsewier, 2002, 197–208 (English). Co-authors: L.L. Sabinina, L.V. Sbitneva. [127] Loop-theoretic foundations of Diﬀerential Geometry and Relativity, Webs and Quasigroups, Tver University Press, 2002, 67–72 (English). [128] Mirror Geometry of Lie Groups, Lie Algebras and Homogeneous Spaces, Monograph. Kluwer Academic Publishers, Dordrecht. The Netherlands. 2004, xvii+312 (English). ISBN 1-4020-2544-0 (HB) [129] Nonassociative geometry: Friedmann-Robertson-Walker universe, hep-th/0406229.

CONTRIBUTORS

Ferm´ın Aceves de la Cruz

Alberto Elduque

Physics Department, Guadalajara University Guadalajara, Jalisco, Mexico

Departamento de Matem´ aticas Universidad de Zaragoza, Spain

Yu.A. Bahturin

V.T. Filippov

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

Mathematics Institute, Siberian Branch of the Russian Academy of Sciences, Novosibirsk

L.Yu. Glebsky L. Bokut Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences, Novosibirsk

Universidad de San Luis Potosi M´ exico

E.I. Gordon

Thomas B. Bouetou

Universidad de San Luis Potosi M´ exico

´ Ecole Nationale Sup´erieure Polytechnique Yaound´ e, Cameroon

H. Guzzo Jr.

Murray R. Bremner Research Unit in Algebra and Logic University of Saskatchewan, Canada

Universidade de S˜ ao Paulo, Instituto de Matem´ atica e Estat´ıstica, S˜ ao Paulo, Brazil

Irvin Roy Hentzel Antonio J. Calder´ on Mart´ın Departamento de Matem´ aticas Universidad de C´ adiz Spain

Department of Mathematics Iowa State University, Ames

A. Janˇ caˇ r´ık

Ramiro Carrillo-Catal´ an

Charles University Prague, Czech Republic

Department of Mathematics, Graduate School of Science, Tokyo Metropolitan University

Noriaki Kamiya

E.S. Chibrikov

Center for Mathematical Sciences University of Aizu, Japan

Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences, Novosibirsk

Jos´ e de Jes´ us Cruz Guzm´ an Universidad Nacional Aut´ onoma de M´exico Facultad de Estudios Superiores Cuautitl´ an

Ivan Correa

T. Kepka Charles University Prague, Czech Republic

V.K. Kharchenko Universidad Nacional Aut´ onoma de M´exico Facultad de Estudios Superiores Cuautitl´ an

Departamento de Matem´ atica Universidad Metropolitana de Ciencias de la Educaci´ on, Santiago, Chile

Jaak L˜ ohmus

Piroska Cs¨ org¨ o

Yu. Lyubich

Department of Algebra and Number Theory E¨ otv¨ os Lor´ and University, Budapest, Hungary

Department of Mathematics Technion–Israel Institute of Technology, Haifa

Valeri V. Dvoeglazov

O.A. Matveyev

Universidad de Zacatecas M´ exico

Institute of Physics University of Tartu, Estonia

Moscow State Regional University, Russia

xxv

xxvi C´ andido Mart´ın Gonz´ alez

Elisabeth Remm

Departamento de Algebra, Geometr´ıa y Topolog´ıa, Universidad de M´ alaga Spain

Laboratoire de Math´ematiques et Applications Universit´ e de Haute, France

M. Rosenbaum Kevin McCrimmon Department of Mathematics University of Virginia, Blacksburg

Instituto de Ciencias Nucleares Universidad Nacional Aut´ onoma de M´exico

C.J. Rubio Kurt Meyberg Zentrum Mathematik der Technischen Universit¨ at Munich, Germany

Guillermo Moreno Departamento de Matem´ aticas, CINVESTAV del IPN, M´ exico, D.F.

Jacob Mostovoy Instituto de Matem´ aticas (Unidad Cuernavaca) Universidad Nacional Aut´ onoma de M´exico

E.L. Nesterenko Friendship of Nations University Russia

Universidad de San Luis Potosi M´ exico

Lev V. Sabinin Universidad Autonoma del Estado de Morelos Cuernavaca, M´exico

Liudmila Sabinina Facultad de Ciencias Universidad Aut´ onoma del Estado de Morelos, M´ exico

Larissa V. Sbitneva Universidad Autonoma del Estado de Morelos Cuernavaca, Mexico

I.P. Shestakov Alexander I. Nesterov Department of Physics, Guadalajara University Guadalajara, Mexico

Mathematics Institute, Siberian Branch of the Russian Academy of Sciences, Novosibirsk

A.I. Shirshov Susumu Okubo Department of Physics and Astronomy University of Rochester, New York

Mathematics Institute, Siberian Branch of the Russian Academy of Sciences, Novosibirsk

Leo Sorgsepp Zbigniew Oziewicz Universidad Nacional Aut´ onoma de M´exico Facultad de Estudios Superiores Cuautitl´ an, M´ exico

Eugen Paal Department of Mathematics Tallinn University of Technology, Estonia

M.M. Parmenter

Tartu Observatory, T˜ oravere Estonia

Earl J. Taft Department of Mathematics, Rutgers University Piscataway, New Jersey

T.V. Tvalavadze Department of Mathematics and Statistics Memorial University of Newfoundland, Canada

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

P. Vicente

Luiz Antonio Peresi

G.P. Wene

Departamento de Matem´ atica Universidade de S˜ ao Paulo, Brazil

Department of Applied Mathematics The University of Texas at San Antonio

C´ esar Polcino Milies

Hao Zhifeng

Instituto de Matem´ atica e Estat´ıstica Universidade de S˜ ao Paulo, Brazil

Department of Applied Mathematics South China University of Technology,

J.L. Quintanar Gonz´ alez Universidad de Zacatecas M´ exico

Universidad de Le´ on, Campus de Vegazana Departamento de Matem´ aticas, Le´ on, Spain

Contents

Preface

ix

Lev Vasilievich Sabinin

xi

Mathematical research of Professor Lev Sabinin

xiii

1 Inﬁnite-dimensional representations of the rotation group and magnetic monopoles Ferm´ın Aceves de la Cruz and Alexander I. Nesterov 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Indeﬁnite metric Hilbert space . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Inﬁnite-dimensional representations of the rotation group . . . . . . . . . . 1.4 Quantum mechanics of Dirac monopole with arbitrary magnetic charge . . 1.4.1 Representation unbounded from above and below . . . . . . . . . . . 1.4.2 Representations bounded above or below . . . . . . . . . . . . . . . . 1.5 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6 Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Generalized Lie nilpotence in integral group rings Yu.A. Bahturin and M.M. Parmenter 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . 2.2 Preliminary results . . . . . . . . . . . . . . . . . . 2.3 Nilpotence . . . . . . . . . . . . . . . . . . . . . . . 2.4 Solvability . . . . . . . . . . . . . . . . . . . . . . . 2.5 Acknowledgments . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . .

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and free Lie algebras

17

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17 18 23 26 27 29 29 34 34

4 Classiﬁcation of solvable 3-dimensional Lie triple systems Thomas B. Bouetou 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

41

3 Lyndon-Shirshov words, Gr¨ obner-Shirshov bases, L. Bokut and E.S. Chibrikov 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . 3.2 A short history of free Lie algebras . . . . . . . . 3.3 A right-normed basis of a free Lie algebra . . . . . 3.4 Another bracketing of Lyndon-Shirshov words . . 3.5 A right-normed basis of a free Lie superalgebra . . 3.6 Free Lie conformal algebras . . . . . . . . . . . . . 3.7 Simple Lie algebras . . . . . . . . . . . . . . . . . 3.8 Acknowledgments . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . .

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xxviii 4.2

4.3 4.4 4.5 4.6

Lie triple systems . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 Solvable and semisimple Lie triple system . . . . . . 4.2.2 Problem setting . . . . . . . . . . . . . . . . . . . . . Classiﬁcation of Lie triple system of dimension 2 . . . . . . Classiﬁcation of solvable Lie triple systems of dimension 3 Classiﬁcation of splitting tree-dimensional Lie triple systems Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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5 Alternating triple systems with simple Lie algebras of derivations Murray R. Bremner and Irvin R. Hentzel 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Representations of the Lie algebra sl(2) . . . . . . . . . . . . . . . . . 5.2.1 The Lie algebra sl(2) . . . . . . . . . . . . . . . . . . . . . . . . 5.2.2 The irreducible representation V (n) . . . . . . . . . . . . . . . 5.2.3 Exterior cubes . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.4 Relation between binary and ternary structures . . . . . . . . . 5.3 General multiplicity formula . . . . . . . . . . . . . . . . . . . . . . . 5.4 Ternary operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Representation V (3): Dimension 4 . . . . . . . . . . . . . . . . . . . . 5.5.1 Identities of degree 5 . . . . . . . . . . . . . . . . . . . . . . . . 5.5.2 Identities of degree 7 . . . . . . . . . . . . . . . . . . . . . . . . 5.5.3 Another alternating triple system of dimension 4 . . . . . . . . 5.6 Representation V (5): Dimension 6 . . . . . . . . . . . . . . . . . . . . 5.6.1 The V (9) summand . . . . . . . . . . . . . . . . . . . . . . . . 5.6.2 The V (5) summand . . . . . . . . . . . . . . . . . . . . . . . . 5.6.3 The V (3) summand . . . . . . . . . . . . . . . . . . . . . . . . 5.6.4 The change of basis matrix . . . . . . . . . . . . . . . . . . . . 5.6.5 The structure constants for the alternating triple system . . . . 5.6.6 Identities of degree 5 . . . . . . . . . . . . . . . . . . . . . . . . 5.6.7 Identities of degree 7 . . . . . . . . . . . . . . . . . . . . . . . . 5.7 Representation V (6): Dimension 7 . . . . . . . . . . . . . . . . . . . . 5.7.1 Structure constants . . . . . . . . . . . . . . . . . . . . . . . . . 5.7.2 Identities of degrees 5 and 7 . . . . . . . . . . . . . . . . . . . . 5.8 Representation V (7): Dimension 8 . . . . . . . . . . . . . . . . . . . . 5.8.1 Structure constants . . . . . . . . . . . . . . . . . . . . . . . . . 5.8.2 Identities of degree 5 . . . . . . . . . . . . . . . . . . . . . . . . 5.8.3 Identities of degree 7 . . . . . . . . . . . . . . . . . . . . . . . . 5.9 Representation V (8): Dimension 9 . . . . . . . . . . . . . . . . . . . . 5.9.1 Structure constants for the triple system . . . . . . . . . . . . . 5.9.2 Identities of degree 5 . . . . . . . . . . . . . . . . . . . . . . . . 5.9.3 Identities of degree 7 . . . . . . . . . . . . . . . . . . . . . . . . 5.10 Representation V (10): Dimension 11 . . . . . . . . . . . . . . . . . . 5.10.1 Structure constants for the triple system . . . . . . . . . . . . . 5.10.2 Identities of degree 5 . . . . . . . . . . . . . . . . . . . . . . . . 5.10.3 Identities of degree 7 . . . . . . . . . . . . . . . . . . . . . . . . 5.10.4 Open problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.11 Other simple Lie algebras . . . . . . . . . . . . . . . . . . . . . . . . . 5.11.1 Tensor products . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.11.2 Exterior cubes . . . . . . . . . . . . . . . . . . . . . . . . . . .

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56 57 57 57 57 58 59 62 63 64 64 65 66 66 67 68 68 69 70 70 71 71 72 72 73 73 73 73 74 75 75 75 76 77 78 78 78 78 78 79

xxix 5.11.3 Special linear: type A . 5.11.4 Orthogonal: type B . . . 5.11.5 Symplectic: type C . . . 5.11.6 Orthogonal: type D . . 5.11.7 Exceptional: types E, F, 5.12 Acknowledgments . . . . . . . References . . . . . . . . . . . .

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6 The Lie group S3 in absolute valued structures Antonio J. Calder´ on Mart´ın and C´ andido Mart´ın Gonz´ alez 6.1 Introduction and preliminaries . . . . . . . . . . . . . 6.1.1 On absolute valued algebras . . . . . . . . . . . 6.1.2 On absolute valued triple systems . . . . . . . 6.2 Main results . . . . . . . . . . . . . . . . . . . . . . . 6.2.1 Previous works . . . . . . . . . . . . . . . . . . 6.2.2 A classiﬁcation of four-dimensional a.v.t.s. . . . 6.2.3 The conjugation techniques . . . . . . . . . . . 6.3 Acknowledgments . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . 7 The theory of Kikkawa spaces Ramiro Carrillo-Catal´ an and Liudmila Sabinina 7.1 Introduction . . . . . . . . . . . . . . . . . . . 7.2 Preliminaries and deﬁnitions . . . . . . . . . . 7.3 Some geometric properties of a Kikkawa space 7.4 Comparison with other results . . . . . . . . . 7.5 Acknowledgments . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . 8 Flexible right-nilalgebras satisfying Ivan Correa 8.1 Introduction . . . . . . . . . . . . 8.2 Notation and preliminary results . 8.3 Proof of the main result . . . . . . 8.4 Acknowledgments . . . . . . . . . References . . . . . . . . . . . . . .

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9 Nijenhuis-Richardson algebra and Fr¨ olicher-Nijenhuis Jos´e de Jes´ us Cruz Guzm´ an and Zbigniew Oziewicz 9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 9.2 Universal Grassmann module . . . . . . . . . . . . . . . 9.3 Leibniz-Loday and Gerstenhaber algebra . . . . . . . . 9.4 Fr¨ olicher and Nijenhuis decomposition . . . . . . . . . . 9.4.1 Universal property of derivation . . . . . . . . . . ´ 9.4.2 Slebodzi´ nski-Lie derivation . . . . . . . . . . . . 9.4.3 Fr¨ olicher and Nijenhuis decomposition . . . . . . 9.5 Main deﬁnition . . . . . . . . . . . . . . . . . . . . . . . 9.5.1 Fr¨ olicher-Nijenhuis Lie M ∧ -module . . . . . . . . 9.5.2 Consequence: module derivation . . . . . . . . . 9.6 Bianchi identity . . . . . . . . . . . . . . . . . . . . . . 9.7 Frobenius algebra . . . . . . . . . . . . . . . . . . . . . 9.8 Frobenius subalgebra of Nijenhuis-Richardson algebra .

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xxx 9.9 Frobenius algebra of two idempotents 9.10 Conclusion . . . . . . . . . . . . . . . 9.11 Acknowledgments . . . . . . . . . . . References . . . . . . . . . . . . . . . . 10 Generalized capable abelian groups Piroska Cs¨ org¨ o, A. Janˇcaˇr´ık and T. Kepka 10.1 Introduction . . . . . . . . . . . . . . 10.2 Preliminaries . . . . . . . . . . . . . . 10.3 Technical results (a) . . . . . . . . . . 10.4 Technical results (b) . . . . . . . . . . 10.5 Main results . . . . . . . . . . . . . . 10.6 Acknowledgments . . . . . . . . . . . References . . . . . . . . . . . . . . . .

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11 Helicity basis for spin 1/2 and 1 Valeri V. Dvoeglazov and J. L. Quintanar Gonz´ alez 11.1 Introduction . . . . . . . . . . . . . . . . . . . 11.2 The (1/2, 0) ⊕ (0, 1/2) case . . . . . . . . . . . 11.3 The (1, 0) ⊕ (0, 1) case . . . . . . . . . . . . . . 11.4 Conclusions . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . .

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12 A New Look at Freudenthal’s magic square Alberto Elduque 12.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2 Symmetric composition algebras and Freudenthal’s magic square 12.3 Okubo algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.4 Invariant form . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.5 Real forms of the simple exceptional Lie algebras . . . . . . . . . 12.6 Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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13 A survey on approximation of locally compact groups by ﬁnite groups, semigroups and quasigroups 167 L.Yu. Glebsky, E.I. Gordon, and C.J. Rubio 13.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 13.2 Approximation of locally compact groups by ﬁnite groups . . . . . . . . . . 169 13.2.1 Positive results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169 13.2.2 Negative results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 13.3 Approximation of locally compact groups by ﬁnite semigroups and quasigroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177 13.3.1 Approximation by ﬁnite semigroups and its application . . . . . . . 177 13.3.2 Approximation of locally compact groups by ﬁnite loops . . . . . . . 179 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181 14 The transformation algebras of Bernstein H. Guzzo Jr. and P. Vicente 14.1 The Bernstein graph algebra . . . . . . . 14.2 The associative algebra Γ(G) . . . . . . . 14.3 The associative algebra T (G) . . . . . . . 14.4 Acknowledgments . . . . . . . . . . . . .

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xxxi References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 Central elements of minimal degree in Irvin Roy Hentzel and Luiz Antonio Peresi 15.1 Introduction . . . . . . . . . . . . . . 15.2 Representation techniques . . . . . . 15.3 Proof of Theorem 15.1 . . . . . . . . 15.4 Annihilators . . . . . . . . . . . . . . 15.5 Discussion . . . . . . . . . . . . . . . 15.6 Further work . . . . . . . . . . . . . . 15.7 Acknowledgments . . . . . . . . . . . References . . . . . . . . . . . . . . . .

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16 Composition, quadratic, and some triple systems Noriaki Kamiya and Susumu Okubo 16.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 16.2 Relationship between triple systems and bilinear algebras 16.3 Composition triple systems . . . . . . . . . . . . . . . . . 16.4 Final comments . . . . . . . . . . . . . . . . . . . . . . . 16.5 Acknowledgments . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . .

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17 An introduction to associator quantization 233 Jaak L˜ ohmus and Leo Sorgsepp 17.1 Nonassociativity and associator quantization . . . . . . . . . . . . . . . . . 234 17.2 Octonion algebra and its regular bimodule representation . . . . . . . . . . 235 17.2.1 Regular birepresentation of octonions . . . . . . . . . . . . . . . . . . 235 17.3 The Dirac equation and its “octonionization”: Introduction of color charges 236 17.3.1 The Dirac equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236 17.3.2 “Octonionization” of the Dirac equation . . . . . . . . . . . . . . . . 237 17.4 The spectrum of fundamental fermions from double octonion structure . . 238 17.5 Evolution of observables in Hamiltonian quantum mechanics . . . . . . . . 242 17.6 Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243 18 A theorem on Bernstein quadratic integral operators 245 Yu. Lyubich References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251 19 The real prosymmetric spaces 253 O.A. Matveyev and E.L. Nesterenko References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259 20 Deep matrices and their Frankenstein actions Kevin McCrimmon 20.1 Prolegomenon . . . . . . . . . . . . . . . . . . 20.2 Heads and Bodies . . . . . . . . . . . . . . . . 20.3 The Deep Matrix Algebra . . . . . . . . . . . . 20.4 The Scalar Multiple Theorem . . . . . . . . . 20.5 Frankenstein Actions . . . . . . . . . . . . . . 20.6 Irreducible Actions . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . .

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xxxii 21 Trace reduction on superspaces Kurt Meyberg 21.1 Superspaces . . . . . . . . . . 21.2 The tensorproduct v ⊗ u∗ . . . 21.3 The supertrace . . . . . . . . . 21.4 The reduction map S → S . References . . . . . . . . . . . .

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22 Monomorphisms between Cayley-Dickson Guillermo Moreno 22.1 Introduction . . . . . . . . . . . . . . . . 22.2 Pure and doubly pure elements in An+1 . 22.3 Monomorphism from Am to An . . . . . References . . . . . . . . . . . . . . . . . .

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23 The notion of lower central series for loops Jacob Mostovoy 23.1 The lower central series . . . . . . . . . . . . . . . . . . . . . . . 23.2 Commutators, associators, and associator deviations . . . . . . . 23.3 The commutator-associator ﬁltration . . . . . . . . . . . . . . . 23.4 The lower central series and the commutator-associator subloops 23.5 Further comments . . . . . . . . . . . . . . . . . . . . . . . . . . 23.6 Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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24 Gravity within the framework of nonassociative geometry 299 Alexander I. Nesterov 24.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299 24.2 Introduction to nonassociative geometry . . . . . . . . . . . . . . . . . . . 300 24.3 Nonassociative discrete geometry . . . . . . . . . . . . . . . . . . . . . . . . 304 24.4 Discrete de Sitter spacetime . . . . . . . . . . . . . . . . . . . . . . . . . . 306 24.5 Toy model of discrete (2+1)-spacetime . . . . . . . . . . . . . . . . . . . . 307 24.5.1 Nonassociative discrete geometry of S 2 and (2+1)-spacetime evolution 308 24.6 Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 310 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 310 25 Algebras satisfying symmetric triality relations Susumu Okubo 25.1 Symmetric triality algebra . . . . . . . . . . . . 25.2 Examples of normal STA . . . . . . . . . . . . . 25.3 Acknowledgments . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . 26 Operads and nonassociative deformations Eugen Paal 26.1 Introduction and outline of the chapter . 26.2 Operad (composition system) . . . . . . . 26.3 Gerstenhaber brackets and associator . . 26.4 Coboundary operator . . . . . . . . . . . 26.5 Generalized Maurer-Cartan equation . . 26.6 Operadic Sabinin principle . . . . . . . . 26.7 Bianchi identity . . . . . . . . . . . . . .

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323 323 324 325 326 327 327

xxxiii 26.8 Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 Units of alternative loop rings: A survey C´esar Polcino Milies 27.1 Introduction . . . . . . . . . . . . . . . . 27.2 Deﬁnitions and basic facts . . . . . . . . 27.3 Finiteness conditions . . . . . . . . . . . 27.4 Normal complements . . . . . . . . . . . 27.5 Unit loops torsion over the center . . . . 27.6 Acknowledgments . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . .

328 328 329

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329 330 334 339 341 344 344

28 Vinberg algebras associated to some nilpotent Lie algebras Elisabeth Remm 28.1 Vinberg algebras associated with Lie algebras . . . . . . . . . . . . . . . . 28.1.1 Generalities: Aﬃne connections . . . . . . . . . . . . . . . . . . . . . 28.1.2 Aﬃne structures on Lie algebras . . . . . . . . . . . . . . . . . . . . 28.1.3 Vinberg algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28.1.4 Classical examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28.2 Vinberg algebras associated with graded ﬁliform complex Lie algebras . . . 28.2.1 Filiform Lie algebras, characteristically nilpotent ﬁliform Lie algebras 28.2.2 Aﬃne structures on ﬁliform complex Lie algebras . . . . . . . . . . 28.2.3 Noncomplete aﬃne structures on ﬁliform Lie algebras . . . . . . . . 28.3 Nilpotent Lie algebras with a contact form . . . . . . . . . . . . . . . . . . 28.3.1 Nilpotent contact Lie algebras . . . . . . . . . . . . . . . . . . . . . . 28.3.2 Aﬃne structures on nilpotent contact Lie algebras . . . . . . . . . . 28.4 Application: Aﬃne structures on 7-dimensional characteristically nilpotent contact Lie algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28.4.1 Algebra η74 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28.4.2 Algebra η712 (λ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28.4.3 Algebra η714 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28.4.4 Algebra η721 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28.4.5 Algebra η728 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28.5 Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

347 348 348 348 349 349 349 350 351 352 355 355 356 360 360 360 361 362 362 363 363

29 Algebraic and diﬀerential structures in renormalized perturbation quantum ﬁeld theory 365 M. Rosenbaum 29.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365 29.2 Rooted trees and Feynman graphs . . . . . . . . . . . . . . . . . . . . . . . 367 29.2.1 Toy model of renormalization . . . . . . . . . . . . . . . . . . . . . . 368 29.3 The Hopf algebra of rooted trees . . . . . . . . . . . . . . . . . . . . . . . . 370 29.3.1 Toy model from the Hopf algebra point of view . . . . . . . . . . . . 373 29.4 Hopf algebra of renormalization in ﬁeld theory . . . . . . . . . . . . . . . . 373 29.5 The diﬀerential calculus of renormalization, the Birkhoﬀ algebraic decomposition, and the forest formula . . . . . . . . . . . . . . . . . . . . . . . . . . 377 29.5.1 The forest formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . 379 29.5.2 The Birkhoﬀ algebraic decomposition . . . . . . . . . . . . . . . . . . 379 29.6 Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 380

xxxiv References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

380

30 Survey on smooth quasigroups development 383 Lev V. Sabinin and Larissa V. Sbitneva 30.1 The development of investigations . . . . . . . . . . . . . . . . . . . . . . . 383 30.2 Smooth quasigroups and loops and their applications in geometry . . . . . 384 30.2.1 Smooth quasigroups and loops: General theory . . . . . . . . . . . . 384 30.2.2 Smooth P L-loops and transsymmetric spaces . . . . . . . . . . . . . 385 30.3 Non-associative geometry and discrete structure of space-time . . . . . . . 388 30.4 Smooth loops action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 389 30.5 Some applications of the smooth quasigroup and loop theory to generalized geometric algebra, mechanics, and physics . . . . . . . . . . . . . . . . . . 390 30.6 Mirror symmetries of Lie algebras, Lie groups, and homogeneous spaces . . 391 30.7 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 391 30.8 Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 392 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 392 31 The Lie multiplication in the Lie duals of the Witt Earl J. Taft and Hao Zhifeng 31.1 Lie bialgebras . . . . . . . . . . . . . . . . . . . . . 31.2 Lie bialgebra structures on the Witt algebras . . . . 31.3 Continuous Lie duals . . . . . . . . . . . . . . . . . 31.4 The continuous duals of the Witt algebras . . . . . (i) 31.5 Lie multiplication in (W1 )0 and (W (i) )0 . . . . . . 31.6 Examples . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . .

algebras . . . . . . .

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395 396 396 397 398 398 399

32 Simple decompositions of simple Lie superalgebras 401 T.V. Tvalavadze 32.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 401 32.2 Decomposition of simple Lie superalgebras . . . . . . . . . . . . . . . . . . 402 32.3 Decompositions of simple superalgebra of the type sl(m, n) into the sum of basic Lie subalgebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 404 32.3.1 General properties of subalgebras in the decomposition . . . . . . . 404 32.3.2 Explicit form of the ﬁrst subalgebra in the decomposition . . . . . . 409 32.3.3 Explicit form of the second subalgebra in the decomposition . . . . 413 32.3.4 Decompositions of sl(m, n) into the sum of osp(p, k) and osp(l, q) . 415 32.3.5 Main theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415 32.4 Decompositions of simple superalgebras of the type sl(m, n) into the sum of basic and strange Lie subalgebras . . . . . . . . . . . . . . . . . . . . . . . 416 32.4.1 General properties of subalgebras in the decomposition . . . . . . . 416 32.4.2 Explicit form of the even part of a strange subalgebra . . . . . . . . 421 32.4.3 Main theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 426 32.5 Decompositions of simple superalgebras of the type sl(m, n) into the sum of two strange Lie subalgebras . . . . . . . . . . . . . . . . . . . . . . . . . . . 427 32.6 Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 429 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 429 33 The structure and classiﬁcation of ﬁnite division rings 431 G.P. Wene 33.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 431

xxxv 33.2 Preliminaries . . . . . . 33.3 Structure . . . . . . . . 33.3.1 Power-associative 33.3.2 Three examples . 33.4 Conclusion . . . . . . . References . . . . . . . .

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A Some problems in the theory of rings that are nearly associative A. I. Shirshov A.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.1.1 Associative and nonassociative rings . . . . . . . . . . . . . . A.1.2 Origins of the theory of nonassociative rings . . . . . . . . . . A.1.3 General results on nonassociative rings . . . . . . . . . . . . . A.1.4 Free rings and algebras . . . . . . . . . . . . . . . . . . . . . . A.1.5 Identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.2 Alternative rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.2.1 The doubling process . . . . . . . . . . . . . . . . . . . . . . . A.2.2 Identities in alternative rings . . . . . . . . . . . . . . . . . . A.2.3 Subrings of alternative rings . . . . . . . . . . . . . . . . . . . A.2.4 Free alternative rings . . . . . . . . . . . . . . . . . . . . . . . A.2.5 Cayley-Dickson algebras . . . . . . . . . . . . . . . . . . . . . A.2.6 Right alternative rings . . . . . . . . . . . . . . . . . . . . . . A.3 J-rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.3.1 J-rings and special J-rings . . . . . . . . . . . . . . . . . . . . A.3.2 An exceptional J-algebra . . . . . . . . . . . . . . . . . . . . A.3.3 Identities in J-rings . . . . . . . . . . . . . . . . . . . . . . . A.3.4 J-rings and alternative rings . . . . . . . . . . . . . . . . . . . A.4 Lie rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.4.1 Lie rings and Lie algebras . . . . . . . . . . . . . . . . . . . . A.4.2 Relations between Lie rings and groups . . . . . . . . . . . . A.4.3 The Engel condition and the Burnside problem . . . . . . . . A.4.4 Free Lie rings . . . . . . . . . . . . . . . . . . . . . . . . . . . A.4.5 Binary-Lie rings and Moufang-Lie rings . . . . . . . . . . . . A.5 Some wider classes of rings . . . . . . . . . . . . . . . . . . . . . . . A.5.1 Power-associative rings . . . . . . . . . . . . . . . . . . . . . . A.5.2 Decomposition with respect to an idempotent . . . . . . . . . A.5.3 Noncommutative J-rings and power-commutative rings . . . . A.6 Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B Dniester Notebook: Unsolved problems in the theory of rings and modules Edited by V.T. Filippov, V.K. Kharchenko, and I.P. Shestakov B.1 Translators’ introduction . . . . . . . . . . . . . . . . . . . . . . B.2 Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.3 Part one . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.4 Part two . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.5 Part three . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.6 Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Chapter 1 Inﬁnite-Dimensional Representations of the Rotation Group and Magnetic Monopoles Ferm´ın Aceves de la Cruz and Alexander I. Nesterov Physics Department, Guadalajara University, Guadalajara, Jalisco, Mexico 1.1 1.2 1.3 1.4

1.5 1.6

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Indeﬁnite metric Hilbert space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Inﬁnite-dimensional representations of the rotation group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Quantum mechanics of Dirac monopole with arbitrary magnetic charge . . . . . . . . . . . . . . 1.4.1 Representation unbounded from above and below . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.2 Representations bounded above or below . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 2 3 5 6 6 7 7 7

Abstract The Dirac monopole problem is studied within the framework of inﬁnite-dimensional representations of the rotation group, and a consistent pointlike monopole theory with an arbitrary magnetic charge is deduced. Key words: monopole, nonassociativity, inﬁnite-dimensional representations, indeﬁnite metric Hilbert space 2000 MSC 20G05, 20C35, 81S99

1.1

Introduction

In [1, 2] Dirac showed that a proper description of the quantum mechanics of a charged particle in the ﬁeld of the magnetic monopole of the charge q requires the quantization condition 2μ = n, n ∈ Z (we set = c = 1 and eq = μ, e being an electric charge). But in spite of numerous eﬀorts, the Dirac monopole has not been found. Recently renewed interest in the Dirac monopole has grown in connection with the “ﬁctitious” monopoles that are similar to the “real” magnetic monopoles appearing in the context of the Berry phase [3]. These type of magnetic monopoles emerge in the anomalous Hall eﬀect of ferromagnetic materials, trapped Λ-type atoms, anisotropic spin systems, noncommutative quantum mechanics, etc., and may carry an arbitrary “magnetic” charge. It is known that in the presence of the magnetic monopole the operator of the total angular momentum J, which includes the contribution of the electromagnetic ﬁeld, obeys the standard commutation relations of the Lie algebra of the rotation group [Ji , Jj ] = iijk Jk .

1

2

Ferm´ın Aceves de la Cruz and Alexander I. Nesterov

The requirement that Ji ’s generate a ﬁnite-dimensional representation of the rotation group yields 2μ ∈ Z [4, 5, 6, 7, 8]. Thus, one should give up ﬁnite-dimensional representations of the rotation group to allow an arbitrary magnetic charge. Here we review the Dirac monopole problem within the framework of inﬁnite-dimensional representations of the rotation group (for details see [9, 10, 11]).

1.2

Indeﬁnite Metric Hilbert Space

Conventional quantum mechanics is realized in Hilbert H, with the norm ¯ ψψdx >0 being positive deﬁnite. Here we treat the more general situation when the integral ¯ ψψdμ(x),

(1.1)

(1.2)

dμ being a suitable measure, is not necessarily positive, and the bilinear form given by ¯ dμ(x), (1.3) ψψ may be divergent. Its value is given by a regularization procedure [13]. Hereafter the regularized integral will be denoted by . Following the notations introduced by Pauli [12], we consider an inner product in the indeﬁnite metric Hilbert space Hη deﬁned by the bilinear form of the type ¯ dμ(x), (1.4) (ψ, ψ )η = (ψ, ηψ ) = ψηψ in which the operator η is only restricted by the condition that it has to be Hermitian and ¯ ψηψdμ(x) > 0. (1.5) Let functions ψm (x) form the basis such that ψ m (x)ψm (x)dμ(x) = ηmm where ηmm is an indeﬁnite diagonal metric. The set {ψm } forms the orthonormal basis with respect to the inner product given by (1.6) (ψm , ψp )η = ψ m (x)ηpm ψm (x)dμ(x) = δmp and an arbitrary function ψ(x) ∈ Hη can be expanded as ψ(x) = cηm ψm ,

(1.7)

m

where cηm

= (ψm , ψ)η = ηmm ψ m (x)ψ(x)dμ(x)

(1.8)

Inﬁnite-dimensional representations of the rotation group and magnetic monopoles

3

In particular, we ﬁnd ¯ (ψ, ψ )η = ψ(x)ηψ (x)dμ(x) = cηm cη m.

(1.9)

m

that implies

¯ (ψ, ψ)η = ψ(x)ηψ(x)dμ(x) = |cηm |2 > 0. m

Thus we see that the inner product in the indeﬁnite metric Hilbert space is a positively deﬁned scalar product. This provides the standard probabilistic interpretation of the quantum mechanics. The expectation value of a physical observable A represented by the linear operator acting in Hη is deﬁned by ¯ Aη = ψ(x)ηAψ(x)dμ(x), (1.10) A generalization of the Hermitian conjugate operator, being denoted as A†η , is deﬁned as follows: A†η = η −1 A† η, (1.11) where A† is the Hermitian conjugate operator. Since the observables are real, one can see see that the related operators have to be selfadjoint in the indeﬁnite metric Hilbert space: A†η = A. Indeed, assuming that the matrix A has a normal form Aψn = an ψn (1.12) we ﬁnd that (ψ, Aψ)η =

an |cηn |2 .

(1.13)

n

This leads to the conclusion that the operator with only positive eigenvalues cannot have negative expectation values. In other words in our approach the standard probabilistic interpretation of the quantum mechanics is preserved.

1.3

Inﬁnite-Dimensional Representations of the Rotation Group

As is known the three-dimensional rotation group is locally isomorphic to the group SU (2): SO(3) = SU (2)/Z2 . In what follows the diﬀerence between SO(3) and SU (2) is not essential and actually we will consider G = SU (2). The Lie algebra su(2) has three generators and we adopt the basis J± = J1 ± iJ2 , J3 . The commutation relations read [J+ , J− ] = 2J3 , 2

[J , J± ] = 0,

[J3 , J± ] = ±J± 2

[J , J3 ] = 0,

(1.14) (1.15)

where 1 J 2 = J32 + (J− J+ + J+ J− ) 2 is the Casimir operator.

(1.16)

4

Ferm´ın Aceves de la Cruz and Alexander I. Nesterov Let ψνj be an eigenvector of the operators J3 and J 2 such that J3 ψνj = (ν + n)ψνj ,

J 2 ψνj = j(j + 1)ψνj ,

(1.17)

where n = 0, ±1, ±2, . . . , and ν, just like j, is a certain complex number. There are four distinct classes of representations and each irreducible representation is characterized by an eigenvalue of Casimir operator and the spectrum of the operator J3 [14, 15, 16, 17, 19]: • Representations unbounded from above and below, in this case neither j + ν nor j − ν can be integers. • Representations bounded below, with j + ν being an integer, and j − ν not equal to an integer. • Representations bounded above, with j − ν being an integer, and j + ν not equal to an integer. • Representations bounded from above and below, with j − ν and j + ν both being integers, that yields j = k/2, k ∈ Z+ . The nonequivalent representations in each series of irreducible representations will be denoted, respectively, by D(j, ν), D+ (j, ν), D− (j, ν), and D(j). The representations D(j, ν), D+ (j, ν), and D− (j, ν) are inﬁnite-dimensional; D(j) is (2j +1)-dimensional representation. The irreducible representations D± (j, ν) and D(j, ν), including the case of j, ν being complex numbers, are discussed in detail in [14, 15, 16, 17, 18]. Further we restrict ourselves by the real eigenvalues of J3 and the Casimir operator J 2 . The inﬁnite-dimensional representations considered here are given by linear unbounded operators in inﬁnite-dimensional linear topological spaces, supplied with a weak topology [16, 17, 18]. By introducing the inner product, it becomes the indeﬁnite metric Hilbert space Hη , and for the orhonormal basis |j, m, m being the eigenvalue of the operator J3 , one has [11] j, m |j, mη = δmm , where ηmm = (−1)σ(m) δmm is the indeﬁnite metric, and (−1)σ(m) = sgn (Γ(j − m + 1)Γ(j + m + 1) . Algebra su(2) acts on the eigenstates |j, m as follows: J+ |j, m = J− |j, m =

j(j + 1) − m(m + 1)|j, m + 1 j(j + 1) − m(m − 1)|j, m − 1

(1.18)

J3 |j, m = m|j, m. As we can easily see, the operator J3 is the self-adjoint operator in the indeﬁnite Hilbert space Hη , J3 = (J3 )†η , and for J± one has J± = (J∓ )†η .

Inﬁnite-dimensional representations of the rotation group and magnetic monopoles

1.4

5

Quantum Mechanics of Dirac Monopole with Arbitrary Magnetic Charge

The magnetic ﬁeld of the Dirac monopole is given by B=q

r . r3

(1.19)

This implies that the deﬁnition B = rotA, where A is the vector potential, holds only locally. Thus, the vector potential must have singularity (the so-called Dirac string), and one can write B=∇×A+h where h is the magnetic ﬁeld of Dirac’s string. In particular, the vector potential can be chosen as r×n , (1.20) An (r) = q r(r − n · r) and one can see that the potential is singular on the axis r = rn. For a non-relativistic charged particle in the ﬁeld of a magnetic monopole the total angular momentum, r J = r × (p − eA) − μ , (1.21) r having the same properties as a standard angular momentum, obeys the following commutation relations: [H, J2 ] = 0, [J2 , Ji ] = 0,

[H, Ji ] = 0, [Ji , Jj ] = iijk Jk ,

(1.22) (1.23)

where H is the Hamiltonian. These commutation relations fail on the string [6]. However, it is found that H and J may be extended to the self-adjoint operator satisfying the commutation relations of the rotation group, with H an invariant, and this is true for any value of μ. However, requiring that the Ji generate a representation of the rotation group and not just the Lie algebra, one obtains that μ must be quantized and only the values 2μ = 0, ±1, ±2, . . . , are allowed [8]. Choosing n = (0, 0, −1), we obtain A=q

1 − cos θ ˆϕ , e r sin θ

where (r, θ, ϕ) are the spherical coordinates. This yields ∂ 1 − cos θ 1 ∂ 2iμ ∂ 1 ∂2 + μ2 + μ2 J2 = − + sin θ − sin θ ∂θ ∂θ 1 + cos θ ∂ϕ 1 + cos θ sin2 θ ∂ϕ2 ∂ μ sin θ ∂ + i cot θ − J± = e±iϕ ± ∂θ ∂ϕ 1 + cos θ ∂ − μ. J3 = − i ∂ϕ Let us consider the Schr¨ odinger’s equation: ˆ = Eψ Hψ

(1.24)

(1.25) (1.26) (1.27)

6

Ferm´ın Aceves de la Cruz and Alexander I. Nesterov

In the spherical coordinates it is written as follows: 1 ∂ 2 ∂ J2 − μ2 Ψ(r) = EΨ(r) r + − 2m ∂r ∂r 2mr2 This equation admits separation of variables and setting Ψ = R(r)Y (θ, ϕ), we get for the angular part the following equation: J2 Y (θ, ϕ) = j(j + 1)Y (θ, ϕ).

(1.28)

In terms of hypergeometric function the solution Yjmμ of Eq. (1.28) is found to be Yjmμ ∝ ei(m+μ)ϕ z (m+μ)/2 (1 − z)(m−μ)/2 F (a, b; c; z),

(1.29)

where z = (1−cos θ)/2, m is an eigenvalue of J3 and a = m−j, b = m+j +1, c = 1+m+μ. The obtained solutions form the basis of the representations of the rotation group realized over S 2 .

1.4.1

Representation unbounded from above and below

Let m be an eigenvalue of the operator J3 with the spectrum being deﬁned by m = n + ν,

n = 0, ±1, ±2, . . . ,

(1.30)

where ν is an arbitrary real number, and we assume that neither j + ν nor j − ν can be integers. Identifying ν as −μ, we obtain the inﬁnite-dimensional irreducible representation D(j, −μ) being realized over the two-dimensional sphere S 2 . The basis of the representation (μ,m) is given by the functions Yj such that (μ,m)

Yj

j = Cμm eimϕ z (m+μ)/2 (1 − z)(m−μ)/2 F (a, b, c; z)

m = n − μ,

n = 0, ±1, ±2, . . . ,

(1.31)

j Cμm

being a normalization constant, form the complete orthonormal canonical basis of representation D(j, −μ) in the indeﬁnite-metric Hilbert space Hη [11, 17, 18]. The indeﬁnite metric is given by (1.32) ηmm = δmm sgn Γ(j − m + 1)Γ(j + m + 1) , and there is no any restriction on μ.

1.4.2

Representations bounded above or below

The representation D(j, −μ) becomes the representation D+ (j, −μ) bounded below, if j − μ is an integer, and j + μ is not equal to an integer. In a similar way, D(j, −μ) reduces to the representation D− (j, −μ) bounded above, with j + μ being an integer, and j − μ not equal to an integer. Here, as above, μ is an arbitrary parameter. In this case, the solution of Eq. (1.28) is of the form Yjμn (θ, ϕ) = eiαϕ Cn (1 − u)δ/2 (1 + u)γ/2 Pn(δ,γ) (u), n ∈ Z+ (2n + δ + γ) Γ(n + 1) Γ(n + δ + γ + 1) Cn = 2π 2δ+γ+1 Γ(n + δ + 1) Γ(n + γ + 1) (δ,γ)

(1.33)

where α = m + μ and u = cos θ, Pn (u) being the Jacobi polynomials [11, 17, 18]. Setting β = m − μ, one can see that the basis of the representation bounded above/below is written as follows:

Inﬁnite-dimensional representations of the rotation group and magnetic monopoles • Representations bounded above

− −μ,n D (j, −μ) : m = j − n, j + μ ∈ Z+ iαϕ (α,β) (u) − Y j±μ = e Yn D (j, −μ) : m = −j − n − 1, j − μ ∈ Z+

7

(1.34)

• Representations bounded below +μ,n

Y j±μ = e

iαϕ

Yn(−α,−β) (u)

D+ (j, −μ) : m = n − j, j − μ ∈ Z+ + D (j, −μ) : m = j + n + 1, j + μ ∈ Z+

(1.35)

Note, that while the representations D− (j, −μ) and D+ (, ∓μ) are irreducible, the repre + (, ∓μ)are partially reducible [19]. − (j, −μ) and D sentations D Finite-dimensional representation occurs when j + μ and j − μ, both are positive integers. This leads immediately to the celebrated Dirac quantization condition, 2μ ∈ Z.

1.5

Concluding remarks

We have shown that a consistent pointlike monopole theory with an arbitrary magnetic charge requires inﬁnite-dimensional representations of the rotation group, which in general case are multivalued [16, 17, 18]. Note, that in Dirac theory “quantization of magnetic charge” follows from the requirement of the wave function be single-valued. However, the requirement of single-valuedness for a wave function is not one of the fundamental principles of quantum mechanics, and having multi-valued wave functions may be allowed until it does not aﬀect the algebra of observables.

1.6

Acknowledgments

This work was partly supported by SEP-PROMEP (Grant No. 103.5/04/1911).

References [1] P. A. M. Dirac, Proc. Roy. Soc. Lond. A180 (1931), 1. [2] P. A. M. Dirac, The theory of magnetic poles, Phys. Rev. 74, (1948), 817. [3] M. V. Berry, Quantul phase factors accompanying adiabatic changes, Proc. Roy. Soc. Lond. A 392 (1984), 45. [4] A. S. Goldhaber, Role of spin in the monopole problem, Phys. Rev. B 140 (1965), 1407. [5] A. S. Goldhaber, Connection of spin and statistics for charge-monopole composites, Phys. Rev. Lett. 36 (1976), 1122.

8

Ferm´ın Aceves de la Cruz and Alexander I. Nesterov [6] D. Zwanziger, Quantum ﬁeld theory of particles with both electric and magnetic charges, Phys. Rev. 176 (1968), 1489. [7] D. Zwanziger, Local-Lagrangian quantum ﬁeld theory of electric and magnetic charges, Phys. Rev. D 3 (1971), 880. [8] A. Hurst, Charge quantization and nonintegrable Lie algebras, Ann. Phys. 50 (1968), 51. [9] A. I. Nesterov and F. Aceves de la Cruz, Magnetic monopoles with generalized quantization condition, Phys. Lett. A 302 (2002), 253, hep-th/0208210.

[10] A. I. Nesterov and F. Aceves de la Cruz, On representations of the rotation group and magnetic monopoles, Phys. Lett. A 324 (2004), 9, hep-th/0402226. [11] A. I. Nesterov, F. Aceves de la Cruz, Inﬁnite-dimensional representation of the rotation group and Dirac monopole with arbitrary magnetic charge, hep-th/0503040 (2005). [12] W. Pauli, On Dirac’s new method of ﬁeld quantization, Rev. Mod. Phys. 15 (1943), 175. [13] I. M. Gel’fand and G. E. Shilov, Generalized Functions, Vol. I (1964), Academic Press, New York. [14] E. G. Beltrami and G. Luzatto, Rotation matrices corresponding to complex angular momenta, Nuov. Cim. bf 5 (1963), 1003. [15] M. Andrews and J. Gunson, Complex angular momenta and many-particle states. I. Properties of local representsations of the rotation group, J. Math. Phys. bf 5 (1964), 1391. [16] S. S. Sannikov, Representations of the rotation group with complex spin, Sov. J. Nucl. Phys. 2 (1966), 407. [17] S. S. Sannikov, Inﬁnite-dimensional representations of the rotation group, Sov. J. Nucl. Phys. 6 (1968), 788. [18] S. S. Sannikov, New representations of the Lie algebra of the rotation group, Sov. J. Nucl. Phys. 6 (1968), 939. [19] B. G. Wybourne, Classical Groups for Physicists (1974) Wiley, New York. [20] J. Schwinger, K. A. Milton, W.-Y. Tsai, L. L. DeRaad Jr, D. C. Clark, Non-relativistic dyon-dyon scattering, Ann. Phys. 101 (1976), 451.

Chapter 2 Generalized Lie Nilpotence in Integral Group Rings Yu.A. Bahturin and M.M. Parmenter Department of Mathematics and Statistics, Memorial University of Newfoundland, St. John’s, Canada

2.1 2.2 2.3 2.4 2.5

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Preliminary results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Nilpotence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Solvability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9 10 13 14 15 15

Abstract We consider group algebras of ﬁnite groups. Our main result holds that such algebra R can be made generalized Lie nilpotent, with respect to a bicharacter on a ﬁnite abelian group G, which grades R, if and only if R is commutative. Key words: Group rings, Graded rings, modules 2000 MSC: 20C05, 16W50

2.1

Introduction

Let A be an associative algebra over a commutative ring R with 1 and assume A is graded by a ﬁnite abelian group H with a skew-symmetric bicharacter β : H × H → R∗ (see [1]). Deﬁne a β-bracket on A by the formula [a, b]β = ab − β(g, h)ba for any homogeneous a ∈ Ag , b ∈ Ah , and extend the bracket to all of A by linearity. In this case (A, [a, b]β ) becomes a color Lie superalgebra, in the sense of [2], or β-Lie algebra in the sense of [3]; that is, we have the following identities satisﬁed for all homogeneous a ∈ Ag , b ∈ Ah , and all c ∈ A: [b, a] = −β(g, h)[a, b]

(β-anticommutativity)

[[a, b], c] = [a, [b, c]] − β(g, h)[b, [a, c]]

(β-Jacobi identity).

A is said to be β-commutative if [a1 , a2 ]β = 0 for all a1 , a2 in A. Similarly the usual notions of Lie nilpotence and Lie solvability (see, for example, [8, p. 142]) can be extended in an obvious way to β-nilpotence and β-solvability on A (see [4, p. 3]). These concepts were investigated for group algebras over ﬁelds in [1, 4, 5]. Recall that it was shown in [5] that over an algrebraically closed ﬁeld F of characteristic zero, the group

9

10

Yu.A. Bahturin and M.M. Parmenter

algebra F G of a ﬁnite group G can always be graded by a ﬁnite abelian group G, with a skew-symmetric bicharacter β, so that F G becomes β-commutative. In the same paper we have also shown that if F is any ﬁeld of characteristic p > 0 and G a ﬁnite p-group then F G can be made commutative only if G is abelian. When A = ZG is an integral group ring, it is very easily seen [5, Proposition 5.4] that ZG can be made β-commutative (i.e., there exists a grading of ZG by a ﬁnite abelian group H and a bicharacter β such that ZG is β-commutative) only in the obvious case where G is abelian. The main purpose of this note is to show that a similar result holds for βnilpotence; the integral group ring ZG of a ﬁnite group can be made β-nilpotent only when G is abelian. The analogous result for β-solvability is not true. In fact, ZG can be made β-solvable when graded by the cyclic group C2 of order 2 if G has an abelian subgroup of index 2. Exactly what the situation is when the grading group H is more general is still an open question. We begin by assembling some necessary preliminary results.

2.2

Preliminary Results

The ﬁrst lemma is well known but we include it for completeness. LEMMA 2.1 Assume that an R-algebra A (R a commutative ring with 1) is graded by a ﬁnite abelian group H. Then (a) If 1 is the multiplication identity of A then 1 is in Ae (where e is the identity of H). (b) If z ∈ Z(A), the center of A, and z = zg where zg is in Ag for all g in H, then zg is in Z(A) for all g in H. PROOF (a) Assume that 1 = a g where ag is in Ag for each g. Let y ∈ Ah be any homogeneous element of A. Then y = ag y and ag y is in Agh for each g. It follows that y = ae y, and similarly y = yae . We conclude that ae is a multiplicative identity for A, as desired. yzg = yz = zy = zg y. (b) Let y ∈ Ah be any homogeneous element of A. Then Since H is abelian and yzg is in Ahg = Agh , and zg y is in Agh , we have yzg = zg y for all g ∈ H. Hence zg is in Z(A) as desired. Several times, beginning with Lemma 2.2 we will need the following important result about central units in integral group rings. THEOREM 2.1 ([7, Corollary 7.3.3]) then α belongs to ±Z(G).

If α is a central unit of ﬁnite order in ZG,

Our ﬁrst lemma about gradings by ﬁnite elementary abelian 2-groups is as follows. LEMMA 2.2 abelian 2-group.

Let G be a ﬁnite group and assume ZG is graded by a ﬁnite elementary

Generalized Lie nilpotence in integral group rings

11

(a) If g ∈ G is central of order 2, then g is homogeneous. (b) If g ∈ G is central of order 2 and g = g02 for some g0 ∈ G then g is in the identity component. PROOF (a) First assume A = ZG is graded by the cyclic group C2 of order 2, and, say, C2 = {e, x}. Let g = αe + αx where αe ∈ Ae , αx ∈ Ax . Note αe , αx are in Z(A) by Lemma 2.1(b). Then 1 = g 2 = (αe2 + αx2 ) + 2αe αx , where αe2 + αx2 is in Ae and 2αe αx is in Ax . Since 1 is in Ae by Lemma 2.1(a), we have αe2 + αx2 = 1 and αe αx = 0. Observe now that (αe −αx )2 = αe2 +αx2 −2αe αx = 1, so αe −αx is a central unit of order 2 in ZG. It follows from Theorem 2.1 that αe − αx belongs to ±Z(G). But αe − αx = g − 2αx , so we must have either αx = 0 or αx = g. In either case, g is homogeneous. We will now assume ZG is graded by the direct product of n copies of C2 and proceed by induction on n. The case n = 1 is settled. Assume the result is true if n = k (where k ≥ 1) and let ZG be graded by H = H1 × H2 × · · · × Hk+1 where Hi is isomorphic to C2 for all i. We have g = αx where αx is in Ax for each x. x∈H

If y is any nonidentity element of H, then K = {e, y} is isomorphic to C2 and H/K is an elementary abelian 2-group of rank k. Viewing ZG as graded by H/K, the induction hypothesis tells us that g is homogeneous. But this means that if αx1 = 0 and αx2 = 0 where x1 = x2 , then we must have x2 = x1 y. Since this holds for any y = e in H, we must have g homogeneous as desired. αx , where αx is in Ax for each x. (b) Assume ZG is graded by H and, say, g0 = x∈H

Then g = g02 =

x∈H

where

x∈H

αx2 is in Ae and

αx2

+

y∈H\{e}

αw αyw−1

,

w∈H

αw αyw−1 is in Ay for each y = e.

w∈H

From (a) we know that g is homogeneous. So if g is not in Ae , we must have g = αw αyw−1 for some y = e. However this latter sum is a multiple of 2 since for any w∈H

w ∈ H, it contains both αw αyw−1 and αyw−1 αw (note w = yw−1 since H is of exponent 2), so this case cannot occur. We conclude that g is in Ae . Lemma 2.2 will be applied in the following setting. LEMMA 2.3 If G is a ﬁnite 2-group of nilpotency class 2, then G contains an element g of the type described in Lemma 2.2(b). PROOF If Z(G) is not of exponent 2 we are done, so assume that it is. Since G is not abelian, we can choose g0 in G such that go4 = 1 but go2 = 1. For any h in G we have [g02 , h] = [g0 , h]2 (since G is of nilpotency class 2) = 1 (since Z(G) is of exponent 2). Hence g02 is in Z(G) and we are done.

12

Yu.A. Bahturin and M.M. Parmenter The following corollary of Lemma 2.2 will also be used in the next section.

COROLLARY 2.1 Let G and H be ﬁnite elementary abelian 2-groups. If ZG is graded by H, then every element g ∈ G is homogeneous. The next lemma is initially stated in more generality than we require. Given an R-algebra A graded by a ﬁnite abelian group H we deﬁne Λ(H) = Hom (H, Z∗ ). For each λ in Λ(H) we can deﬁne λ∗ : A → A by

λ∗ ag = λ(g)ag where ag is in Ag for all g ∈ H. It is easily seen that λ∗ is a ring automorphism of A. Let K = {h ∈ H|λ(h) = 1 for all λ ∈ Λ(H)} (note λ(h) is always 1 or −1). A subgroup S of (A, +) is called 2-isolated if and only if the quotient group A/S has no 2-torsion. LEMMA 2.4 Let I be an ideal of A such that (I, +) is a 2-isolated subgroup of (A, +). The following are equivalent: (a) I is an H/K-graded ideal of A. (b) λ∗ (I) = I for all λ ∈ Λ(H). PROOF First we assume (a) and let s ∈ I. We have s = sgK , where the sum runs sgk with sgk in Agk . over all distinct left cosets of K in H and for each g, sgK = k∈K

Since I is H/K-graded, sgK is in I for all gK in H/K. Note that sgk ) = λ(gk)sgk = λ(g)sgk = λ(g)sgK . λ∗ (sgK ) = λ∗ ( k∈K

k∈K

k∈K

∗

It follows that λ (s) is in I, and we have (b). Next assume (b). If (a) is not true, then using the same notation as above, we can choose s = sg1 K + · · · + sgt K in I with t minimal such that none of sgi K are in I. Since g1 K = g2 K, there exists λ ∈ Λ(H) with λ(g1 ) = λ(g2 ). But then (b), together with the minimality assumption, tells us that 2sg1 K is in I. Since I is 2-isolated, we have sg1 K in I, which is a contradiction. Note that if H is an elementary abelian 2-group, K = {e} in the above. We will apply Lemma 2.4 to the following case. Recall [8] that if N G, Δ(G, N ) is the (2-sided) ideal of ZG generated by {1 − n|n ∈ N }. Since ZG ∼ = Z(G/N ), Δ(G, N ) we conclude that Δ(G, N ) is a 2-isolated subgroup of ZG. COROLLARY 2.2 Assume that G is ﬁnite and ZG is graded by a ﬁnite elementary abelian 2-group H. Let N be a normal subgroup of G, which is invariant under all ring automorphisms of ZG. Then Δ(G, N ) is an H-graded ideal of ZG.

Generalized Lie nilpotence in integral group rings

2.3

13

Nilpotence

If G is a ﬁnite nonabelian group, it can easily be deduced from [8, Theorem 4.4, p. 151] that ZG is not Lie nilpotent. Our main result generalizes this to β-nilpotence. THEOREM 2.2 Assume that G is ﬁnite and ZG is graded by a ﬁnite abelian group H. If there exists a skew-symmetric bicharacter β : H × H → Z∗ such that ZG is β-nilpotent, then G is abelian. PROOF Assume to the contrary that G is a minimal counterexample. In other words, G is a nonabelian group of minimal order with the property that ZG has been graded by a ﬁnite abelian group H that has been assigned a skew-symmetric β so that ZG is β-nilpotent. First note that we may assume the grading group H is an elementary abelian 2-group. To see this, observe that for any x, y in H we have β(x, y 2 ) = [β(x, y)]2 = 1 since β(x, y) = ±1. So H 2 ⊆ Kerβ, and if we set H = H/H 2 , then β induces a skew-symmetric bicharacter β on H and ZG is still β-nilpotent. ZG ∼ Clearly the ideal 2ZG is a graded ideal, and so = Z2 G is still β-nilpotent. But this 2ZG just means Z2 G is Lie nilpotent, and it follows from a result of Passi-Passman-Sehgal (see, for example [8, Theorem 4.4, p. 151]), that G is 2-abelian and nilpotent. So G ∼ = K×P where K is a 2-group and P is an abelian group of odd order. Since P is precisely the set of central units of odd order in ZG (see Theorem 2.1), Corollary 2.2 tells us Δ(G, P ) is an H-graded ideal of ZG, so Z (G/P ) ∼ =

ZG Δ(G, P )

is β-nilpotent. Minimality of |G| forces P = {e}, i.e., G is a 2-group. Letting U denote the group of units of ZG, we set N = Z(G) ∩ U 2 where U 2 is the subgroup of U generated by {u2 |u ∈ U }. Since G is nilpotent and not of exponent 2, and G2 G, N ⊇ Z(G) ∩ G2 = {e}. If α is any ring automorphism of ZG (not necessarily augmentation preserving), then Theorem 2.1 tells us that α(Z(G)) ⊆ ±Z(G). Since α(U 2 ) = U 2 , augmentation considerations allow us to conclude that α(N ) = N . It follows from Corollary 2.2 that Δ(G, N ) is an H-graded ideal of ZG, and so minimality of |G| forces G/N to be abelian. Since we now know G is nilpotent of class two, Lemmas 2.3 and 2.2(b) allow us to choose z in Z(G) such that z is of order 2 and z is in Ae . It follows easily that the ideal (1 − z)ZG is a graded ideal, and so Z (G/z) is also β-nilpotent. Minimality tells us that G/z must be abelian. Let g ∈ G be arbitrary. Since z is the only nonidentity commutator in G, g 2 ∈ Z(G). Hence g 2 ∈ N (using the same notation as before), and so G/N is an elementary abelian 2-group. We saw earlier that Δ(G, N ) is a graded ideal, and so our grading of ZG induces a grading on Z (G/N ). But Corollary 2.1 says that all elements of G/N must be homogeneous with respect to this grading. It follows that for any g ∈ G, there exists α in Δ(G, N ) ⊆ Δ(G, Z(G)) such that g + α is homogeneous in ZG.

14

Yu.A. Bahturin and M.M. Parmenter

Now let a and b be any pair of noncommuting elements of G. We have just shown that there exist α1 , α2 in Δ(G, Z(G)) such that a1 = a + α1 and b1 = b + α2 are homogeneous in A = ZG. Say a1 ∈ Ax , b1 ∈ Ay . Since β takes values ±1, two of β(x, y), β(x, xy), β(y, xy) must be equal. We may assume β(x, y) = β(x, xy) (otherwise it may be necessary to interchange a1 and b1 or replace one of them by a1 b1 ). Assume ﬁrst that β(x, y) = β(x, xy) = −1. Then [a1 , b1 ]β = a1 b1 +b1 a1 has augmentation 2 and is in Axy . Next [a1 , [a1 , b1 ]β ]β = a1 [a1 , b1 ] + [a1 , b1 ]a1 (since β(x, xy) = −1) has augmentation 4 and is in Ay . In general, we get [a1 , [a1 , · · · , [a1 , b1 ]β · · · ]β has augmentation 2n (where n is the number of a1 ’s) and hence can never be 0. The other possibility is that β(x, y) = β(x, xy) = 1. Then [a1 , b1 ]β = a1 , b1 − b1 a1 = ab − ba + aα2 − α2 a + α1 b − bα1 + α1 α2 − α2 α1 . Using the fact that α1 , α2 ∈ Δ(G, Z(G)), and also that z is the only nonidentity commutator in G, we get [a1 , b1 ]β = ab(1 − z) + t1 where t1 is in (1 − z)Δ(G, Z(G)). Next we have [a1 , [a1 , b1 ]β ]β = a1 [a1 , b1 ]β − [a1 , b1 ]β a1 (since β(x, xy) = 1) = a2 b(1 − z) + at1 + α1 ab(1 − z) + α1 t1 − ab(1 − z)a −t1 a − ab(1 − z)α1 − t1 α1 = a2 b(1 − z)2 + l2 (where l2 ∈ (1 − z)2 Δ(G, Z(G))) = 2b(1 − z) + 2t2 (where t2 ∈ (1 − z)Δ(G, Z(G))) (since (1 − z)2 = 2(1 − z)). This continues. In general we get [a1 , [a1 , · · · , [a1 , b1 ]β · · · ]β 2n−1 c(1 − z) + 2n−1 tn , where tn is in (1 − z)Δ(G, Z(G)), c is either b or ab, and n is the number of a1 ’s. But note that 2n−1 c(1 − z) + 2n−1 tn = 0 implies 1 − z annihilates an element of augmentation 1 in ZG, and this is impossible.

2.4

Solvability

If G is a ﬁnite nonabelian group, it can easily be deduced from [8, Theorem 3.1, p. 147] that ZG is not Lie solvable. However this result does not not extend to β-solvability as ZG can be made β-solvable whenever G has an abelian subgroup of index 2. PROPOSITION 2.1 Assume G is ﬁnite and has an abelian subgroup K of index at most 2.Then ZG can be graded by the cyclic group C2 of order 2 in such a way that there exists a skew-symmetric bicharacter β on C2 making ZG β-solvable. PROOF Assume G has an abelian subgroup K of index 2, so G = K, b, and let C2 = {e, x}. Grade ZG by setting Ae = ZK, Ax = bZK, and deﬁne β by β(e, e) = β(e, x) = β(x, e) = 1, β(x, x) = −1. Note that [Ae , Ae ]β = [Ae , Ae ] = 0 since K is abelian.

Generalized Lie nilpotence in integral group rings

15

Next observe that if k1 , k2 are in K then [bk1 , bk2 ]β = b2 k1b k2 + b2 k2b k1 = b2 k1b k2 + b2 k1 k2b , which is central in ZG. We conclude that [Ax , Ax ]β is central in ZG. It follows that [[Ax , Ax ]β , ZG]β = 0 since [Ax , Ax ]β ⊆ Ae . Finally note that [Ae , Ax ]β ⊆ Ax , so [[[Ae , Ax ]β , [Ae , Ax ]β ]β , ZG]β = 0. This completes the proof. The general situation with respect to β-solvability is still unclear. We only mention some reductions. If we assume ZG is β-solvable, then arguing in the same way as in the last section we see that Z2 G is Lie solvable and hence ([8], as noted above) G has a 2-abelian subgroup of index at most 2. In addition, we can show [4] that if ZG is β-solvable then the dimensions of all complex irreducible representations of G must be 2-powers. Moreover if the grading group is of order 2 then these dimensions are all less than or equal to 2, and this means [6] either G has an abelian subgroup of index 2 or [G : Z(G)] = 8. Another remark is as follows. One can see from the proof of Proposition 2.1 that, in fact, ZG is β-center-by-metabelian. It follows from [4] that whenever a semiprime R is β-solvable, it must be center-by-metabelian, that is, satisfy [[R, R]β , [R, R]β ], R]β = 0.

2.5

Acknowledgments

This research was supported in part by NSERC.

References [1] Yu.A. Bahturin, D. Fischman, and S. Montgomery, On the generalized Lie structure of associative algebras, Israel J. Math. 96 (1996), 27-48. [2] Y. Bahturin, A. Mikhalev, V. Petrogradsky, and M. Zaicev, Inﬁnite dimensional Lie superalgebras, Expos. Math. vol. 7, Walter de Gruyter, Berlin, 1992. [3] Y. Bahturin and S. Montgomery, PI-envelopes of Lie superalgebras, Proc. AMS 127 (1999), 2829-2939. [4] Yu.A. Bahturin, S. Montgomery, and M. Zaicev, Generalized Lie solvability of associative algebras, Proceedings of the International Workshop on Groups, Rings, Lie and Hopf Algebras (St. John’s), Kluwer, 1-23, 2003. [5] Yu. A. Bahturin and M.M. Parmenter, Generalized commutativity in group algebras, Canad. Math. Bull. 46 (2003), 14-25.

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Yu.A. Bahturin and M.M. Parmenter [6] I.M. Isaacs and D.S. Passman, A characterization of groups in terms of degrees of their characters, Paciﬁc J. Math. 15 (1965), 877-903. [7] Cesar Polcino Milies and Sudarshan K. Sehgal, An Introduction to Group Rings, Algebras and Applications, Kluwer, 2002. [8] S.K. Sehgal, Topics in Group Rings, Marcel Dekker, New York, 1978.

Chapter 3 Lyndon-Shirshov Words, Gr¨ obner-Shirshov Bases, and Free Lie Algebras L. Bokut and E.S. Chibrikov Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences, Novosibirsk

3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A short history of free Lie algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A right-normed basis of a free Lie algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Another bracketing of Lyndon-Shirshov words . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A right-normed basis of a free Lie superalgebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Free Lie conformal algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Simple Lie algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

17 18 23 26 27 28 29 33 34

Abstract This is a survey of some old and new results on free Lie algebras. We start with a more or less detailed description of Shirshov’s papers on the subject that were done in the 1950s to the beginning of the 1960s. These papers had been published in Russian and have not been translated into English with the exception of his paper, Certain algorithmic problems for Lie algebras, Siberian Math. Z., 1962, which was translated in the ACM SIGSAM Bulletin, 1999; it was the beginning of what is now called the Gr¨ obner-Shirshov bases theory. Section 3.2 deals with this history. Sections 3.3–3.6 are devoted to new bases of a free Lie (super)algebra and to some information on the basis of a free Lie conformal algebra. These results are due to E. S. Chibrikov. Section 3.7 focuses on the Gr¨ obnerShirshov bases of simple Lie (super)algebras and Kac-Moody algebras. Key words: free Lie (conformal) (super)algebra, Hall-Shirshov bases, Lazard-Shirshov elimination, Lyndon-Shirshov word, Gr¨ obner-Shirshov bases, composition (diamond) lemma, algorithmic problems, simple Lie (super)algebras, Kac-Moody algebras 2000 MSC: 17-02, 17B01, 17B20, 17B65, 17B67, 17B69

3.1

Introduction

This is a survey of some old and new results on free Lie algebras. We start with a more or less detailed description of Shirshov’s papers on the subject that were done in the 1950s to the beginning of the 1960s. These papers had been published in Russian and have not been translated into English with the exception of his paper Certain algorithmic problems of Lie algebras, which appeared in Siberian Math. Z. 1962 [84] and was translated in the ACM SIGSAM Bulletin, 1999; it was the beginning of what is now called the Gr¨ obner-Shirshov

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bases theory. Details of these papers are still poorly known in the west. Let us state some of the most striking examples of that. 1) What is called the “(Lazard) elimination process” (Lazard, 1960 [58], see [6], [74]) had been invented much earlier in the supposedly very well known Shirshov’s paper (Mat. Sb., 1953, [81]) (this paper is about subalgebras of free Lie algebras). 2) The series of bases of a free Lie algebra were invented by Shirshov in his thesis (1953, MSU; published in Algebra Logika, 1962 [83]), were also rediscovered many years later. They are called “Hall Bases” (see [74]). 3) What is called the “Lyndon basis” of a free Lie algebra (see [59], [74]) had been published in the same year (Shirshov, Mat. Sb., 1958, [82], and Chen-Fox-Lyndon, Ann. Math., 1958 [29]). 4) What is now called the “noncommutative Gr¨ obner bases theory” was started in the Shirshov’s SMZ, 1962 paper, mentioned above. 5) Shirshov’s composition lemma for Lie polynomials (his SMZ, 1962 paper above) is essentially more general then Bergman’s diamond lemma (1978, [5]). Section 3.2 deals with this history. Sections 3.3–3.6 are devoted to new bases of a free Lie (super) algebra and to some information on a basis of a free Lie conformal algebra. These results are due to E. S. Chibrikov. Section 3.7 is on the Gr¨ obner-Shirshov bases of simple Lie (super)algebras.

3.2

A Short History of Free Lie Algebras

Free Lie algebras theory has the prehistory and the history. As for the prehistory one can mention papers by Poincar´e, Campbell, Baker, Hausdorﬀ, Magnus, Ph. Hall, Zassenhaus, G. Birkhoﬀ, Witt, Thrall, Brandt (see Reutenauer’s book [74]). The current history started with the paper by M. Hall (1950 [39]) and followed with Shirshov’s thesis (1953 [80]) and his papers (1953 [81]), (1958 [82]), (1962 [83]), (1962, [84]), (1962 [85]). In these works the theory took on the modern form including diﬀerent linear bases, descriptions of subalgebras (an analog of Nielsen-Schreier theorem on subgroups of free groups), the theory of Gr¨ obner-Shirshov bases in free Lie algebras, the theory of onerelator Lie algebras (an analog of the celebrated Magnus theory of one-relator groups), the theory of the free product of Lie algebras (the linear basis of the free product and negative solution of Kurosh-Witt conjecture on subalgebras of the free product), a theorem on embedding any countably generated (restricted) Lie algebra into 2-generated (restricted) Lie algebra with the same number of deﬁning relations (an analog of Higman-NeumannNeumann’s theorem for groups and Malcev’s theorem for associative algebras). Some works parallel to Shirshov’s had been done independently by Witt [90], Chen-Fox-Lyndon [29], Lazard [58], and Sch¨ utzenberger [78]. In M. Hall’s paper [39], the ﬁrst linear basis of a free Lie algebra has been found. It consists of the so-called Hall (nonassociative) words in a linearly ordered alphabet X. Let us designate letters xi ∈ X as Hall words. Suppose that we deﬁne the ordered set of Hall words of length < n such that the order agrees with the degree (length) function. Then a word (w) = ((u)(v)) of degree n is called a Hall word if (u), (v) are Hall words, (u) > (v), and if (u) = ((u1 )(u2 )) then (u2 ) ≤ (v). Let us order Hall words of the degree ≤ n in an arbitrary way that agree with the degree function and with the order of Hall words of degree < n. We will call any of such order as a deg-order. Then the set of Hall words forms a linear basis of the free Lie algebra Liek (X) over X and a ﬁeld (or any commutative ring) k.

Lyndon-Shirshov words, Gr¨ obner-Shirshov bases, and free Lie algebras

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In Shirshov’s thesis (1953 [80]), two principal results have been proved. The ﬁrst one, published in (1953 [81]), is a theorem that any subalgebra of a free Lie algebra is again free. Later on, this theorem was also published by Witt [90] and now it is known as the Shirshov-Witt theorem. Shirhshov’s thesis and the paper above also include the following result of some importance (see [81] Lemma 3): Words [xi xn1 ] = ((xi x1 )...x1 )(n − times), where x1 , xi ∈ X, i > 1, n ≥ 0, are independent in the free Lie algebra Lie(X) (it means that they freely generate a subalgebra of Lie(X)). See also Theorem 0.6 of Reutenauer [74], where this result was credited to Lazard [58] and associated with the notions “Lazard set” and “Lazard elimination process”. We call them the “Lazard-Shirshov set” and the “Lazard-Shirshov elimination process.” A further principal result of Shirshov’s thesis [80], published in [83], was a series of linear bases of a free Lie algebra. Shirshov words diﬀer from Hall words in one crucial point: to design Shirshov words by induction, one can use any order of (nonassociative) words that possesses the property ((v)(w)) > (w) instead of deg-order. Shirshov’s bases are more general than bases by Meier-Wunderly (1951), Witt (1956), Sch¨ utzenberger (1958), and were rediscovered later by Michel (1974) and Viennot (1978) (see [74]). Shirshov words are called Hall words in Reutenauer [74]. We call them Hall-Shirshov words. One of the ﬁrst application of Shirshov’s bases scheme has been done in (Bokut, 1963 [8]). It gives a basis of L = Lie(X) that agrees with poly-nilpotent series, L ⊃ Ln1 ⊃ (Ln1 )n2 ⊃ · · · ⊃ (. . . (Ln1 )n2 . . . )nk ⊃ . . . , where all ni ≥ 2. For ni = 2, i ≥ 1, this is the derived series, and the basis gives rise a basis of free solvable Lie algebra (see also Reutenauer [73]). In the paper (1958 [82]), Shirshov proved the following main results: Let L = Lie(xi , i = 1, 2, . . . | fk (xi ) = 0, k = 1, 2, . . . ) be a countably (or ﬁnitely) generated Lie algebra over a ﬁeld. Then it is embeddable into two-generated Lie algebra, M (L) = Lie(a, b | fk ([aabi ab]) = 0, k = 1, 2, . . . ), with the same number of deﬁning relations, where [aabi ab] = ((a(. . . (ab) . . . b))(ab)) (itimes). A monomorphism we look for has the form xi → [aabi ab]. The same result is proved to be valid for restricted Lie algebras in the sense of Jacobson [43]. In fact, these results are analogous to several results for rings and groups. First, it is an analog of Higman-Neumann-Neumann’s result [41] that any countable group is embeddable into a two-generated group with the same number of deﬁning relations. Secondly, it is an analog of Malcev’s result [63] that any countable generated associative algebra is embeddable into a two generated associative algebra with the same number of deﬁning relations. To prove that outstanding results, Shirshov [82] created a new linear base of a free Lie algebra, invented a notion of elimination of the leading monomial of a Lie polynomial g in a Lie polynomial f , and began the theory of Gr¨ obner-Shirshov bases for Lie and associative algebras. Let us give some deﬁnitions and formulate some results [82]. Let X = {ai , i ∈ I} be a totally ordered alphabet, 1, u, v, . . . be associative X-words, (u), [u], (v), . . . be nonassociative X-words, where u, v . . . are results of omission brackets in (u), [u], (v) . . . . Deﬁne

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u > v in the lexicographical way assuming that u is not a beginning of v. A word u is called regular if for any proper presentation u = u1 u2 , we have u > u2 u1 . Note: Shirshov did not know at the time that this notion had been introduced a bit earlier by Lyndon [61]. Now regular words are called Lyndon words or Lyndon-Shirshov words. Let us extend the order, assuming v > vw for regular words v, vw, where w = 1. We have u = u1 u2 > u2 for any regular u and its proper end u2 (see Remark, Ch. 2 [82]). Deﬁne regular a nonassociative word [u] by induction on the degree function using the following rules: 1) Letters ai = [ai ] are regular words, 2) u is a regular associative word, 3) If [u] = [[v][w]], then [v], [w] are regular (clearly v > w), 4) If in addition [v] = [[v1 ][v2 ]], then v2 ≤ w. There is 1-1 correspondence u ↔ [u] between regular associative and nonassociative words. Here [u] is standard bracketing of u. Shirshov used the “Lazard-Shirshov elimination process” above (down-to-up process: rewrite an associative regular word in a new alphabet {xi xni0 , i > i0 , n ≥ 0} ordering as regular words, put brackets xi xni0 → [xi xni0 ], and do the same with the new word). Chen-Fox-Lyndon [29] used up-to-down process, [u] = [[v][w]], where w is the longest proper regular end of u (then v is a regular word as well). It is proved that regular nonassociative words form a linear basis of Liek (X) (here k may be any commutative ring). The crucial observation was that the leading associative word of [u], as a Lie polynomial in the free associative algebra kX, is equal to u. Now it follows from (see [82]) the famous Witt dimension formula for dim Lie(x1 , . . . , xn )k [89], as well as the famous Friedrichs criterion to be Lie polynomial [36]. It is worth mentioning that in the same paper, Shirshov gave a new conceptual proof of the Poincare-Birkhoﬀ-Witt theorem (see below), and probably the ﬁrst explicit proof that the algebra of Lie polynomials of free associative algebra over any commutative ring k is the free Lie algebra over k. Clearly the above basis is in the Hall-Shirshov series of bases (with the order [u] > [v] iﬀ u > v). For many years this basis was known as Shirshov regular words basis, see, for example, P.M. Cohn (1964 [33]), Bokut (1972 [10]), Bakhturin (1982 [1]), BakhturinMikhalev-Petrogradskij-Zaizev (1992 [2]). After Lothair [59] we knew that essentially the same basis had been done by Chen-Fox-Lyndon in their paper (1958 [29]). This basis is called the Lyndon basis in Lothair [59] and Reutenauer [74]. We call it the Lyndon-Shirshov basis. Let us deﬁne as in [82] the special Lie bracketing of a regular word u = avb with a ﬁx regular subword v, [u]v . Here a, b are some words. The standard bracketing [u] has a form [a[vc]d], where cd = b (it means that a bracket in [u] has a form [vc] for some (possibly empty) associative word c; here we omit all another brackets of [u]). For example, let u = a3 a2 a2 a1 a1 a1 a2 , v = a2 a2 a1 . Then [u] = [a3 [a2 a2 a1 a1 a1 ]a2 ], where [vc] = [a2 a2 a1 a1 a1 ] = [a2 [[[a2 a1 ]a1 ]a1 ]], c = a1 a1 . The associative word c can be uniquely presented in the form c = c1 . . . cs , where c1 ≤ · · · ≤ cs are regular words. Let us have in [u] the following substitution: [vc] → [. . . [[v][c1 ]] . . . [cs ]]. The resulting word [u]v is the special Lie bracketing of u relative to the subword v (see Lemma 4 [82]). The main property of this bracketing is that the leading associative word of [u]v as polynomial in kX is equal to u (see the proof of Lemma 10 [82]). It follows that for any monic Lie polynomials f, g such that f = u = avb, g = v, the leading associative word of the Lie polynomial f − [agb]v

Lyndon-Shirshov words, Gr¨ obner-Shirshov bases, and free Lie algebras

21

is less than u (here [agb]v = [a[v]b]v |[v]→g ). This result is essentially used in the proof of the main Lemma 10 [82]. Actually, Lemma 10, i.e., the proof of the main theorem of [82], is the beginning of Shirshov’s Gr¨ obner-Shirshov bases theory for associative and Lie polynomials (see below). The transformation f → f − [agb]v is called the Lie elimination of the leading word of g in f . The proof of the above main results gives rise to proof of the Poincare-Birkhoﬀ-Witt theorem and the Poincare-Birkhoﬀ-Witt-Jacobson theorem [43] for Lie and restricted Lie algebras correspondingly. In his paper (1962 [84]), Shirshov proved the analog of the Magnus theory of one-relator groups [64], [65]: Let L = Lie(xi , i = 1, . . . n| s(x1 , . . . , xn ) = 0) be any one-relator Lie algebra over a ﬁeld. Then the word problem for L is algorithmically decidable (using the above-mentioned elimination of the leading word of the relation). If xn is entered into s, then a subalgebra of L, generated by x1 , . . . , xn−1 , is free. To prove these important results, Shirshov (1962 [84]) invented a new (Gr¨ obner-Shirshov, see [20], [21]) theory for free Lie algebras Lie(X) based on Lyndon-Shirshov words, on a notion of composition of Lie polynomials, and on the composition lemma. Later on, an analogous theory for commutative polynomials with s-polynomial and the main theorem was created by B. Buchberger (thesis (1964 [24]), publication [25]). Note that Shirshov was dealing with Lie polynomials via noncommutative (associative) polynomials kX, Lie(X) ⊂ kX (recall that Lie(X) is generated by X under a Lie bracket [xy] = xy − yx). To be more precise making the composition (s-polynomial by Buchberger’s later terminology [24], [25]) of two monic Lie polynomials f, g relative to some word (ambiguity by Bergman’s later terminology [5]) w, where w = f¯v = u¯ g , deg(f¯) + deg(¯ g ) > deg(w) (f¯ is the leading associative monomial of f as noncommutative polynomial), he makes ﬁrst the associative composition (f, g)w = f v − ug, and then he puts the special Lie bracketing [f v]f¯ − [ug]g¯ in order to get the Lie composition. The same applied to a notion of the elimination of the leading monomial of one monic Lie polynomial in another: as we have mentioned above, he ﬁrst makes the elimination in an associative sense (f → f − ugv, where f = ugv) and then puts the special Lie bracketing (f → f − [ugv]g ). He proved the composition lemma. Let S be a set of Lie polynomials, S ∗ be a completion of S under all possible multiple compositions. If f ∈ Id(S) then f contains as a subword s for some s ∈ S ∗ . We are using Shirshov’s original notation for S ∗ . The process of adding to S nontrivial multiple compositions is now called the Shirshov algorithm for ﬁnding the Gr¨ obner-Shirshov basis or Gr¨ obner-Shirshov set S ∗ . It is an algorithm of the Knuth-Bendix type [51]. A diﬀerence is that in the Knuth-Bendix algorithm for Lie algebras one should present a Lie algebra as a quotient of an absolutely free nonassociative algebra rather than as a quotient of a free Lie algebra. To make a comparison, in the case of group to apply the Knuth-Bendix algorithm one should present a group as a quotient of an absolutely free nonassociative groupoid rather than as a quotient of a free group. Shirshov had further assumed that S is stable in the sense that for any f, g ∈ S ∗ degree of a composition (f, g)w (after eliminations of leading words) is greater then degrees of f, g (or (f, g)w is zero after the elimination of leading words). He did not use this condition in the proof. On the other hand, stability of S guaranties that S ∗ is a recursive set (for a obner-Shirshov basis of Id(S). ﬁnite S). Set S ∗ is now known as the Gr¨ Of course the composition (diamond) lemma for associative polynomials ([11], [5]) is just a simple version of the case of Lie polynomials (we do not need extra Lie bracketing). In 1972 ([10]), the ﬁrst author formulated Shirshov’s composition lemma for Lie polynomials in the modern form.

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Let S ⊂ Lie(X) be a set of Lie polynomials that is closed under compositions (i.e., any composition, including f − [ugv]g¯ , of polynomials of S goes to zero under elimination of the leading words of S, or S = S ∗ ). If f ∈ Id(S), then f¯ contains a subword s for some s ∈ S. As the result, the set {[u], u = v¯ sw, s ∈ S} of S-irreducible (regular) words is a linear basis of the quotient algebra Lie(X)/Id(S) = Lie(X|S), the so-called irreducible basis of the algebra. A set S ⊂ Lie(X), that is closed under composition, is called a Gr¨ obner-Shirshov set in Lie(X) or a Gr¨ obner-Shirshov basis in Lie(X). Of course, S is not a basis in Lie(X) in any sense; it is a special set of generators of the ideal Id(S). Such S is now called a Gr¨ obnerShirshov basis of the algebra Lie(X)/Id(S) = Lie(X|S), too. Note that a Gr¨ obner-Shirshov “basis” of an algebra L and corresponding irreducible (linear) basis of L both depend on a ﬁxed set of generators of L (and on an order of generators). An irreducible linear basis of obner-Shirshov basis of an algebra Lie(X)/Id(S) = Lie(X|S) can be deﬁned without a Gr¨ Id(S). Namely, it is the set of regular words [u] in X such that u does not contain leading subwords s for s ∈ Id(S). For commutative polynomials Shirshov’s composition lemma is essentially the same as Buchberger’s theorem. As we mentioned above, for noncommutative polynomials this lemma is essentially the same as Bergman’s diamond lemma [5]. Shirshov’s composition lemma for noncommutative polynomials was explicitly formulated in (Bokut, [11]): Let S ⊂ kX be a set of noncommutative polynomials that is closed under compositions (any composition of polynomials of S including f − ugv goes to zero under the elimination of leading words of S, or S = S ∗ ). If f ∈ Id(S), then f¯ contains a subword s for some s ∈ S. As the result, the set {u, u = v¯ sw, s ∈ S} of S-irreducible words is a linear basis of the quotient algebra kX/Id(S) = k < X|S >. Set S ⊂ kX that is closed under composition is called a Gr¨ obner-Shirshov set in kX or a Gr¨ obner-Shirshov basis in kX though again it is not a basis in kX in any sense. It seems the only reason to call such a set basis is Hilbert’s basis theorem and that S ia a special basis of the Id(S). All in all, Shirshov’s paper [84] must be viewed as the very beginning of the Gr¨ obner bases theory not only for Lie polynomials, but also for noncommutative polynomials. In his paper (1962 [85]), Shirshov deﬁned a notion of the free product of Lie algebras, found a linear basis of the free product, and gave an example showing that the Kurosh subalgebra theorem [56] is not valid for subalgebras of the free product of Lie algebras. Again, it was just an application of his Gr¨ obner-Shirshov bases theory for Lie algebras. The ﬁrst author in (1962 [7]) used the Shirshov method to prove the following result. Any Lie algebra A is embeddable into an algebraically closed (a.c.) Lie algebra L = L(A) (meaning that any Lie equation f (x1 , . . . , xn ) = 0 with coeﬃcients in algebra L has a solution in it; here f is a nontrivial element of the free product of L and the free Lie algebra Lie(x1 , . . . , xn , . . . )). By the way, at the same time the ﬁrst author posed problems of the existence of a.c. associative algebras and groups (see [9]) (in the case of groups, the nontriviality of an equation f = 1 means that f is not conjugate to a constant). The ﬁrst problem was solved positively by L. Makar-Limanov [62]. Recently P. Kolesnikov [52] has found a diﬀerent example of this sort. The second problem was also solved positively by G. Brodskii [23]. In the paper mentioned above (Bokut [10] 1972), the Shirshov method had been used to prove the following result: Let M be a recursively enumerable set of natural numbers, and let LM = Lie(a, b, c, a1 , b1 , c1 | [abn c] = [a1 bn1 c1 ], n ∈ M ) be the recursively presented Lie algebra. Then LM is eﬀectively embeddable into a ﬁnitely

Lyndon-Shirshov words, Gr¨ obner-Shirshov bases, and free Lie algebras

23

presented Lie algebra LM . If M is not recursive, then LM has the algorithmically undecidable word (equality) problem, so LM does. It was a weak analog of the Higman theorem for groups [40] (that any recursively presented group is embeddable into a ﬁnitely presented one) and a solution of Shirshov’s problem [83] on the undecidability of the word problem for ﬁnitely presented Lie algebras. In the same paper, the ﬁrst author posed problems as to whether the Higman embedding theorem is valid for Lie algebras and associative algebras. The second problem was solved positively by V. Ya. Belyaev [4]. The ﬁrst problem is still open. An algebra K has a property (α), where α is an inﬁnite cardinal, if the cardinality of K is α, and K is a union of an increasing series of the length ω of subalgebras, such that all factors of the series have dimension α. Using the Shirshov method, the ﬁrst author proved the following results ([11], [12]): Let A, K1 , K2 , K3 , K4 be some Lie (associative) algebras such that the cardinality of A does not exceed α and all Ki have the property (α). Then A is embeddable into a simple Lie (associative) algebra, which is a sum of K1 , K2 , K3 , K4 . This is an analog of Ph. Hall’s embedding theorems for groups [38]. S. Shelah [79], in connection with [11], gave an example of associative algebra of the cardinality ℵ1 that does not possess the property (ℵ1 ). In particular, algebras Ki from the previous theorem can be nilpotent. So, any Lie (associative) algebra is embeddable into a simple Lie (associative) algebra, which is a sum of four (actually this number can be decreased to three [11], [12]) nilpotent subalgebras. Kegel’s result [49] shows that this number cannot be decreased to 2: an associative algebra that is a sum of two nilpotent subalgebras is nilpotent. For Lie algebras, the analog result on solvability of an algebra that is a sum of two nilpotent subalgebras is only known for ﬁnite dimensional case over a ﬁeld of a characteristic that is not 2 (V. V. Panyukov [67], P. Zusmanovich [92]; for the characteristic 2 there is a counter example (A. P. Petravchuk [68]). On the other hand, recently A. Smoctunowicz [88] (see also [60]) has proved that there exists a simple nil associative (Lie) algebra (over a countable ﬁeld; for uncountable ﬁelds the problem is still open). This is the solution for the longstanding Levitski’s problem.

3.3

A Right-Normed Basis of a Free Lie Algebra

In this chapter, according to [30] and [31] we construct a basis of a free Lie algebra that consists of right-normed words, i.e., the words that have the following form: [ai1 [ai2 [. . . [ait−1 ait ] . . .]]], where aip are free generators of the Lie algebra. Let X = {ai |i ∈ I} be a linearly ordered set. We order the free monoid X of all associative words in X lexicographically: u < 1 for every nonempty word u, and, by induction, u < v if u = ai u , v = aj v and either ai < aj or ai = aj and u < v . In particular vw < v for w = 1. We will also use another order u ≺ v, u just less v: u < v such that u = u b, where b is a letter, and u ≥ v. ∗ Let w = ai1 ai2 . . . ait belong to X. Denote by w = ait . . . ai2 ai1 the inverse of w and w = [ai1 [ai2 [. . . [ait−1 ait ] . . .]]] the right normed word of Lie(X). We say that a is the leading letter of w, if a ≥ aij for all 1 ≤ j ≤ t and a = aip for some 1 ≤ p ≤ t. LEMMA 3.1 Let v be some word of X and w1 = (av)m1 au1 , w2 = (av)m2 au2 be words with the same leading letter a, where uj ≺ v, mj ≥ 0 and v, uj do not contain the letter a for j = 1, 2. Then the condition w1 > w2 implies that w1 is not a preﬁx of w2 .

24

L. Bokut and E. Chibrikov

We will often use an alphabet Az , where z ∈ { (av)nj auj | uj ≺ v }. Lemma 3.1 lets us introduce a linear order for words of the alphabet Az so that the inverse replacing Az by z holds this order. LEMMA 3.2 (i) Every associative Lyndon-Shirshov word w of X has a unique representation in the following form: w = (av)(av)n1 au1 u1 (av)n2 au2 u2 . . . (av)nt aut ut ,

(3.1)

where v, uj , uj do not contain the leading letter a of w, nj ≥ 0, uj ≺ v for all 1 ≤ j ≤ t. (ii) Let w(1) = A(av)n1 +1 au1 u1

(1)

A(av)n2 au2 u2

(1)

. . . A(av)nt aut ut

(1)

(3.2)

be the result of replacing in (3.1) all the subwords (av)nj auj by A(av)nj auj (here n1 := n1 + 1), and for all nonempty uj = aj1 . . . ajk , where ajs ∈ X, replacing ajs by Aajs . If we order the set Y = Az | z ∈ {(av)nj auj , ajs } as follows: Az1 ≥ Az2 if and only if z1 ≥ z2 , then the word w(1) is an associative Lyndon-Shirshov word in the alphabet Y . (iii) Let u(1) = Ax1 Ax2 . . . Axk be an associative Lyndon-Shirshov word in the alphabet Y . Then the word u = x1 x2 . . . xk is an associative Lyndon-Shirshov word in the alphabet X. DEFINITION 3.1 An associative word w of X is a normal word if w∗ > w for even |w| and w∗ ≥ w for odd |w|, where |w| denotes the length of w. PROPOSITION 3.1

A word w is a normal word if and only if w = u1 v ∗ ,

(3.3)

where u1 ≺ v. REMARK 3.1

The condition on u1 determines the presentation (3.3) uniquely.

DEFINITION 3.2 Using the induction on the number of occurrences of the leading letter in words we deﬁne a subset TX of X as follows. If w ∈ X has one occurrence of the leading letter a and w = va for some word v, then we assume that w belongs to TX . Let w be a word of X in which the leading letter occurs more than once and w may be presented in the following form: w = ut+1 (av)nt aut ut . . . (av)n2 au2 u2 (av)n1 au1 v ∗ a,

(3.4)

where ut+1 , v, uj , uj do not contain the leading letter a of w, nj ≥ 0, uj ≺ v for all 1 ≤ j ≤ t. In particular, the word u1 v ∗ is a normal word.

Lyndon-Shirshov words, Gr¨ obner-Shirshov bases, and free Lie algebras

25

The conditions on uj determine the presentation (3.4) uniquely. All words of X that do not have the presentation (3.4) do not belong to TX . Suppose that for any alphabet Z we have already deﬁned all words of TZ with the number of occurrences of the leading letter less than in w. Let w ˜ = ut+1 (av)nt aut ut . . . (av)n2 au2 u2 (av)n1 +1 au1 . We rewrite w ˜ by replacing all the subwords (av)nj auj by A(av)nj auj (here n1 := n1 + 1) and replacing ais by Aais for all nonempty uj = aj1 . . . ajk(j) . Then we obtain the new word (1)

w(1) = ut+1 A(av)nt aut ut

(1)

. . . A(av)n2 au2 u2

(1)

A(av)n1 +1 au1

(3.5)

(1) = Aaj1 . . . Aajk(j) . We order of the alphabet Y = Az | z ∈ {(av)nj auj , ais } , where uj the set Y in the same way as before. Hence the number of occurrences of the leading letter in w(1) is less than in w and we deﬁne that w belongs to TX if and only if w(1) belongs to TY . For example, the word w2 (av)n1 au1 v ∗ a of (3.4) belongs to TX . Let w = w3 (av)n2 au2 u2 (av)n1 au1 v ∗ a be a word of (3.4) for t = 2. Then w belongs to TX if and only if (av)n1 +1 au1 > (av)n2 au2 or (av)n1 +1 au1 = (av)n2 au2 and u2 is a normal word. PROPOSITION 3.2 Let LS(X) be the set of all associative Lyndon-Shirshov words in the alphabet X, and let TX be the set of words from Deﬁnition 3.2. Then there is a bijective map ψ : TX → LS(X) that does not change the content of words. Let us construct the map ψ. If a word w of TX has one occurrence of the leading letter a, then by Deﬁnition 3.2 w = ua for some u ∈ X and we deﬁne ψ(w) = au∗ . Pick w ∈ TX in which the leading letter occurs more than once. Hence by Deﬁnition 3.2, w may be written in the form (3.4). Suppose that for any alphabet Z the map ψ : TZ → LS(Z) is deﬁned for words with the number of occurrences of the leading letter less than in w and ψ is a content-preserving map. Let w(1) be the word (3.5) of the alphabet Y = Az | z ∈ {(av)nj auj , ais } . Since w(1) ∈ TY we have that A(av)n1 +1 au1 is the leading letter of w(1) . Hence the number of occurrences of the leading letter in w(1) is less than in w. By the induction hypothesis ψ(w(1) ) of LS(Y ) is deﬁned. Let ψ(w(1) ) = Az1 Az2 . . . Azk , where Azi ∈ Y for all 1 ≤ i ≤ k. The condition ψ(w(1) ) ∈ LS(Y ) implies that Az1 is the leading letter of ψ(w(1) ), therefore, z1 = (av)n1 +1 au1 . We deﬁne ψ(w) = z1 z2 . . . zk . For example, if w2 (av)n1 au1 v ∗ a ∈ TX , then ψ w2 (av)n1 au1 v ∗ a = (av)n1 +1 au1 w2∗ . For w3 (av)n2 au2 u2 (av)n1 au1 v ∗ a ∈ TX we have ∗ ψ w3 (av)n2 au2 u2 (av)n1 au1 v ∗ a = (av)n1 +1 au1 u2 (av)n2 au2 w3∗ ,

26

L. Bokut and E. Chibrikov

if (av)n1 +1 au1 > (av)n2 au2 , and ˜2 w3∗ , ψ w3 (av)n2 au2 u2 (av)n1 au1 v ∗ a = (av)n1 +1 au1 v˜(av)n2 au2 u ˜2 v˜∗ is a normal word, u ˜2 ≺ v˜. if (av)n1 +1 au1 = (av)n2 au2 , and u2 = u REMARK 3.2 The deﬁnition of TX may be given as the image of the map ψ −1 : LS(X) → TX . We construct ψ −1 as follows. Suppose that w ∈ LS(X) has one occurrence of the leading letter a, then w = av for some word v and we obtain that ψ −1 (w) = v ∗ a. Let w be a word of LS(X) in which the leading letter occurs more than once. By Lemma 3.2 (i) w has a presentation (3.1). Let w(1) be a word (3.2). Since the number of occurrences of the leading letter in w(1) is less than in w ψ −1 (w(1) ) of TX is already deﬁned by induction hypothesis. If ψ −1 (w(1) ) = Az1 . . . Azk−1 Azk , where zk = (av)n1 +1 au1 is the leading letter of ψ −1 (w(1) ) by induction, then we obtain that ψ −1 (w) = z1 . . . zk−1 (av)n1 au1 v ∗ a. THEOREM 3.1 Let Lie(X) be the free Lie algebra over a ﬁeld k generated by X, and let TX be the set of words from Deﬁnition 3.2. Then the following words, [ai1 [ai2 [. . . [ait−1 ait ] . . .]]], where ai1 ai2 . . . ait belong to TX , form a linear basis of Lie(X). Theorem 3.1 is proved by induction on the number of occurrences of the leading letter in words. To prove the linear independence we use a bijective map ψ : TZ → LS(Z). Importantly, the map ψ does not change the content of words. REMARK 3.3

3.4

Theorem 3.1 is valid for a commutative ring k with unit.

Another Bracketing of Lyndon-Shirshov Words

In this section we assume as above that Lie(X) is the subspace of the free associative algebra kX, which is generated by X under the Lie bracketing [xy] = xy − yx. Deﬁne the bracketing [[u]] of associative Lyndon-Shirshov words u. For any word of LS(X) we have the presentation (3.1). If a word w of LS(X) has one occurrence of the leading letter a, then w = av for some v ∈ X and we take [[w]] = {av}, where {b1 . . . bk } = [. . . [b1 b2 ] . . . bk ] is a left normed bracketing. Let w be a word of LS(X) written in the form (3.1) in which the leading letter a occurs more than once and w(1) will be the word (3.2) of LS(Y ). By induction on the number of occurrences of the leading letter in Lyndon-Shirshov that [[w]] is the result of replacing in [[w(1) ]] all words, [[w(1) ]] is deﬁned and we suppose the letters A(av)nj auj by the words ({av})nj {auj } (here n1 := n1 + 1) and Aajs by ajs . Observe that the bracketing [[u]] is not equal to the standard Lyndon-Shirshov bracketing [u]. For example, let a, b ∈ X and a > b. Then w = aabbb ∈ SX and [w] = [a[[[ab]b]b]] = [[w]] = [[[a[ab]]b]b].

Lyndon-Shirshov words, Gr¨ obner-Shirshov bases, and free Lie algebras

27

Recall that for any f ∈ kX we denote by f¯ the maximal associative word of f under the deg-lex order. PROPOSITION 3.3

For any word w of LS(X) we have [[w]] = w.

LEMMA 3.3 Let TX be the set of words from Deﬁnition 3.2 and ψ : TX → LS(X) be the map deﬁned in Proposition 3.2. Then for any w ∈ TX we have w = ±[[ψ(w)]] + αi vi , i

where αi ∈ k and vi < w. From Lemma 3.3 and Proposition 3.3 we obtain the following. THEOREM 3.2 algebra Lie(X). REMARK 3.4

The set

[[w]] | w ∈ LS(X) forms a linear basis for the free Lie

Theorem 3.2 is valid for a commutative ring k with unit.

The next result is a new version of Shirshov’s composition lemma for Lie polynomials. THEOREM 3.3 Let X be a well-ordered alphabet. The set S ⊂ Lie(X) is a Gr¨ obnerShirshov set if and only if the set [[w]] | w is an associative Lyndon-Shirshov S − irreducible word is a linear basis for Lie(X)/Id(S) = Lie(X|S).

3.5

A Right-Normed Basis of a Free Lie Superalgebra

In this section we will construct a right-normed basis for free Lie superalgebras that is a generalization of a right-normed basis of a free Lie algebra. Let X = X0 X1 , where X0 and X1 are sets of even and odd letters correspondingly. We determine that the word u is strongly just less than the word v ( u ≺s v ), if u < v and u=u b, for some word u > v and some letter b, i.e., the word u is a proper preﬁx of v. DEFINITION 3.3 Using the induction on the number of occurrences of the leading letter in words, we deﬁne a subset QX of X as follows. If a word w of X has one occurrence of the leading letter a and w = va for some word v, then we assume that w belongs to QX . Let us consider the word w2 aw1 a with two occurrences of the leading letter a, where w1 = u1 v ∗ for some words u1 , v ∗ . The word w2 aw1 a belongs to QX if and only if either 1) av is an even word and u1 ≺ v or 2) av is an odd word and u1 = v or u1 ≺s v.

28

L. Bokut and E. Chibrikov

Let w be a word in which the leading letter a occurs more than twice and w may be presented in the following form: w = ut+1 (av)nt aut ut . . . (av)n2 au2 u2 (av)n1 au1 v ∗ a,

(3.6)

uj ,

u1 do not contain the leading letter a for all where uj ≺ v, nj ≥ 0, the words v, uj , 2 ≤ j ≤ t + 1 and either 1) av is an even word, n1 ≥ 0, u1 ≺ v or 2) av is an odd word, n1 = 0, u1 = v or u1 ≺s v. The conditions on uj determine the presentation (3.6) uniquely. All words of X that do not have the presentation (3.6) do not belong to QX . Suppose that for any alphabet Z = Z0 ∪ Z1 , where Z0 and Z1 are sets of even and odd words, respectively, we have already deﬁned all words of QZ with the number of occurrences of the leading letter less than in w. Assume that u1 = v ( in this case n1 = 0 and av is an odd word ) in the presentation (3.6) of w. If u2 = 1, then w ˜ = ut+1 (av)nt aut ut . . . (av)n3 au3 u3 (av)n2 +2 au2 ,

(3.7)

¯i2 , where a ¯i2 ∈ X, then if u2 = u2 a ai2 ). w ˜ = ut+1 (av)nt aut ut . . . (av)n2 au2 u2 (av)(av¯

(3.8)

w ˜ = ut+1 (av)nt aut ut . . . (av)n2 au2 u2 (av)n1 +1 au1 .

(3.9)

If u1 = v, then

We rewrite w ˜ by replacing all the subwords (av)mj auj by A(av)mj auj (in Eq.(3.8) we ai2 ), for all nonempty uj = aj1 . . . ajk(j) replacing ais by consider that u2 := u2 and u1 := v¯ Aais . Then we obtain a new word w(1) of the alphabet Y = Az | z ∈ {(av)mj auj , ajs } . Let (1) uj = Aaj1 . . . Aajk(j) . If w ˜ is one of Eqs. (3.7), (3.8) or (3.9), then (1)

w(1) = ut+1 A(av)nt aut ut (1)

(1)

w(1) = ut+1 A(av)nt aut ut (1)

w(1) = ut+1 A(av)nt aut ut

. . . A(av)n3 au3 u3

(1)

(1)

(1)

. . . A(av)n2 au2 u2

. . . A(av)n2 au2 u2

A(av)n2 +2 au2 ,

(1)

(1)

(3.10)

A(av)av¯ai2 ,

(3.11)

A(av)n1 +1 au1 ,

(3.12)

respectively. If z is an even word, then Az is called an even letter, otherwise Az is called an odd letter. We order the set Y in the following way: Az1 ≥ Az2 if and only if z1 ≥ z2 . Hence the number of occurrences of the leading letter in w(1) is less than in w and we deﬁne that w belongs to QX if and only if w(1) belongs to QY . THEOREM 3.4 Let Lie(X 0 ; X1 ) be the free Lie superalgebra over a ﬁeld k (chark = 2, 3) generated by X = X0 X1 , QX be the set of words from Deﬁnition 3.3. Then the following words, [ai1 [ai2 [. . . [ait−1 ait ] . . .]]], where ai1 ai2 . . . ait belong to QX , form a linear basis of Lie(X0 ; X1 ).

Lyndon-Shirshov words, Gr¨ obner-Shirshov bases, and free Lie algebras

3.6

29

Free Lie Conformal Algebras

The origin of vertex and conformal algebras is in physics (see [3], [45], [46]). R. Borcherds [22] gave the formal deﬁnition of vertex algebras and used it for resolving the moonshine conjecture in the ﬁnite groups (see [35]). Conformal algebras were introduced, along with others, by V. Kac in his book [45]. It is a valuable tool in studying vertex algebras. R. Borcherds [22] announced an existence of free vertex algebras. This result was proved by M. Roitman in his paper [75]. M. Roitman [75], [76] also constructs the basis of a free vertex algebra. The linear basis of a free Lie conformal algebras is unknown. In this chapter we follow [32] to construct the basis for the subspace of a free Lie conformal algebra generated by words of length two. Let us recall some deﬁnitions and results. DEFINITION 3.4 A Lie conformal algebra is a linear space A over ﬁeld k (chark = 0) with a linear operator D : A → A and a sequence of bilinear products (n) : (A, A) → A, n ∈ Z+ = {0, 1, 2, ...} such that for any a, b ∈ A the following properties hold: C1. (Locality axiom) a(n)b = 0 for any n ≥ N (a, b); C2. D{a(n)b} = {Da}(n)b + a(n){Db}; C3. {Da}(n)b = −na(n − 1)b; C4. (Jacobi identity) {a(n)b}(m)c =

n

(−1)s

s=0

n {a(n − s){b(m + s)c} − b(m + s){a(n − s)c}}; s

C5.(skew-commutative) a(n)b = −

n+s 1 s≥0 (−1) s!

Ds {b(n + s)a}.

The minimal N (a, b) with property C1 is called the locality number of a, b ∈ A. In fact (see, for example [75]), that every free Lie conformal algebra is embedded into a formal Laurant series L[[z, z −1 ]], where L is a Lie algebra. THEOREM 3.5 Let C(N, B) be a free Lie conformal algebra generated by B with respect to the constant locality function N (a, b) ≡ N for a, b ∈ B. Then the following words form a linear basis for the subspace of C(N, B) generated by words of length two: a0 (n0 ){Di b}, a0 , b ∈ B and either 1) n0 + i + 1 ≤ 2) n0 + i + 1 >

3.7

N −1, a0 >b N, a0 ≤b

or

N −1, a0 >b N, a0 ≤b

and 0 ≤ n0 <

N 2.

Simple Lie Algebras

In this section we survey the results of the Gr¨ obner–Shirshov bases of the simple ﬁnite dimensional Lie (super)algebras and the (simple inﬁnite dimensional) aﬃne Kac–Moody Lie algebras of the untwisted types. Recall that by deﬁnition, a Gr¨ obner-Shirshov basis of

30

L. Bokut and E. Chibrikov

a Lie algebra Lie(X)/Id(S) = Lie(X|S) presented by generators and deﬁning relations is a Gr¨ obner-Shirshov basis of the ideal Id(S) in the free Lie algebra Lie(X). This topic is closely related to investigations begun by Lalonde and Ram, who found [57] bases of simple complex ﬁnite dimensional Lie algebras that consist of Lyndon–Shirshov words, which they call Lyndon words, in the Weyl generators. They obtain bases of the simple ﬁnite dimensional Lie algebras, which is irreducible in the sense of Shirshov’s composition lemma, which they call standard. It turns out that one can ﬁnd an irreducible basis of these algebras without having to ﬁnd their Gr¨ obner–Shirshov basis. Lalonde and Ram work with the presentation of a simple ﬁnite dimensional Lie algebra by its Weyl generators and its Serre relations; their methods rely on the properties of simple root systems. In other words, the combined techniques involve both the combinatorics of free Lie algebras and the classical combinatorics of root systems. Another pleasant feature of [57] is that its authors present the ﬁnal result, the basis Lyndon–Shirshov words, in a graphical form (as graphs with marked vertices), making it easier to understand and use. Bokut and Klein [13, 14, 16, 18] have found the Gr¨ obner–Shirshov bases of the simple ﬁnite dimensional Lie algebras in Weyl generators. Note that if we ﬁx a linear basis A of a Lie algebra L, then the multiplication table of L in A is a Gr¨ obner-Shirshov basis of L in generators A. So, the problem was to ﬁnd Gr¨ obner-Shirshov bases of the simple algebras in Weyl generators. In the above papers, the authors use solely the methods of free Lie algebras, in essence, adding nontrivial compositions to the Serre relations by Shirshov’s algorithm (which remained behind the scenes though, only its ﬁnal results were written out). Simultaneously, irreducible bases of the simple ﬁnite dimensional Lie algebras have been found. Note that to prove that those systems of relations are closed under the compositions, the authors need the converse statement in Shirshov’s composition lemma for Lie algebras, or actually in the main corollary to it. We formulate it as the compositiondiamond lemma to stress that the converse statement was explicitly given for the ﬁrst time by G. Bergman [5]. LEMMA 3.4 (Composition-Diamond Lemma) (Cf. Prop. 2 in [13]) A set S ⊂ Lie(X) is a Gr¨ obner–Shirshov set if and only if Irr(S) = {[u] | u = v¯ sw, s ∈ S} is a linear basis of the quotient algebra Lie(X)/Ideal(S) = Lie(X | S). In addition, for the classical series An , Bn , Cn , and Dn of Lie algebras, Bokut and Klein use the easy fact that sln+1 (k), so2n+1 (k), sp2n (k), and so2n (k) satisfy Serre relations (in the Weyl generators). For the exceptional Lie algebras, they use a much more diﬃcult fact: the precise values of the dimensions of the algebras G2 , F4 , E6 , E7 , and E8 . The explicit form of the Gr¨ obner–Shirshov bases of the simple ﬁnite dimensional Lie algebras has the following corollaries: 1. The Lie algebras deﬁned by the Serre relations of types An , Bn , Cn , and Dn over a ﬁeld k of characteristic diﬀerent from 2 for An and Dn and diﬀerent from 2 and 3 for Bn and Cn are isomorphic, respectively, to sln+1 (k), so2n+1 (k), sp2n (k), and so2n (k). 2. The dimensions of the exceptional Lie algebras G2 , F4 , E6 , E7 , and E8 can be computed directly by the combinatorial method of Shirshov. It suﬃces to check that the resulting systems of relations are closed under the compositions, which is easy to do manually or with the aid of a computer program, see [37].

Lyndon-Shirshov words, Gr¨ obner-Shirshov bases, and free Lie algebras

31

Let us give the results of [13, 14] for the classical types. Let X = {x1 . . . xn }, Y = {x1 . . . yn }, and H = {h1 . . . hn } be Weil generators, C = aij n×n be a Cartan matrix. The Serre relations S(C) are [hi hj ] = 0 for i > j, [xi yi ] = hi , [xi yj ] = 0 for i = j, [hi xj ] = aji xj , [hi yj ] = −aji yj , [(ad xi )1−aji (xj )] = 0, for i = j, [(ad yi )1−aji (yj )] = 0, for i = j, where ad xi (xj ) = [xi xj ]. We will use the notation xii = xi and xij = [xi . . . xj ] for n ≥ i > j ≥ 1, and similarly for yij . THEOREM 3.6

([14]) The Gr¨ obner–Shirshov basis of the Lie algebra LS(C) = Lie(X ∪ H ∪ Y | S(C))

over a ﬁeld k of characteristics diﬀerent from 2 for C = An , Dn , and diﬀerent from 2 and 3 for C = Bn , Cn , consists of the Serre relations together with the following relations among xij , i ≥ j, and the same relations among yij : An :[xij xi−1 ] = 0, [xij xi,j−1 ] = 0 for n ≥ i > j ≥ 1, Bn :[xij xi−1 ] = 0 for n ≥ i > j ≥ 1, [xij xi,j−1 ] = 0 for n − 1 ≥ i ≥ j ≥ 2, [xni [xnj xn,j−1 ]] = 0 for n ≥ i ≥ j ≥ 2, [[xni xnj ]xn,i−1 ] = 0 for n ≥ i > j ≥ 1, [[xni xnj ][xni xn,j−1 ]] = 0 for n ≥ i > j ≥ 2, Cn :[xij xi−1 ] = 0 for n − 1 ≥ i > j ≥ 2, [xij xi,j−1 ] = 0 for n − 1 ≥ i ≥ j ≥ 2, [xni [xn−1,j xn,i−1 ]] = 0 for n − 1 ≥ j ≥ i ≥ 1, [xni [xn−1,j xn,i−1 ]] = 0 for n − 1 ≥ j ≥ i ≥ 2, [xni [xni xn−1 ] = 0 for n ≥ i ≥ 1, [[xni xn−1,j ][xni xn−1,j−1 ]] = 0 for n ≥ j > i ≥ 1, Dn (let xni = [xn xn−2 . . . xi ], n − 2 ≥ i) : [xij xi−1 ] = 0 for n − 1 ≥ i > j, [xi,j xn−2 ] = 0 for n − 2 ≥ j, [xij xi,j−1 ] = 0 for n ≥ i ≥ j ≥ 2 with j ≤ n − 2 when i = n, [xni xn−1,i ] = 0 for n − 2 ≥ i, [[xni xn−1,j ]xn−1 ] = 0 for n − 1 ≥ j > i, [xni [xni xn−1 ] = 0 for n − 2 ≥ i, [[xni xn−1,j ]xn,i−1 ] = 0 for n − 1 ≥ j > i ≥ 2, [[xni xn−1,j ][xni xn−1,j−1 ]] = 0 for n − 1 ≥ j > i + 1.

32

L. Bokut and E. Chibrikov

The (irreducible) bases of Lalonde and Ram and of Bokut and Klein use diﬀerent orderings of the generators (the simple roots), and thus it is not easy to compare them. Let us note right away that in an unpublished and still incomplete series of papers by A. N. Koryukin [53, 54] and A. N. Koryukin and K. P. Shum [55], Gr¨ obner–Shirshov bases and respective irreducible bases of the simple ﬁnite dimensional Lie algebras have been found for arbitrary orderings of the simple roots (the generators X). In particular, this uniﬁes and generalizes the results of [57] and [13, 14, 16, 18]. Like Lalonde and Ram, Koryukin and Shum use the techniques both of free Lie algebras and of root systems. Bokut and Malcolmson have noticed [19] that a set S ⊂ Lie(X) ⊂ kX of Lie polynomials is a Gr¨ obner–Shirshov set in Lie(X) iﬀ S is a Gr¨ obner–Shirshov set in kX. This statement applies to universal enveloping algebras. Take a Lie algebra L = Lie(X | S), and let U (L) = X | S be its associative enveloping algebra. By the previous statement, if S is a Gr¨ obner–Shirshov basis for L, then S considered as a set of associative polynomials is a Gr¨ obner–Shirshov basis for U (L), and vice versa. Therefore, Bokut and Klein have simultaneously found the Gr¨ obner–Shirshov bases for the universal enveloping algebras of the simple ﬁnite dimensional Lie algebras. Bokut and Malcolmson [15] have obtained the following information about the Gr¨ obner– Shirshov bases of the universal enveloping algebras of Kac–Moody Lie algebras. If C = aij n×n is a symmetrizable Cartan matrix (aii = 2, aij ≤ 0 for i = j, and di aij = dj aji for some nonzero d1 . . . dn ), let g(C) be the Kac–Moody Lie algebra [44] deﬁned by the Serre relations corresponding to C; the only diﬀerence from the usual Serre relations given above is that [hi xj ] = di aij xj and [hi yj ] = −di aij yj . Thus, we may write g(C) = Lie(X ∪ H ∪ Y | K ∪ T ∪ S + ∪ S − ), where K are the Serre relations involving the highest powers of hi ; T are the Serre relations involving the lowest powers of hi , namely, [xi yj ] = δij hi ; S + are the relations among xi ; and S − are the relations among yi . Let g(C)+ = Lie(X | S + ) and g(C)− = Lie(X | S − ). Let S +c be a Gr¨ obner–Shirshov basis of g(C)+ , and similarly S −c for g(C)− . A theorem in [15] states that if C is a generalized Cartan matrix, and U (C) is the universal enveloping algebra of the Kac–Moody Lie algebra g(C), then S +c ∪ K ∪ T ∪ S −c is the Gr¨ obner–Shirshov basis of U (C). This implies that U (C) = U − (C) ⊗ k[H] ⊗ U+ (C), where U + (C) = U (g(C)+ ), or g(C) = g(C)− ⊕ k[H] ⊕ g(C)+ . The last property of Kac–Moody algebras is well known, but it is usually proved by an entirely diﬀerent method [44]. The other main result of [15] concerns ﬁnding the Gr¨ obner–Shirshov basis of the quantum enveloping algebra Uq (An ) of type An for q 8 = 1. It is shown that the Jimbo relations of Uq (An )+ (see [91]) are closed under compositions. Consequently, this provides another proof of the Yamane [91] and Rosso [77] result on a linear basis of Uq (An )+ . Note that to

Lyndon-Shirshov words, Gr¨ obner-Shirshov bases, and free Lie algebras

33

save space, Bokut and Malcolmson use Yamane’s lemma, but as in the case of exceptional Lie algebras, the closedness of Jimbo relations can be proved directly. Thus, Yamane’s results [91] follow from the closedness of the Jimbo relations. Bokut, Kang, Lee, and Malcolmson have generalized [17] the results of [13, 14, 15, 19] to the case of Lie superalgebras; A. A. Mikhalev [66] had earlier proved the Lie superalgebra analog of Shirshov’s composition lemma. The main results of [17] are the following: 1. Theorem 2.8: A subset S ⊂ Lie(X0 ∪ X1 ) ⊂ kX0 ∪ X1 of the free Lie superalgebra obner–Shirshov basis in on the sets X0 and X1 of even and odd generators is a Gr¨ obner–Shirshov basis in kX0 ∪ X1 . Lie(X0 ∪ X1 ) iﬀ S is a Gr¨ 2. Theorem 3.5: If C is a generalized Cartan matrix, g(C) is the Kac–Moody Lie superalgebra with deﬁning relations S(C) = S + ∪K∪T ∪S − , and S +c and S −c are the closures of S + and S − with respect to taking compositions, then S(G)c = S +c ∪ K ∪ T ∪ S −c is a Gr¨ obner–Shirshov basis of g(C) and U (g(C)). In particular, U (g) = U (g− ) ⊗ U (H) ⊗ U (g+ ), and g = g− ⊕ H ⊕ g+ . 3. Explicit Gr¨ obner–Shirshov bases for the classical Lie superalgebras: A(m, n), m, n > 0; B(m, n), m, n > 0; B(0, n), n > 0; C(n), n ≥ 2; D(m, n), m ≥ 2, n > 0. E. N. Poroshenko have found [69, 70, 71, 72] the Gr¨ obner–Shirshov bases of the untwisted (1) (1) (1) (1) aﬃne Kac–Moody Lie algebras An , Bn , Cn , and Dn , oﬀering a new algorithm for ﬁnding the Gr¨ obner–Shirshov bases for these algebras diﬀerent from Shirshov’s method of adding nontrivial compositions. Poroshenko’s algorithm uses isomorphisms of the untwisted Kac–Moody Lie algebras with the corresponding loop algebras [44]; the explicit Gr¨ obner– Shirshov bases for the simple ﬁnite dimensional Lie algebras from [13, 14] are also essential. By the results of [15, 17] mentioned above it suﬃces to ﬁnd the Gr¨ obner–Shirshov bases of the positive parts of the Kac–Moody algebras in question. It turns out that the Gr¨ obner– Shirshov bases of these algebras have complicated structure and include many dozens of (usually inﬁnite) series of relations: , 72 series for A(1)+ n

108 series for

Bn(1)+ ,

54 series for Cn(1)+ ,

140 series for

Dn(1)+ .

The restrictions on the characteristics of the ground ﬁeld here are the same as in [13, 14]: (1) (1) The characteristic is diﬀerent from 2 for An and Dn , and it is diﬀerent from 2 and 3 (1) (1) for Bn and Cn . V. K. Kharchenko [50] has extended the results of Yamane [91] and Rosso [77] on the PBW basis for the quantum universal enveloping algebra Uq (An ), ﬁnding a basis for Uq (An ), Uq (Bn ), Uq (Cn ), and Uq (Dn ). It consists of the Lalonde–Ram words [57] in which skew commutators replace the Lie bracket. Moreover, in the same paper Kharchenko has introduced the notion of a quantum universal enveloping algebra for every Lie algebra deﬁned by generators and relations. It generalizes the Drinfeld–Jimbo quantization of the simple ﬁnite dimensional Lie algebras and the Kang quantization of Kac–Moody Lie algebras [47]. Kharchenko has also found a basis of his quantum enveloping algebras, which he calls the Poincar´e–Birkhoﬀ–Witt basis. This basis admits a monomial crystallization along the lines of Kashiwara.

34

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3.8

Acknowledgments

We are grateful to the referee for many valuable remarks.

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[68] A. P. Petravchuk. Lie algebras decomposable as sum of an Abelian and a nilpotent subalgebras. Ukr. Mat. Zh. 40 (1988), No. 3, 385-388. (1)

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Chapter 4 Classiﬁcation of Solvable 3-Dimensional Lie Triple Systems Thomas B. Bouetou ´ Ecole Nationale Sup´erieure Polytechnique, Yaound´e, Cameroon

4.1 4.2

4.3 4.4 4.5 4.6

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Lie triple systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 Solvable and semisimple Lie triple system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.2 Problem setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Classiﬁcation of Lie triple system of dimension 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Classiﬁcation of solvable Lie triple systems of dimension 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . Classiﬁcation of splitting tree-dimensional Lie triple systems . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

41 42 43 45 45 46 51 52 53

Abstract We intend to give the classiﬁcation of solvable and splitting Lie triple systems and ﬁnd the solvable Lie algebras associated. It turns out that, up to isomorphism, there exist 7 nonisomorphic canonical Lie triple systems, and 6 nonisomorphic splitting canonical Lie triple systems Key words: Lie triple systems, solvable and splitting Lie triple systems, Lie algebras 2000 MSC: 17A40, 17B30, 17B35, 17B40, 17D99

4.1

Introduction

A Lie triple system LTS is a ﬁnite-dimensional vector space in which is deﬁned a ternary operation, verifying some conditions, namely, the Jacobi identity and the derivation identity. They were ﬁrst introduced by Jacobson [13]. Later on Lister [17] presented a structure theory and the classiﬁcation of simple LTS. Yamaguty [35] obtained from a total geodesic space triple algebras that were the generalization of LTS. Loos [19] showed that a symmetric space can be seen as a quasigroup, and Sabinin [27, 28] showed that a quasigroup can be seen as a homogeneous space. In particular, any Bol loop under the left action derivative gives a LTS, i.e., the description of the inﬁnitesimal structure of a smooth Bol loop contains an LTS. This fact gives the idea of investigation of LTS since the use of LTS also appears in the ordinary diﬀerential equation, functional analysis. In this paper our main objective is to give the classiﬁcation of solvable and splitting LTS right up to isomorphism. Our approach is based on enveloping Lie algebras of an LTS. The Lie algebra obtained from the standard embedding of an LTS is an enveloping Lie algebra. If an LTS is solvable its enveloping Lie algebra is solvable; conversely if a Lie algebra is solvable then the LTS obtained is solvable. Considering the classiﬁcation of solvable Lie algebras, we will carry out the classiﬁcation of

41

42

Thomas B. Bouetou

LTS of small dimension. We will organize this chapter as follows: The ﬁrst section is the introduction; in the second section we will give the deﬁnitions and some results about LTS. In the third section we will give the classiﬁcation of LTS of dimension two; in the fourth section we will give the classiﬁcation of solvable LTS and ﬁnally, in the last section, we will give the classiﬁcation of splitting LTS.

4.2

Lie Triple Systems

DEFINITION 4.1 The vector space M (ﬁnite-dimensional over the ﬁeld of real numbers R) with trilinear operation (x,y,z) is called an LTS if the following identities are veriﬁed: (x, x, y) = 0 (x, y, z) + (y, z, x) + (z, x, y) = 0 (x, y, (u, v, w)) = ((x, y, u), v, w) + (u, (x, y, v), w) + (u, v, (x, y, w)). Let M be an LTS, a subspace D ⊂ M is called a subsystem if (D, D, D) ⊂ D, and is called an ideal, if (D, M, M) ⊂ D. The ideal is the kernel of the homomorphism of the LTS (see [19, 32]). Example 4.1

For a typical way of constructing of an LTS, see [19, 32].

Let G be a Lie algebra (ﬁnite-dimensional over the ﬁeld of real numbers R) and σ be its involutive automorphism, then G = G + G− , where σ|G+ = Id and σ|G− = −Id, as any element x from G can be written in the form: x=

1 1 (x + σx) + (x − σx), 2 2

where x + σx ∈ G+ , x − σx ∈ G− and G+ ∩ G− = 0. The following inclusions hold: [G+ , G+ ] ⊂ G+ ,

[G+ , G− ] ⊂ G− ,

[G− , G− ] ⊂ G+ .

Then the subspace G− turns into an LTS with respect to the operation (x, y, z) = [[x, y], z]. The inverse construction [19]. Let M be an LTS deﬁned by h(X, Y ) : Z −→ (X, Y, Z), a linear transformation of the space M into itself where X, Y, Z ∈ M. Let H be a subspace of the space of linear transformations of the LTS M whose elements are the transformations of the form h(X, Y ). The vector space, G = M H, becomes a Lie algebra relative to the commutator [A, B] = AB − BA, [A, X] = −[X, A] = AX; [X, Y ] = h(X, Y ) where A, B ∈ H, X, Y ∈ M.

Classiﬁcation of solvable 3-dimensional Lie triple systems

43

Let us deﬁne the mapping σ with the condition σ(A) = A, if A ∈ H and σ(X) = −X, X ∈ M, then σ is an involutive automorphism of a Lie algebra G = M H. The algebra G constructed above from the LTS, is called a universal enveloping Lie algebra of the LTS M. DEFINITION 4.2 M −→ M such that

The derivation of the LTS M, is the linear transformation d :

(X, Y, Z)d = (Xd, Y, Z) + (X, Y d, Z) + (X, Y, Zd). One can verify that the set d(M) of all the derivations of the LTS M is a Lie algebra of the linear transformations acting on M. DEFINITION 4.3 The embedding of an LTS M into a Lie algebra G is the linear injection R : M −→ G such that (X, Y, Z) = [[X R , Y R ], Z R ]. The embedding R of the LTS M into the Lie algebra G is called canonical, if the envelope of the image of the set MR in the Lie algebra G coincides with G and H does not contain trivial ideals of Lie algebra G. Let us note that if the LTS M is a subset of the Lie algebra G, then (X, Y, Z) = [[X, Y ], Z] and [M, M] is a subalgebra of the Lie algebra G, hence M + [M, M] is a Lie subalgebra of G and the initial embedding R can be considered as canonical in MR + [MR , MR ]; this leads us to formulate the following proposition: PROPOSITION 4.1 For any ﬁnite-dimensional LTS M over R, there exists one and only one, up to automorphism, canonical embedding to the Lie algebra.

4.2.1

Solvable and semisimple Lie triple system

Following [17]: let Ω be an ideal of the LTS M, we assume Ω(1) = (M, Ω, Ω) and, Ω = (M, Ω(k−1) , Ω(k−1) ). (k)

PROPOSITION 4.2 [17] For all natural numbers k, the subspace Ω(k) is an ideal of M and we have the following inclusions: Ω ⊇ Ω(1) ⊇ ........ ⊇ Ω(k) PROOF (Ω(1) , M, M) = ((M, Ω, Ω), M, M) ⊆ ((M, Ω, M), Ω, M) + [[[M, Ω], [,Ω]], M] according to the deﬁnition of an LTS, (Ω(1) , M, M) ⊆ (Ω, Ω, M) + [(M, Ω, M), [M, Ω]] ⊆ (M, Ω, Ω) + (M, Ω, Ω) = Ω(1) , that means Ω(1) is an ideal of M, furthermore Ω(k) = (Ω(k−1) )(1) , hence each Ω(i) is an ideal in M. DEFINITION 4.4 The ideal Ω of an LTS M is called solvable, if there exists a natural number k such that Ω(k) = 0.

44

Thomas B. Bouetou

PROPOSITION 4.3 [17] If Ω and Θ are two solvable ideals of an LTS M then Ω + Θ is also a solvable ideal in M. PROOF Using the deﬁnition of an LTS, the following inclusion holds: (Θ + Ω)(1) ⊆ (M, Θ, Θ) + (M, Ω, Ω) + (M, Θ, Ω) + (M, Ω, Θ) ⊆ Θ(1) + Ω(1) + Θ ∩ Ω. Assume for every natural number k the following inclusion holds: (Θ + Ω)(k) ⊆ Θ(k) + (k) Ω + Θ ∩ Ω. By induction let us prove that it holds for (k + 1): (Θ + Ω)(k+1) = (M, (Θ + Ω)(k) , (Θ + Ω)(k) ) ⊆ (M, Θ(k) + Ω(k) + Θ ∩ Ω, (Θ ∩ Ω)) ⊆ Θ(k+1) + Ω(k+1) + Θ ∩ Ω; hence the result. DEFINITION 4.5 ideal of the LTS M.

The radical of an LTS denoted by R(M) is the maximal solvable

An LTS M is called semisimple if R(M) = 0. THEOREM 4.1 [17] If R is a radical in M then (M/R) is semisimple. And if Ω is an ideal in M such that (M/R) is semisimple then Ω ⊃ R. PROPOSITION 4.4 [17] The enveloping Lie algebra of a solvable LTS is solvable. And if an LTS has some solvable enveloping Lie algebra, it is solvable. THEOREM 4.2 G is semisimple.

If M is a semisimple LTS, then the universal enveloping Lie algebra

THEOREM 4.3 [2] Let M be an LTS and G = M h be its canonical enveloping Lie algebra and let r be the radical of the Lie algebra G. In G there exists a semisimple subalgebra P supplementary to r such that M = M M

(direct sum of vectors spaces),

where M = M ∩ r − radical of the LT S M M = M ∩ P − semisimple subalgebra of LT S M h = h h

(direct sum of vector spaces) h = h ∩ r

and h = h ∩ P are subalgebra in h r = M h P = M h .

Classiﬁcation of solvable 3-dimensional Lie triple systems

4.2.2

45

Problem setting

Let M be an LTS and dim M = 3. To be consistent with the above theorem the following cases are possible: 1. Semisimple case: M is a semisimple LTS (in fact simple). About the classiﬁcation of such LTS see [1, 7, 17] 2. Solvable case: M is a solvable LTS. The classiﬁcation of such a system is given in Section 4.5. 3. Splitting case: M = M1 M2 , where M ≡ R is a solvable ideal of dimension 1 in R and M2 is a semisimple LTS of dimension 2. This type of LTS is considered in the last section.

4.3

Classiﬁcation of Lie Triple System of Dimension 2

For a better survey of such LTS, we will write their trilinear operation in a special form. Let M be a 2-dimensional LTS, we write the trilinear operation in the form (X, Y, Z) = β(X, Y )Y − β(Y, Z)X, where β : V × V −→ R is a symmetric form. The choice of the basis V =< e1 , e2 > can reduce the symmetric form to the corresponding matrix form: α0 , 0ν where α, ν = ±1; 0. By introducing the notation of the derivation, Dx,y : M −→ M z −→ (x, y, z) h = {Dx,y }x,y∈M , G = M h is a canonical enveloping Lie algebra of the LTS M. Let M =< e1 , e2 > then, h = {tDx,y }t∈R , e1 D = (e1 , e2 , e1 ) = β(e1 , e1 )e2 e2 D = (e1 , e2 , e2 ) = −β(e2 , e2 )e1 G =< e1 , e2 , e3 >, where [e1 , e2 ] = e3 , [e1 , e3 ] = −e1 D, [e2 , e3 ] = −e2 D. Therefore we can have, up to isomorphism, the following ﬁve cases: 1. (Spherical Geometry)

G/h ∼ = so(3)/so(2)

10 01

,

46

Thomas B. Bouetou 2. (Lobatchevski Geometry)

−1 0 0 −1

,

G/h ∼ = sl(2, R)/so(2) 3. LTS with noncompact subalgebra h

1 0 0 −1

,

G/h ∼ = sl(2, R)/R 4. Solvable case • a)

β=

10 00

,

e1 · e2 = e3 , e1 · e3 = e2 (This is a Lie algebra G of type g3,5 (p = 0) in [21]) • b)

β=

−1 0 0 0

,

e1 · e2 = e3 , e1 · e3 = −e2 (This is a Lie algebra G of type g3,4 (h = −1) in [21]) 5. Abelian case β = 0 G/h ∼ = (R)2 / {0}

4.4

Classiﬁcation of Solvable Lie Triple Systems of Dimension 3

Let M be a solvable LTS of dimension 3, and G h its canonical enveloping Lie algebra, then G is solvable, in particular, G possesses a characteristic ideal G = [G, G] G, σG = G , G ∩ M = M = (M, M, M); furthermore h ⊂ G since h = [M, M], then G = [G, G] = M + h, where M M. Possible situations: 1. dim M = 0. Then [h, M] = M = {O}, that means h G is an ideal; that is why h = {O} (since G is an enveloping Lie algebra) and M = R ⊕ R ⊕ R. In this case, the LTS is Abelian and we denote it (type I). 2. dim M = 1. Choosing the base e1 , e2 , e3 in M such that M =< e1 > and M = M + < e2 , e3 >, we will introduce in consideration the linear transformations A, B, C : M −→ M, deﬁned as ⎛ ⎞ ⎛ ⎞ abc αβγ A = (e1 , e2 , −) = ⎝ 0 0 0 ⎠ , B = (e2 , e3 , −) = ⎝ 0 0 0 ⎠ , 000 0 00

Classiﬁcation of solvable 3-dimensional Lie triple systems ⎛ ⎞ x −α − c y 0 0⎠, C = (e3 , e1 , −) = ⎝ 0 0 0 0

47

and a skew-symmetric form deﬁned as Φ(−, −) : M × M −→ R, such that (x, y, e1 ) = Φ(x, y)e1 . The dimension of M is 3, that is why there exists z ∈ M, z = 0, such that Φ(−, z) = 0. The following cases are possible: • b.I. The skew-symmetric form Φ is nonzero and z is parallel to e1 (z e1 ), then in the base e1 , e2 , e3 the skew-symmetric form Φ has the corresponding matrix: ⎛

⎞ 0 0 0 ⎝0 0 δ ⎠, 0 −δ 0 where δ = 0. Adjusting e3 to α = 1, a = x = 0 and

1 δ e3 ,

then Φ(e2 , e3 ) = 1, Φ(e3 , e2 ) = −1, so that

⎛

⎞ 0bc A = (e1 , e2 , −) = ⎝ 0 0 0 ⎠ , 000

⎛

⎞ 1βγ B = (e2 , e3 , −) = ⎝ 0 0 0 ⎠ , 000 ⎛ ⎞ 0 −1 − c y 0 0⎠. C = (e3 , e1 , −) = ⎝ 0 0 0 0

The veriﬁcation of the deﬁning relations of LTS shows that, with accuracy as to the choice of the vector basis e2 and e3 , it is possible to aﬀord the following realization of the operators A, B, C as ⎛

⎞ 100 A = 0, B = (e2 , e3 , −) = ⎝ 0 0 0 ⎠ , 000 ⎛ ⎞ 0 −1 0 C = (e3 , e1 , −) = ⎝ 0 0 0 ⎠ . 0 0 0 (type VII) • b.II. The skew-symmetric form Φ is nonzero and z is not parallel to e1 ; let z = e2 , then ⎛

⎞ 0bc A = (e1 , e2 , −) = ⎝ 0 0 0 ⎠ , 000

⎛

⎞ 0βγ B = (e2 , e3 , −) = ⎝ 0 0 0 ⎠ , 000 ⎛ ⎞ −1 −c y C = (e3 , e1 , −) = ⎝ 0 0 0 ⎠ . 0 0 0

The veriﬁcation of the deﬁning relations of LTS shows that the indicated case has no realization.

48

Thomas B. Bouetou • b.III. The skew-symmetric form Φ is trivial. By completing the vector e1 with the arbitrarily chosen vector e2 and e3 up to the base, it is possible to realize the operators A, B, and C as ⎛

⎞ 0bc A = (e1 , e2 , −) = ⎝ 0 0 0 ⎠ , 000

⎛

⎞ 0βγ B = (e2 , e3 , −) = ⎝ 0 0 0 ⎠ , 000 ⎛ ⎞ 0 −c y C = (e3 , e1 , −) = ⎝ 0 0 0 ⎠ . 0 0 0

The veriﬁcation of the deﬁning relations of LTS shows that by a suitable choice of basis vectors e2 , e3 , the following realization of operators A, B, C is possible: – i) Abelian Type (Type above) – ii) A = C = 0,

⎛

⎞ 001 B = (e2 , e3 , −) = ⎝ 0 0 0 ⎠ . 000

(Type II) This LTS is obtained by a direct multiplication of a LTS of dimension two < e1 , e2 >, by an Abelian one-dimensional < e3 >. – iii) ⎛ ⎞ 0 ±1 0 A = (e1 , e2 , −) = ⎝ 0 0 0 ⎠ , 0 0 0 B=C=0. (Type III) – iv)

⎛

⎞ 0 ±1 1 A = (e1 , e2 , −) = ⎝ 0 0 0 ⎠ , 0 0 0

B=0,

⎛

⎞ 0 −1 ∓1 C = (e3 , e1 , −) = ⎝ 0 0 0 ⎠ . 0 0 0 (Type IV)

3. dim M = 2 in particular, M is a subsystem of dimension two in M. One can consider (refer to Section 4.4 ) ∀a, b, c ∈ M (a, b, c) = β(a, c)b − β(b, c)a,

where β=

±1 0 0 0

and M is a two-dimensional Abelian ideal in M. In the ﬁrst case, choosing the base M =< e1 , e2 , e3 > such that M =< e1 , e2 >, the operations of the LTS are reduced to

Classiﬁcation of solvable 3-dimensional Lie triple systems ⎛

⎞ 0 ±1 x A = (e1 , e2 , −) = ⎝ 0 0 y ⎠ , 0 0 0

49

⎛

⎞ αγμ B = (e2 , e3 , −) = ⎝ β δ ν ⎠ , 0 00 ⎛ ⎞ κ −x − α ξ C = (e3 , e1 , −) = ⎝ χ −y − β β ⎠ . 0 0 0

The veriﬁcation of the deﬁning relations of LTS leads to the contradiction with the condition that dim M = 2. Let M =< e1 , e2 > be a two-dimensional Abelian ideal, and e3 the vector completing e1 , e2 up to the basis. Then ⎛

⎞ 00a A = (e1 , e2 , −) = ⎝ 0 0 b ⎠ , 000

⎛

B ⎛

κ C = (e3 , e1 , −) = ⎝ χ 0

⎞ αγμ = (e2 , e3 , −) = ⎝ β δ ν ⎠ , 0 00 ⎞ −a − α ξ −b − β β ⎠ . 0 0

Deforming the vector e1 of the subspace < e1 , e2 > accordingly, the matrix A can be reduced to the form a = b = 0 or a = 1, b = 0. The veriﬁcation of the deﬁning relations of the LTS in the second case leads to the following realization of the operators A, B, C: ⎛

⎞ 001 A = 0, B = (e2 , e3 , −) = ⎝ 0 0 0 ⎠ , 000 ⎛ ⎞ 000 C = (e3 , e1 , −) = ⎝ 0 0 1 ⎠ . 000 ( Type V) ⎛

A = 0,

⎞ 01 0 B = (e2 , e3 , −) = ⎝ 0 0 ±1 ⎠ , 00 0

C = 0.

(Type VI) In conclusion the conducted examination we have the following. theorem: THEOREM 4.4 Let M =< e1 , e2 , e3 > be a solvable LTS of dimension 3, G its canonical enveloping Lie algebra (solvable), and let A, B, C : M −→ M the linear transformations of the form: A = (e1 , e2 , −), A = (e1 , e2 , −), B = (e2 , e3 , −), C = (e3 , e1 , −). Up to isomorphism, one can ﬁnd the possibility of the following types: • Type I. M is Abelian Lie triple system.

50

Thomas B. Bouetou • Type II

⎛

A = 0,

⎞ 001 B = (e2 , e3 , −) = ⎝ 0 0 0 ⎠ . 000

C = 0,

G =< e1 , e2 , e3 , e4 > four-dimensional nondecomposable nilpotent Lie algebra with deﬁning relations, [e2 , e3 ] = e4 , [e3 , e4 ] = −e1 . (This is g4,1 algebra in the Mubaraczyanov classiﬁcation [21]). • Type III. M is a direct product of a two-dimensional solvable LTS < e1 , e2 >, and a one-dimensional Abelian < e3 > : ⎛

⎞ 0 ±1 1 A = (e1 , e2 , −) = ⎝ 0 0 b ⎠ , 0 0 0

B = 0,

C = 0.

G =< e1 , e2 , e3 , e4 > four-dimensional solvable and decomposable Lie algebra, with deﬁning relations: [e1 , e2 ] = e4 , [e2 , e4 ] = ±e1 . Moreover, G =< e1 , e2 , e4 > ⊕ < e3 >, where < e1 , e2 , e4 > is three-dimensional solvable Lie algebra (g3,4\5 in Mubaraczyanov classiﬁcation [21]). • Type IV ⎛

⎞ 0 ±1 1 A = (e1 , e2 , −) = ⎝ 0 0 0 ⎠ , 0 0 0

⎛

⎞ 0 −1 ±1 C = (e3 , e1 , −) = ⎝ 0 0 0 ⎠ . 0 0 0

B = 0,

G =< e1 , e2 , e3 , e4 > is four-dimensional solvable and nondecomposable Lie algebra, with deﬁning relations: [e1 , e2 ] = e4 , [e1 , e3 ] = ±e4 ,

[e2 , e4 ] = ±e1 [e3 , e4 ] = −e1

(algebra g4,5\6 in Mubaraczyanov classiﬁcation [21]). • Type V

⎛

⎞ 01 0 B = ⎝ 0 0 ±1 ⎠ , 00 0

A = C = 0.

G =< e1 , e2 , e3 , e4 > four-dimensional solvable nondecomposable Lie algebra with deﬁning relations: [e2 , e3 ] = e4 , [e2 , e4 ] = −e1 [e3 , e4 ] = ∓e2 (algebra g8\9 in Mubaraczyanov classiﬁcation [21]).

Classiﬁcation of solvable 3-dimensional Lie triple systems

51

• Type VI ⎛

A = 0,

⎞ 001 B = (e2 , e3 , −) = ⎝ 0 0 0 ⎠ , 000

⎛

⎞ 000 C = (e3 , e1 , −) = ⎝ 0 0 1 ⎠ . 000

G =< e1 , e2 , e3 , e4 , e5 > is ﬁve-dimensional solvable nondecomposable Lie algebra, with deﬁning relations: [e1 , e2 ] = e4 ,

[e1 , e3 ] = −e5

[e3 , e4 ] = −e1 ,

[e3 , e5 ] = −e2 .

(As a result we obtain an extension of four-dimensional Abelian ideal G =< e1 , e2 , e4 , e5 > by means of < e3 >, algebra g4,13 in Mubaraczyanov classiﬁcation [21]). • Type VII ⎛

A = 0,

⎞ 100 B = (e2 , e3 , −) = ⎝ 0 0 0 ⎠ , 000

⎛

⎞ 0 −1 0 C = (e3 , e1 , −) = ⎝ 0 0 0 ⎠ . 0 0 0

Lie algebra G =< e1 , e2 , e3 , e4 , e5 > is a ﬁve-dimensional solvable nondecomposable Lie algebra, with deﬁning relations: [e2 , e3 ] = e4 , [e1 , e4 ] = −e1 ,

[e1 , e3 ] = e5

[e2 , e5 ] = −e1 ,

[e4 , e5 ] = e5

(algebra g4,11 in Mubaraczyanov classiﬁcation [22, 23]).

4.5

Classiﬁcation of Splitting Three-Dimensional Lie Triple Systems

Let M = M1 M2 be a splitting 3-dimensional LTS, where M1 ∼ = R is a one dimensional solvable ideal in M, and M2 be a 2-dimensional simple LTS. Introduce in consideration a basis (e1 , e2 , e3 ) in M such that M1 =< e1 > and M2 =< e2 , e3 > and linear operators A, B, C : M −→ M such that A = (e1 , e2 , −), A = (e1 , e2 , −), B = (e2 , e3 , −), C = (e3 , e1 , −) using the process applied in the previous case one can obtain the following theorem. THEOREM 4.5

The following situations are possible and nonisomorphic:

• Type 1. M = R ⊕ M2 is a direct sum of one-dimensional Abelian ideal and 2dimensional simple ideal in M, where M2 is a simple 2-dimensional LTS of the form so(3)/so(2),

sl(2, R)/so(2),

sl(2, R)/R

52

Thomas B. Bouetou • Type 2. ⎛

⎞ 0 −1 0 A = ⎝0 0 0⎠, 0 0 0

⎛

⎞ 00 0 B = ⎝ 0 0 −1 ⎠ , 01 0

⎛

⎞ 001 C = ⎝0 0 0⎠. 000

M2 is a simple 2-dimensional LTS of the form so(3)/so(2) • Type 3. ⎛

⎞ 010 A = ⎝0 0 0⎠, 000

⎛

⎞ 0 0 0 B = ⎝0 0 1⎠, 0 −1 0

⎛

⎞ 0 0 −1 C = ⎝0 0 0 ⎠. 00 0

M2 is a simple 2-dimensional LTS of the form sl(2, R)/so(2) • Type 4. ⎛

⎞ 0 −1 0 A = ⎝0 0 0⎠, 0 0 0

⎛

⎞ 000 B = ⎝0 0 1⎠, 010

⎛

⎞ 0 0 −1 C = ⎝0 0 0 ⎠. 00 0

M2 is a simple 2-dimensional LTS of the form sl(2, R)/so(2) • Type 5.

⎞ 0 − 14 41 A = −C = ⎝ 0 0 0 ⎠ , 0 0 0 ⎛

⎛

⎞ − 12 0 0 B = ⎝ 0 0 1⎠. 0 10

M2 is a simple 2-dimensional LTS of the form sl(2, R)/so(2) • Type 6.

⎞ 0 − 41 − 41 A = C = ⎝0 0 0 ⎠, 0 0 0 ⎛

⎛1

⎞ 00 B = ⎝ 0 0 1⎠. 010 2

M2 is a simple 2-dimensional LTS of the form sl(2, R)/so(2) The proof is a somewhat intricate calculation as has been done in the section above.

4.6

Acknowledgments

This paper was able to be written thanks to the scholarship obtained from the Agence Universitaire de la Francophonie.

Classiﬁcation of solvable 3-dimensional Lie triple systems

53

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Thomas B. Bouetou Amsterdam, 1969.

[20] Mubarakzianov G.M. Classiﬁcation of real structure of Lie algebras of ﬁfth order. Izv. Vusov: seri. Math., 1963, T. 34, N 99 (Russian). [21] Mubarakzianov G.M. About solvable Lie algebras. Izv. Vusov: ser. Math., 1963, T. 114 (Russian). [22] Mubarakzianov G.M. Classiﬁcation of solvable Lie algebras of dimension 6 with one non nilpotent basic element. Izv. Vusov: ser. Math., 1963, T. 35, N 104, (Russian). [23] Mubarakzianov G.M. Somes problems about solvable Lie algebras. Izv. Vusov: ser. Math., 1966; T. 32, N 95, (Russian). [24] Nono T. Sur les familles triples inﬁnit´esimales attach´es aux familles triples de Lie. J. Sci. Hiroshima Univ., ser. A, 1960, vol. 24, N. 3, 573–578. [25] Rashevsky P.K. About the geometry of homogeneous spaces. Work of seminar in vectorial analysis, M., 1952, T. 9, 49–74, (Russian). [26] Rosenfeld B.A. Theory of symmetric space of rang I. Math. Review, 1957, 41/83, 373–380, (Russian). [27] Sabinin L.V. About the geometry of loop. Math. Zamet., 1972, t.5, 605-616, (Russian). [28] Sabinin L.V. Odules as a new approach to a geometry with a connection, (Russian). Report of Ac. of Sci of the USSR (Math.), 233 (1977), N. 5, 800–803. English translation. Soviet. Math. Dokl. 18 (1977), N. 2, 515–518, Amer. Math. Soc. [29] Sabinin L.V. Classiﬁcation of trisymmetrical spaces. Sov. Math Dokl., USSR, 1970, N 3, 194, (Russian). [30] Stitzinger E.L. On derivation algebras of Malcev algebras and Lie triple systems. Proc. Amer. Math. Soc. 55 (1976), N. 1,9-13. [31] Taniguchu Y. On a kind of pairs of Lie triple systems. Math. Japon. 24 (1979/80), N. 6, 605–608. [32] Troﬁmov V.V. Introduction to geometry of manifolds with symmetry. Translated from the 1989 Russian original. Mathematics and its Applications, 270, Kluver Academic Publishers Group, Dordrecht, 1994. [33] Turkowski P. Low-dimensional real Lie algebras. J. Geom. Phys., T. 4, 1978. [34] Turkovski P. Solvable Lie algebra of dimension six. J. Math. Phys., T. 6, vol. 31, 1990. [35] Yamaguti K. On the Lie triple systems and its generalization. J. Sci. Hiroshima Univ., vol. A-21, 1958, 155–160. [36] Yamaguti K. On cohomology groups of general Lie triple system. Kumamoto J. Sci., ser. A 8, 1967/1969, 107–114. [37] Zhang Z.X., Shi Y.Q., Zhao L.N. Invariant symmetric bilinear form on Lie triple systems. Commm. Algebra 30 (2002), N. 11, 5563–5578.

Chapter 5 Alternating Triple Systems with Simple Lie Algebras of Derivations Murray R. Bremner Research Unit in Algebra and Logic, University of Saskatchewan, Canada Irvin R. Hentzel Department of Mathematics, Iowa State University Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Representations of the Lie algebra sl(2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.1 The Lie algebra sl(2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.2 The irreducible representation V (n) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.3 Exterior cubes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.4 Relation between binary and ternary structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 General multiplicity formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Ternary operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Representation V (3): Dimension 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.1 Identities of degree 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.2 Identities of degree 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.3 Another alternating triple system of dimension 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6 Representation V (5): Dimension 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6.1 The V (9) summand . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6.2 The V (5) summand . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6.3 The V (3) summand . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6.4 The change of basis matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6.5 The structure constants for the alternating triple system . . . . . . . . . . . . . . . . . . . . . . . 5.6.6 Identities of degree 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6.7 Identities of degree 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.7 Representation V (6): Dimension 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.7.1 Structure constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.7.2 Identities of degrees 5 and 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.8 Representation V (7): Dimension 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.8.1 Structure constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.8.2 Identities of degree 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.8.3 Identities of degree 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.9 Representation V (8): Dimension 9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.9.1 Structure constants for the triple system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.9.2 Identities of degree 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.9.3 Identities of degree 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.10 Representation V (10): Dimension 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.10.1 Structure constants for the triple system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 5.2

56 57 57 57 57 58 59 62 63 64 64 65 66 66 67 68 68 69 70 70 71 71 72 72 73 73 73 73 74 75 75 75 76 77

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5.10.2 Identities of degree 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.10.3 Identities of degree 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.10.4 Open problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.11 Other simple Lie algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.11.1 Tensor products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.11.2 Exterior cubes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.11.3 Special linear: type A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.11.4 Orthogonal: type B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.11.5 Symplectic: type C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.11.6 Orthogonal: type D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.11.7 Exceptional: types E, F, G . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.12 Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

78 78 78 78 78 79 80 81 81 81 81 82 82

Abstract We prove a formula for the multiplicity of the irreducible representation V (n) of sl(2, C) as a direct summand of its own exterior cube Λ3 V (n). From this we determine that V (n) occurs exactly once as a summand of Λ3 V (n) if and only if n = 3, 5, 6, 7, 8, 10. These representations admit a unique sl(2)-invariant alternating ternary structure obtained from the projection Λ3 V (n) → V (n). We calculate the structure constants for each of these alternating triple systems and use computer algebra to determine their polynomial identities of degree ≤ 7. We discover a remarkable 14-term identity in degree 7. The variety deﬁned by this identity contains V (3), V (5), and V (7). Key words:triple systems, polynomial identities, representations of simple Lie algebras, computer algebra 2000 MSC: Primary 17A40; Secondary 17A30, 17A36, 17B10, 17B60, 17D10, 17-04

5.1

Introduction

An irreducible representation of a simple Lie algebra can be a direct summand of its own exterior cube. In this case, the representation admits the structure of an alternating triple system that is invariant in the sense that the Lie algebra acts as ternary derivations. This chapter studies this situation in detail for the Lie algebra sl(2, C). In Section 5.2 we review the basic representation theory of sl(2). In Section 5.3 we prove a general formula for the multiplicity of an irreducible representation in its own exterior cube. From this we determine all representations for which the multiplicity equals 1; such representations admit an sl(2)-invariant alternating ternary structure, which is unique up to a scalar multiple. In Section 5.4 we review basic material about ternary operations. In Sections 5.5–5.10 we describe computer searches for polynomial identities satisﬁed by the six representations, which admit a unique alternating ternary structure; we determine all their identities of degree ≤ 7. A detailed discussion of our computational methods for discovering identities satisﬁed by nonassociative algebras may be found in three previous articles by the authors [3], [4], [5]. These methods involve expressing the identities as the nullspace of a large linear system, and then solving the system by using a computer algebra system to compute the row canonical form of the coeﬃcient matrix. In Section 5.11 we go beyond sl(2) and use the computer algebra package LiE [6] to determine all fundamental representations of simple Lie algebras of rank ≤ 8 that occur as

Alternating triple systems with simple Lie algebras of derivations

57

summands of their own exterior cubes. This demonstrates the existence of a large number of new alternating triple systems, with simple Lie algebras in their derivation algebras, which deserve further study.

5.2

Representations of the Lie Algebra sl(2)

We ﬁrst recall some standard facts about sl(2) and its representations. All vector spaces and tensor products are over F, an algebraically closed ﬁeld of characteristic zero. Our basic reference is Humphreys [8], especially §II.7.

5.2.1

The Lie algebra sl(2)

As an abstract Lie algebra, sl(2) has a basis {E, H, F } and commutation relations [H, E] = 2E,

[H, F ] = −2F,

[E, F ] = H.

All other relations between basis elements follow from anticommutativity. Since the Lie bracket is bilinear these relations determine the product [X, Y ] for all X, Y ∈ sl(2).

5.2.2

The irreducible representation V (n)

For any nonnegative integer n, there is an irreducible representation of sl(2) containing a nonzero vector vn (called the highest weight vector) satisfying the conditions H.vn = nvn ,

E.vn = 0.

This representation is unique up to isomorphism of sl(2)-modules. It is denoted V (n) and is called the representation with highest weight n. Its dimension is n + 1; a basis of V (n) consists of the n + 1 vectors vn−2i where vn−2i =

1 i F .vn , i!

0 ≤ i ≤ n.

The action of sl(2) on V (n) is then as follows: E.vn−2i = (n − i + 1)vn−2i+2 , H.vn−2i = (n − 2i)vn−2i , F.vn−2i = (i + 1)vn−2i−2 . The basis vectors vn−2i are called weight vectors since they are eigenvectors for H.

5.2.3

Exterior cubes

In this chapter we are primarily concerned with the multiplicity of V (n) as a direct alternating summand of its exterior cube Λ3 V (n). A basis of Λ3 V (n) consists of the n+1 3 sums, vp ⊗vq ⊗vr = alt

vp ⊗vq ⊗vr + vq ⊗vr ⊗vp + vr ⊗vp ⊗vq − vp ⊗vr ⊗vq − vq ⊗vp ⊗vr − vr ⊗vq ⊗vp ,

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Murray R. Bremner and Irvin R. Hentzel

where p, q, r are decreasing distinct weights of V (n); that is, n ≥ p > q > r ≥ −n,

p, q, r ≡ n (mod 2).

If D ∈ sl(2) then D acts on these alternating sums by the derivation rule: D.

alt

vp ⊗vq ⊗vr = alt

(D.vp )⊗vq ⊗vr +

vp ⊗(D.vq )⊗vr +

alt

vp ⊗vq ⊗(D.vr ),

alt

and this action extends linearly to all of Λ3 V (n). If V (n) occurs as a summand of Λ3 V (n) then the multiplicity, m = dim Homsl(2) (Λ3 V (n), V (n)) will be positive. Any sl(2)-module homomorphism p : Λ3 V (n) → V (n) gives V (n) the structure of an alternating triple system in the sense that p(uσ , v σ , wσ ) = (σ)p(u, v, w), where σ ∈ S3 is any permutation of u, v, w and (σ) is its sign. This alternating ternary product is sl(2)-invariant in the sense that the elements of sl(2) act as ternary derivations: D.p(u, v, w) = p(D.u, v, w) + p(u, D.v, w) + p(u, v, D.w). Two projections p1 and p2 , which diﬀer by a nonzero scalar multiple, deﬁne isomorphic triple systems on V (n) and so the family of nonisomorphic alternating ternary structures on V (n) has m − 1 projective parameters.

5.2.4

Relation between binary and ternary structures

By the Clebsch-Gordan Theorem (see Bremner and Hentzel [5] for an explicit version of this classical result) we know that V (n) occurs as a summand of its exterior square if and only if n ≡ 2 (mod 4). In this case the multiplicity is 1, and so V (n) has an sl(2)invariant anticommutative binary product. For n = 2 we recover sl(2) and for n = 6 we recover the simple non-Lie Malcev algebra. For n ≥ 10 we obtain less familiar anticommutative algebras; the n = 10 case is studied in detail by Bremner and Hentzel [5]. On any anticommutative algebra, the Jacobian J(x, y, z) = [[x, y], z] + [[y, z], x] + [[z, x], y] deﬁnes an alternating ternary operation, which is identically zero if and only if the algebra is a Lie algebra.

Alternating triple systems with simple Lie algebras of derivations

5.3

59

General Multiplicity Formula

In this section we prove a formula for the multiplicity of V (n) as a summand of its exterior cube Λ3 V (n): see Theorem 5.1. We ﬁrst prove four lemmas. LEMMA 5.1 Let M be a ﬁnite dimensional sl(2)-module. For every integer n deﬁne Mn = {v ∈ M | H.v = nv}; that is, the subspace of M consisting of all vectors of weight n (together with 0). Then for any nonnegative integer n the multiplicity of V (n) as a direct summand of M equals dim Mn − dim Mn+2 . PROOF We know that M is completely reducible; this follows from Weyl’s theorem (any ﬁnite dimensional module over any semisimple Lie algebra is completely reducible), but we only need the special case when the Lie algebra is sl(2). So we can write M = m1 V (n1 ) ⊕ m2 V (n2 ) ⊕ · · · ⊕ mk V (nk ), where m1 , . . . , mk are positive integers and n1 > n2 > · · · > nk ≥ 0; furthermore, the integers k, mi , ni are uniquely determined. (The notation mV (n) abbreviates “the direct sum of m copies of V (n).”) We know that V (n) has a one-dimensional weight space for any weight n with n ≥ n ≥ −n and n ≡ n (mod 2). Therefore, we need to separate the “even” and “odd” parts of M , so we deﬁne mi V (ni ), M1 = mi V (ni ). M0 = ni ≡n1 (mod 2)

ni ≡n1 (mod 2)

Now assume that n ≡ n1 (mod 2); equivalently, we will assume that M 0 = M and M 1 = {0}. The direct sum decomposition of M now gives dim Mn = m1 for n1 ≥ n > n2 , dim Mn = m1 + m2 for n2 ≥ n > n3 , ··· dim Mn = m1 + m2 + · · · + mi for ni ≥ n > ni+1 , ··· dim Mn = m1 + m2 + · · · + mk for nk ≥ n ≥ 0. From this it is obvious that mi = (m1 + · · · + mi ) − (m1 + · · · + mi−1 ) = dim Mni − dim Mni +2 . The proof in the case n ≡ n1 (mod 2) is similar. In order to apply Lemma 5.1 to M = Λ3 V (n) we need to know the dimensions of the weight spaces Mn . We state the following result for any exterior power. LEMMA 5.2 Let M = Λk V (n) and assume that nk ≥ n ≥ −nk with n ≡ nk (mod 2). Then the dimension of the weight space Mn equals the number of sequences (n1 , n2 , . . . , nk ) satisfying the conditions n ≥ n1 > n2 > · · · > nk ≥ −n,

n1 + n2 + · · · + nk = n .

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PROOF

A basis of Λk V (n) consists of the alternating sums: (σ) vnσ(1) ⊗ vnσ(2) ⊗ · · · ⊗ vnσ(k) , n ≥ n1 > n2 > · · · > nk ≥ −n,

σ∈Sk

where (σ) is the sign of the permutation σ and vni is a ﬁxed vector of weight ni in V (n) (such a vector is unique up to a nonzero scalar multiple). Since H.v = (n1 + n2 + · · · + nk ) v,

for

v = vnσ(1) ⊗ vnσ(2) ⊗ · · · ⊗ vnσ(k) ,

the result follows. To compute the dimensions of the weight spaces in M = Λ3 V (n), by Lemma 5.2 we need to determine the number of triples (n1 , n2 , n3 ) satisfying the conditions n ≥ n1 > n2 > n3 ≥ −n,

n1 + n2 + n3 = n ,

for the cases n = n and n = n + 2; then we can apply Lemma 5.1. We ﬁrst simplify the notation. Let n be a nonnegative integer, let n be an integer satisfying 3n ≥ n ≥ −3n, n ≡ 3n (mod 2), and let p, q, r be integers satisfying n ≥ p > q > r ≥ −n,

p + q + r = n ,

p, q, r ≡ n (mod 2).

Now deﬁne

1 1 (p + n), Q = (q + n), 2 2 Then (P, Q, R) is a triple of integers satisfying P =

n ≥ P > Q > R ≥ 0,

R=

P + Q + R = N,

1 (r + n). 2

N=

1 (n + 3n). 2

So we need to count the number of partitions of N into two or three distinct parts less than or equal to n. We only need to consider the cases N = 2n (n = n) and N = 2n + 1 (n = n + 2). LEMMA 5.3

The number of triples (P, Q, R) of integers satisfying the conditions n ≥ P > Q > R ≥ 0,

is given by the formula

n/3−1

i=0

PROOF

P + Q + R = 2n,

n+i−1 − 2i . 2

We enumerate the triples as follows. First, those with P = n: (n, n − 1, 1),

(n, n − 2, 2),

....

The last triple with P = n is (n, n2 + 1, n2 − 1) (n even);

n−1 (n, n+1 2 , 2 ) (n odd).

Alternating triple systems with simple Lie algebras of derivations

61

For P = n there are altogether n−1 2 triples. Second, the triples with P = n − 1: (n − 1, n − 2, 3),

(n − 1, n − 3, 4),

....

The last triple with P = n − 1 is (n − 1, n2 + 1, n2 ) (n even);

n−1 (n − 1, n+3 2 , 2 ) (n odd).

For P = n − 1 there are altogether n2 − 2 triples. In the general case, when P = n − i, we have (n − i, n − (i + 1), 2i + 1),

(n − i, n − (i + 2), 2i + 2),

....

The last triple with P = n − i is i n i 2 + 1, 2 + 2 − 1) (n even, i even); n i+1 n i+1 2 + 2 , 2 + 2 − 1) (n even, i odd); n+1 i n+1 i 2 + 2 , 2 + 2 − 1) (n odd, i even); n+1 i+1 n+1 i+1 2 + 2 , 2 + 2 − 2) (n odd, i odd).

(n − i, n2 + (n − i, (n − i, (n − i,

These four triples can be reduced to two cases: , n+i−1 ) (n − i, n+i+1 2 2

when n and i have opposite parity;

(n − i,

when n and i have the same parity.

n+i 2

+ 1,

n+i 2

− 1)

For P = n − i the total number of triples is n+i−1 − 2i. 2 Since P > Q we obtain the condition n ≥ 3(i + 1), or equivalently, n! − 1. 0≤i≤ 3 Summing the number of triples for each i over this range gives the result. LEMMA 5.4

The number of triples (P, Q, R) of integers satisfying the conditions n ≥ P > Q > R ≥ 0,

is given by the formula

(n−1)/3−1

i=0

PROOF

P + Q + R = 2n + 1,

n+i − (2i + 1) . 2

This proof is very similar to the proof of Lemma 5.3.

THEOREM 5.1 Let V (n) denote the simple sl(2)-module with highest weight n. Write n = 6k + where 0 ≤ ≤ 5. Then " k, if = 0, 1, 2, 4; dim Homsl(2) (Λ3 V (n), V (n)) = k + 1, if = 3, 5.

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PROOF By Lemma 5.1, the multiplicity is the diﬀerence between the formulas of Lemmas 5.3 and 5.4. Since the expressions n3 and n2 occur, we need to distinguish 6 cases, depending on the congruence class of n modulo 6. So we write n = 6k + where 0 ≤ ≤ 5. First assume that = 0. Then n3 = 2k and n−1 3 = 2k − 1. Therefore the multiplicity is 2k−2 2k−1 6k + i − 1 6k + i − 2i − − (2i + 1) . 2 2 i=0 i=0 Separating the last term of the ﬁrst sum, and combining the other terms in one sum, gives 2k−2 i=0

6k + i − 1 6k + i − 2i − + (2i + 1) 2 2 6k + (2k − 1) − 1 + − 2(2k − 1) . 2

Simplifying this, we obtain 2k−2 i=0

i−1 i − 3k − + (2k − 1) + (4k − 1) − (4k − 2) 2 2 2k−2 i − 1 i = − + 2k. 2 2 i=0 3k +

Splitting the remaining sum into two parts for i even and i odd gives

2k−2 i=0, even

=

2k−3 i − 1 i i−1 i − + − + 2k 2 2 2 2 i=1, odd

2k−2

(−1) +

i=0, even

2k−3

(0) + 2k = −k + 0 + 2k = k.

i=1, odd

This completes the proof for = 0. The other cases (1 ≤ ≤ 5) are similar. COROLLARY 5.1 The simple sl(2)-module V (n) admits a unique sl(2)-invariant alternating ternary structure if and only if n = 3, 5, 6, 7, 8, 10. PROOF By Theorem 5.1 the module V (n) occurs exactly once as a summand of Λ3 V (n) if and only if n = 3, 5, 6, 7, 8, 10. Hence (up to a nonzero scalar multiple) there is a unique sl(2)-module homomorphism from Λ3 V (n) to V (n) exactly in these cases.

5.4

Ternary Operations

Let A be a vector space over a ﬁeld F, together with a trilinear map p : A × A × A → A. We call the pair (A, p) a triple system (or ternary algebra) over F. We often suppress p

Alternating triple systems with simple Lie algebras of derivations

63

and write [a, b, c] for p(a, b, c). We are interested in the (ternary) polynomial identities satisﬁed by A. To simplify the discussion, we will assume initially that the base ﬁeld F has characteristic 0. This implies that any polynomial identity over F is equivalent to a family of homogeneous multilinear identities. See Bremner [1, 2] for the following results. PROPOSITION 5.1 Up to alternating equivalence, there is one association type in degree 5 for an alternating ternary product [a, b, c]: [[a, b, c], d, e]. There are 10 distinct multilinear monomials: [[a, b, c], d, e], [[a, b, d], c, e], [[a, b, e], c, d], [[a, c, d], b, e], [[a, c, e], b, d], [[a, d, e], b, c], [[b, c, d], a, e], [[b, c, e], a, d], [[b, d, e], a, c], [[c, d, e], a, b]. The decomposition of the 10-dimensional representation of S5 with these monomials as basis is [221] ⊕ [2111] ⊕ [11111]. Here [λ] denotes the irreducible representation of S5 corresponding to the partition λ. PROPOSITION 5.2 degree 7:

Up to alternating equivalence, there are two association types in [[[a, b, c], d, e], f, g],

[[a, b, c], [d, e, f ], g].

For the ﬁrst association type the number of distinct multilinear monomials is 7! = 210, 3!2!2! and for the second association type the number is 1 7! · = 70; 2 3!3! altogether there are 280 multilinear monomials in degree 7. The decomposition of the 280dimensional representation of S7 with these monomials as basis is [322] ⊕ [3211] ⊕ 3[31111] ⊕ [2221] ⊕ 3[22111] ⊕ 4[211111] ⊕ 2[1111111]. Here m[λ] denotes the direct sum of m copies of the irreducible representation of S7 corresponding to the partition λ.

5.4.1

Notation

Many of the identities we present in this chapter contain alternating sums over all permutations of certain sets of variables. We will use the notation [ai1 ai2 · · · aid ], alt(S)

to denote the alternating sum over all permutations of the variables as for s ∈ S. (The other variables remain in the same position in each term of the sum.) The term [ai1 ai2 · · · aid ]

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Murray R. Bremner and Irvin R. Hentzel

denotes a multilinear alternating ternary monomial of degree d in some association. If S = {1, . . . , k} where 2 ≤ k ≤ d then we will use the notation [ai1 ai2 · · · aid ]. alt(k)

If |S| = k then such an alternating sum contains k! terms each with a coeﬃcient +1 or −1. The alternating ternary laws will cause many of these terms to be equal. So the actual identity will have many fewer terms when expressed as a linear combination of the basic monomials.

5.5

Representation V (3): Dimension 4

The decomposition of the exterior cube is Λ3 V (3)∼ =V (3). The exterior cube, which is this case is isomorphic to the original representation, has a basis consisting of the 4 alternating sums corresponding to the triples of weights: [p, q, r] = [3, 1, −1], [3, 1, −3], [3, −1, −3], [1, −1, −3]. Any highest weight vector in Λ3 V (3) is a nonzero scalar multiple of z3 = v3 ⊗v1 ⊗v−1 . alt

Applying F ∈ sl(2) repeatedly we obtain the other weight vectors forming a basis of V (3): z1 = 3v3 ⊗v1 ⊗v−3 , z−1 = 3v3 ⊗v−1 ⊗v−3 , z−3 = v1 ⊗v−1 ⊗v−3 . alt

alt

alt

Let A be the change of basis matrix in which the ij entry is the coeﬃcient of the ith alternating sum in the jth weight vector. Then ⎞ ⎛ ⎛ ⎞ 10 00 1000 ⎜0 3 0 0⎟ ⎜0 1 0 0⎟ 3 ⎟ ⎟ A=⎜ A−1 = ⎜ ⎝0 0 3 0⎠ , ⎝0 0 1 0 ⎠ . 3 0001 00 01 Therefore the structure constants for the alternating ternary product on V (3) are [v3 , v1 , v−1 ] = v3 , [v3 , v−1 , v−3 ] = 13 v−1 ,

[v3 , v1 , v−3 ] = 13 v1 , [v1 , v−1 , v−3 ] = v−3 ,

and all others follow from the alternating property.

5.5.1

Identities of degree 5

THEOREM 5.2 Every identity in degree 5 satisﬁed by the alternating ternary structure on V (3) is a consequence of the identity [[abc]de] − [[abd]ce] + [[abe]cd] + [[acd]be] − [[ace]bd] + [[ade]bc] − [[bcd]ae] + [[bce]ad] − [[bde]ac] + [[cde]ab].

Alternating triple systems with simple Lie algebras of derivations

65

1 -th of ) the alternating sum over all permutations of the arguments in This identity is ( 12 the basic monomial [[abc]de].

PROOF We create a matrix of size 14 × 10 in which the columns are labelled by the ordered basis of multilinear monomials in degree 5 for an alternating ternary operation. We generate ﬁve random elements of V (3) (random vectors with 4 components) and evaluate the 10 monomials on these ﬁve elements. We put the 10 resulting 4 × 1 vectors into the bottom of the matrix (in rows 11 through 14). Each of the last four rows of the matrix now contains a linear relation that must be satisﬁed by the coeﬃcients of any identity for the triple system. We compute the row canonical form of the matrix. Now the last four rows are zero, so we can repeat the “ﬁll and reduce” process. We keep repeating this process until the rank of the matrix stabilizes. In this case the rank reached 9 and did not increase further. The nullspace of the matrix at this point is one-dimensional. A basis for the nullspace is the identity in the theorem. The identity of Theorem 5.2 can also be written using our alternating sum notation in the more compact form: 1 [[a1 a2 a3 ]a4 a5 ]. 12 alt(5)

We will use this notation in the rest of this chapter.

5.5.2

Identities of degree 7

THEOREM 5.3 Every identity in degree 7 satisﬁed by the alternating triple system V (3) follows from the identity in degree 5 from Theorem 5.2 and the 14-term identity displayed in Theorem 5.4. PROOF Using the random element method described in our previous papers (Bremner and Hentzel [3, 4, 5]) we found that the vector space of all identities for V (3) in degree 7 is a subspace of dimension 210 in the 280-dimensional space of all possible degree 7 identities. We need to determine which of these identities are consequences of the known identity in degree 5 given in Theorem 5.2. We write I = I(a, b, c, d, e) for the identity of Theorem 5.2. This is an alternating function of its ﬁve arguments, so there are only two inequivalent ways of lifting this identity to degree 7: I([a, f, g], b, c, d, e), [I(a, b, c, d, e), f, g]. 7 There are 3 = 35 diﬀerent forms of the ﬁrst lifting, and 72 = 21 diﬀerent forms of the second. We put these 56 diﬀerent liftings into a matrix of size 56 × 280. (The ij entry is the coeﬃcient of the jth monomial in degree 7 in the ith lifting.) We computed the rank of this matrix: the rank is 56, so the diﬀerent liftings are linearly independent. They form a basis for the subspace of the degree 7 identities, which are consequences of the degree 5 identity of Theorem 5.2. We took the 14-term identity in Theorem 5.4 and added it as the last row of a matrix of size 211 × 280 in which the ﬁrst 210 rows are the basis vectors of the identities for V (3) in degree 7. We reduced this matrix and found that its rank is 210. It follows that the 14-term identity is satisﬁed by the alternating ternary structure on V (3). We then took the 14-term identity and determined a basis for the submodule of identities in degree 7 that it generates. (For the method we used see the proof of Theorem 5.5.) This

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Murray R. Bremner and Irvin R. Hentzel

gives 189 linearly independent vectors of length 210. We stacked two matrices: the ﬁrst matrix of size 56 × 280 contains a basis for the lifted identities from degree 5; the second matrix of size 189×280 contains a basis for the submodule generated by the 14-term identity. This gives a matrix of size 245 × 280. We reduced this matrix and found that its rank is 210. This implies that the lifted identities together with the 14-term identity generate the entire 210-dimensional space of identities satisﬁed by V (3) in degree 7.

5.5.3

Another alternating triple system of dimension 4

Let V be a vector space over the ﬁeld F with basis I, J, K, L. Let X = (x1 , x2 , x3 , x4 ),

Y = (y1 , y2 , y3 , y4 ),

Z = (z1 , z2 , z3 , z4 ),

be any three elements of V expressed as quadruples with respect to the given basis. Deﬁne the ternary cross product on V by the formal determinant: % % %I J K L% % % %x x x x % [X, Y, Z] = %% 1 2 3 4 %% % y1 y2 y3 y4 % % z1 z2 z3 z4 .% This gives the following triple products of basis vectors: [I, J, K] = −L,

[I, J, L] = K,

[I, K, L] = −J,

[J, K, L] = I.

It is shown in Bremner and Hentzel [3] that every identity of degree ≤ 7 satisﬁed by this triple system follows from the alternating property and the ternary Jacobi identity [V, W, [X, Y, Z]] = [[V, W, X], Y, Z] + [X, [V, W, Y ], Z] + [X, Y, [V, W, Z]]. In terms of the standard association type this identity can be written as [[X, Y, Z], V, W ] − [[X, V, W ], Y, Z] + [[Y, V, W ], X, Z] − [[Z, V, W ], X, Y ]. It is shown in Bremner [1] (Theorem 2, page 83) that this identity implies the identity of Theorem 5.2 but is not implied by that identity. (This proves that these two fourdimensional alternating triple systems are not isomorphic.) The ternary Jacobi identity and its n-ary generalization were introduced by Filippov [7] in his study of n-Lie algebras.

5.6

Representation V (5): Dimension 6

The decomposition of the exterior cube is Λ3 V (5)∼ =V (9) ⊕ V (5) ⊕ V (3). In this case, to illustrate our computational methods, we will present the calculations in greater detail than in the following sections. The multiplicities of the submodules in this decomposition can be read oﬀ by applying Lemma 5.1 to Table 5.1, which contains the ordered triples [p, q, r] of distinct weights of

Alternating triple systems with simple Lie algebras of derivations

67

V (5): that is, p, q, r are odd integers satisfying 5 ≥ p > q > r ≥ −5. The triple [p, q, r] represents the alternating sum vp ⊗vq ⊗vr . The

6 3

alt

= 20 alternating sums in Table 5.1 form a basis of Λ3 V (5). Table 5.1. Alternating sums forming a basis of Λ3 V (5) Weight 9 7 5 3 1 −1 −3 −5 −7 −9

5.6.1

Triple 1

Triple 2

Triple 3

[5, 3, 1] [5, 3, −1] [5, 3, −3] [5, 3, −5] [5, 1, −5] [5, −1, −5] [5, −3, −5] [3, −3, −5] [1, −3, −5] [−1, −3, −5]

[5, 1, −1] [5, 1, −3] [3, 1, −3] [3, 1, −5] [3, −1, −5] [1, −1, −5]

[3, 1, −1] [3, 1, −3] [3, −1, −3] [1, −1, −3]

The V (9) summand

A highest weight vector for the V (9) summand of Λ3 V (5) is x9 = v5 ⊗v3 ⊗v1 . alt

Applying F repeatedly we obtain the other basis vectors for the V (9) summand: x7 = 3v5 ⊗v3 ⊗v−1 , x5 = (6v5 ⊗v3 ⊗v−3 + 3v5 ⊗v1 ⊗v−1 ) , alt

x3 =

alt

(10v5 ⊗v3 ⊗v−5 + 8v5 ⊗v1 ⊗v−3 + v3 ⊗v1 ⊗v−1 ) ,

alt

x1 =

(15v5 ⊗v1 ⊗v−5 + 6v5 ⊗v−1 ⊗v−3 + 3v3 ⊗v1 ⊗v−3 ) ,

alt

x−1 =

(15v5 ⊗v−1 ⊗v−5 + 6v3 ⊗v1 ⊗v−5 + 3v3 ⊗v−1 ⊗v−3 ) ,

alt

x−3 =

(10v5 ⊗v−3 ⊗v−5 + 8v3 ⊗v−1 ⊗v−5 + v1 ⊗v−1 ⊗v−3 ) ,

alt

x−5 =

(6v3 ⊗v−3 ⊗v−5 + 3v1 ⊗v−1 ⊗v−5 ) ,

alt

x−7 =

alt

3v1 ⊗v−3 ⊗v−5 ,

x−9 =

alt

v−1 ⊗v−3 ⊗v−5 .

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Murray R. Bremner and Irvin R. Hentzel

5.6.2

The V (5) summand

A highest weight vector for the V (5) summand of Λ3 V (5) has the form v5 ⊗v3 ⊗v−3 + b v5 ⊗v1 ⊗v−1 . y5 = a alt

alt

Applying E to this vector we obtain E.y5 = (2a + 4b)

v5 ⊗v3 ⊗v−1 .

alt

Therefore 2a + 4b = 0 so we take a = 2, b = −1 to get the highest weight vector: (2v5 ⊗v3 ⊗v−3 − v5 ⊗v1 ⊗v−1 ) . y5 = alt

Applying F repeatedly we obtain the other basis vectors for the V (5) summand: (10v5 ⊗v3 ⊗v−5 − v3 ⊗v1 ⊗v−1 ) , y3 = alt

y1 =

(10v5 ⊗v1 ⊗v−5 − 2v3 ⊗v1 ⊗v−3 ) ,

alt

y−1 =

(10v5 ⊗v−1 ⊗v−5 − 2v3 ⊗v−1 ⊗v−3 ) ,

alt

y−3 =

(10v5 ⊗v−3 ⊗v−5 − v1 ⊗v−1 ⊗v−3 ) ,

alt

y−5 =

(2v3 ⊗v−3 ⊗v−5 − v1 ⊗v−1 ⊗v−5 ) .

alt

5.6.3

The V (3) summand

A highest weight vector for the V (3) summand of Λ3 V (5) has the form v5 ⊗v3 ⊗v−5 + b v5 ⊗v1 ⊗v−3 + c v3 ⊗v1 ⊗v−1 . z3 = a alt

alt

alt

Applying E to this vector we obtain v5 ⊗v3 ⊗v−3 + (2b + 5c) v5 ⊗v1 ⊗v−1 . E.z3 = (a + 4b) alt

alt

Therefore a + 4b = 2b + 5c = 0 so we take a = 20, b = −5, c = 2 to get the highest weight vector: (20v5 ⊗v3 ⊗v−5 − 5v5 ⊗v1 ⊗v−3 + 2v3 ⊗v1 ⊗v−1 ) . z3 = alt

Applying F repeatedly we obtain the other basis vectors for the V (3) summand: z1 = (15v5 ⊗v1 ⊗v−5 − 15v5 ⊗v−1 ⊗v−3 + 3v3 ⊗v1 ⊗v−3 ) , alt

z−1 =

(−15v5 ⊗v−1 ⊗v−5 + 15v3 ⊗v1 ⊗v−5 − 3v3 ⊗v−1 ⊗v−3 ) ,

alt

z−3 =

alt

(−20v5 ⊗v−3 ⊗v−5 + 5v3 ⊗v−1 ⊗v−5 − 2v1 ⊗v−1 ⊗v−3 ) .

Alternating triple systems with simple Lie algebras of derivations

5.6.4

69

The change of basis matrix

We now consider two ordered bases of the exterior cube Λ3 V (5). The tensor basis consists of the 20 alternating sums listed in Table 5.1 in lexicographical order: v5 ⊗v3 ⊗v1 , v5 ⊗v3 ⊗v−1 , v5 ⊗v3 ⊗v−3 , v5 ⊗v3 ⊗v−5 , alt

alt

v5 ⊗v1 ⊗v−1 ,

alt

v5 ⊗v−1 ⊗v−5 ,

alt

v5 ⊗v1 ⊗v−3 ,

v3 ⊗v1 ⊗v−5 ,

v5 ⊗v−3 ⊗v−5 ,

alt

v5 ⊗v1 ⊗v−5 ,

v3 ⊗v−1 ⊗v−3 ,

v3 ⊗v1 ⊗v−1 ,

alt

alt

v5 ⊗v−1 ⊗v−3 ,

v3 ⊗v1 ⊗v−3 ,

alt

v3 ⊗v−1 ⊗v−5 ,

alt

v1 ⊗v−1 ⊗v−5 ,

alt

alt

alt

v1 ⊗v−1 ⊗v−3 ,

alt

alt

alt

alt

alt

alt

v3 ⊗v−3 ⊗v−5 ,

alt

v1 ⊗v−3 ⊗v−5 ,

v−1 ⊗v−3 ⊗v−5 .

alt

The module basis consists of the weight vectors in the summands computed above, in the following order: x9 , x7 , x5 , x3 , x1 , x−1 , x−3 , x−5 , x−7 , x−9 , y5 , y3 , y1 , y−1 , y−3 , y−5 , z3 , z1 , z−1 , z−3 . We now create a 20 × 20 matrix A in which the rows are labelled by the tensor basis sums and the columns are labelled by the module basis vectors; the ij entry is the coeﬃcient of the ith tensor basis sum in the jth module basis vector. The matrix A is displayed in Table 5.2. Table 5.2. Change of basis matrix for Λ3 V (5) ⎛

1 ⎜0 ⎜ ⎜0 ⎜ ⎜0 ⎜ ⎜0 ⎜ ⎜0 ⎜ ⎜0 ⎜ ⎜0 ⎜ ⎜0 ⎜ ⎜0 ⎜ ⎜0 ⎜ ⎜0 ⎜ ⎜0 ⎜ ⎜0 ⎜ ⎜0 ⎜ ⎜0 ⎜ ⎜0 ⎜ ⎜0 ⎜ ⎝0 0

0 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 6 0 0 0 0 0 0 0 0 10 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 8 0 0 0 0 0 0 0 0 15 0 0 0 0 0 0 0 6 0 0 0 0 0 0 0 0 15 0 0 0 0 0 0 0 0 10 0 0 0 0 1 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 6 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 8 0 0 0 0 0 0 0 0 6 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 1

0 0 0 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 10 0 0 0 0 −1 0 0 0 0 0 0 0 0 0 0 0 0 0 10 0 0 0 0 0 0 0 0 0 0 0 0 10 0 0 0 0 0 0 10 0 0 −1 0 0 0 0 0 0 −2 0 0 0 0 0 0 0 0 0 0 0 0 −2 0 0 0 0 0 0 0 0 0 0 0 0 0 2 0 0 0 0 −1 0 0 0 0 0 0 −1 0 0 0 0 0 0 0 0 0 0 0 0

⎞ 0 0 0 0 0 0 0 0⎟ ⎟ 0 0 0 0⎟ ⎟ 20 0 0 0⎟ ⎟ 0 0 0 0⎟ ⎟ −5 0 0 0⎟ ⎟ 0 15 0 0⎟ ⎟ 0 −15 0 0⎟ ⎟ 0 0 −15 0⎟ ⎟ 0 0 0 −20 ⎟ ⎟ 2 0 0 0⎟ ⎟ 0 3 0 0⎟ ⎟ 0 0 15 0⎟ ⎟ 0 0 −3 0⎟ ⎟ 0 0 0 5⎟ ⎟ 0 0 0 0⎟ ⎟ 0 0 0 −2 ⎟ ⎟ 0 0 0 0⎟ ⎟ 0 0 0 0⎠ 0 0 0 0

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5.6.5

The structure constants for the alternating triple system

The columns of the inverse matrix A−1 give the coeﬃcients of the tensor basis sums in terms of the module basis vectors. From this we can read oﬀ the matrices representing the projection maps from Λ3 V (5) onto its simple submodules. Rows 11–16 of A−1 give the structure constants for the alternating triple system obtained from the projection p : Λ3 V (5) → V (5). These rows are displayed in Table 5.3. Table 5.3. Structure constants for V (5) triple system ⎛

0 ⎜0 ⎜ 1 ⎜ ⎜0 20 ⎜ ⎜0 ⎝0 0

0 0 0 0 0 0

5 0 0 0 0 0

0 −10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 −10 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 −5 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 −5 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 −10 0 0 0 0 0 0 0 0 0 0 0 0 0 5 0 −10

0 0 0 0 0 0

⎞ 0 0⎟ ⎟ 0⎟ ⎟ 0⎟ ⎟ 0⎠ 0

Let h be the isomorphism from the original copy of V (5) (with basis vectors vp ) to the V (5) summand (with basis vectors yp ). Then h−1 ◦ p is the alternating ternary product on V (5). If we write [vp , vq , vr ] = (h

−1

◦ p)

vp ⊗vq ⊗vr

,

alt

then the structure constants for the sl(2)-invariant alternating triple system obtained from V (5) are as follows; here we have scaled the basis vectors to clear the denominators: [v5 , v3 , v1 ] = 0,

[v5 , v3 , v−1 ] = 0,

[v5 , v3 , v−3 ] = 5v5 ,

[v5 , v3 , v−5 ] = v3 , [v5 , v1 , v−5 ] = v1 ,

[v5 , v1 , v−1 ] = −10v5 , [v5 , v−1 , v−3 ] = 0,

[v5 , v1 , v−3 ] = 0, [v5 , v−1 , v−5 ] = v−1 ,

[v5 , v−3 , v−5 ] = v−3 , [v3 , v1 , v−5 ] = 0,

[v3 , v1 , v−1 ] = −10v3 , [v3 , v−1 , v−3 ] = −5v−1 ,

[v3 , v1 , v−3 ] = −5v1 , [v3 , v−1 , v−5 ] = 0,

[v3 , v−3 , v−5 ] = 5v−5 , [v1 , v−1 , v−3 ] = −10v−3 , [v1 , v−3 , v−5 ] = 0, [v−1 , v−3 , v−5 ] = 0.

[v1 , v−1 , v−5 ] = −10v−5 ,

All other triple products follow from the alternating property.

5.6.6

Identities of degree 5

Computations with p = 101 show that this triple system satisﬁes no identity in degree 5. By the discussion in Bremner and Hentzel [5] this implies that it also satisﬁes no identities of degree 5 in characteristic 0.

Alternating triple systems with simple Lie algebras of derivations

5.6.7

71

Identities of degree 7

THEOREM 5.4 The space of identities in degree 7 for the alternating ternary structure on V (5) has this decomposition into irreducible representations of S7 : [322] ⊕ 2[3211] ⊕ [31111] ⊕ 2[2221] ⊕ 3[22111] ⊕ 2[211111] ⊕ 2[1111111]. This space of identities is generated by the 14-term identity: ([[[aef ]bc]dg] − [[[abc]dg]ef ]) − ([[[aef ]bd]cg] − [[[abd]cg]ef ]) − ([[[aef ]bg]cd] − [[[abg]cd]ef ]) + ([[[aef ]cd]bg] − [[[acd]bg]ef ]) + ([[[aef ]cg]bd] − [[[acg]bd]ef ]) − ([[[aef ]dg]bc] − [[[adg]bc]ef ]) + 2[[[bcd]ef ]ag] + 2[[bcd][ef g]a], together with the identity,

[[[a1 a2 a3 ]a4 a5 ]a6 a7 ].

alt(7)

The ﬁrst identity has exactly one term in the second association type. It shows that any term in the second association type can be written as a linear combination of terms in the ﬁrst association type. Computations with p = 101 show that this triple system satisﬁes 190 linearly independent identities in degree 7. These computations were repeated in characteristic 0 and conﬁrmed the existence of 190 linearly independent identities. One of these identities generates a 189-dimensional S7 -submodule of identities. The ﬁrst identity in the last theorem implies the identity, [[[a1 a2 a3 ]a4 a5 ]a6 a7 ] − [[a1 a2 a3 ][a4 a5 a6 ]a7 ]. alt(7)

alt(7)

This is the diﬀerence of the alternating sums over the two association types in degree 7. The second identity is the alternating sum over the ﬁrst association type. Thus the triple system V (5) satisﬁes both alternating sums over the two association types in degree 7. The ﬁrst identity is the same as the generating identity for the degree 7 identities of the alternating triple system V (7). (For details of the method by which this identity was discovered, see Section 5.8.3.) It is remarkable that the two alternating triple systems V (5) and V (7) satisfy almost the same identities in degree 7. They both satisfy all the identities in the 189-dimensional space generated by the ﬁrst identity in Theorem 5.4. In addition, the system V (5) satisﬁes the second identity in Theorem 5.4.

5.7

Representation V (6): Dimension 7

The decomposition of the exterior cube is Λ3 V (6)∼ =V (12) ⊕ V (8) ⊕ V (6) ⊕ V (4) ⊕ V (0). Before stating the highest weight vectors we introduce the notation vp ⊗vq ⊗vr . [p, q, r] = alt

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Murray R. Bremner and Irvin R. Hentzel

With this convention, the highest weight vectors for the summands on the right side of the isomorphism are w8 = 5 [6, 4, −2] − 3 [6, 2, 0], u12 = [6, 4, 2], x6 = 5 [6, 4, −4] − 2 [6, 2, −2] + [4, 2, 0], y4 = 10 [6, 4, −6] − 2 [6, 2, −4] + [6, 0, −2], z0 = 30 [6, 0, −6] − 20 [6, −2, −4] − 20 [4, 2, −6] + 5 [4, 0, −4] − 2 [2, 0, −2].

5.7.1

Structure constants

The structure constants for the alternating triple system on V (6) are presented in the following list. Here we use the compact notation [p, q, r] = c to abbreviate the equation [vp , vq , vr ] = cvp+q+r : [6, 4, 2] = 0, [6, 4, −6] = 1, [6, 2, −6] = 1,

[6, 4, 0] = 0, [6, 2, 0] = 0, [6, 0, −2] = −10,

[6, 4, −2] = 0, [6, 2, −2] = −15, [6, 0, −4] = −4,

[6, 4, −4] = 6, [6, 2, −4] = 0, [6, 0, −6] = 0,

[6, −2, −4] = −3, [4, 2, −2] = 15,

[6, −2, −6] = −1, [4, 2, −4] = 6,

[6, −4, −6] = −1, [4, 2, −6] = 3,

[4, 2, 0] = 60, [4, 0, −2] = 0,

[4, 0, −4] = 0, [4, −4, −6] = −6,

[4, 0, −6] = 4, [2, 0, −2] = 0,

[4, −2, −4] = −6, [2, 0, −4] = 0,

[4, −2, −6] = 0, [2, 0, −6] = 10,

[2, −2, −4] = −15, [0, −2, −6] = 0,

[2, −2, −6] = 15, [0, −4, −6] = 0,

[2, −4, −6] = 0, [−2, −4, −6] = 0.

[0, −2, −4] = −60,

5.7.2

Identities of degrees 5 and 7

The anticommutative binary structure on V (6) obtained from the projection Λ2 V (6) → V (6) produces an algebra isomorphic to the 7-dimensional simple non-Lie Malcev algebra; see Bremner and Hentzel [5]. Since this is not a Lie algebra, the Jacobian of the binary Malcev product gives a nontrivial alternating ternary product on V (6). Since V (6) occurs only once as a summand of Λ3 V (6), the alternating ternary product on V (6) obtained from the projection Λ3 V (6) → V (6) must be equal (up to a nonzero scalar multiple) to the Jacobian of the Malcev product. In any alternative algebra (over a ﬁeld of characteristic not 2 or 3) the associator is a multiple of the Jacobian. Therefore the Jacobian on the 7-dimensional non-Lie Malcev algebra equals (up to a scalar multiple that does not aﬀect the identities) the associator on the 7-dimensional subspace of pure imaginary Cayley numbers. The alternating ternary structure on V (6) is therefore isomorphic to that obtained from the associator on the Cayley numbers (after factoring out the one-dimensional ideal of scalars). The identities of degree 7 for the associator on the Cayley numbers have been described in Bremner and Hentzel [3]. See especially Theorem 1 (page 262) and Theorem 2 (page 267). There are no identities in degree 5, and seven identities in degree 7. In that paper the results are stated for a ﬁeld of characteristic p = 103 since we used arithmetic modulo 103 for the computations. However, all the coeﬃcients of the identities can be regarded as integers that are small in absolute value. In this way the identities are meaningful over any ﬁeld. As long as the characteristic is greater than 7 (the degree of the identities in question) the results will remain valid.

Alternating triple systems with simple Lie algebras of derivations

5.8

73

Representation V (7): Dimension 8

The decomposition of the exterior cube is Λ3 V (7)∼ =V (15) ⊕ V (11) ⊕ V (9) ⊕ V (7) ⊕ V (5) ⊕ V (3). Highest weight vectors for the summands on the right side are t15 = [7, 5, 3], u11 = 3 [7, 5, −1] − 2 [7, 3, 1], w9 = 14 [7, 5, −3] − 7 [7, 3, −1] + 4 [5, 3, 1], x7 = 15 [7, 5, −5] − 5 [7, 3, −3] + 3 [7, 1, −1], y5 = 210 [7, 5, −7] − 35 [7, 3, −5] + 7 [7, 1, −3] + 5 [5, 3, −3] − 3 [5, 1, −1], z3 = 7 [7, 1, −5] − 7 [7, −1, −3] − 5 [5, 3, −5] + 2 [5, 1, −3] − [3, 1, −1].

5.8.1

Structure constants

The structure constants for the alternating triple system on V (7) are displayed in the following list (again using the compact notation introduced in the previous section): [7, 5, 3] = 0, [7, 5, −5] = 7, [7, 3, −3] = −21,

[7, 5, 1] = 0, [7, 5, −7] = 1, [7, 3, −5] = 0,

[7, 5, −1] = 0, [7, 3, 1] = 0, [7, 3, −7] = 1,

[7, 5, −3] = 0, [7, 3, −1] = 0, [7, 1, −1] = 35,

[7, 1, −3] = 0, [7, −1, −5] = 0,

[7, 1, −5] = 0, [7, −1, −7] = 1,

[7, 1, −7] = 1, [7, −3, −5] = 0,

[7, −1, −3] = 0, [7, −3, −7] = 1,

[7, −5, −7] = 1, [5, 3, −5] = −7,

[5, 3, 1] = 0, [5, 3, −7] = 0,

[5, 3, −1] = 0, [5, 1, −1] = 35,

[5, 3, −3] = −21, [5, 1, −3] = 0,

[5, 1, −5] = −7, [5, −1, −7] = 0,

[5, 1, −7] = 0, [5, −3, −5] = −7,

[5, −1, −3] = 0, [5, −3, −7] = 0,

[5, −1, −5] = −7, [5, −5, −7] = 7,

[3, 1, −1] = 35, [3, −1, −3] = 21, [3, −3, −7] = −21,

[3, 1, −3] = 21, [3, −1, −5] = 0, [3, −5, −7] = 0,

[3, 1, −5] = 0, [3, −1, −7] = 0, [1, −1, −3] = 35,

[3, 1, −7] = 0, [3, −3, −5] = −21, [1, −1, −5] = 35,

[1, −1, −7] = 35, [−1, −3, −5] = 0,

[1, −3, −5] = 0, [−1, −3, −7] = 0,

[1, −3, −7] = 0, [−1, −5, −7] = 0,

[1, −5, −7] = 0, [−3, −5, −7] = 0.

5.8.2

Identities of degree 5

Computations with p = 101 show that this triple system satisﬁes no identity in degree 5; this implies that it also satisﬁes no identities of degree 5 in characteristic 0.

5.8.3

Identities of degree 7

Computations with p = 101 show that this triple system satisﬁes 189 linearly independent identities in degree 7. The simplest of these identities has 14 terms. All of its nonzero coeﬃcients are in the set {1, 50, 51, 100}. These residue classes modulo 101 correspond to

74

Murray R. Bremner and Irvin R. Hentzel

the rational numbers 1, −1/2, 1/2, −1. We can therefore regard this as an identity with rational coeﬃcients. We repeated our computations in characteristic 0 and conﬁrmed that this triple system satisﬁes 189 linearly independent identities. THEOREM 5.5 The space of identities in degree 7 for the alternating ternary structure on V (7) has this decomposition into irreducible representations of S7 : [322] ⊕ 2[3211] ⊕ [31111] ⊕ 2[2221] ⊕ 3[22111] ⊕ 2[211111] ⊕ [1111111]. This space of identities is generated by the single 14-term identity displayed in Theorem 5.4. PROOF To show that this single identity generates the entire 189-dimensional space of identities we proceed as follows. We create a matrix of size 400 × 280. We generate all 5040 permutations of seven letters and divide them into 42 groups of 120 permutations. For each of the 42 groups we apply each of the 120 permutations to the given identity. This gives us 120 new identities, which we place in the bottom of the matrix (in rows 281 through 400). We then compute the row canonical form of the matrix. At this point the last 120 rows are zero. We then repeat this “ﬁll and reduce” process with the next group of 120 permutations. After we have processed all 42 groups the nonzero rows of the row canonical form give us a basis of the S7 -submodule generated by the original identity. When the last group has been processed the rank of the matrix is 189. This shows that the original identity generates the entire space of identities in degree 7 for the triple system V (7). The multiplicities of the irreducible representations in this case are almost the same as for V (5). The only diﬀerence is that here the last representation occurs only once.

5.9

Representation V (8): Dimension 9

The decomposition of the exterior cube is Λ3 V (8)∼ =V (18) ⊕ V (14) ⊕ V (12) ⊕ V (10) ⊕ V (8) ⊕ 2V (6) ⊕ V (2). Highest weight vectors for the summands on the right side are s18 = [8, 6, 4],

t14 = 7 [8, 6, 0] − 5 [8, 4, 2],

u12 = 14 [8, 6, −2] − 8 [8, 4, 0] + 5 [6, 4, 2], w10 = 7 [8, 6, −4] − 3 [8, 4, −2] + 2 [8, 2, 0], x8 = 56 [8, 6, −6] − 16 [8, 4, −4] + 4 [8, 2, −2] + 3 [6, 4, −2] − 2 [6, 2, 0], y6 = 105 [8, 6, −8] − 15 [8, 4, −6] + 5 [8, 2, −4] − 3 [8, 0, −2], y6 = 2352 [8, 6, −8] − 336 [8, 4, −6] + 56 [8, 2, −4] + 42 [6, 4, −4] − 21 [6, 2, −2] + 12 [4, 2, 0], z2 = 280 [8, 2, −8] − 210 [6, 4, −8] − 112 [8, 0, −6] + 35 [6, 2, −6] + 70 [8, −2, −4] − 7 [6, 0, −4] − 5 [4, 2, −4] + 3 [4, 0, −2]. In this case the module V (6) occurs with multiplicity 2, so we have two linearly independent highest weight vectors of weight 6.

Alternating triple systems with simple Lie algebras of derivations

5.9.1

75

Structure constants for the triple system

The structure constants for the alternating triple system on V (8) are displayed in the following list: [8, 6, 4] = 0, [8, 6, −4] = 0, [8, 4, 0] = 0,

[8, 6, 2] = 0, [8, 6, −6] = 16, [8, 4, −2] = 0,

[8, 6, 0] = 0, [8, 6, −8] = 2, [8, 4, −4] = −56,

[8, 6, −2] = 0, [8, 4, 2] = 0, [8, 4, −6] = 0,

[8, 4, −8] = 2, [8, 2, −6] = −6,

[8, 2, 0] = 0, [8, 2, −8] = 1,

[8, 2, −2] = 56, [8, 0, −2] = 35,

[8, 2, −4] = −21, [8, 0, −4] = 0,

[8, 0, −6] = −5, [8, −2, −8] = −1,

[8, 0, −8] = 0, [8, −4, −6] = −3,

[8, −2, −4] = 0, [8, −4, −8] = −2,

[8, −2, −6] = −4, [8, −6, −8] = −2,

[6, 4, 2] = 0, [6, 4, −6] = 2, [6, 2, −4] = 0,

[6, 4, 0] = 0, [6, 4, −8] = 3, [6, 2, −6] = −6,

[6, 4, −2] = 168, [6, 2, 0] = −280, [6, 2, −8] = 4,

[6, 4, −4] = 7, [6, 2, −2] = 42, [6, 0, −2] = 70,

[6, 0, −4] = 35, [6, −2, −6] = 6,

[6, 0, −6] = 0, [6, −2, −8] = 6,

[6, 0, −8] = 5, [6, −4, −6] = −2,

[6, −2, −4] = 28, [6, −4, −8] = 0,

[6, −6, −8] = −16, [4, 2, −6] = −28,

[4, 2, 0] = −245, [4, 2, −8] = 0,

[4, 2, −2] = −98, [4, 0, −2] = 0,

[4, 2, −4] = −49, [4, 0, −4] = 0,

[4, 0, −6] = −35, [4, −2, −8] = 21,

[4, 0, −8] = 0, [4, −4, −6] = −7,

[4, −2, −4] = 49, [4, −4, −8] = 56,

[4, −2, −6] = 0, [4, −6, −8] = 0,

[2, 0, −2] = 0, [2, −2, −4] = 98, [2, −4, −8] = 0,

[2, 0, −4] = 0, [2, −2, −6] = −42, [2, −6, −8] = 0,

[2, 0, −6] = −70, [2, −2, −8] = −56, [0, −2, −4] = 245,

[2, 0, −8] = −35, [2, −4, −6] = −168, [0, −2, −6] = 280,

[0, −2, −8] = 0, [−2, −4, −6] = 0,

[0, −4, −6] = 0, [−2, −4, −8] = 0,

[0, −4, −8] = 0, [−2, −6, −8] = 0,

[0, −6, −8] = 0, [−4, −6, −8] = 0.

5.9.2

Identities of degree 5

Computations with p = 101 show that this triple system satisﬁes no identity in degree 5; this implies that it also satisﬁes no identities of degree 5 in characteristic 0.

5.9.3

Identities of degree 7

THEOREM 5.6 In characteristic 0 the alternating triple system on V (8) satisﬁes eight linearly independent multilinear identities. These eight identities span a representation of the symmetric group S7 , which decomposes into the direct sum of irreducible components: [211111] ⊕ 2[1111111]. This representation is generated by these two identities: I1 = 6 [[[a1 a2 a3 ]a4 b]a5 a6 ] − [[a1 a2 a3 ][a4 a5 a6 ]b], alt(6)

I2 =

alt(7)

[[[a1 a2 a3 ]a4 a5 ]a6 a7 ].

alt(6)

76

Murray R. Bremner and Irvin R. Hentzel

The ﬁrst identity along with the second identity implies that the alternating sum over the second association type is also an identity. The computational methods used in the proof of this result are very similar to those discussed elsewhere in this chapter, so we omit the details.

5.10

Representation V (10): Dimension 11

The decomposition of the exterior cube is Λ3 V (10)∼ =V (24) ⊕ V (20) ⊕ V (18) ⊕ V (16) ⊕ V (14) ⊕ 2V (12) ⊕ V (10) ⊕ 2V (8) ⊕ V (6) ⊕ V (4) ⊕ V (0). Highest weight vectors for the summands on the right side are o24 = [10, 8, 6],

p20 = 9 [10, 8, 2] − 7 [10, 6, 4],

q18 = 15 [10, 8, 0] − 10 [10, 6, 2] + 7 [8, 6, 4], r16 = 36 [10, 8, −2] − 20 [10, 6, 0] + 15 [10, 4, 2], s14 = 36 [10, 8, −4] − 16 [10, 6, −2] + 5 [10, 4, 0] + 4 [8, 6, 0] − 3 [8, 4, 2], t12 = 42 [10, 8, −6] − 14 [10, 6, −4] + 7 [10, 4, −2] − 5 [10, 2, 0], t12 = 360 [10, 8, −6] − 120 [10, 6, −4] + 30 [10, 4, −2] + 24 [8, 6, −2] − 15 [8, 4, 0] u10

+ 10 [6, 4, 2], = 630 [10, 8, −8] − 140 [10, 6, −6] + 35 [10, 4, −4] + 14 [8, 6, −4]

− 10 [10, 2, −2] − 7 [8, 4, −2] + 5 [8, 2, 0], w8 = 252 [10, 8, −10] − 28 [10, 6, −8] + 7 [10, 4, −6] − 3 [10, 2, −4] w8

+ 2 [10, 0, −2], = 18900 [10, 8, −10] − 2100 [10, 6, −8] + 315 [10, 4, −6] + 168 [8, 6, −6] − 45 [10, 2, −4] − 63 [8, 4, −4] + 18 [8, 2, −2] + 14 [6, 4, −2] − 10 [6, 2, 0],

x6 = 315 [10, 4, −8] − 252 [8, 6, −8] − 180 [10, 2, −6] + 63 [8, 4, −6] + 75 [10, 0, −4] + 9 [8, 2, −4] − 28 [6, 4, −4] − 30 [8, 0, −2] + 16 [6, 2, −2] − 10 [4, 2, 0], y4 = 1890 [10, 4, −10] − 1512 [8, 6, −10] − 540 [10, 2, −8] + 189 [8, 4, −8] + 240 [10, 0, −6] − 36 [8, 2, −6] − 14 [6, 4, −6] − 162 [10, −2, −4] + 9 [8, 0, −4] + 6 [6, 2, −4] − 4 [6, 0, −2], z0 = 1890 [10, 0, −10] − 1134 [8, 2, −10] + 882 [6, 4, −10] − 1134 [10, −2, −8] + 378 [8, 0, −8] − 126 [6, 2, −8] + 882 [10, −4, −6] − 126 [8, −2, −6] − 14 [6, 0, −6] + 42 [4, 2, −6] + 42 [6, −2, −4] − 21 [4, 0, −4] + 12 [2, 0, −2]. In this case the modules V (12) and V (8) occur with multiplicity 2, so for weights 12 and 8 we have two linearly independent highest weight vectors. The anticommutative binary structure on V (10) obtained from the projection Λ2 V (10) → V (10) has been studied in detail in Bremner and Hentzel [5]. Since this is not a Lie algebra, the Jacobian of this binary product gives a nontrivial alternating ternary product on V (10).

Alternating triple systems with simple Lie algebras of derivations

77

Since V (10) occurs only once as a summand of Λ3 V (10), the alternating ternary product on V (10) obtained from the projection Λ3 V (10) → V (10) must be equal (up to a nonzero scalar multiple) to the Jacobian of the binary product.

5.10.1

Structure constants for the triple system

The structure constants for the alternating triple system on V (10) are displayed in the following list: [10, 8, 6] = 0, [10, 8, 0] = 0,

[10, 8, 4] = 0, [10, 8, −2] = 0,

[10, 8, 2] = 0, [10, 8, −4] = 0,

[10, 8, −6] = 0, [10, 6, 4] = 0, [10, 6, −2] = 0,

[10, 8, −8] = 30, [10, 6, 2] = 0, [10, 6, −4] = 0,

[10, 8, −10] = 3, [10, 6, 0] = 0, [10, 6, −6] = −135,

[10, 6, −8] = 0, [10, 4, 0] = 0,

[10, 6, −10] = 3, [10, 4, −2] = 0,

[10, 4, 2] = 0, [10, 4, −4] = 240,

[10, 4, −6] = −36, [10, 2, 0] = 0,

[10, 4, −8] = −8, [10, 2, −2] = −210,

[10, 4, −10] = 2, [10, 2, −4] = 84,

[10, 2, −6] = 0, [10, 0, −2] = −126,

[10, 2, −8] = −7, [10, 0, −4] = 0,

[10, 2, −10] = 1, [10, 0, −6] = 0,

[10, 0, −8] = −6, [10, −2, −6] = 0, [10, −4, −6] = 0,

[10, 0, −10] = 0, [10, −2, −8] = −5, [10, −4, −8] = −4,

[10, −2, −4] = 0, [10, −2, −10] = −1, [10, −4, −10] = −2,

[10, −6, −8] = −3, [8, 6, 4] = 0,

[10, −6, −10] = −3, [8, 6, 2] = 0,

[10, −8, −10] = −3, [8, 6, 0] = 0,

[8, 6, −2] = 0, [8, 6, −8] = −6,

[8, 6, −4] = 360, [8, 6, −10] = 3,

[8, 6, −6] = −27, [8, 4, 2] = 0,

[8, 4, 0] = 0, [8, 4, −6] = 0,

[8, 4, −2] = −840, [8, 4, −8] = −16,

[8, 4, −4] = 192, [8, 4, −10] = 4,

[8, 2, 0] = 1260, [8, 2, −6] = 63, [8, 0, −2] = −252,

[8, 2, −2] = −168, [8, 2, −8] = −8, [8, 0, −4] = 0,

[8, 2, −4] = 168, [8, 2, −10] = 5, [8, 0, −6] = 54,

[8, 0, −8] = 0, [8, −2, −6] = 45,

[8, 0, −10] = 6, [8, −2, −8] = 8,

[8, −2, −4] = 0, [8, −2, −10] = 7,

[8, −4, −6] = 36, [8, −6, −8] = 6,

[8, −4, −8] = 16, [8, −6, −10] = 0,

[8, −4, −10] = 8, [8, −8, −10] = −30,

[6, 4, 2] = 0, [6, 4, −4] = −144, [6, 4, −10] = 0,

[6, 4, 0] = 0, [6, 4, −6] = −54, [6, 2, 0] = 1134,

[6, 4, −2] = −756, [6, 4, −8] = −36, [6, 2, −2] = −126,

[6, 2, −4] = 0, [6, 2, −10] = 0,

[6, 2, −6] = 27, [6, 0, −2] = −378,

[6, 2, −8] = −45, [6, 0, −4] = −216,

[6, 0, −6] = 0, [6, −2, −4] = −180,

[6, 0, −8] = −54, [6, −2, −6] = −27,

[6, 0, −10] = 0, [6, −2, −8] = −63,

78

5.10.2

Murray R. Bremner and Irvin R. Hentzel [6, −2, −10] = 0,

[6, −4, −6] = 54,

[6, −4, −8] = 0,

[6, −4, −10] = 36, [6, −8, −10] = 0,

[6, −6, −8] = 27, [4, 2, 0] = 1008,

[6, −6, −10] = 135, [4, 2, −2] = 504,

[4, 2, −4] = 288, [4, 2, −10] = 0,

[4, 2, −6] = 180, [4, 0, −2] = 0,

[4, 2, −8] = 0, [4, 0, −4] = 0,

[4, 0, −6] = 216, [4, −2, −4] = −288, [4, −2, −10] = −84,

[4, 0, −8] = 0, [4, −2, −6] = 0, [4, −4, −6] = 144,

[4, 0, −10] = 0, [4, −2, −8] = −168, [4, −4, −8] = −192,

[4, −4, −10] = −240, [4, −8, −10] = 0,

[4, −6, −8] = −360, [2, 0, −2] = 0,

[4, −6, −10] = 0, [2, 0, −4] = 0,

[2, 0, −6] = 378, [2, −2, −4] = −504,

[2, 0, −8] = 252, [2, −2, −6] = 126,

[2, 0, −10] = 126, [2, −2, −8] = 168,

[2, −2, −10] = 210, [2, −4, −10] = 0, [2, −8, −10] = 0,

[2, −4, −6] = 756, [2, −6, −8] = 0, [0, −2, −4] = −1008,

[2, −4, −8] = 840, [2, −6, −10] = 0, [0, −2, −6] = −1134,

[0, −2, −8] = −1260, [0, −4, −8] = 0,

[0, −2, −10] = 0, [0, −4, −10] = 0,

[0, −4, −6] = 0, [0, −6, −8] = 0,

[0, −6, −10] = 0, [−2, −4, −8] = 0,

[0, −8, −10] = 0, [−2, −4, −10] = 0,

[−2, −4, −6] = 0, [−2, −6, −8] = 0,

[−2, −6, −10] = 0, [−4, −6, −10] = 0,

[−2, −8, −10] = 0, [−4, −8, −10] = 0,

[−4, −6, −8] = 0, [−6, −8, −10] = 0.

Identities of degree 5

Computations with p = 101 show that this triple system satisﬁes no identity in degree 5; this implies that it also satisﬁes no identities of degree 5 in characteristic 0.

5.10.3

Identities of degree 7

Computations with p = 101 show that this triple system satisﬁes no identity in degree 7; this implies that it also satisﬁes no identities of degree 7 in characteristic 0.

5.10.4

Open problem

Determine the lowest degree for which the alternating ternary structure on V (10) has nontrivial identities. In particular, are there any identities of degree 9?

5.11 5.11.1

Other Simple Lie Algebras Tensor products

Let L be a semisimple (ﬁnite dimensional) Lie algebra over an algebraically closed ﬁeld F of characteristic 0. Let U, V, W be three ﬁnite dimensional representations of L. By Weyl’s

Alternating triple systems with simple Lie algebras of derivations

79

Table 5.4. Alternating ternary structures in type B B2

10

Ω1

5

0

Ω2

4

1

B3

21

Ω1

7

0

Ω2

21

0

Ω3

8

1

B4

36

Ω1 Ω4

9 16

0 0

Ω2

36

0

Ω3

84

1

B5

55

Ω1 Ω4

11 330

0 1

Ω2 Ω5

55 32

0 1

Ω3

165

1

B6

78

Ω1 Ω4

13 715

0 1

Ω2 Ω5

78 1287

0 2

Ω3 Ω6

286 64

1 2

B7

105

Ω1 Ω4 Ω7

15 1365 128

0 1 1

Ω2 Ω5

105 3003

0 2

Ω3 Ω6

455 5005

1 3

B8

136

Ω1 Ω4 Ω7

17 2380 19448

0 1 4

Ω2 Ω5 Ω8

136 6188 256

0 2 1

Ω3 Ω6

680 12376

1 3

theorem we know that any ﬁnite dimensional representation of L is completely reducible; that is, it decomposes as the direct sum of irreducible representations. In particular, this holds for the tensor product U ⊗ V ⊗ W . In the special case U = V = W we are interested in the multiplicity of V as a direct summand of its own tensor cube. If this multiplicity is nonzero then there exists a nonzero L-module homomorphism p : V ⊗3 → V , which gives V the structure of a triple system, and this structure is L-invariant in the sense that L is contained in the derivation algebra. This structure is alternating when V occurs as a summand of the exterior cube Λ3 (V ). The simple Lie algebras are characterized by their Dynkin diagrams. We follow the conventions of Humphreys [8] for the labelling of these diagrams.

5.11.2

Exterior cubes

A simple Lie algebra of rank has fundamental representations, which we will denote by Ωi for 1 ≤ i ≤ . In this section we list, for each simple Lie algebra of rank 2 ≤ ≤ 8 and each fundamental representation, the multiplicity dim HomL (Λ3 Ωi , Ωi ) of Ωi as a direct summand of its exterior cube. This multiplicity is (one more than) the number of (projective) parameters that occur in the classiﬁcation of the L-invariant alternating ternary structures on Ωi . To perform these calculations we used the computer algebra package LiE [6].

80

Murray R. Bremner and Irvin R. Hentzel Table 5.5. Alternating ternary structures in type C C3

21

Ω1

6

1

Ω2

14

0

Ω3

14

1

C4

36

Ω1 Ω4

8 42

1 0

Ω2

27

0

Ω3

48

2

C5

55

Ω1 Ω4

10 165

1 1

Ω2 Ω5

44 132

0 1

Ω3

110

1

C6

78

Ω1 Ω4

12 429

1 1

Ω2 Ω5

65 572

0 2

Ω3 Ω6

208 429

3 1

C7

105

Ω1 Ω4 Ω7

14 910 1430

1 1 1

Ω2 Ω5

90 1638

0 3

Ω3 Ω6

350 2002

3 2

C8

136

Ω1 Ω4 Ω7

16 1700 7072

1 1 3

Ω2 Ω5 Ω8

119 3808 4862

0 4 1

Ω3 Ω6

544 6188

3 2

Table 5.6. Alternating ternary structures in type D

5.11.3

D4

28

Ω1

8

0

Ω2

28

0

D5

45

Ω1 Ω4

10 16

0 0

Ω2

45

0

Ω3

120

1

D6

66

Ω1 Ω4

12 495

0 1

Ω2 Ω5

66 32

0 1

Ω3

220

2

D7

91

Ω1 Ω4

14 1001

0 2

Ω2 Ω5

91 2002

0 3

Ω3 Ω6

364 64

1 0

D8

120

Ω1 Ω4 Ω7

16 1820 128

0 1 0

Ω2 Ω5

120 4368

0 4

Ω3 Ω6

560 8008

1 4

Special linear: type A

The only fundamental representation of a simple Lie algebra of type A (with 1 ≤ ≤ 8) which occurs as a direct summand of its own exterior cube, is the 20-dimensional representation Ω3 of the Lie algebra A5 . The multiplicity is 1, and so Ω3 admits an A5 -invariant alternating ternary operation, which is unique up to a scalar multiple.

Alternating triple systems with simple Lie algebras of derivations

81

Table 5.7. Alternating ternary structures in types E, F, G

5.11.4

E6

78

Ω1 Ω3 Ω5

27 351 351

0 0 0

Ω2 Ω4 Ω6

78 2925 27

0 10 0

E7

133

Ω1 Ω3 Ω5 Ω7

133 8645 27664 56

0 7 23 1

Ω2 Ω4 Ω6

912 365750 1539

1 209 1

E8

248

Ω1 Ω3 Ω5 Ω7

3875 6696000 146325270 30380

1 214 4087 6

Ω2 Ω4 Ω6 Ω8

147250 6899079264 2450240 248

9 173159 128 0

F4

52

Ω1 Ω3

52 273

0 4

Ω2 Ω4

1274 26

7 0

G2

14

Ω1

7

1

Ω2

14

0

Orthogonal: type B

The results are displayed in Table 5.4. In this and the following tables the results are presented in this format: the name of the Lie algebra is followed by its dimension; and the name of each fundamental representation is followed by its dimension and its multiplicity in its own exterior cube.

5.11.5

Symplectic: type C

The results are displayed in Table 5.5.

5.11.6

Orthogonal: type D

By the symmetry of the Dynkin diagram the dimensions for Ω −1 and Ω are the same, so we omit the latter. By the triality symmetry of the Dynkin diagram of D4 the dimensions of Ω1 , Ω3 , and Ω4 are the same so we omit the latter two. The results are displayed in Table 5.6.

5.11.7

Exceptional: types E, F, G

The results are displayed in Table 5.7. The computation of the largest exterior cube in this list, Λ3 Ω4 for E8 , took 860 seconds; altogether 1436 distinct representations occur in the decomposition.

82

5.12

Murray R. Bremner and Irvin R. Hentzel

Acknowledgments

We thank the organizers of NONAA-V for the invitation to present the results of our earlier paper [5] at that conference. The current chapter extends those results to the ternary case. Murray Bremner thanks NSERC (the Natural Sciences and Engineering Research Council of Canada) for supporting this research with a Discovery Grant, the University of Saskatchewan for a sabbatical leave from January to June 2004, and the Department of Mathematics at Iowa State University for its hospitality. He also thanks Lauren Bains, recipient of an NSERC Undergraduate Summer Research Award, for proofreading parts of an earlier version of this chapter and making helpful suggestions for improving the exposition.

References [1] M. Bremner, Varieties of anticommutative n-ary algebras, Journal of Algebra 191 (1997), 76–88. [2] M. Bremner, Identities for the ternary commutator, Journal of Algebra 206 (1998), 615–623. [3] M. Bremner, I. Hentzel, Identities for the associator in alternative algebras, Journal of Symbolic Computation 33 (2002), 255–273. [4] M. Bremner, I. Hentzel, Identities for algebras of matrices over the octonions, Journal of Algebra, 277 (2004) 73-95. [5] M. Bremner, I. Hentzel, Invariant nonassociative algebra structures on irreducible representations of simple Lie algebras, Experimental Mathematics, 13 (2004) 231-256. [6] A. Cohen, M. van Leeuwen, B. Lisser, LiE: A computer gebra package for Lie group computations, available on-line http://young.sp2mi.univ-poitiers.fr/~marc/LiE/

alat

[7] V. T. Filippov, n-Lie algebras [Russian], Sibirskii Matematicheskii Zhurnal 26 (1985), 126–140; English translation in Siberian Mathematical Journal 26 (1985), 879–891. [8] J. Humphreys, Introduction to Lie Algebras and Representation Theory, SpringerVerlag, New York, 1972.

Chapter 6 The Lie Group S3 in Absolute Valued Structures Antonio J. Calder´ on Mart´ın Departamento de Matem´aticas, Universidad de C´ adiz, Spain C´ andido Mart´ın Gonz´ alez Departamento de Algebra, Geometr´ıa y Topolog´ıa, Universidad de M´ alaga, Spain 6.1

6.2

6.3

Introduction and preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.1 On absolute valued algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.2 On absolute valued triple systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.1 Previous works . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.2 A classiﬁcation of four-dimensional a.v.t.s. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.3 The conjugation techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

83 83 84 85 85 85 86 90 90

Abstract We develop techniques of conjugation, with respect to a certain maximal tori in the Lie group S 3 , in the framework of absolute valued structures. As a consequence, we give a classiﬁcation of four-dimensional absolute-valued triple systems attending just to three reals and one quaternion in the nonexceptional case and one more quaternion in the exceptional case. The isomorphism classes among them are also stated. Key words: Absolute values, triple systems 2000 MSC: 17A80, 17A40, 17Axx

6.1 6.1.1

Introduction and Preliminaries On absolute valued algebras

Let K denote the ﬁeld of real or complex numbers. An absolute-valued algebra over K is a nonzero algebra A over K, endowed with a norm | · | satisfying |xy| = |x||y| for all x, y ∈ A. The most natural examples of absolute-valued algebras are R, C, H (the algebra of Hamilton quaternions), and O (the algebra of Cayley numbers), with norms equal to their usual absolute values. Since the early paper of A. Albert ([1]) where it is proved that the only ﬁnite dimensional absolute-valued algebra is C in the complex case and R, C, H and O in the real one, absolute-valued algebras have been intensively studied by many authors (see, for instance, the excellent survey [2] and [1, 3, 4, 5, 6, 7, 8, 9, 10, 11]). Clearly, any ﬁnite dimensional absolute-valued algebra is a division algebra, conversely, absolute-valued division algebras are ﬁnite dimensional ([12]). It is easy to see that if

83

84

Antonio J. Calder´ on Mart´ın and C´ andido Mart´ın Gonz´ alez

two norms on a ﬁnite dimensional algebra convert it into an absolute-valued algebra, then they must coincide (see for instance [4]). From here, it is also clear that any isomorphism between two ﬁnite dimensional absolute-valued algebras f : A → A is isometric. Indeed, we can deﬁne a new norm on A by |x|A = |f (x)|A that makes A an absolute-valued algebra, ﬁnally the uniqueness of the absolute value gives us |x|A = |x|A = |f (x)|A . The precise determination of isomorphism classes for absolute-valued real algebras of dimensions 1 and 2 is contained in ([10]), where the number of classes reduces to 1 and 4 respectively, while a detailed determination for the four-dimensional ones appears in ([9]).

6.1.2

On absolute valued triple systems

Let T be a vector space over K. We say that T is a triple system if it is endowed with a trilinear map,

·, ·, · : T × T × T → T,

called the triple product of T. Let T , T be triple systems, a bijective linear map f : T → T is called an isomorphism of triple systems if it satisﬁes

f (x, y, z) = f (x), f (y), f (z)

for any x, y, z ∈ T . Triple systems appear in the literature as the natural ternary extension of algebras and have been studied in the associative ([13, 14, 15]), nonassociative ([16, 17, 18, 19, 20, 21]) and general context ([22]). An absolute-valued triple system is deﬁned as follows

DEFINITION 6.1 An absolute-valued triple system, (a.v.t.s.), is a nonzero triple system T over K, K = R or C, endowed with a norm | · | that satisﬁes |x, y, z| = |x||y||z| for any x, y, z ∈ T . A fundamental reference in this framework is [23]. Any absolute-valued algebra A can be seen as an a.v.t.s., with the same norm, by deﬁning, for instance, the triple product as x, y, z = (xy)z. Then we have that the class of absolute-valued algebras is related to the class of a.v.t.s. Moreover, given u ∈ T with |u| = 1, we can construct an absolute-valued algebra, denoted by T u , by deﬁning xy = x, u, y and with the same norm as T . Since dim(T ) = dim(T u ), Albert’s result in 6.1.1 gives us that any ﬁnite dimensional a.v.t.s. has dimension 1 in the complex case and 1, 2, 4, or 8 in the real one, and that the absolute values are the usual euclidean norms. By considering T u and taking into account the observations in 6.1.1, we also obtain that if we have two norms on T such that convert it into an a.v.t.s., then they must coincide and, that any isomorphism between two ﬁnite dimensional a.v.t.s. is isometric.

The Lie group S 3 in absolute valued structures

6.2 6.2.1

85

Main Results Previous works

In [24] we study some aspects of the theory of a.v.t.s. and exhibit the isomorphisms classes of a.v.t.s. of dimension 1, 2, and 4. We also relate the theory of a.v.t.s. to the one of two-graded absolute-valued algebras in [25]. We are interested in the present chapter in investigating deeply the four-dimensional a.v.t.s.

6.2.2

A classiﬁcation of four-dimensional a.v.t.s.

If T is any four-dimensional (real) a.v.t.s. we will identify T with H as euclidean vector spaces. From now on, given x, y ∈ H, the juxtaposition xy shall mean the usual product in H. Let us introduce some of notation: We will denote by (6.1) Tσp ,σq ,σr (a, b, c, d)μ the four dimensional a.v.t.s. with product < x1 , x2 , x3 >= aσp (xμ(1) )bσq (xμ(2) )cσr (xμ(3) )d, where any σi , i ∈ {p, q, r} is either the identity or the Cayley involution on H, μ ∈ S3 a permutation of the set {1, 2, 3} and a, b, c ∈ S 3 elements of the unit sphere in R4 . We shall denote σi = 1 if σi is the identity, and σi = −1 if σi is the Cayley involution on H. In [24, Theorem 2.10] it is shown that any four-dimensional a.v.t.s. T is isomorphic to one of the form (6.1) with μ ∈ {1, (2, 3), (1, 2)}. In case (σp , σq , σr ) = (1, −1, 1) we say that T is exceptional. This term is a consequence of being the only case, among the eight possible cases attending to the values of σp , σq , σr , which cannot be reduced, by isomorphism, to depend on just three quaternions, (see [24]). The remaining cases, that can be written depending of three quaternions following Table 6.1 for suitable x, y, z ∈ S 3 , are called nonexceptional, (see also [24]). Table 6.1. Nonexceptional four-dimensional a.v.t.s. (σp , σq , σr ) 1, 1, 1 1, 1, −1 1, −1, −1 −1, 1, 1 −1, 1, −1 −1, −1, 1 −1, −1, −1

∼ = T1,1,1 (x, 1, y, z)μ T1,1,−1 (x, 1, y, z)μ T1,−1,−1 (x, y, 1, z)μ T−1,1,1 (1, x, y, z)μ T−1,1,−1 (1, x, y, z)μ T−1,−1,1 (1, x, y, z)μ T−1,−1,−1 (1, x, y, z)μ

From now on, as any nonexceptional four dimensional a.v.t.s. T is determined, following the above table, with just three quaternions x, y, z in S 3 (and the σi ’s and μ), we will denote T by Tσp ,σq ,σr (x, y, z)μ . The isomorphism classes among four dimensional a.v.t.s. are given in [24, Theorem 2.10 and Corollary 2.11]. They are summarized in the following paragraphs.

86

Antonio J. Calder´ on Mart´ın and C´ andido Mart´ın Gonz´ alez

THEOREM 6.1 1. Two nonexceptional a.v.t.s. Tσp ,σq ,σr (x, y, z)μ and Tσp ,σq ,σr (x , y , z )μ ,

μ, μ ∈ {1, (2, 3), (1, 2)},

are isomorphic if and only if σi = σi for any i ∈ {p, q, r}, μ = μ and there exist q ∈ S 3 and , δ ∈ ±1 such that x = qxq −1 , y = δqyq −1 and z = δqzq −1 .

2. Two exceptional a.v.t.s. T1,−1,1 (x, y, z, t)μ and T1,−1,1 (x , y , z , t )μ , where μ, μ ∈ {1, (2, 3), (1, 2)}, are isomorphic if and only if μ = μ and there exist p, q ∈ S 3 and , δ, τ ∈ ±1 such that x = pxp−1 , y = δqyq −1 , z = τ δpzp−1 and t = τ qtq −1 .

6.2.3

The conjugation techniques

As we observe in 6.2.2, any four dimensional a.v.t.s. Tσp ,σq ,σr (a, b, c, d)μ is given depending, up to the σi ’s and μ, on either three or four quaternions of norm 1. We will show how techniques of conjugation with respect to a certain maximal tori in the Lie group S 3 allow us to reﬁne the above classiﬁcation so as to give a classiﬁcation of four-dimensional a.v.t.s. depending just on three reals and one quaternion in S 3 in the nonexceptional case and of one more quaternion in S 3 in the exceptional case. This will give some advantages from the viewpoint of the isomorphism condition. Although some of this ideas have been introduced in [25], for the convenience of the reader we develop all of them in detail. If we consider the standard basis {1, i, j, k} of H, with i2 = j 2 = k 2 = −1,

ij = −ji = k,

jk = −kj = i

and ki = −ik = j,

then H = R1 ⊕ W where W = 1⊥ = span({i, j, k}). Taking S 3 = {x ∈ H : |x| = 1}, this is obviously a compact connected Lie group, it is easy to prove directly that one of its maximal tori is the subgroup {exp(θi) : θ ∈ R} ∼ = S 1 (see also [26, (3.7) Theorem, p. 173]). It is also a well-known result that in a compact connected Lie group, any element is conjugated to some element in a preﬁxed maximal torus (see [26, (1.7) main lemma, p.159], or [27]). In particular, for any x ∈ S 3 there is some q ∈ S 3 such that qxq −1 = exp(θi) for some θ ∈ [0, 2π). Let us consider a nonexceptional four-dimensional a.v.t.s. Tσp ,σq ,σr (x, y, z)μ . Taking into account Theorem 6.1, if x = 1, we have Tσp ,σq ,σr (1, y, z)μ ∼ = Tσp ,σq ,σr (1, qyq −1 , qzq −1 )μ , and for a suitable q we can write qyq −1 = exp(θi) for some θ ∈ [0, 2π). Then we obtain Tσp ,σq ,σr (1, y, z)μ ∼ = Tσp ,σq ,σr (1, exp(θi), qzq −1 )μ . If x = 1, we also have an isomorphism Tσp ,σq ,σr (x, y, z)μ ∼ = Tσp ,σq ,σr (exp(θi), y , z )μ . We can now consider an arbitrary element q1 = exp(si) for some s ∈ R and so Tσp ,σq ,σr (exp(θi), y , z )μ ∼ = Tσp ,σq ,σr (q1 exp(θi)q1−1 , q1 y q1−1 , q1 z q1−1 )μ . But q1 exp(θi)q1−1 = exp(θi) for any such q1 . As a consequence, we still have a certain freedom degree to simplify y by conjugating it with some q1 chosen as above. So if y = y0 + y1 i + w for some w ∈ span({j, k}), then q1 y q1−1 = y0 + y1 i + q1 w q1−1 = y0 + y1 i + q12 w but q12 w = exp(2si)w can have any value in span({j, k}) with the same norm as w .

The Lie group S 3 in absolute valued structures

87

+ |w |j and For instance, q12 w = |w |j for a suitable q1 . Therefore q1 y q1−1 = y0 + y1 i 2 2 2 y0 + y1 + |w | = 1. Thus we may write y0 = ρ cos φ, y1 = ρ sin φ, and |w | = 1 − ρ2 for −1 some φ, ρ ∈ R, 0 ≤ ρ ≤ 1. In this way q1 y q1 = ρ exp(φi) + 1 − ρ2 j and Tσp ,σq ,σr (x, y, z)μ ∼ = Tσp ,σq ,σr (exp(θi), ρ exp(φi) +

1 − ρ2 j, z )μ

with θ, φ ∈ [0, 2π), ρ ∈ [0, 1] and z ∈ S 3 . Analogously, Tσp ,σq ,σr (1, y, z)μ ∼ = Tσp ,σq ,σr (1, exp(θi), ρ exp(φi) +

1 − ρ2 j)μ ,

with the same restrictions on θ, φ and ρ. We shall denote the a.v.t.s. Tσp ,σq ,σr (exp(θi), ρ exp(φi) + 1 − ρ2 j, z )μ by Tσp ,σq ,σr (θ, φ, ρ, z )μ . The dependence of Tσp ,σq ,σr (x, y, z)μ on the quaternions x, y and z is replaced by the dependence of Tσp ,σq ,σr (θ, φ, ρ, z )μ on the three real numbers θ, φ, and ρ and the quaternion z . Let us now bound the possible values of the three parameters θ, φ, and ρ. We already know that ρ ∈ [0, 1]. Taking into account the relation: j exp(θ i)j −1 = exp(−θ i), we can limit θ to range on [0, π], and since j exp(θi)j = exp((π − θ)i), we can take θ ∈ [0, π2 ]. On the other hand, the relation −i(ρ exp(φi) + 1 − ρ2 j)i = −(ρ exp((φ + π)i) + 1 − ρ2 j), implies that we can take φ ∈ [0, π). We can claim the following: THEOREM 6.2 Any nonexceptional four-dimensional a.v.t.s. is isomorphic to one of the triple systems Tσp ,σq ,σr (θ, φ, ρ, z)μ with any σi , i ∈ {p, q, r} the identity or the Cayley involution, θ ∈ [0, π2 ], φ ∈ [0, π), ρ ∈ [0, 1], z ∈ S 3 and μ ∈ {1, (2, 3), (1, 2)}. If θ = 0 then ρ = 1 and, with the above conditions for the parameters, Tσp ,σq ,σr (θ, φ, ρ, z)μ ∼ = Tσp ,σq ,σr (θ , φ , ρ , z )μ

if and only if σi = σi for i ∈ {p, q, r}, μ = μ, θ = θ, ρ = ρ and (a) If ρ = ρ = 0, 1, then φ = φ and z = z, except in the cases that follow. 1. θ = π/2. We have Tσp ,σq ,σr (π/2, φ, ρ, z)μ ∼ = Tσp ,σq ,σr (π/2, φ , ρ, z )μ , with (φ , z ) ∈ {(φ, z), (π − φ, −kzk)}. 2. θ = π/2, φ = 0. We have Tσp ,σq ,σr (π/2, 0, ρ, z)μ ∼ = Tσp ,σq ,σr (π/2, φ , ρ, z )μ , with (φ , z ) ∈ {(0, z), (0, jzj), (π, −kzk)}. (b) If ρ = ρ = 1, then φ = φ and z = exp(si)z exp(−si), s ∈ [0, 2π), except in the cases that follow. 1. θ = 0. We have Tσp ,σq ,σr (0, φ, 1, z)μ ∼ = Tσp ,σq ,σr (0, φ , 1, z )μ , with (φ , z ) ∈ {(φ, exp(si)z exp(−si)), (π − φ, −q2 zq2−1 )}, where s ∈ [0, 2π) and q2 ∈ span{j, k} ∩ S 3 .

88

Antonio J. Calder´ on Mart´ın and C´ andido Mart´ın Gonz´ alez 2. θ = π/2. We have Tσp ,σq ,σr (π/2, φ, 1, z)μ ∼ = Tσp ,σq ,σr (π/2, φ , 1, z )μ , with (φ , z ) ∈ {(φ, exp(si)z exp(−si)), (π − φ, q2 zq2−1 )} where s ∈ [0, 2π) and q2 ∈ span{j, k} ∩ S 3 . 3. θ = π/2, φ = 0. We have Tσp ,σq ,σr (π/2, 0, 1, z)μ ∼ = Tσp ,σq ,σr (π/2, φ , 1, z )μ , with −1 (φ , z ) ∈ {(0, exp(si)z exp(−si)), (0, −q2 zq2 ), (π, q2 zq2−1 )}, where s ∈ [0, 2π) and q2 ∈ span{j, k} ∩ S 3 . 4. θ = φ = 0. We have Tσp ,σq ,σr (0, 0, 1, z)μ ∼ = Tσp ,σq ,σr (0, φ , 1, z )μ , with (φ , z ) ∈ {(0, qzq −1 ), (π, −q2 zq2−1 )}, where s ∈ [0, 2π) and q ∈ S 3 . (c) If ρ = ρ = 0, then φ, φ are under no restriction and z ∈ {z, izi}, except in the case that follows. 1. θ = π/2. We have Tσp ,σq ,σr (π/2, φ, 0, z)μ ∼ = Tσp ,σq ,σr (π/2, φ , 0, z )μ , with z ∈ {z, izi, jzj, −kzk} and where φ, φ are under no restriction.

PROOF The only thing we have to prove is the isomorphism condition. Suppose Tσp ,σq ,σr (θ, φ, ρ, z)μ ∼ = Tσp ,σq ,σr (θ , φ , ρ , z )μ , with θ, θ ∈ [0, π2 ], φ, φ ∈ [0, π) and ρ, ρ ∈ [0, 1]. By Theorem 6.1, there exists q ∈ H, |q| = 1, such that

ρ exp(φ i) +

&

exp(θ i) = q exp(θi)q −1 , 1 − ρ 2 j = δq(ρ exp(φi) +

(6.2)

1 − ρ2 j)q −1 ,

z = δqzq −1 ,

(6.3) (6.4)

with , δ ∈ {−1, 1}. Thus by writing q = q1 + q2 with q1 ∈ span({1, i}), q2 ∈ span({j, k}), we conclude exp(θ i)qr = qr exp(θi) for r = 1, 2. ρ exp(φ i)q1 + ρ exp(φ i)q2 +

& &

1 − ρ 2 jq2 = δq1 ρ exp(φi) + δq2 1 − ρ 2 jq1 = δq2 ρ exp(φi) + δq1

(6.5)

1 − ρ2 j.

(6.6)

1 − ρ2 j.

(6.7)

z = δqzq −1 ,

(6.8)

Since q = q1 + q2 = 0, we shall distinguish three cases: (a) If q1 = 0 and q2 = 0, as q1 commutes with exp(θi), we have by Eq. (6.5) exp(θi) = exp(θ i). For = 1 this implies θ = θ . For = −1, we get θ = θ + π which is impossible given the restrictions on the parameters. Then necessarily = 1 and by applying again Eq. (6.5), exp(θ i)q2 = q2 exp(θi) = exp(−θi)q2 . We conclude θ = θ = 0 and so ρ = ρ = 1. From here, Eqs. (6.6) and (6.7) imply exp(φ i)qr = δqr exp(φi), r = 1, 2. Arguing as above we also get that necessarily δ = 1 and φ = φ = 0. This is in exception (b)-4 of the theorem.

(b) If q1 = 0 and q2 = 0, we obtain as in (a), = 1 and θ = θ . By taking norms in Eq. (6.6) we obtain ρ = ρ .

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1. If ρ = ρ = 0, 1. By Eq. (6.6), exp(φ i) = δ exp(φi) and for δ = 1 we conclude that φ = φ. For δ = −1 we have φ = φ + π, which is impossible if φ, φ ∈ [0, π). Then we have by Eq. (6.7) jq1 = q1 j and so q1 = ±1. This is case (a) of the theorem. 2. If ρ = ρ = 1. As in 1, δ = 1 and φ = φ. However, in this case q1 take any value in {exp(si) : s ∈ [0, 2π)}. This is case (b) of the theorem. 3. If ρ = ρ = 0, we will distinguish two possibilities: (i) If δ = 1. By Eq. (6.7), q1 = ±1 and φ is not determined by φ. (ii) If δ = −1, by applying again Eq. (6.7), jq1 = −q1 j and so q1 = ±i. We also know φ is not determined by φ. Items (i) and (ii) are case (c) of the theorem. (c) If q1 = 0 and q2 = 0, that is, q = q2 ∈ span({j, k}). From Eq. (6.5) exp(θ i)q2 = q2 exp(θi) = exp(−θi)q2 . This implies that exp(θ i) = exp(−θi).

(6.9)

Hence, 1. For = 1 we have θ = −θ, and for θ, θ ∈ [0, π2 ] this is only possible if θ = θ = 0 and so ρ = ρ = 1. We now consider two possibilities: (i) If δ = 1. Eq. (6.7) now gives φ = φ = 0 and we have q2 takes any value in span{j, k} ∩ S 3 . This have been contemplated in exception (b)-4 of the theorem. (ii) If δ = −1. We obtain, as above, φ+φ = π and q2 takes any value in span{j, k}∩S 3 . This is exception (b)-1 of the theorem. 2. For = −1, Eq. (6.9) gives exp(θ i) = exp((π − θ)i) and so θ = θ = π/2. By taking norms in Eq. (6.7) we also have ρ = ρ . We will distinguish three cases: (i) If ρ = ρ = 0, 1. Then we have by Eq. (6.7), exp(φ i)q2 = δq2 exp(φi), so φ = φ = 0 if δ = 1 and φ + φ = π if δ = −1. By Eq. (6.6), jq2 = δq2 j and so q2 = ±j if δ = 1 and q2 = ±k if δ = −1. We have the exceptions (a)-1 and (a)-2 of the theorem. (ii) If ρ = ρ = 1. Then we have as in (i) that φ = φ = 0 if δ = 1 and φ + φ = π if δ = −1. However, q2 can take any value in span{j, k} ∩ S 3 . We have the exceptions (b)-2 and (b)-3 of the theorem. (iii) If ρ = ρ = 0. Then by Eq. (6.6), q2 = ±j if δ = 1 and q2 = ±k if δ = −1. We also know φ is not determined by φ. This is the exception (c)-1.

We see as above that any exceptional a.v.t.s. can be determined by μ ∈ {1, (2, 3), (1, 2)}, θ ∈ [0, π2 ], φ ∈ [0, π), ρ ∈ [0, 1] and z, t ∈ S 3 . We shall denote it by T1,−1,1 (θ, φ, ρ, z, t)μ . We also have that by arguing as in the previous theorem we can prove the following. THEOREM 6.3 Any exceptional four-dimensional a.v.t.s. is isomorphic to one of the triple systems T1,−1,1 (θ, φ, ρ, z, t)μ , with μ ∈ {1, (2, 3), (1, 2)}, θ ∈ [0, π2 ], φ ∈ [0, π), ρ ∈ [0, 1] and z, t ∈ S 3 . With the above conditions for the parameters, T1,−1,1 (θ, φ, ρ, z, t)μ ∼ = T1,−1,1 (θ , φ , ρ , z , t )μ

90

Antonio J. Calder´ on Mart´ın and C´ andido Mart´ın Gonz´ alez

if and only if&μ = μ, θ = θ and φ, φ , ρ, ρ , z, z , t and t are related by the equations 2 ρ exp(φ i) + 1 − ρ j = q(ρ exp(φi) + 1 − ρ2 j)q −1 , z = δpzp−1 and t = τ qtq −1 for , δ, τ ∈ ±1 and p, q ∈ S 3 .

6.3

Acknowledgments

The authors are supported in part by the PCI of the UCA “Teor´ıa de Lie y teor´ıa de espacios de Banach,” by the PAI project numbers FQM-298 and FQM-900, by the Spanish Ministerio de Educaci´ on y Ciencia project MTM2004-06580-C02-02 and with the funds FEDER

References [1] Albert, A. A. (1947). Absolute valued real algebras. Ann. Math. 48: 495–501. [2] Rodr´ıguez, A. (2004). Absolute-valued algebras and absolute-valuable Banach spaces. In: Advanced Courses of Mathematical Analysis I. Proceedings of the First International School, World Scientiﬁc Publishing, pp. 99–155. [3] Albert, A. A. (1949). Absolute valued algebraic algebras, Bull. Amer. Math. Soc. 55: 763–768. [4] Cuenca, J. A., Rodr´ıguez, A. (1995). Absolute values on H ∗ -algebras. Comm. In Algebra. 23: 1709–1740. [5] El-Amin, K., Ram´ırez, M. I., Rodr´ıguez, A. (1997). Absolute-Valued Algebraic Algebras are Finite-Dimensional. J. Algebra. 195: 295–307. [6] El-Mallah, M. L. (1990). Absolute valued algebras containing a central idempotent. J. Algebra. 128 (1): 180–187. [7] El-Mallah, M. L. (1998). Absolute valued algebras containing a central element. Italian J. Pure Appl. Math. 3: 103–105. [8] El-Mallah, M. L. (2001). Absolute valued algebras satisfying (x, x, x2 ) = 0. Arch. Math. 77: 378–382. [9] Ram´ırez, M. I. (1999). On four dimensional absolute-valued algebras. In: Proceedings of the International Conference on Jordan Structures, Univ. M´ alaga, M´ alaga. pp. 169–173. [10] Rodr´ıguez, A. Absolute valued algebras of degree two (1994). In: Non-Associative Algebra and Its Applications, Kluwer Academic Publishers, pp. 350–357. [11] Urbanik, K., Wright, F. B. (1960). Absolute valued algebras. Proc. Amer. Math. Soc.

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11: 861–866. [12] Wright, F. B. (1953). Absolute valued algebras. Proc. Nat. Acad. Sci. U.S.A. 39: 330– 332. [13] Castell´on, A., (1992). Cuenca, J. A. In: Associative H ∗ -triple Systems. Workshop on Nonassociative Algebraic Models, Nova Science Publishers, New York, pp. 45–67. [14] Shaw, R. (1990). Ternary composition algebras. I. Structure theorems: deﬁnite and neutral signatures. Proc. Soc. London Ser. A 431: 1–9. [15] Shaw, R. (1990). Ternary composition algebras. II. Automorphism groups and subgroups. Proc. Soc. London Ser. A. 431: 21-36. [16] Calder´ on, A. J., Mart´ın, C. (2001). On L∗ -triples and Jordan H ∗ -pairs. In: Ring Theory and Algebraic Geometry, Marcel Dekker, Inc., pp. 87–94. [17] Calder´ on, A. J., Mart´ın, C. (2001). Hilbert space methods in the theory of Lie triple systems. In: Recent Progress in Functional Analysis, North-Holland Math. Studies, pp. 309–319. [18] Castell´on, A., Cuenca, J. A., Mart´ın, C. (2000). Special Jordan H ∗ -triple systems. Comm. Alg. 28(10): 4699–4706. [19] Faulkner, J. R. (1980). Dynkin diagrams for Lie triple systems. J. Algebra. 62: 384– 392. [20] Hopkins, N.C. (1985). Some structure theory for a class of triple systems. Trans. Amer. Math. Soc. 1: 203–212. [21] W. G. Lister. (1952). A structure theory of Lie triple systems. Trans. Amer. Math. Soc. 72: 217-242. [22] Castell´on, A., Cuenca, J. A. (1993). The Centroid and Metacentroid of an H ∗ -triple system. Bull. Soc. Math. Belg. 45: 85–93. [23] McCrimmon, K. (1983). Quadratic forms permitting triple composition. Trans. Amer. Math. Soc. 275(1): 107–130. [24] Calder´ on, A. J., Mart´ın, C. (2004). Absolute valued triple systems. International Mathematical Journal 5 (8), 741–753. [25] Calder´ on, A. J., Mart´ın, C. Two-graded absolute valued algebras. J. Algebra 292 492– 515. [26] Br¨ ocker, T., Dieck, T. (1985). Representations of Compact Lie Groups. SpringerVerlag, New York-Berlin-Heidelberg-Tokyo. [27] Adams, J. F. (1996). Lectures on Exceptional Lie Groups. Edited by Zafer Mahmud and Mamoru Mimura. xiv, Chicago Lectures in Mathematics Series.

Chapter 7 The Theory of Kikkawa Spaces Ramiro Carrillo-Catal´ an Department of Mathematics, Graduate School of Science, Tokyo Metropolitan University Liudmila Sabinina Facultad de Ciencias, Universidad Aut´ onoma del Estado de Morelos, M´exico 7.1 7.2 7.3 7.4 7.5

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Preliminaries and deﬁnitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Some geometric properties of a Kikkawa space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Comparison with other results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

93 94 96 99 100 100

Abstract We discuss some properties of Kikkawa spaces (families of geodesic rightmonoalternative loops) in the context of the theory of smooth loops and smooth odules. Key words: Geodesic loop; Al -loop; Moufang loop; Preparallel relation; Kikkawa space 2000 MSC: Primary: 53B05; Secondary: 20N05

7.1

Introduction

Right-monoalternativity has long been recognized as one of the key properties of smooth loops. Kikkawa in his paper [5] introduced the concept of a geodesic loop and studied the case of a right-monoalternative geodesic loop without torsion. He established that for such a loop the curvature of the associated connection vanishes. Mikheev and Sabinin (see [12]) obtained a diﬀerential equation deﬁning right-monoalternative smooth loops. In particular, they showed that for every right-monoalternative smooth loop there exists a smooth manifold with an aﬃne connection (which can be chosen to be ﬂat) with the property that the geodesic loop of such a manifold is isomorphic to the initial loop. Using the results on right-monoalternative smooth loops they established the identities for the tangent algebra of a smooth Bol loop (called a Bol algebra), and proved that the correspondence between smooth Bol loops and their Bol algebras is analogous to the correspondence between Lie groups and Lie algebras. Moreover, their study of smooth right-monoalternative loops permitted them to describe the tangent algebra of a general smooth loop. They called this object a hyperalgebra, now known as a Sabinin algebra. The theory of smooth Bol loops (in particular, smooth Bruck loops) and Bol algebras is in the intersection of several areas: diﬀerential geometry, algebra, and physics, as these structures describe symmetric spaces and their generalizations in an algebraic way (ﬁrst suggested by O. Loos). Shestakov and Umirbaev [15] discovered the fact that Sabinin algebras appear as generalizations of Lie algebras in the context of the theory of bialgebras; this stimulated further research on the subject. In our paper [3], as part of the study of the property of right-monoalternativity of

93

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geodesic loops, we described the class of Malcev algebras that appear as tangent algebras to geodesic Moufang loops of reductive homogeneous spaces. In this chapter we present some geometric properties of right-monoalternativity of geodesic loops and compare the results from [3] with other known results.

7.2

Preliminaries and Deﬁnitions

In order to make this chapter more self-contained, many deﬁnitions that can be found in [1] or [3] or [12] are included. DEFINITION 7.1 A loop is a set Q with three binary operations: multiplication (·), left division \, and right division /, and with the neutral element e such that the following identities, x · e = e · x = x, (x/y) · y = y · (y\x) = (x · y)/y = y\(y · x) = x hold for all x, y ∈ Q. DEFINITION 7.2 Two loops < Q, ·, \, /, e > and < Q, ◦, \\, //, > are isotopic if there exists a triple of permutations (α, β, γ) of the set Q such that a · b = γ −1 (α(a) ◦ β(b)) for all a, b ∈ Q. Two isotopic loops are called isotopes of each other. DEFINITION 7.3 a subgroup of Q.

A loop Q is called diassociative if every two elements of Q generate

We shall consider the following property of loops. DEFINITION 7.4

A loop is a right-monoalternative loop if the identity (b · al ) · ak = b · al+k

holds for all elements a, b of the loop and for all rational numbers l, k. If l, k are integers, this property is known as right-power-alternativity. For smooth loops these two properties are equivalent, and for analytic loops these properties are equivalent to the right-alternative property, i.e., l = k = 1 (see [12]). Left-monoalternative loops are deﬁned similarly. Left-monoalternative and right-monoalternative loop is called monoalternative loop. It is evident that a diassociative loop is power-alternative. DEFINITION 7.5

A loop with the identity

The theory of Kikkawa spaces

95

x · (y · x) = (x · y) · x is called ﬂexible loop. We remark that a diassociative loop satisﬁes the property of ﬂexibility. DEFINITION 7.6

A loop with the identity ((xy)x)z = x(y(xz))

is called a Moufang loop. It is known that a Moufang loop is diassociative. DEFINITION 7.7 Let Q be a loop. With every element a ∈ Q one can associate a left translation, La : Q → Q, such that La x = a · x, and a right translation, Ra : Q → Q, such that Ra x = x · a. The group generated by the left and right translations is called the multiplication group of the loop Q and is denoted by M lt(Q). The subgroup Inn(Q) of M lt(Q) deﬁned as {φ ∈ M lt(Q)|φe = e}, where e is the neutral element of Q, is called the inner mapping group of Q. It is known that Inn(Q) is generated by the following mappings: lx,y = L−1 x·y ◦ Lx ◦ Ly , −1 rx,y = Rx·y ◦ Ry ◦ Rx ,

Tx = L−1 x ◦ Rx for all x, y in Q.

DEFINITION 7.8 A loop Q is called an A-loop if Inn(Q) is a subgroup of Aut(Q)— the group of all automorphisms of the loop Q. A loop Q is called an Al -loop if the group < lx,y > generated by all mappings lx,y is a subgroup of Aut(Q). A loop Q is an AT -loop if the group < Tx > is a subgroup of Aut(Q). The Ar - property is deﬁned analogously. DEFINITION 7.9 Let us consider a manifold with an aﬃne connection (M, ∇) endowed locally with a binary operation at the point p given by the formula: u · v = Expu ◦ τp,u ◦ Exp−1 p (v), where Expp : Tp (M ) → M is the exponential mapping, and τp,u is the parallel displacement of tangent vectors at p along the geodesic arc joining p to the point u. Left and right translations deﬁned by this binary operation are local smooth diﬀeomorphisms and p is the neutral element. This binary operation determines a geodesic loop. A geodesic loop with multiplication by scalars t ∈ R : tx = Expp (t Exp−1 p (x)) is called a geodesic odule (see [12]) and is denoted by < Up , ·, (t)t∈R , p >.

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DEFINITION 7.10 A smooth odule < Up , ·, (t)t∈R , ε > is a smooth loop with multiplications by scalars, such that the following conditions hold: (t + u)x = tx · ux,

(tu)x = t(ux),

1x = x,

where t, u ∈ R, x ∈ Up . A geometric odule is a smooth odule < Up , ·, (t)t∈R , ε > with the identity: lx,a ta = tlx,a a where t ∈ R, x ∈ Up . It is known that a geometric odule can be realized as a geodesic odule in some smooth manifold M with an aﬃne connection ∇ (see [10], [12]). In [11] we introduced the concept of a Kikkawa space. The name Kikkawa spaces was suggested to us by L.V. Sabinin. DEFINITION 7.11 Let M be a C ∞ –manifold with an aﬃne connection ∇. Let Up be a restricted normal neighborhood of the point p ∈ M . Suppose that for every point of Up the corresponding geodesic loop is right-monoalternative. A neighborhood Up ⊂ M with this property is called a Kikkawa space. DEFINITION 7.12

A loop Q is a G-loop if Q is isomorphic to each of its loop isotopes.

Example 7.1 Let us consider a local analytic Moufang loop < Up , ·, p > on a manifold M . There exists an aﬃne connection ∇ on M , such that Lp is a geodesic loop with respect to ∇ (see [10]). At every point of the neighborhood Up of the point p we can construct a collection of loops isotopic to the loop < Up , ·, p >, using for example the family of isotopes (Rq−1 , id, id), for all q ∈ Up . The analytic Moufang loop is a G-loop (see, for example [7]), so every loop in the collection is a local analytic Moufang loop and so we obtain an analytic Kikkawa space < Up , ∇ >.

7.3

Some Geometric Properties of a Kikkawa Space

I. In [5] Kikkawa notes that the property of monoalternativity of a smooth geodesic loop Up implies the following property: ∇X X = 0 for an arbitrary adapted vector ﬁeld X in Up . We will show that the inverse statement also holds. We will use the techniques of the diﬀerential equations of a geometric odule that were developed by L. Sabinin [12]. Let < Up , ·, (t)t∈R , ε > be a smooth odule. Denote by ∂(a · b)α Aα (a) = . [l(a, b)]∗,ε = ˜l(a, b), β ∂bβ b=ε Then the following diﬀerential equations hold in a geometric odule: dϕi d˜ μi = Aij (ϕ)˜ = λipq (ϕ)˜ μj , μp μ ˜q dt dt with initial values

The theory of Kikkawa spaces ϕi (a, b, 0) = ai ,

97

μ ˜i (a, b, 0) = (Exp−1 b)i

and conditions λipr (c) = λirp (c),

Aij (ε) = δji ,

λipq (c)(Exp−1 c)p (Exp−1 c)q = 0. These equations determine the operations of a geometric odule in the following way: x · y = ϕ(x, y, 1), tb = ϕ(ε, b, t). Then ϕ(x, y, t) = x · ty. Here λirp (c)

( ' 1 ∂ ˜lpi (c, z) ∂ ˜lri (c, z) = + 2 ∂z r ∂z p

. z=ε

LEMMA 7.1 (Sbitneva [13]) The geometric odule deﬁned by ϕ(x, y, t) = x · ty is rightmonoalternative, i.e., (x · ty) · uy = x · (t + u)y, if and only if λirp (c) = 0. For a geometric odule the following formula holds (see [12], formula 4.71): ∇Aβ Aγ = −Aσ (λσγβ − λσγ μβ + λσβ μγ ). If λσγβ = 0, then we have ∇Aβ Aγ = 0. An arbitrary adapted vector ﬁeld X can be expressed in the base of adapted vector ﬁelds Aα and hence we obtain the formula: ∇X X = 0. Now, suppose that for every adapted vector ﬁeld X the condition ∇X X = 0 holds, in particular, ∇Aβ +Aγ (Aβ + Aγ ) = 0, which implies ∇Aβ Aγ + ∇Aγ Aβ = 0. Therefore, we obtain the condition λσγβ + λσβγ = 0 and, by considering the initial condition λσβγ = λσγβ , we get λσβγ (c) = 0. From the lemma we obtain the following. PROPOSITION 7.1 A geodesic odule is right-monoalternative if and only if for an arbitrary adapted vector ﬁeld X, the condition ∇X X = 0 holds. Equivalently, for a geodesic loop we have the following. PROPOSITION 7.2 Let Φp be a family of all adapted vector ﬁelds deﬁned by vectors at a point p. A geodesic loop < Up , ·, p > is right-monoalternative if and only if for all X ∈ Φp , the property ∇X X = 0 holds. II. Let M be a smooth manifold with an aﬃne connection ∇. A geodesic arc of ordered )b. In the set of pair of points (a, b) in some restricted neighborhood Up we will denote by a, all arcs in Up , we introduce a binary relation , which we call the preparallelism relation, in the following way: )b c, )d a,

if and only if

c · b = d. a

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In the Euclidean addition of vectors. )d means that and c,

space the geodesic loop at each point is just an abelian group with )b The preparallelism relation (parallelism and congruence) between a, * ) a, d = a, b + a, )c. In this case preparallelism is an equivalence relation.

N. Castaneda posed the following problem: Find properties of a geodesic odule such that the preparallelism relation is an equivalence relation. He showed that for any geodesic odule the preparallelism relation is reﬂexive and symmetric, and that the preparallelism relation is transitive if and only if (Up , ∇) is ﬂat. He also asserted that the property of right-monoalternativity of all geodesic loops of Up is a necessary condition. PROPOSITION 7.3 (Castaneda [4]) Let the neighborhood (Up , ∇) be of zero curvature. Then (Up , ∇) is a Kikkawa space. REMARK 7.1 In [3] we established the relation between the curvature tensor and torsion tensor in a Kikkawa space, that is,

R(X, Y )Z =

, 1+

S ∇X T (Y, Z) − S T ([X, Y ], Z) 3

for all vector ﬁelds X, Y, Z ∈ T (Up ). Now, we will follow some of Castaneda’s ideas. The pair (a, b) determines a unique )b and c, )d be preparallel and γ1 and γ2 be the corresponding geodesic curve in Up . Let a, geodesics. In this case we will call γ1 and γ2 parallel. Let us denote by γi the set of all arcs in the geodesic γi . Now, using the results of [4], we can describe the following characteristic property for a Kikkawa space. PROPOSITION 7.4 Let (Up , ∇) be a Kikkawa space, γ1 and γ2 be two parallel geodesics.Then the preparallel relation on the set γ1 ∪ γ2 is an equivalence relation. If preparallelism is an equivalence relation on the set γi ∪ γj of every two parallel geodesics in some neighborhood Up of a smooth manifold M with an aﬃne connection ∇, then (Up , ∇) is a Kikkawa space. In [3] we studied a Kikkawa space (Up , ∇) with a reductive aﬃne connection ∇R = 0, ∇T = 0. We showed that a geodesic loop < Up , ·, p > of a reductive Kikkawa space (Up , ∇) is an Al -Moufang loop. Combining the results of [4] and [3] we get the following. PROPOSITION 7.5 Let (Up , ∇) be a reductive Kikkawa space. Let (x1 , y1 , z1 ) and (x2 , y2 , z2 ) be two triples of points of Up such that x 1 , y1 x 2 , y2 and y 1 , z1 y 2 , z2 . Then the property x 1 , z1 x 2 , z2 holds. REMARK 7.2 The class of Kikkawa spaces with the property that, for any two geodesic triangles, preparallelism between any two pairs of sides of triangles implies preparallelism of the third pair of sides, is larger than the class of reductive Kikkawa spaces. Example 7.1 gives us another such Kikkawa space.

The theory of Kikkawa spaces

7.4

99

Comparison with Other Results

I. It is well known that for linear algebras the property of left-alternativity implies the property of diassociativity of the operation. For analytic loops the following result is known. PROPOSITION 7.6 (Mikheev [8]) An analytic alternative loop with the property of ﬂexibility is diassociative. It is clear that ﬂexibility is also a necessary condition. From this result it is easy to state diﬀerent suﬃcient conditions, in particular, the following.: COROLLARY 7.1 PROOF

An analytic alternative AT -loop is diassociative.

Let Tx be an automorphism of a loop. Then, we have x\((uv)x) = (x\(ux)) · (x\(vx)).

With u = x we obtain (xv)x = x(vx), which is the property of ﬂexibility. COROLLARY 7.2

An analytic right-alternative commutative loop is diassociative.

By using our result from [3] we obtain another suﬃcient condition for an alternative analytic loop to be diassociative. PROPOSITION 7.7 An analytic alternative geodesic loop deﬁned in some analytic Kikkawa space (Up , ∇) with conditions ∇R = 0, ∇T = 0, is diassociative. In terms of identities of analytic loops we can assert: PROPOSITION 7.8 An analytic alternative loop < Up , ·, p > realized as a geodesic loop of a Kikkawa space (Up , ∇) with left-alternativity and Al -identity for all geodesic loops in Up , is diassociative. II. In 1956 Bruck and Paige [2] noted that diassociative A-loops have many common properties with Moufang loops. A natural problem arises: under what conditions are diassociative A-loops Moufang loops? Osborn [9] gave a partial answer to this question. He showed that commutative diassociative A-loops are Moufang loops. Finally, this problem was solved by Kinyon, Kunen, and Phillips [6]. Moreover, they showed that every alternative A-loop is a Moufang loop. To prove this assertion it is essential that all generators of Inn(G) are automorphisms. Considering analytic loops we showed in [3] that every analytic alternative Al -loop realized as a geodesic loop of a Kikkawa space with left-alternativity and Al -property is a Moufang

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loop. From the theory of Moufang loops, it is clear that such a loop possesses Al - and Ar -properties, but not necessarily the AT -property. PROPOSITION 7.9

(Shelekhov [14]) An analytic AT -Moufang loop is a group.

So, as a corollary, we have the following assertion: COROLLARY 7.3 An analytic loop, realized as a geodesic loop of an analytic Kikkawa space with left-alternativity and the A-property, is a group.

7.5

Acknowledgments

Liudmila Sabinina was partially supported by the CONACyT grant SEP-2003-CO244100.

References [1] Bruck H.R. A Survey of Binary Systems. Springer-Verlag, Berlin, 1971. [2] Bruck H.R. and Paige L.J. Loops whose inner mappings are automorphisms. Annals of Mathematics, Vol 63, 2, March, pp. 308–323 (1956). [3] Carrillo Catalan R., Sabinina L. On the smooth power-alternative loops. Communications in Algebra, Vol. 32, N 8, pp. 2969-2976 (2004). [4] Castaneda N. Sabinin spaces and Moufang loops. Algebras Groups Geom. 18, pp. 259-280 (2001). [5] Kikkawa M. On local loops in aﬃne manifolds. J. Sci. Hiroshima Univ., Ser. A–I, 28, pp. 199–207 (1964). [6] Kinyon M.K., Kunen K., Phillips J.D. Every diassociative A-loop is Moufang. Proc. Amer. Math. Soc. 130, 3, pp. 619-624 (2002). [7] Mikheev P.O. On the G-property of local analytic Bol loops (in Russian). Webs and Quasogroups, pp. 54-59, Kalinin (1986). [8] Mikheev P.O. On analytic diassociative composition laws (in Russian). Mat. Zametki, 54(1993), no. 5, pp. 72-77, 158, translations in Math. Notes 54 (1993), no. 5-6, pp. 1131-1133 (1994). [9] Osborn M. A theorem on A-loops. Proc. Amer. Math. Soc. 9, pp. 347-349 (1958). [10] Sabinina L. On a problem of the theory of smooth loops. Mathem. Notes, 74 (5–6), pp. 897–898 (2003).

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[11] Sabinina L. Kikkawa spaces. Russian Math. Surveys 58, no. 4, 796–797 (2003). [12] Sabinin L.V. Smooth Quasigroups and Loops. Kluver Academic Publishers. Dordrecht/Boston/London (1999). [13] Sbitneva L.V. On the conditions of right-monoalternativity of a geometric odule (in Russian). Webs and Quasigroups. Kalinin, pp. 46–48 (1988). [14] Shelekhov A.M. On a subclass of Bol loops. Webs and Quasigroups, pp. 27–33, Tver Gos. Univ., Tver (1994). [15] Shestakov I.P., Umirbaev U.U., Free Akivis algebras, primitive elements and hyperalgebras. J. Algebra, 250, pp. 533–548 (2002).

Chapter 8 Flexible Right-Nilalgebras Satisfying x(yz)=y(zx) Ivan Correa Departamento de Matem´atica, Universidad Metropolitana de Ciencias de la Educaci´ on, Santiago, Chile

8.1 8.2 8.3 8.4

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Notation and preliminary results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Proof of the main result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

103 104 105 106 106

Abstract We prove that every ﬁnite-dimensional ﬂexible right-nilalgebra satisfying x(yz) = y(zx) is nilpotent. Key words: nilpotency, power-associative nilalgebras, ﬂexible right-nilalgebra 2000 MSC: 17A05, 17A30

8.1

Introduction

The problem of the nilpotency of commutative ﬁnite-dimensional power-associative nilalgebras was ﬁrst presented by A. Albert in [1]. We study in [2] the noncommutative case and we prove that noncommutative ﬁnite-dimensional power-associative nilalgebras are not necessarily nilpotent but they are solvable in some cases. Looking for an example of a class of noncommutative nilpotent algebras, we study the nilpotency of ﬂexible algebras, that is, algebras satisfying the identity (xy)x = x(yx). (8.1) In this chapter we prove that a suﬃcient condition for the nilpotency of a ﬂexible rightnilalgebra is the additional identity x(yz) = y(zx).

(8.2)

Algebras satisfying (8.2) have been studied by M. Kleinfeld in [3]. Let A be a noncommutative and nonassociative algebra. For an element x in A we deﬁne the right-powers of x by x1 = x and xn = xn−1 x for n > 1. We say that x is right-nilpotent if there exists k such that xk = 0. The smallest k with this property is called index of rightnilpotency of x. A is called right-nilalgebra if each element of A is right-nilpotent. When there is a bound on the indices of right-nilpotency, the right-nilindex of A is the smallest k suchthat xk = 0 for all x in A. We deﬁne the principal powers of A by A1 = A and k=n−1 An = k=1 An−k Ak for n > 1. We say that A is nilpotent when Ak = 0 for some k. The smallest k with this property is called the index of nilpotency of A.

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8.2

Notation and Preliminary Results

Let A be an arbitrary algebra. The linear transformations Lx : A −→ A and Rx : A −→ A, x ∈ A, deﬁned, respectively, by yLx = xy and yRx = yx, generate an algebra called the multiplication algebra of A. This algebra is denoted by M (A). The transformations Rx and Lx are called the right and left multiplication by x, respectively. Let S be a subset of A. We will denote by S ∗ the subalgebra of M (A) generated by the set {Ls , Rs |s ∈ S} and, by < S > the subalgebra of A generated by S. We remark that in terms of right and left multiplications, identity (8.1) becomes Rx Lx = Lx Rx . PROPOSITION 8.1 (Schafer [4], Theorem 2.4, p. 18). An algebra A is nilpotent if and only if M (A) is nilpotent. LEMMA 8.1

Let A be a ﬂexible algebra. Then xi x = xxi for every x in A and i ≥ 1.

PROOF The proof is by induction on i. For i = 1 the result is immediate. Assume that xi x = xxi . Then, using (8.1) and the hypothesis of induction we obtain that xxi+1 = x(xi x) = (xxi )x = xi+1 x = xi+2 . This proves Lemma 8.1. LEMMA 8.2 have

Let A be a ﬂexible algebra satisfying (8.2). Then for every x in A we

a) Lix = Rxi for every i ≥ 2. b) Rxi = Rxi for every i ≥ 3. PROOF

c) Lix = Lxi for every i ≥ 3. d) Lix = Rxi for every i ≥ 3.

Substituting y by x in (8.2) we obtain x(xz) = x(zx) = zx2 . It follows that L2x = Rx Lx = Rx2 .

(8.3)

In particular we obtain a) for i = 2. Using induction on i, we assume that Lix = Rxi (i ≥ 2). Then, using (8.2) and the hypothesis of induction we obtain zLi+1 = zLix Lx = zRxi Lx = x(zxi ) = z(xi x) = zxi+1 = zRxi+1 , x for every z in A. This proves a). Linearizing (8.1) we obtain: (zy)x + (xy)z = z(yx) + x(yz).

(8.4)

Substituting y by x in (8.4) we get (zx)x + x2 z = zx2 + x(xz). Hence, we obtain that Rx2 + Lx2 = Rx2 + L2x . Using part a) it follows that Rx2 = 2L2x − Lx2 .

(8.5)

Applying Rx to (8.5) by the left side we obtain Rx3 = 2Rx L2x − Rx Lx2 .

(8.6)

On the other hand, using (8.2), we have that for every z in A : zRx L2x = zRx Rx2 = (zx)(xx) = x(x(zx)) = x(z(xx)) = z((xx)x) = zRx3 , that is, Rx L2x = Rx3 . Furthermore, zRx Lx2 = x2 (zx) = z(xx2 ) = zRx3 , that is, Rx Lx2 = Rx3 . Hence, from (8.6) we obtain

Flexible right-nilalgebras satisfying x(yz)=y(zx)

105

that Rx3 = Rx3 . Assume now as hypothesis of induction that Rxi = Rxi (i ≥ 3). Then, using (8.3) and part a) we obtain that for every z in A; zRxi+1 = zRx Rxi = zRx Rxi = zRx Lix = = zL2x Li−1 = zLi+1 = zRxi+1 . This proves part b). Now replacing y by xi in z(Rx Lx )Li−1 x x x i i+1 i+1 + x(xi z). Using (8.2) and Lemma 8.1, we obtain that (8.4) we get (zx )x + x z = zx i i+1 i+1 i+1 whence x z = 2zx − (zxi )x. Hence there follows the identity x(x z) = zx Lxi+1 = 2Rxi+1 − Rxi Rx .

(8.7)

Replacing i = 2 in (8.7) and using a) and (8.3) we get that Lx3 = 2Rx3 − Rx2 Rx = 2L3x − L2x Rx = 2L3x − Lx (Lx Rx ) = 2L3x − Lx L2x = L3x , that is, Lx3 = L3x . Assume as a hypothesis of induction that Lxi = Lix (i ≥ 3). From (8.7), (a) and (8.3) we obtain i i+1 i−1 i+1 i−1 2 i+1 Lxi+1 = 2Rxi+1 − Rxi Rx = 2Li+1 x − Lx Rx = 2Lx − Lx (Lx Rx ) = 2Lx − Lx Lx = Lx . This proves c). d) is an immediate consequence of parts a) and b). COROLLARY 8.1 Let A be as in Lemma 8.2 and x in A. Then: a) < x >∗ = < {Lx , Rx } > . b) < x >∗ is commutative. c) If x is right-nilpotent then Lx and Rx are nilpotent. d) If x is right-nilpotent then < x >∗ is nilpotent. PROOF a) is a direct consequence of a) and c) of Lemma 8.2 and identity (8.5). b) is the consequence of a) and that Lx Rx = Rx Lx . c) is the consequence of a) and c) of Lemma 8.2. d) Notice that from Lemma 8.2 and identity (8.5), we can assume that every element of < x >∗ is a ﬁnite sum of elements of the form T1 · · · Tp−1 Tp , where each Ti is equal to Lx or Rx . Since Lx Rx = Rx Lx , we can assume that T1 · · · Tp−1 Tp = Lix Rxj with i + j = p. Therefore, for p ≥ 3, we have that T1 · · · Tp−1 Tp = Lpx . Hence, if the index of right-nilpotency of x is r, it follows from part a) of Lemma 8.2 that < x >∗ is nilpotent of index r.

8.3

Proof of the Main Result

Let F be the set of all subalgebras B of A such that B ∗ is nilpotent. Since the algebra {0} is nilpotent, it follows that F is a nonempty set. Therefore, since A is ﬁnite-dimensional, from Zorn’s lemma it follows that there exists a maximal element B in F. We will prove that B = A. Assume the contrary. Let k be the index of nilpotency of B ∗ . Then, 0 = A(B ∗ )k ⊆ B. Let m be the smallest integer with A(B ∗ )m ⊆ B. If m = 1 we take x in A − B. If m > 1 we take x in A(B ∗ )m−1 − B. A(B ∗ )m ⊆ B implies that BLix ⊆ B and BRxi ⊆ B for every i ≥ 1. Then, using part a) of Corollary 8.1, it follows that B < x >∗ ⊆ B.

(8.8)

This implies that C = B + < x > is a subalgebra of A strictly containing B. Our next step is to prove that C ∗ is nilpotent, which contradicts the maximality of B with respect to this property. Notice that C ∗ = B ∗ + < x >∗ B ∗ + B ∗ < x >∗ + < x >∗ . We will prove that ∗ B < x >∗ ⊆ B ∗ + < x >∗ B ∗ , whence we conclude that C ∗ = B ∗ + < x >∗ B ∗ + < x >∗ . In order to prove our claim, we only need to prove that for every b in B the elements

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Lb Lx , Rb Lx , Rb Rx and Lb Rx are in B ∗ + < x >∗ B ∗ . Let b in B, xb = b and bx = b . We remark that from (8.8), b and b are in B. Using identity (8.2) we get that for any y in A; yLb Lx = x(by) = b(yx) = y(xb) = yLb and yRb Lx = x(yb) = y(bx) = yLb . Now, substituting y by b and z by y in (8.4) and, using (8.2) we get yRb Rx = −(xb)y + y(bx) + x(by) = y(−Lb + Rb + Rb ). Finally, substituting z by b in (8.4) we obtain yLb Rx = −(xy)b + b(yx) + x(yb) = −(xy)b + y(xb) + y(bx) = y(−Lx Rb + Rb + Rb ). We have proved that Lb Lx , Lx Lb , Rb Rx , and Lb Rx are in < x >∗ B ∗ + B ∗ , which proves the claim. Now, we will prove that (C ∗ )n ⊆ B ∗ + < x >∗ B ∗ + (< x >∗ )n , for every n ≥ 1. For n = 1 the result is immediate. Assume the result is true for n. Direct calculations give (C ∗ )n+1 ⊆ C ∗ [B ∗ + < x >∗ B ∗ + (< x >∗ )n ] = [B ∗ + < x >∗ B ∗ + < x >∗ ][B ∗ + < x >∗ B ∗ + (< x >∗ )n ] ⊆ B ∗ + < x >∗ B ∗ + (< x >∗ )n+1 . This proves the claim. We will prove now that (B ∗ + < x >∗ B ∗ )n ⊆ (B ∗ )n + < x >∗ (B ∗ )n , for every n ≥ 1. Assume the result is true for n. Then: (B ∗ + < x >∗ B ∗ )n+1 ⊆ (B ∗ + < x >∗ B ∗ )((B ∗ )n + < x >∗ (B ∗ )n ) ⊆ ((B ∗ + < x >∗ B ∗ )(B ∗ )n + < x >∗ (B ∗ )n+1 + (B ∗ )n+1 ⊆ (B ∗ )n+1 + < x >∗ (B ∗ )n+1 . Finally, if the right-nilindex of A is r, we have (C ∗ )rk = ((C ∗ )r )k ⊆ (B ∗ + < x >∗ B ∗ + (< x >∗ )r )k = (B ∗ + < x >∗ B ∗ )k ⊆ (B ∗ )k + < x >∗ (B ∗ )k = 0. Therefore C ∗ is nilpotent, which contradicts the maximality of B. It follows that A∗ is nilpotent. From Proposition 8.1 we obtain that A is nilpotent.

8.4

Acknowledgments

This work was supported by FONDECYT 1010196 and FIBAS-UMCE 10-05.

References [1] Albert, A. A. (1948). On Power-Associative Rings, Trans. Amer. Math. Soc. 64:552-593. [2] Correa, I. and Hentzel, I. R. (2001). On Solvability of Noncommutative PowerAssociative Nilalgebras, Journal of Algebra 240:98-102.

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[3] Kleinfeld, M. (1995). Rings with x(yz)=y(zx), Comm. in Algebra 23(13):5085-5093. [4] Schafer, R. D. (1966). An Introduction to Nonassociative Algebras. Academic Press, New York, London.

Chapter 9 Nijenhuis-Richardson Algebra and Fr¨ olicher-Nijenhuis Lie Module Jos´ e de Jes´ us Cruz Guzm´ an and Zbigniew Oziewicz Universidad Nacional Aut´ onoma de M´exico, Facultad de Estudios Superiores Cuautitl´ an, M´exico

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Universal Grassmann module . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Leibniz-Loday and Gerstenhaber algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fr¨ olicher and Nijenhuis decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4.1 Universal property of derivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ´ 9.4.2 Slebodzi´ nski-Lie derivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4.3 Fr¨ olicher and Nijenhuis decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.5 Main deﬁnition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.5.1 Fr¨ olicher-Nijenhuis Lie M ∧ -module . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.5.2 Consequence: module derivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.6 Bianchi identity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.7 Frobenius algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.8 Frobenius subalgebra of Nijenhuis-Richardson algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.9 Frobenius algebra of two idempotents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.10 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.11 Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9.1 9.2 9.3 9.4

109 111 112 113 113 114 115 115 115 117 117 118 120 121 122 123 124

Abstract Nonassociative Nijenhuis-Richardson graded algebra on universal module over Grassmann algebra of diﬀerential forms allows a novel and algorithmic deﬁnition of the Fr¨ olicher-Nijenhuis Lie R-algebra. Some consequences are derived. The signature of the ﬁve-dimensional Frobenius subalgebra of the Nijenhuis-Richardson algebra is calculated. Keywords: universal Grassmann module, Nijenhuis-Richardson algebra, Fr¨ olicher-Nijenhuis Lie module, Leibniz algebra, Frobenius algebra. 2000 MSC: Primary 16W25 derivation; Secondary 17A32 Leibniz algebra, 16W30 coalgebra, bialgebra.

9.1

Introduction

In 1956 Fr¨ olicher and Nijenhuis discovered the Lie R-algebra implicit structure on a Grassmann module of vector-valued diﬀerential forms. More on this was presented in Nijenhuis’ contribution to the Edinburgh Congress in 1958. Since 1985 Peter Michor, together with several collaborators published many papers and a monograph [Kol´ ar, Michor, Slov´ ak

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1993] investigating all aspects of the Fr¨ olicher and Nijenhuis Lie bracket. In 1995 DuboisViolette and Michor found a common generalization of the Fr¨ olicher-Nijenhuis bracket and the Schouten bracket for symmetric algebra of multivector ﬁelds. The Fr¨ olicher and Nijenhuis Lie module and Lie R-operation found important applications and interpretations in diﬀerential geometry of connections (and in particular the Nijenhuis tensor that describes the curvature of an almost product structure) [Gray 1967, Gancarzewicz 1987, Kocik 1997, Krasil’shchik and Verbovetsky 1998, Wagemann 1998], in algebraic geometry, in cohomology of Lie algebras [Wagemann 1999], in special relativity theory, in Maxwell’s theory of electromagnetic ﬁeld [Fecko 1997, Kocik 1997, Cruz and Oziewicz 2003], in Einstein’s gravity theory [Minguzzi 2003], in classical mechanics for symplectic structure [Gruhn and Oziewicz 1983, Gozzi and Mauro 2000, Chavchanidze 2003]. From the point of view of applications there is a need for the algorithmic deﬁnition of the Fr¨ olicher and Nijenhuis Lie operation, in order to be implemented for a symbolic program. In the present chapter we recall the basic concepts, and we are proposing a novel and algorithmic deﬁnition of the Fr¨ olicher and Nijenhuis Lie R-operation in terms of the primary nonassociative (Lie-admissible) F-algebra structure on universal Grassmann module of vector-valued diﬀerential forms, which was introduced by Nijenhuis and Richardson in 1967. The nonassociative Nijenhuis-Richardson algebra that we need in order to deﬁne Fr¨ olicher and Nijenhuis Lie operation, is a natural extension of the associative algebra of endomorphisms, trace-class (1, 1)-ﬁelds, to algebra of (any, 1)-ﬁelds with generalized Grassmann-valued “trace.” The main objective of this chapter is to rethink the basic concepts, introduce a novel/algorithmic deﬁnition of the diﬀerential Fr¨ olicher and Nijenhuis Lie-module over Grassmann module, present some consequences of this deﬁnition, and provide detailed proofs of some statements that otherwise are hard to ﬁnd in available literature. The Nijenhuis-Richardson nonassociative algebra possesses the associative subalgebra that is the Frobenius algebra. For Frobenius algebra we refer to [Frobenius 1903, Curtis and Reiner 1962, Kauﬀman 1994, Voronov 1994, Beidar et al. 1997, Kadison 1999, Baez 2001, Caenepeel et al. 2002]. In the last sections we brieﬂy deﬁne the Frobenius algebra, and study a ﬁve-dimensional Frobenius associative subalgebra of the Nijenhuis-Richardson nonassociative algebra. Table 9.1. Some notation F

Denotes the associative, unital and commutative ring, e.g. R-algebra.

derR F ≡ derR (F, F)

Denotes the Lie F-module of the derivations, Lie F-module of the vector ﬁelds.

M = FM

Denotes the projective F-module of the diﬀerential 1-forms (the Pfaﬃan forms), dimF M < ∞, with a derivation d ∈ derR (F, M ). M = (derR F)∗ ≡ modF (derR F, F).

M∗

Denotes “dual of dual” F-module of the vector ﬁelds, M ∗ ≡ modF (M, F) = (derR F)∗∗ ! derR F.

(−)AB

Is an abbreviation for (−1)(grade A)(grade B) .

Nijenhuis-Richardson algebra and Fr¨ olicher-Nijenhuis Lie module

9.2

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Universal Grassmann Module

In the sequel the Grassmann factor-F-algebra of diﬀerential multiforms is denoted by M ∧ ≡ M ⊗ /I, where I < M ⊗ is an ideal in a free tensor F-algebra, generated by α ⊗ α ∀ α ∈ M. A left M ∧ -module M ∧ ⊗F M ∗ ! modF (M, M ∧ ) is said to be an M ∗ -universal Grassmann module, known variously as the module of “vector-valued diﬀerential forms” or module of “vector-forms.” An R-linear or F-linear homogeneous endomorphism, D ∈ End (M ∧ ) with

grade D ∈ Z,

is said to be a Z2 -graded derivation (skew derivation, antiderivation), D ∈ der(M ∧ ), if the graded Leibniz axiom holds. A Z-graded Lie F-algebra of F-derivations of the Grassmann F-algebra, derF (M ∧ ), is a left M ∧ -module. We are going to describe an M ∧ -module isomorphism that Nijenhuis and Richardson in 1967 extended to an isomorphism of graded commutators (actually this is an isomorphism of Gerstenhaber algebras, see Lemma 9.2), i ∈ modM ∧ (M ∧ ⊗F M ∗ , derF (M ∧ )).

(9.1)

Every derivation of a Grassmann algebra, D ∈ der(M ∧ ), is uniquely determined by values of D on generating R-algebra F and on F-module M : D|F ∈ derR (F, M ∧ ) and D|M ∈ derR (M, M ∧ ), if and only if (D|M )⊗ I ⊂ I. Therefore a Z-homogeneous derivation D with a grade D ≤ −2 must be the trivial zero derivation. A F-module map p ∈ modF (M, M ∧ ) lifts to the unique Z2 -graded F-derivation ip with grade (i) = 0, such that ip |F = 0 and ip |M = p, modF (M, M ∧ ) ! M ∧ ⊗F M ∗ " p

i

−→

ip ∈ derF (M ∧ ),

(9.2)

Let α, β ∈ M ∧ , X ∈ M ∗ ! derR F and p ≡ α ⊗F X ∈ (M ∧ ⊗F M ∗ ). We abbreviate β ∧ p = βp. Then [e.g., Dubois-Violette and Michor 1995] i(α ⊗F X) ≡ eα ◦ iX , iX ∈ derF (M ∧ ), iαp = α ip , grade(eα ◦ iT ) = −1 + grade α.

(9.3)

If p ∈ derF (M ∧ ), then (i ◦ |M )p = p. Therefore the restriction “|M ” is the inverse of (9.1), (9.2), (9.3), i−1 = |M, and there is a bijection, |M =i−1

derF (M ∧ ) −−−−−→ M ∧ ⊗F M ∗ .

(9.4)

Example 9.1 A vector ﬁeld T ∈ M ∗ ! modF (M, F) lifts to an algebraic derivation T → iT ∈ derF (M ∧ ) with grade(iT ) = −1. Consider p, q ∈ M ∧ ⊗F M ∗ ! modF (M, M ∧ ). In 1967 under this identiﬁcation Nijenhuis and Richardson deﬁned nonassociative F-algebra as follows. DEFINITION 9.1 (Nijenhuis-Richardson algebra)

Let α, β ∈ M ∧ .

{modF (M, M ∧ )} ⊗F {modF (M, M ∧ )}−→{modF (M, M ∧ )}, p ⊗F q −→ pq ≡ (ip) ◦ q ∈ {modF (M, M ∧ )}.

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Jos´e de Jes´ us Cruz Guzm´ an and Zbigniew Oziewicz If p = α ⊗F P then

and q = β ⊗F Q,

pq = (α ∧ (iP β)) ⊗F Q.

Clearly (αp)q = α(pq) and pα ≡ (ip)α. Moreover vector-valued diﬀerential form is his own M ∧ -module derivation, e.g. [Dubois-Violette and Michor 1994], p(αq) = (pα)q + (−)pα α(pq).

(9.5)

The Nijenhuis-Richardson Z-graded F-algebra is nonassociative, nonunital, and noncommutative: (pq)r ≡ i(ip ◦ q) ◦ r = ip ◦ (iq ◦ r) ≡ p(qr). (9.6) If grade q = −1 then ∀ p, pq = 0.

9.3

Leibniz-Loday and Gerstenhaber Algebra

Let F be a ring. A category of F-bimodules is a monoidal abelian category. DEFINITION 9.2 (Leibniz-Loday algebra, Loday 1993) A pair of binary operations/morphisms, ∩ and [·, ·], is said to be the Leibniz-Loday algebra if [·,·]

(9.7)

carrier −−−−→ der ∩.

A graded Leibniz algebra is a pair of homogeneous binary operations ∩ and [·, ·] on a Zgraded object/carrier such that ∀ a, b ∈ carrier, [a ≡ [a, ·] ∈ der ∩, ([a) ◦ ∩b = ∩[a,b] + (−1)(a+[·,·])(b+∩) · ∩b ◦ ([a)

∈ End A.

(9.8)

DEFINITION 9.3 (Gerstenhaber algebra) The Z-graded Leibniz algebra (∩, [·, ·]) is said to be the graded Poisson algebra or the graded Gerstenhaber algebra if " even : the Poisson algebra, grade[·, ·] + grade∩ = odd : the Gerstenhaber algebra. Deﬁnition 9.3 [Oziewicz and Paal 1995] generalizes the Gerstenhaber [1963] structure carried by the Hochschild cohomology of an associative algebra ∩. In this deﬁnition both binary operations need not be graded commutative, ∩ need not be associative, and [·, ·] need not be Lie-admissible. However a crossing 2 → 2 needs to be the Artin braid [Oziewicz, R´oz˙ a´ nski and Paal 1995]. A concept of the Lie-Cartan pair introduced by Jadczyk and Kastler [1987, 1991] is a generalization of Leibniz algebra to a pair of objects; it is a twosorted Leibniz-Loday algebra. DEFINITION 9.4 (Graded commutator) Let A, B, C be R- or F-linear Z-homogeneous graded endomorphisms A, B, C ∈ End (M ∧ ). We abbreviate (−1)(grade A)(grade B) to (−)AB . The graded commutator needs the Koszul rule of signs, {A, B} ≡ A ◦ B − (−)AB B ◦ A,

(9.9)

Nijenhuis-Richardson algebra and Fr¨ olicher-Nijenhuis Lie module grade {A, B} = grade{·, ·} + grade A + grade B.

113 (9.10)

Because the composition is associative, the above commutator gives an example of the Poisson algebra, {A, (B ◦ C)} = {A, B} ◦ C + (−)AB · B ◦ {A, C}.

(9.11)

An associative Z2 -graded R- and F-algebra End (M ∧ ) with the above commutator is a Z-graded Poisson F-algebra and a Lie ring. The Jacobi identity is a consequence of (9.11): {A, {B, C}} = {{A, B}, C} + (−)AB {B, {A, C}}. LEMMA 9.1 (Lie superalgebra of derivations) Let A, B ∈ der(M ∧ ). A commutator of derivations is a derivation {A, B} ∈ der(M ∧ ). LEMMA 9.2 (Nijenhuis and Richardson 1967) Let p, q ∈ M ∧ ⊗F M ∗ . The Fmodule isomorphism (9.1)–(9.3) is a graded Lie F-algebra map: {p, (αq)} = (ip α)q + (−)pα α{p, q}, {ip , iq } = i{p, q} ∈ derF (M ∧ ). PROOF

9.4 9.4.1

(9.12) (9.13)

An equality of algebraic derivations must be veriﬁed on restriction i−1 ≡ |M.

Fr¨ olicher and Nijenhuis Decomposition Universal property of derivation

The derivation d ∈ derR (F, M ) has the universal property: for D ∈ derR (F, M ∧ ), there is the unique F-module map, jD ∈ modF (M, M ∧ ), such that D = jD ◦ d, grade j = −1, d

F −−−−→ M ⏐ ⏐j /D D

F −−−−→ M ∧ j

i

derR (F, M ∧ ) −−−−→ modF (M, M ∧ ) −−−−→ derF (M ∧ ).

(9.14)

In particular d = jd ◦ d ⇒ jd = idM . The grade operator is a derivation: End F M = modF (M, M ) " idM −→ grade ≡ iid ∈ derF (M ∧ ), {(i ◦ j) d, d} = d. i

(9.15) (9.16)

From the universal property of d ∈ derR (F, M ) it follows the F-module isomorphism of the vector ﬁelds, derR F ≡ derR (F, F) with the F-dual F-module, M ∗ ≡ modF (M, F) ≡ F M . Let T ∈ der F, then ∀ f ∈ F,

T f ≡ (df )T ≡ jT df ∈ F,

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Jos´e de Jes´ us Cruz Guzm´ an and Zbigniew Oziewicz j

derR (F, F) −−−−→ M ∗ d∗

derR (F, F) ←−−−− M ∗ . Therefore derR F " T = jT ◦ d = d∗ (jT ) = (d∗ ◦ j)T.

9.4.2

´ Slebodzi´ nski-Lie derivation

The Grassmann-Hopf F-algebra M ∧ with the unique lifted graded diﬀerential, d ∈ derR (M ∧ ), grade d = +1, d2 = 0 ∈ derR (M ∧ ), is said to be the diﬀerential N-graded algebra (DGA), de Rham complex. ´ The following R-derivation with grade L = +1 is said to be the (right/left) Slebodzi´ nskiLie derivation of the endomorphism algebra: Lr/l ∈ derR (End R (M ∧ )) ≡ derR (◦), r/l

r/l

Lr

End R (M ∧ ) " A −−−−→ LrA ≡ {A, d} ∈ End R (M ∧ ), Lr

derR (M ∧ ) " p −−−−→ d2 = 0

Lrp ≡ {p, d} ∈ derR (M ∧ ),

=⇒

L2 = 0.

(9.17) (9.18)

The last implication follows from graded Jacobi identity L2 A = {A, d2 }. Let A ∈ End (M ∧ ) be a Z-graded F- or R-map, and f ∈ F. Then LrA ≡ {A, d} ≡ A ◦ d − (−)A · d ◦ A ≡ (−)1+A LlA

∈ End R (M ∧ ),

(9.19)

Lf ≡ {f, d} = −edf ≡ −(df ) ∧ . . . ,

(9.20)

LrA

(9.21)

LrA◦B

B

= (−1)

◦B+A◦

LrB .

For multivector ﬁelds X, Y ∈ M ∗∧ , iX∧Y = iY ◦ iX ∈ End (M ∧ ) (for grade X ≥ 2, der(M ∧ )), and LX ≡ {iX , d} ∈ End R (M ∧ ) [Tulczyjew 1974]. iX ∈ For a 1-vector ﬁeld, X ∈ derR F ≡ derR (F, F) ! M ∗ " jX, lifted to F-derivation of the Grassmann algebra (i ◦ j)X ∈ derF (M ∧ ), the 0-grade directional R-derivation along a ´ nski 1-vector ﬁeld X ∈ der F, LX ≡ {(i ◦ j)X, d} ∈ derR (M ∧ ), was invented by Slebodzi´ [1931]. For X ∈ der F, and for f ∈ F, we have LX ≡ L(i◦j)X , (L2 )X = {LX , d} = 0, LX f = (i ◦ j)X df = jX df = Xf.

(9.22) (9.23)

The name “Lie derivation” along the vector ﬁeld X ∈ der F, was introduced in 1932 by ´ D. van Dantzig (a collaborator of Schouten). The Slebodzi´ nski-Lie derivation is implicit in [Cartan 1922]. ´ The Slebodzi´ nski-Lie derivations ‘along’ Grassmann module, LlA ≡ {d, A} = (−)1+A LrA , possess the following Leibniz expressions for α ∈ M ∧ and q ∈ M ∧ ⊗F M ∗ , Lri(αq) = (−)1+α+q (dα) ∧ q + Lli(αq) =

α ∧ Lriq ,

(dα) ∧ q + (−)α α ∧ Lliq .

(9.24)

Nijenhuis-Richardson algebra and Fr¨ olicher-Nijenhuis Lie module

9.4.3

115

Fr¨ olicher and Nijenhuis decomposition

In the sequel we use the universal property (9.4.1), and to simplify notation we write j instead of the composition j ◦ (|M ). In this convention (9.4.1) reads derR (M ∧ )

i ◦ j ◦ (|F )

−−−−−−−→

derF (M ∧ ).

(9.25)

THEOREM 9.1 (Fr¨ olicher and Nijenhuis 1956) Any R-derivation D ∈ derR (M ∧ ) possesses the following unique decomposition D = (L ◦ i ◦ j + i ◦ j ◦ L)D = {ijD , d} + ij{D,d} . PROOF

(9.26)

First we need to remember the deﬁnitions of “vector-forms” (9.4.1), jD,

jLD

∈ M ∧ ⊗F M ∗ .

For D ∈ derR (M ∧ ), D|F ∈ derR (F, M ∧ ). Universality of d ∈ derR (F, M ) gives D|F ≡ (jD) ◦ d,

jD = 0 ⇐⇒ D|F = 0,

LD |F ≡ (jLD ) ◦ d.

(9.27) (9.28)

The Fr¨ olicher and Nijenhuis decomposition (9.26) is an equality of derivations, D = 0 iﬀ D|F = 0 and D|dF = 0. We must check that the F-N decomposition (9.26) is an identity on a ring F and on exact diﬀerential one-forms dF < M.

9.5 9.5.1

Main Deﬁnition Fr¨ olicher-Nijenhuis Lie M ∧-module

Let A, B ∈ End F (M ∧ ), i.e. Af = 0. Then Lr(A◦B) |F = (A ◦ B)|dF,

LlA◦B |F = . . .

{A, LrB }|F = (A ◦ B)|dF. Set an M ∧ -module map (9.4), i−1 ≡ |M : derF (M ∧ )−→M ∧ ⊗F M ∗ . Let p, q, pq, qp ∈ M ∧ ⊗F M ∗ , where pq is the Nijenhuis-Richardson nonassociative product. We have Li(pq) |F = (pq)d|F, {ip , Liq }|F = ip ◦ Liq |F = ip ◦ iq ◦ d|F = (ip ◦ q)d|F.

(9.29) (9.30)

Li(pq) − {ip , Liq } ∈ derF (M ∧ ).

(9.31)

This proves that r/l

r/l

The Fr¨ olicher-Nijenhuis diﬀerential binary operation on the R-Lie M ∧ -module derF (M ∧ ) ! ∧ M ⊗F M ∗ , is denoted by [· ⊗R ·], with grade[· ⊗R ·] = +1, [·⊗R ·]

(M ∧ ⊗F M ∗ ) ⊗R (M ∧ ⊗F M ∗ ) −−−−→ (M ∧ ⊗F M ∗ ).

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DEFINITION 9.5 (Fr¨ olicher-Nijenhuis Lie M ∧ -module) algorithmic form of the Fr¨ olicher-Nijenhuis R-bracket,

We deﬁne the following

∈ derF (M ∧ ), (−)q i[p ⊗R q] ≡ Lri(pq) − {ip , Lriq }

(−)q [p ⊗R q] ≡ i−1 Lri(pq) − {ip , Lriq } ∈ M ∧ ⊗F M ∗ .

(9.32) (9.33)

In particular if p is an idempotent (with respect to the Nijenhuis-Richardson product), p2 = p ∈ M ∧ ⊗F M ∗ , then grade p = 0 and i[p ⊗R p] = Lrip − {ip , Lrip } = 2 ip ◦ d ◦ ip . LEMMA 9.3

The binary R-operation (9.33) is graded commutative: [p ⊗R q] = (−1)p+q+pq · [q ⊗R p].

PROOF

(9.34)

(9.35)

For p, q ∈ M ∧ ⊗F M ∗ , and for A, B ∈ derF (M ∧ ), we have i{p, q} = ipq − (−)pq iqp = {ip , iq },

(9.36)

L{A,B} = {A, LB } + (−) {LA , B},

(9.37)

B

Li(pq) = (−)

pq

Li(qp) + {ip , Liq } + (−) {Lip , iq }. q

(9.38)

All this implies that (−)q [p ⊗R q] = Li(pq) − {ip , Liq } = (−)pq Li(qp) + (−)q {Lip , iq } p+pq

= (−)

(9.39)

[q ⊗R p].

In order to relate Deﬁnition 9.5, (9.32) and (9.33), to implicit deﬁnition by Fr¨ olicher and ´ Nijenhuis [1956], we need to calculate the Slebodzi´ nski-Lie map on (9.32), Li[p⊗R q] = (−)1+q {{ip , Liq }, d} = {Lip , Liq }.

(9.40)

The implicit deﬁnition by Fr¨ olicher-Nijenhuis is as follows. By the Jacobi identity we have L◦L=0

=⇒

{LA , d} = 0

& {{LA , LB }, d} = 0.

(9.41)

The Fr¨ olicher and Nijenhuis decomposition [1956] (9.26) implies that for A, B ∈ derF (M ∧ ) a derivation [A ⊗R B] ∈ derF (M ∧ ) exists (in an implicit way) such that L[A⊗R B] ≡ {LA , LB } ∈ derR (M ∧ ), [A ⊗R B] = (−1)

A+B+AB

· [B ⊗R A].

(9.42) (9.43)

Example 9.2 If grade q = −1 we set q = X ∈ M ∗ . Then ∀ p ∈ M ∧ ⊗F M ∗ , pq = 0 ∈ M ∧ ⊗F M ∗ . In this case the deﬁnition (9.32) and (9.33) is simpliﬁed i[p ⊗R X] = {ip , LiX }.

(9.44)

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117

Evaluating above brackets on the exact 1-form df ∈ M, is showing that the Fr¨ olicher and Nijenhuis Lie M ∧ -module generalizes the Lie F-module of the vector ﬁelds [p ⊗R X]df = ip d(Xf ) − (LiX )pdf.

(9.45)

REMARK 9.1 Vinogradov in 1990, in an attempt of uniﬁcation of the Schouten Lie module of multivector ﬁelds [Schouten 1940, Nijenhuis 1955], with the Fr¨ olicher and Nijenhuis Lie operation, introduced new R-bracket as the sum of double-graded commutator of derivations. The value of the Vinogradov binary bracket do not vanish on a ring of the scalars and therefore is not given by the tensor ﬁeld. Vinogradov proposed the following explicit R-bracket for A, B ∈ End F (M ∧ ): 2[A ⊗R B]V ≡ {LA , B} − (−)B {A, LB }.

(9.46)

´ An evaluation of the Slebodzi´ nski-Lie map gives L[A⊗R B]V = {LA , LB }.

(9.47)

Contrary to the deﬁnition (9.33) where [p ⊗R q] ∈ M ∧ ⊗F M ∗ , the Vinogradov bracket does not deﬁne a tensor ﬁeld, [A ⊗R B]V |F = 0.

9.5.2

Consequence: module derivation

The notion of the Leibniz-Loday algebra can be weakened by relaxing the condition of an algebra derivation to a module derivation. De Rham complex M ∧ with d ∈ derR (M ∧ ) is a DGA. Then an M ∧ -module with a binary operation [· ⊗R ·] is said to be Leibniz-Loday R-algebra if [· ⊗R ·] is M ∧ -module derivation. THEOREM 9.2 (e.g., Dubois-Violette and Michor 1994) Let p, q ∈ M ∧ ⊗F M ∗ and α ∈ M ∧ . We abbreviate α ∧ q to αq. The following Leibniz formula for the M ∧ -module graded derivation holds [p ⊗R (αq)] = (Lip α)q − (−)p(α+q+1) (dα)(qp) + (−)α(p+1) α[p ⊗R q]. The above clue M ∧ -module graded derivation is well known; however, it is frequently presented without proof. We claim that the proof is a trivial consequence of Deﬁnition 9.5, (9.32) and (9.33). Straightforward calculations using (9.24) proves the above theorem. Another important easy consequence of Deﬁnition 9.5, (9.32) and (9.33), is the graded Jacobi relation that is an example of the graded Leibniz derivation. In this respect it is instructive to compare with Kanatchikov [1996], where the graded Jacobi relation was olicher-Nijenhuis derived for “semi-bracket” {ip , Lriq }, which does not coincide with the Fr¨ bracket (9.32) and (9.33).

9.6

Bianchi Identity

In this section p ≡ τ ⊗F P ∈ M ⊗F M ∗ with τ P = 1 ∈ F.

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The composition ip = eτ ◦ iP ∈ derF (M ∧ ) is a zero-grade derivation. This implies iP ◦ eτ |F = idF · τ P, and (ip )2 = ip . P

P

M −−−−→ F ⏐ ⏐ p/ || e

τ M ←−− −− F

i

F ←−−−− M ⏐ ⏐ id/ ||

M ∧ −−−P−→ M ∧ ⏐ ⏐ p/ ||

e

τ M ∧ ←−− −− M ∧

F −−−τ−→ M

(9.48)

e

However iP ◦ eτ does not split on M ∧ , iP ◦ eτ = (τ P )id − eτ ◦ iP

= id.

DEFINITION 9.6 (Angular rotation) Let p2 = p ∈ M ∧ ⊗F M ∗ , then grade p = 0. The angular rotation tensor ω of the (1, 1)-tensor ﬁeld p, is deﬁned as follows: iω ≡ (id − ip ) ◦ d ◦ ip

= −Lip ◦ ip .

We will show that iω ∈ derF (M ∧ ) and therefore ω is a (2, 1)-tensor ﬁeld. The term “angular rotation of idempotent” is motivated in the proof of the next Theorem 9.3. (Anholonomy) Let p2 = p, ip ≡ eτ ◦ iP ∈ derF (M ∧ ). Then

THEOREM 9.3 i.

ω=

1 2

· [p ⊗R p].

ii. iii.

ω = (ωτ ) ⊗F P = (iP (τ ∧ dτ )) ⊗F P. iω = {(d − τ ∧ LiP ), ip }, {(d − τ ∧ LiP ), iP } = 0.

iv.

(d − τ ∧ LiP )2 = (ωτ ) ∧ LiP ! curvature .

PROOF The proof of (i) and (ii) is straightforward, by direct inspection. The equalities (iii) and (iv) are equalities of derivations. The identity (iii) tells us that the tensor ﬁeld ω is “the spatial divergence” of the connection p. This is even more convincing than adopted Deﬁnition 9.6, to interpret ω as the angular rotation tensor ﬁeld. An exact two-form dτ sometimes is called the vortex form of the connection p ∈ M ⊗F M ∗ . The diﬀerential operator, (eτ /(τ P ) ◦LiP )◦(id−ip ), is invariant with respect to the dilation P → f P,

eτ /(τ f P ) ◦ Lf iP = eτ ◦ LiP − edf /f ◦ ip .

(9.49)

THEOREM 9.4 (Bianchi identity) Luigi Bianchi introduced his identity in Lezioni di geometria, three volumes published in 1902–1909. We refer also to [Kol´ ar, Michor, and Slov´ ak 1993]. The Bianchi identity for a connection p ∈ M ⊗F M ∗ tells us that 1 2 [[p

9.7

⊗R p] ⊗R p] = [ω ⊗R p] = {ω, (d − Lip )} = 0.

(9.50)

Frobenius Algebra

Let F denote an associative and commutative unital ring. Let A be F-module (F-Fmodule) and A∗ ≡ modF (A, F) be a dual F-module, together with the right and the left

Nijenhuis-Richardson algebra and Fr¨ olicher-Nijenhuis Lie module

119

evaluations and co-evaluations, also known as the closed/pivotal structures whose axioms are given by the Reidemeister zero moves, A∗ ⊗F A

−−−−−−−−−→

left evaluation

F

A ⊗F A∗

right evaluation

F

−−−−−−−−−−→

(9.51)

left co-evaluation

A∗ ⊗F A ←−−−−−−−−−−− F right co-evaluation

A ⊗F A∗ ←−−−−−−−−−−−− F An F-algebra m = F-algebra,

with a Frobenius covector (a co-unit) ε is said to be co-unit-class ∈ (2 → 1) ≡ modF (A ⊗F A, A), ε ∈ (1 → 0) ≡ modF (A, F) ≡ A∗ .

The composition (co-unit ◦ map hl/r ∈ modF (A, A∗ ),

(9.52)

) is a binary form equivalent to unary left/right F-module ε◦m=hl ◦(evl ⊗id)=(id⊗evr )◦hr

A ⊗ A −−−−−−−−−−−−−−−−−−−−→ F hl ,hr

−−−−→

A

(9.53)

∗

A

If a form hl or/and hr is nondegenerate, ker(h) = 0 ∈ A, then {m, ε} is said to be Frobenius F-algebra [Ferdinand Georg Frobenius 1903]. An F-co-algebra ' = with unit η is said to be unit-class co-algebra, ∈ (1 → 2) ≡ modF (A, A ⊗F A), η/1 ∈ (0 → 1) ≡ modF (F, A) ! A.

(9.54)

The composition ◦ η is a co-binary form that is equivalent to left/right unary F-module map f l/r ∈ modF (A∗ , A), ◦η=(f l ⊗coevl )◦coevl =coevr ◦(id⊗f r )

A ⊗F A ←−−−−−−−−−−−−−−−−−−−−−−−−− F f l ,f r

←−−−−

A

(9.55)

∗

A

If this co-binary form ' ◦ η is nondegenerate, ker(f l /f r ) = 0 ∈ A∗ , then {', (unit = η)} is said to be Frobenius F-co-algebra. A Frobenius F-algebra is both Frobenius algebra and Frobenius co-algebra subject two Frobenius axioms [Frobenius 1903, Curtis and Reiner 1962, Kauﬀman 1994, Voronov 1994, Kadison 1999, Caenepeel et al. 2002, Baez 2001],

∼ ∼ .

The Frobenius axioms do not imply uniqueness of The Frobemius axioms can be rephrased as ∈ bicomod(||, |),

for a given model of

and vice-versa.

∈ bimod(|, ||).

A Cliﬀord algebra is a particular example of a Frobenius algebra where unary “handle” e ◦ = ∈ (1 → 1) is diagonal [Oziewicz 2003, Figure 10]. Such Frobenius algebra is also said to be “canonical.” A Frobenius algebra is antipodeless [Oziewicz 1997, 1998].

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DEFINITION 9.7 (Trace is a co-unit) A trace on F-algebra A is an F-module map, a covector tr ∈ modF (A, F ) ≡ A∗ , i.e., a co-unit/(Frobenius covector), such that ∀ u, v ∈ A, tr(uv) = tr(vu). An F-algebra A with a trace is said to be trace-class F-algebra. A unit η ∈ A ! modF (F, A) is said to be co-trace if ' ◦ η = 'op ◦ η. An F-co-algebra with co-trace, cotr = tr∗ , is said to be co-trace-class co-algebra, ' ◦ cotr = 'op ◦ cotr.

(9.56)

The composition (tr ◦ m) is a symmetric binary form, and (' ◦ cotr) is a symmetric co-binary form. The Nijenhuis-Richardson Z-graded F-algebra restricted to zero-grade endomorphisms M ⊗ M ∗ is associative and unital trace-clase algebra, trace = counit

M ⊗F M ∗ −−−−−−−−−→ F,

tr(pq) = tr(qp).

(9.57)

One can extend F-valued trace to M ∧ -valued counit=“super-trace” over the NijenhuisRichardson nonassociative graded F-algebra, ‘trace’

M ∧ ⊗F M ∗ −−−−→ M ∧ ,

9.8

tr(α ⊗F P ) ≡ iP α ∈ M ∧ .

(9.58)

Frobenius Subalgebra of Nijenhuis-Richardson Algebra

DEFINITION 9.8 (Atomic idempotent) An idempotent p2 = p ∈ A in an algebra ∧2 A is said to be an atom if p ∧ (pAp) = 0 ∈ A [Jones, Statistical Mechanics, 1989]. The Nijenhuis-Richardson nonassociative F-algebra possesses important associative subalgebra of endomorphisms, End F M ≡ modF (M, M ). The endomorphism algebra with a trivial center is said to be the von Neumann factor. The endomorphism subalgebra is not stable under Fr¨ olicher-Nijenhuis Lie diﬀerential Roperation, if p ∈ End F M then [p ⊗R p] ∈ End F M. We consider the unital subalgebra of endomorphism algebra, generated by a ﬁnite set of primitive idempotents (an idempotent p2 = p is said to be primitive if p = a + b for idempotents a and b with ab = ba = 0 imply that a = 0 or b = 0). It appears that in the generic case such subalgebra “of idempotents” is Frobenius. A set n ∈ N of primitives idempotents {p1 , . . . , pn }, tr(pi ) = 1 ∈ F, and unit u, with a ﬁnite trace tru = d ∈ N, generate noncommutative trace-class Frobenius F-algebra Frn (relations are given below) with the symmetric form h ≡ tr ◦ m ∈ modF ((Frn )⊗2 , F). This particular biassociative and bi-unital/bi-trace Frobenius F-algebra Frn is a subalgebra of Nijenhuis-Richardson algebra, Deﬁnition 9.1. A Frobenius F-algebra Frn of atomic/simple idempotents is the subject of the following relations: (pi )2 = pi , i = 1, . . . , n, (9.59) ∀ w ∈ Frn , pi wpj tr(pi pj ) = pi pj tr(pi wpj ).

Nijenhuis-Richardson algebra and Fr¨ olicher-Nijenhuis Lie module

121

Every pair of atomic idempotents p and q with trp = trq = 1 ∈ F, satisfy the Galois connection (name introduced by Ore), a property that is also called a generalized inverse, pqp = tr(pq) p

qpq = tr(pq) q.

and

(9.60)

This reminds the relations of the Jones algebra and of the von Neumann ﬁnite dimensional algebra generated by atoms p and q [Jones 1983, §3]. From this it follows that a length of every word in Frobenius F-algebra Frn must be ≤ 2, and the F-dimensions are dimF (Frn ) = 1 + n2

= 1, 2, 5, 10, 17, 26, . . . .

THEOREM 9.5 (Laplace expansion) The Frobenius covector is given by a trace tr ∈ (Fr)∗ . The following Laplace expansion holds, also called a “weak coalgebra” condition. In the Sweedler notation for three words a, b, c ∈ Frn ), tr(abc) = Σ tr(a1 c)tr(a2 b).

(9.61)

In particular for a = b = c = u ≡ η, N " d ≡ tr ◦ cotr ≡ tr(u) = Σ tr(u1 )tr(u2 ), tr

u = u ⊗ u −→ u −→ d.

(9.62) (9.63)

THEOREM 9.6 (Frobenius coalgebra) Let {ei ∈ Frn } be a basis diagonalizing h = tr ◦ , i.e. h(ei ⊗ ej ) ≡ tr(ei ej ) = hi δij . Then ei = tr(ei ek el )

el ek ⊗ . hl hk

The Frobenius algebra of atomic idempotents is noncocommutative and moreover is antipodeless.

9.9

Frobenius Algebra of Two Idempotents

The bilinear form on 2-dimensional F-algebra Fr1 = spanF {u, p} for 1 < d is positive deﬁnite (++). To see this, let us choose the volume form as z1 ≡ u ∧ p ∈ (Fr1 )∧2 . Then detz h ≡ (h∧ z)z = d − 1. The form, h ≡ tr ◦ , in the basis {u, p} and in the basis {u − p, p} (after Gram-Schmidt orthogonalization) possesses the following basis-dependent matrix presentations: h

∗ u d1 u = , p 11 p∗

h

u−p p

=

d−1 0 u∗ . u ∗ + p∗ 0 1

(9.64)

A coalgebra Fr1 is group-like (no nonzero primitives). The co-unital comultiplication is not unital: (u − p) =

(u − p) ⊗ (u − p) , tr(u − p)

p = p ⊗ p.

(9.65)

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The Lagrange/Sylvester theorem and Gram-Schmidt orthogonalization allows us to calculate the signature for Frobenius F-algebra Frn for any n ∈ N. Here we wish to report a signature for ﬁve-dimensional Frobenius F-algebra Fr2 = gen{p, q} generated by two atomic idempotents. THEOREM 9.7 (Signature) Let t ≡ tr(pq) = {−1, 0, +1}. The signature of the bilinear form h ! tr ◦ m : (Fr2 )⊗2 −→F for ﬁve-dimensional F-algebra Fr2 , dimF (Fr2 ) = 5, depends on d ≡ tr(u) ∈ N only. ⎧ ⎪ ⎨+ + + + − if d > 2, Signature of h = 0 + + + − if d = 2, ⎪ ⎩ − + + + − if d < 2. PROOF Let p and q ∈ Fr2 generate atomic idempotents. The center ZFr2 of Frobenius F-algebra Fr2 is two-dimensional, u, (p − q)2 ∈ ZFr2 ,

dimF (ZFr2 ) = 2, 2

2

(pq + qp) = t(p + q) ,

4

(9.66) 2

(p − q) = −(t − 1)(p − q) .

(9.67)

Let a volume form for a F-module Fr2 be z2 ≡ u ∧ p ∧ q ∧ pq ∧ qp ∈ (Fr2 )∧5 . Then detz (tr ◦ m) = −(d − 2)(t − 1)4 t2 . In the basis {u, p, q, pq, qp} the bilinear form h ≡ tr ◦ m has the following basis-dependent-matrix: ⎛ ⎞ ⎛ d u ⎜ p ⎟ ⎜1 ⎜ ⎟ ⎜ ⎟ ⎜ h⎜ ⎜ q ⎟ = ⎜1 ⎝pq ⎠ ⎝ t qp t

1 1 t t t

1 t t t 1 t t t2 t t

⎞⎛ ∗ ⎞ t u ⎜ p∗ ⎟ t⎟ ⎟⎜ ∗ ⎟ ⎜ ⎟ t⎟ ⎟ ⎜ q ∗⎟ ⎝ ⎠ t (pq) ⎠ 2 (qp)∗ t

(9.68)

For t = {−1, 0, +1}, the particular basis of Fr2 diagonalizing the form h = tr ◦ cotr is u+

(p − q)2 , t−1

qp,

p + tq − (pq + qp),

q−

pq + qp , t+1

In this basis the matrix of the scalar product h is diagonal, t−1 2 2 2 ,t − 1 . h ! diag d − 2, t , (t − 1) , − t+1

9.10

pq −

qp . t

(9.69)

(9.70)

Conclusion

The Fr¨ olicher and Nijenhuis Lie R-algebra structure on the universal Grassmann module of diﬀerential multiforms found an increasing number of important applications/interpretations

Nijenhuis-Richardson algebra and Fr¨ olicher-Nijenhuis Lie module

123

both in pure algebra and in diﬀerential geometry of Ehresmann connections [Kocik 1997, Wagemann 1998], as well as in many branches of mathematical physics, in the special and in the general theory of relativity [Minguzzi 2003], in Maxwell’s theory of the electromagnetic ﬁeld [Fecko 1997, Kocik 1997, Cruz and Oziewicz 2003], in the Hamilton-Jacobi theory in classical mechanics [Gruhn and Oziewicz 1983], in symplectic geometry of the Lagrangian and Hamiltonian mechanics [Chavchanidze 2003] and others. From the point of view of these numerous fundamental applications there is a need for the algorithmic computational programming methods to deal with many structural aspects of this nontrivial Lie R-algebra. The present chapter was motivated by this need of explicit and algorithmic easy-to-handle the deﬁnition of the Fr¨ olicher and Nijenhuis Lie operation. We are proposing here such deﬁnition of the Fr¨ olicher and Nijenhuis Lie operations (9.32) and (9.33). This deﬁnition has a clear advantage that can be implemented for a computational symbolic program in computer algebra. Many identities that hold in Fr¨ olicher and Nijenhuis Lie Grassmann module follow much more easily from the proposed deﬁnition. It is important that Deﬁnition 9.5, (9.32) and (9.33), of Lie R-algebra need nonassociative Fr¨ olicher-Richardson F-operation on universal Grassmann module. The Fr¨ olicherRichardson nonassociative F-algebra deserves future studies in many respects. The Fr¨ olicher-Richardson algebra includes associative endomorphism subalgebra. Of special interests, from fundamental physical theories, quantum mechanics and relativity theory, are endomorphism subalgebras generated by atomic idempotents. Such generic subalgebras are Frobenius algebras, they possess nondegenerate scalar product that gives antipodeless algebra structure. In the last sections the Frobenius algebra is illustrated on an example of the ﬁve-dimensional algebra generated by two atomic idempotents. We believe that the correct environment for these particular Frobenius associative algebras must be nonassociative Fr¨ olicher-Richardson algebra, because the Fr¨ olicher and Nijenhuis diﬀerential Lie operation do not preserve associative endomorphism algebra. If p ∈ M ⊗F M ∗ is an endomorphism, then the Fr¨ olicher and Nijenhuis diﬀerential Lie operations (9.32) and (9.33) give olicher-Richardson algebra. We conjecture [p ⊗R p] ∈ M ⊗F M ∗ , but [p ⊗R p] is inside the Fr¨ that the Frobenius associative algebra could be related/identiﬁed with the kinematics and the Fr¨ olicher-Richardson nonassociative algebra with dynamics. Table 9.2. Kinematics versus dynamics

9.11

Kinematics

Dynamics

Special relativity Frobenius algebra

General relativity = gravity Fr¨ olicher-Richardson algebra

Acknowledgments

This work was Supported by Consejo Nacional de Ciencia y Tecnolog´ıa (CONACyT de M´exico), Grant # U 41214 F; and by Programa de Apoyo a Proyectos de Investigaci´ on e Innovaci´ on Tecnol´ogica, UNAM, Grant # IN 105402. The second author is a member of Sistema Nacional de Investigadores in M´exico, Expediente # 15337.

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Chapter 10 Generalized Capable Abelian Groups Piroska Cs¨ org¨ o Department of Algebra and Number Theory, E¨ otv¨ os Lor´and University, Budapest, Hungary A. Janˇ caˇ r´ık Charles University, Prague, Czech Repubplic T. Kepka Charles University, Prague, Czech Repubplic 10.1 10.2 10.3 10.4 10.5 10.6

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Technical results (a) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Technical results (b) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Abstract

129 129 131 133 134 135 135

Two-generated abelian inner permutation groups of loops are studied.

Key words: capable groups, l-capable groups, inner permutation groups, loops 2000 MSC: 20N05, 20K30

10.1

Introduction

The inner automorphism groups Int(G), G any group, and their isomorphic copies are known as capable groups. Of course, G is capable if Z(G) = 1, and so it is not surprising that the main task is to ﬁnd all capable nilpotent groups. On the other hand, it is somewhat surprising that only a few results are known in this respect. All ﬁnitely generated capable abelian groups are found. Now, in a more general setting, we call a group G l-capable if G is isomorphic to the inner permutation group Int(Q) of a loop Q. The present short chapter may be viewed as a modest contribution to the enormous task of describing l-capable (abelian) groups.

10.2

Preliminaries

The reader is referred to [2], [3], [4], and [6] for prerequisite pieces of information. Anyway, we recall some basic notions and facts in the sequel.

129

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P. Cs¨ org¨ o, A. Janˇcaˇr´ık, and T. Kepka

If Q is a loop and a ∈ Q, then two permutations La and Ra of Q are deﬁned by La (x) = ax and Ra (x) = xa. The permutation group M ul(Q) = La , Ra ; a ∈ Q is called the multiplication group of Q and the stabilizer Int(Q) of the neutral element of Q is called the inner permutation group of Q. Let H be a subgroup of a group G. Then Z(G) is the center of G, NG (H) the normalizer of H and LG (H) the core of H in G. A subset A of G is said to be a (left) pseudotransversal to H in G if G = AH and it is said to be a (left) transversal if, moreover, A−1 A ∩ H = 1. Two subsets A and B of G are called H-connected if [A, B] ⊆ H. PROPOSITION 10.1 ([8, Theorem 4.1]) A group H is l-capable if and only if it is a subgroup of a group G such that LG (H) = 1 and there exist H-connected transversals A,B to H in G with A, B = G. PROPOSITION 10.2 ([7, Theorem 2.4], [8, Corollary 3.3], [5, Theorem 4.2]) Let H be a nontrivial l-capable abelian group. Then: (i) H is not a cyclic group. (ii) H is not a quasicyclic Pr¨ ufer p-group. (iii) If H is ﬁnite, then no nontrivial p-primary component of H is cyclic. Let H be a ﬁnite abelian group. If H is trivial, then we put α(H) = 0. If H is a nontrivial p-group, then H ∼ ≤ rs , and we = Zpr1 × Zpr2 × · · · × Zprs , 1 ≤ s, 1 ≤ r1 ≤ r2 ≤ · · · α(Hp ), Hp put α(H) = rs − rs−1 (r0 = 0). Finally, in the general case we put α(H) = running through all p-primary components of H. PROPOSITION 10.3 ([1]) A ﬁnite abelian group H is capable if and only if α(H) = 0. Deﬁne the following two conditions (υ) and (ϑ) for an abelian group H: If H is a subgroup of a group G such that G = A, B for H-connected pseudotransversals A, B to H in G, then

(υ) (ϑ)

G ⊆ NG (H), LG (H) = 1.

PROPOSITION 10.4 (i) Every nontrivial cyclic group satisﬁes both (υ) and (ϑ). (ii) For any prime p, the group Zp × Zp satisﬁes (υ). (iii) Every nontrivial ﬁnite abelian group with at least one nontrivial primary component that is cyclic satisﬁes condition (ϑ). REMARK 10.1

An abelian group H satisﬁes (ϑ) if and only if H is not l-capable.

Generalized capable abelian groups

10.3

131

Technical Results (a)

In this section, let H be an abelian subgroup of a group G such that there exist Hconnected pseudotransversals A, B to H in G with A, B = G. PROPOSITION 10.5 Then LG (H) = 1.

Assume that H ∼ = Zpm × Zpn for a prime p and 0 ≤ m < n.

PROOF Suppose, on the contrary, that the quadruple (G, H, A, B) is a counterexample with the smallest m + n. Using [7, Theorem 2.2] and [5, Proposition 3.1], we see that m ≥ 1 and that H is subnormal in G. Moreover, Z(G) = 1 by [6, Corollary 5.5] and NG (H) = HZ(G) by [8, Proposition 2.7]. The rest of the proof is divided into eight parts. (i) Since H is ﬁnite, we have H ⊆ C for a ﬁnite subset C of A ∪ B. Further, from LG (H) = 1 and G = AH it follows easily that for every 1 = u ∈ H there exists au ∈ A with a−1 u uau ∈ H. Put G1 = C, au ; u ∈ H, u = 1, A1 = A ∩ G1 , B1 = B ∩ G. Then G1 is a ﬁnitely generated subgroup of G, H ⊆ G1 , LG1 (H) = 1, A1 , B1 are H-connected transversals to H in G1 and G1 = A1 , B1 . (ii) Due to (i), we may assume in the remaining part of the proof that G is ﬁnitely generated. Now, denote by K the set of subgroups K of H such that K = 1 and H/K is not cyclic and consider a subgroup N of G maximal with respect to the following three properties: (a) N G; (b) HN G; (c) KN G for every K ∈ K. If G = G/N , then A, B are H-connected pseudotransversals to H in G = AB, H = HN/N ∼ = H/H ∩ N . Put E = H ∩ N . Then EN = N G, and so E = H and E ∈ K. That is, either E = 1 or H/E is cyclic. In the latter case, we have (G) ⊆ H by [7, 2.2] and hence G ⊆ HN , HN G, a contradiction. We have shown that E = 1 and H ∼ = H. Now, we check that LG (HN ) = N . Firstly, N ⊆ LG (HN ) = L, and so L = F N for a subgroup F of H. Then HN/L ∼ = H/F and, since HN G, we have either F = 1 and L = N or F ∈ K and L = F N G, a contradiction. Thus LG (HN ) = N . Let 1 = M G. Then, using the maximality of N , we get either H · M G or K · M G for some K ∈ K. In both cases, H ∩ LG (HM ) = 1. (iii) With regard to (ii), we assume that H ∩ LG (HM ) = 1 for every nontrivial normal subgroup M of G. In particular, since 1 = Z(G) ⊆ LG (NG (H)), we have H1 = H ∩ LG (NG (H)) = 1. Of course, LG (NG (H)) = H1 Z(G). Let D be a left transversal to H1 Z(G) in G such that 1 ∈ D. Put f = Πfd , d ∈ D, where fd : H1 Z(G) = (H1 Z(G))d = H1d × Z(G) → Z(G) are the natural projection. Then f becomes a homomorphism of H1 Z(G) into a cartesian power of Z(G). If v ∈ Ker(f ), then v ∈ H1d for every d ∈ D, and it follows that v ∈ LG (H) = 1. Thus f is injective and since H1 is a p-group, we conclude that Z(G) contains at least one element of order p.

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(iv) Let Z be a p-element subgroup of Z(G) (see (iii)), V = HZ, W = LG (V ) = H2 Z, H2 ⊆ H. Similarly as above, we have H2 = 1. Moreover, W1 = {up ; u ∈ W } G, W1 ⊆ H, W1 ⊆ LG (H) = 1, W1 = 1, W ⊆ Soc(V ) = Soc(H)Z and, ﬁnally, ˜ = 1, and ˜ ⊆G ˜ = G/W . Of course, L ˜ H H2 ⊆ Soc(H). Now 1 = H/H2 ∼ = V /W = H G ˜ is not cyclic. If H2 = Soc(H), then m ≥ 2, n ≥ 3 and H ˜ ∼ hence H = Zpm−1 × Zpm−1 , a contradiction with the minimality of m + n. Thus H2 = Soc(H) and consequently H2 n−1 is a p-element subgroup of H. Now, put Q = {up ; u ∈ H}, so that Q is a p-element ˜ is not cyclic, it follows that H ˜ ∼ subgroup of H. Since H = Zpr × Zps , 1 ≤ r ≤ s, r + s + 1 = m + n, r ≤ m, s ≤ n. Using the minimality of m + n again, we get ˜ contains an element of order r = s = m and n = m + 1. Further, if Q ⊆ H2 , then H n ∼ m m p , but this is not possible, since H = Zp × Zp . Thus Q ⊆ H2 and it follows that Q = H2 and W = QZ. −1

(v) Let x ∈ G be such that Q ⊆ H ∩ H x . Then Qx Q is ﬁnite, we have Q = Qx and x ∈ NG (Q).

⊆ H ∩ W = Q, Q ⊆ Qx and, since

(vi) Let x ∈ G be such that H ∩ H x = 1. We have x = ua for some u ∈ H and a ∈ A, −1 and so H x = H a , H ∩ H a = 1 = H ∩ H a . Let b ∈ H be such that aH = bH. Then −1 −1 a−1 bH = H = Ha−1 b, H a = H b , [a, b] ∈ H, b−1 a ∈ aHb−1 = bHb−1 = aHa−1 , −1 b−1 a ∈ H ∩ H a = 1, b = a ∈ A ∩ B. Let a1 ∈ A and b1 ∈ B be such that a1 H = b1 H. Then a−1 1 b1 ∈ H, [a, a1 ] ∈ H, −1 −1 −1 −1 a−1 = 1, a a1 ab1 ∈ H, [b1 , a] ∈ H and, ﬁnally, a a1 b1 a ∈ H. Thus a−1 1 b1 ∈ H ∩H a1 = b1 and we conclude that A = B. Next, let c ∈ A be such that cH = a2 H. Then [c, a] ∈ H, a−3 ca = a−2 c·[c, a] ∈ H and c2 ∈ a2 H. It follows that (c−1 a2 )a = (c−1 )a · a2 = (ca )−1 · a2 ∈ H and consequently, −1 c−1 a2 ∈ H ∩ H a = 1, c = a2 and a2 ∈ A. 2

Finally, take d ∈ A arbitrarily. The elements d−1 da and d−1 da are in H and, since 2 (d−1 da )a = (d−1 )a da = a−1 d−1 a−1 da2 = [a, d] · [d, a2 ] ∈ H, we have d−1 da ∈ H ∩ −1 H a = 1 and d = da . Thus d ∈ CG (a) and this implies that A ⊆ CG (a). On the other hand, if z ∈ CG (a), z = ev, e ∈ A, v ∈ H, then v = e−1 z ∈ CG (a) ∩ H, −1 v ∈ H ∩ H a = 1 and z = e ∈ A. We see that A = CG (a) is a subgroup G. Then [A, A] ⊆ A ∩ H = 1 means [A, A] = 1, i.e., A is abelian. Thus G = A, B = A = A is abelian a contradiction with LG (H) = 1. (vii) Let x ∈ G be such that Q ⊆ H ∩ H x = P . According to (vi), P = 1 and since Q ⊆ P , −1 the group R = Soc(P ) is of order p. Further, S = Rx ⊆ H, R = S x and R ∪ S ⊆ x x Soc(H). If S = Q, then R = Q ⊆ W = W = QZ, R ⊆ QZ ∩ H = Q, Qx = Q and Q ⊆ H ∩ H x = P , a contradiction. Thus S = Q = R and SW = Soc(V ) = RW . But then Soc(V )x = S x W x = RW = Soc(V ) and x ∈ NG (Soc(V )). (viii) It follows from (v) and (vii) that G = NG (Q) ∪ NG (T ), T = Soc(V ). On the other G, NG (T ) = G and we hand, since LG (V ) = W ⊆ T ⊆ V and W = T , we have T take w ∈ G \ NG (T ). Then w ∈ NG (Q) and wv ∈ NG (T ) for every v ∈ NG (T ). Consequently, wv ∈ NG (Q) and v ∈ NG (Q). Thus NG (T ) ⊆ NG (Q), NG (Q) = G and Q G, a contradiction with LG (H) = 1.

PROPOSITION 10.6 Assume that H is subnormal of depth at most 2 in G, H is torsion and, for at least one prime p, the p-primary component of H is the product P × Q,

Generalized capable abelian groups

133

n where P ∼ = Zpm , 1 ≤ m ≤ ∞ and there exist n such that 0 ≤ n < m and up = 1 for every u ∈ Q. Then LG (H) = 1.

PROOF We proceed by induction on n. If n = 0, then the result follows from [4, Proposition 2.1]. If m = 1, then n = 0. Now, let LG (H) = 1. Then m ≥ 2, n ≥ 1 and HZ(G) = NG (H) G. Consequently, S = Socp (HZ(G)) = Socp (H)Socp (Z(G)) G and S ⊆ LG (K) = L, where K = HSocp (Z(G)). On the other hand, L = RSocp (Z(G)), R = H ∩ L, L1 = {up ; u ∈ L} G, L1 ⊆ LG (H) = 1, L1 = 1 and R ⊆ Socp (H). Thus R = Socp (H) ∼ = Hp /Socp (H) ∼ = H = K/L. Further, H P ∼ = P/Soc(P ) × Q/Soc(Q) = P × Q, pn−1 ∼ P = Zpm−1 and u = 1 for every u ∈ Q. Using induction, we get a contradiction with LG (H) = 1.

10.4

Technical Results (b)

PROPOSITION 10.7 Let H be a nontrivial ﬁnite abelian group such that every nontrivial proper factor of H satisﬁes at least one of the conditions (υ) and (ϑ). If H does not satisfy (υ), then H is a p-group for a prime p. PROOF Assume that H is a subgroup of a group G such that G ⊆ NG (H) and there exists H-connected pseudotransversals A, B to H in G with G = A, B. The rest of the proof is divided into seven parts. (i) It follows easily from our assumptions that LG (H) = 1. Further, Z(G) = 1 and K = NG (H) = HZ(G). Put L = NG (K), K1 = LG (K) = H1 Z(G), H1 = H ∩K1 = H. (ii) If H1 = 1 and G = G/K1 , then H ∼ = H and K1 = Z(G). Moreover, NG (H) = H · Z(G) = K 2 , where K2 = HZ2 (G). Put C = A ∩ K2 and D = B ∩ K2 , so that CH = K2 = DH. Since Z(G) ⊆ A ∩ B, we have A ∩ K 2 = Z(G) = B ∩ K 2 . Of course, C ⊆ A ∩ K 2 , and hence C ⊆ Z(G) and C ⊆ CK1 ⊆ Z2 (G)K1 = Z2 (G); quite similarly, D ⊆ Z2 (G). On the other hand Z(G) ∩ H = 1, Z2 (G) = K1 Z2 (G), Z2 (G) ∩ H ⊆ K1 ∩ H = Z(G) ∩ H = 1 and consequently, C = Z2 (G) = D. Thus C = D G and C ⊆ A ∩ B implies C ⊆ (A, B) = Z(G). We have shown that Z(G) = Z2 (G) and Z(G) = 1, a contradiction. (iii) According to (i) and (ii), we have 1 = H1 = H. Now, H = HK1 /K1 ∼ = H/H1 , LG (H) = 1 and it follows that H satisﬁes (υ). That is, G ⊆ L and L G. (iv) Let E be a subgroup of H such that E ∩ H1 = 1 and E L. We wish to show that E = 1. Indeed, since E L, we get [E, L] ⊆ E. On the other hand, [E, L] ⊆ L ⊆ K1 , and so [E, L] ⊆ E ∩ K1 = E ∩ H1 Z(G) = 1. This implies that E ⊆ Z(L). But Z(L) ⊆ NL (H) ⊆ NG (H) = K, Z(L) G and Z(L) ⊆ K1 . Consequently, E ⊆ K1 and E = 1. (v) Assume, for a short moment, that Z(G) is a p-group for a prime p. We claim that H is a p-group, too. H = P × Q, where P is a p-group and Q has no elements of order p. Since H1 is embedded into a cartesian product of copies of Z(G), H1 is a p-group

134

P. Cs¨ org¨ o, A. Janˇcaˇr´ık, and T. Kepka and H1 ⊆ P . Further, Q is characteristic in K, and hence Q L. Now, Q = 1 by (iv).

(vi) Let a ∈ A, a = 1, and let Ma be the set of normal subgroups M of G such that a ∈ LG (HM ). Then Ma = ∅ and we choose a maximal member Ma of Ma . It is easy to check that Ma = LG (HMa ), a ∈ Ma and Z(G/Ma ) is a cyclic or quasicyclic pa -group for a prime pa . Moreover, if N is a normal subgroup of G with N ⊆ Ma and N = Ma , then a ∈ Z(G), then Ma can be chosen such that K1 ⊆ Ma . (vii) For every a ∈ A, a = 1, let Ma be as in vi). Now, ∩NG (HMa ) ⊆ NG (∩HMa ) = NG (H) = K. Since G ⊆ K, there exists at least one a1 ∈ A such that a1 = 1 ˜ ⊆G ˜ = G/Ma , and G ⊆ NG (HMa1 ). Then Ma1 ∩ H = 1, H ∼ = HMa1 /Ma1 = H 1 ˜ = 1, (G) ˜ ⊆ N ˜ (H) ˜ and Z(G) ˜ is a p-group, p = pa . According to (v), H is a LG˜ (H) 1 G p-group, too.

COROLLARY 10.1 Let H be a ﬁnite abelian group such that every p-factor group of H satisﬁes (υ). Then H satisﬁes (υ), too.

10.5

Main Results

THEOREM 10.1 Let p be a prime and H an (at most) two-generated abelian p-group. The following conditions are equivalent: (i) H is capable. (ii) H is l-capable. (iii) α(H) = 0. (iv) H ∼ = Zpm × Zpm for some m ≥ 0. PROOF

Combine Propositions 10.1, 10.3 and 10.5.

THEOREM 10.2 Let p1 < p2 < · · · < pr , r ≥ 1, be prime numbers, m1 , . . . , mr nonnegative integers and let H ∼ 1 × · · · × Zpr × Zpmr . Then H satisﬁes (υ) and, = Zp1 × Zpm r 1 if mi ≥ 2 for at least one 1 ≤ i ≤ r, then H is not l-capable. PROOF If r = 1, then the result follows from propositions 10.5 and 10.4(ii). Now, assume r ≥ 2. Firstly, we prove that H satisﬁes (υ). We proceed by induction on s = r + m1 + · · · + mr . If s = 2, then H ∼ = Zp1 × Zp2 is cyclic and H satisﬁes (υ) by [7, Theorem 2.2]. Next, let K be a proper nontrivial subgroup of H such that the factor H/K does not satisfy (ϑ). According to [5, Theorem 4.2], no nontrivial primary component of H/K is cyclic. Consequently, if H/K is a p-group for a prime p, then H/K ∼ = Zp × Zpn , n ≥ 1, and using [9, Lemma 4.2] and Proposition 10.5, we conclude that n = 1 and H/K satisﬁes

Generalized capable abelian groups

135

(υ). If H/K is not a p-group, then H/K satisﬁes (υ) by induction on s. We have checked that every nontrivial proper factor of H satisﬁes at least one of the conditions (υ) and (ϑ). Now, by Proposition 10.7, H satisﬁes (υ). If mi ≥ 2 for at least one i, then H satisﬁes (ϑ) by Proposition 10.6. THEOREM 10.3 Every ﬁnite two-generated abelian group satisﬁes (υ) if and only if all the groups Zpk × Zpk , p a prime, k ≥ 1, satisfy (υ). PROOF We prove the converse implication, the direct one being trivial. Now, let H be the smallest counterexample. By Proposition 10.7, H is a p-group, so that H ∼ = Zpm × Zpn , 0 ≤ m ≤ n. According to our assumption and proposition 10.4(i), we have 1 ≤ m < n. Assume that H is a subgroup of a group G = A, B, where A, B are H-connected pseudotransversals to H in G. By Proposition 10.5, L = LG (H) = 1. Then H = H/L satisﬁes (υ), (G) ⊆ NG (H) and G ⊆ NG (H). That is, H satisﬁes (υ), contradiction. THEOREM 10.4 Assume that every ﬁnite two-generated abelian group satisﬁes (υ) (see Theorem 10.3). Then the following conditions are equivalent for every ﬁnite two-generated abelian group H: (i) H is l-capable. (ii) H is capable. (iii) α(H) = 0. PROOF 10.6.

10.6

The result follows through easy combination of Theorem 10.3 and Proposition

Acknowledgments

ˇ This work was supported by the Grant Agency of the Czech Republic, Grants GACR ˇ No. 406/05/P561 and by the institutional grant No. 201/02/0594 and GACR MSM 0021620839, and partly supported by the Hungarian National Foundation for Scientiﬁc Research, Grant No. TO34878 and No. TO38O59.

References [1] R. Baer: Erweiterungen von Gruppen und ihren Isomorphismes, Math. Z. 38, 1934, 375–416.

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[2] R.H. Bruck: Contributions to the theory of loops, Trans. Amer. Math. Soc. 60, 1946, 245–354. [3] R.H. Bruck: A Survay of Binary Systems, Springer Verlag: Berlin, 1971 (third printing). [4] P. Cs¨ org¨ o and T. Kepka: On loops whose inner permutations commute, Comment. Math. Univ. Carolinae 45, 2004, 213–221. [5] T. Kepka: On the abelian inner permutation groups of loops, Commun. Alg. 26, 1998, 857–861. [6] T. Kepka and A. Janˇcaˇr´ık: Multiplication groups of quasigroups and loops III, A cta Univ. Carol. Math. Phys. 38/1, 1993, 53–62. [7] T. Kepka and M. Niemenmaa: On loops with cyclic inner mapping groups, Arch. Math. 60, 1993, 233–236. [8] M. Niemenmaa and T. Kepka: On the multiplication groups of loops, J. Alg. 135, 1990, 112–122. [9] M. Niemenmaa and T. Kepka: On connected transversals to abelian subgroups, Bull. Austral. Math. Soc. 49, 1994, 121–128.

Chapter 11 Helicity Basis for Spin 1/2 and 1 Valeri V. Dvoeglazov and J. L. Quintanar Gonz´ alez Universidad de Zacatecas, M´exico

11.1 11.2 11.3 11.4

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The (1/2, 0) ⊕ (0, 1/2) case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The (1, 0) ⊕ (0, 1) case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

137 137 141 145 146

Abstract We study the theory of the (1/2, 0) ⊕ (0, 1/2) and (1, 0) ⊕ (0, 1) representations in the helicity basis. The helicity eigenstates are not the parity eigenstates. This is in accordance with the idea of Berestetski˘ı, Lifshitz, and Pitaevski˘ı. The behavior of the helicity eigenstates with respect to the charge conjugation, CP - conjugation is also diﬀerent compared to the parity eigenstates. Key words: helicity spin basis, parity eigenstates, 2000 MSC: 17XX, 81XX

11.1

Introduction

Recently we generalized the Dirac formalism [1, 2, 3, 4] and the Bargmann-Wigner formalism [5, 6, 7]. On this basis we proposed a set of equations for antisymmetric tensor (AST) ﬁeld. Some of them imply parity-violating transitions. In this chapter we are going to study transformations from the standard basis to the helicity basis in the Dirac theory and in the (1, 0) ⊕ (0, 1) Sankaranarayanan-Good theory [8, 9]. The spin basis rotation changes properties of the corresponding states with respect to parity. The parity is a physical quantum number; so, we try to extract corresponding physical contents from considerations of the various spin bases.

11.2

The (1/2, 0) ⊕ (0, 1/2) Case

Beginning the consideration of the helicity basis, we observe that it is well known that ˆ 3 = σ3 /2 ⊗ I2 does not commute with the Dirac hamiltonian unless the the operator S 3-momentum is aligned along with the third axis and the plane-wave expansion is used: ˆ 3 ]− = (γ 0 γ k × ∇i )3 ˆ S [H,

(11.1)

137

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Valeri V. Dvoeglazov and J.L. Quintanar Gonz´ alez

Moreover, Berestetski˘ı, Lifshitz, and Pitaevski˘ı wrote [10]: “... the orbital angular momentum l and the spin s of a moving particle are not separately conserved. Only the total angular momentum j = l + s is conserved. The component of the spin in any ﬁxed direction (taken as the z-axis) is therefore also not conserved, and cannot be used to enumerate the polarization (spin) states of a moving particle.” A similar conclusion has been given by /2 ⊗ I, p = p/|p|, Novozhilov in his book [11]. On the other hand, the helicity operator σ · p commutes with the Hamiltonian (more precisely, the commutator is equal to zero when acting on the one-particle plane-wave solutions). So, it is a bit surprising, why the 4-spinors have been studied so well when the basis has ˆ 3 operator: been chosen in such a way that they were eigenstates of the S ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ 1 0 1 0 ⎜ ⎜ ⎜ ⎜ ⎟ ⎟ ⎟ ⎟ 0 1 0 ⎜ ⎟ , u 1 ,− 1 = N +1 ⎜ ⎟ , v 1 , 1 = N − ⎜ ⎟ , v 1 ,− 1 = N −1 ⎜ 1 ⎟ ,(11.2) u 12 , 12 = N + 1 1 ⎝1⎠ 2 2 ⎝−1⎠ 2 2 − 2 ⎝0⎠ −2 ⎝ 0 ⎠ 2 2 2 2 0 1 0 −1 and, in opposition, the helicity basis case has been studied very little (see, however, refs. [11, 12, 13]. Let me remind the reader that the boosted 4-spinors in the “commonly used” basis are ⎞ ⎛ + ⎛ ⎞ p +m pl + + N1 N− 1 ⎜ pr ⎟ ⎜p− + m⎟ 2 2 ⎟ , u 1 ,− 1 = ⎜ − ⎜ ⎟ , (11.3) u 12 , 12 = ⎠ ⎝ 2 2 2m(E + m) p + m 2m(E + m) ⎝ −pl ⎠ p+ + m −pr ⎞ ⎛ + ⎛ ⎞ p +m pl − − N1 N− 1 ⎟ ⎜ ⎜ p− + m ⎟ 2 2 ⎟ , v 1 ,− 1 = ⎜ −pr ⎜ ⎟ , (11.4) v 12 , 12 = ⎠ ⎝ ⎠ 2 2 pl 2m(E + m) −p − m 2m(E + m) ⎝ + −p − m pr p± = E ± pz ,pr,l = px ± ipy . They are the parity eigenstates with eigenvalues of ±1. The 0 11 is used in the parity operator. matrix γ0 = 11 0 Let me now turn your attention to the helicity spin basis. The 2-eigenspinors of the helicity operator, 1 1 cos θ sin θe−iφ = σ·p , (11.5) 2 2 sin θe+iφ − cos θ can be deﬁned as follows [14, 15]: cos θ2 e−iφ/2 , φ 12 ↑ = sin θ2 e+iφ/2

φ 12 ↓ =

sin θ2 e−iφ/2 − cos θ2 e+iφ/2

,

(11.6)

for ±1/2 eigenvalues, respectively. We start from the Klein-Gordon equation, generalized for describing the spin-1/2 particles (i.e., two degrees of freedom); c = = 1:1 (E + σ · p)(E − σ · p)φ = m2 φ .

(11.7)

It can be rewritten in the form of the set of two ﬁrst-order equations for 2-spinors. Simultaneously, we observe that they may be chosen as eigenstates of the helicity operator that 1 The

following method follows the work of van der Waerden, Sakurai, and Gersten, see [16].

Helicity basis for spin 1/2 and 1

139

is present in (11.7):2 (E − (σ · p))φ↑ = (E − p)φ↑ = mχ↑ ,

(11.8)

(E + (σ · p))χ↑ = (E + p)χ↑ = mφ↑ ,

(11.9)

(E − (σ · p))φ↓ = (E + p)φ↓ = mχ↓ ,

(11.10)

(E + (σ · p))χ↓ = (E − p)χ↓ = mφ↓ .

(11.11)

and

If the φ spinors are deﬁned by the Eq. (11.6), then we can construct the corresponding uand v- 4-spinors: ⎛& ⎞ E+p φ 1 ↑ φ ↑ ⎠, u↑ = N↑+ E−p = √ ⎝& m m 2 φ m φ↑ ↑ E+p (11.12) ⎛& ⎞ m φ ↓ 1 E+p φ↓ ⎠, u↓ = N↓+ E+p = √ ⎝& E+p φ ↓ 2 m φ↓ m

⎛&

⎞ E+p φ 1 ↑ φ↑ ⎠, v↑ = N↑− = √ ⎝ & mm − E−p φ ↑ 2 − φ m E+p ↑ ⎞ ⎛& m φ ↓ 1 E+p φ↓ ⎠, v↓ = N↓− =√ ⎝ & − E+p φ ↓ 2 m − E+p φ↓

(11.13)

m

3

where the normalization to the unit (±1) was used:

u ¯λ uλ = δλλ , v¯λ vλ = −δλλ , u ¯λ vλ = 0 = v¯λ uλ One can prove that the matrix P = γ0 =

0 11 11 0

(11.14) (11.15)

(11.16)

can also be used in the parity operator as well as in the original Dirac basis. Indeed, the 4-spinors (11.12) and (11.13) satisfy the Dirac equation in the spinorial representation of the γ-matrices (see clearly from (11.7)). Hence, the parity-transformed function Ψ (t, −x) = P Ψ(t, x) must satisfy (11.17) [iγ μ ∂μ − m]Ψ (t, −x) = 0 , with ∂μ = (∂/∂t, −∇i ). This is possible when P −1 γ 0 P = γ 0 and P −1 γ i P = −γ i . The matrix (11.16) satisﬁes these requirements, as in the textbook case. Next, it is easy to prove that one can form the projection operators P+ = +

λ

2 This

uλ (p)¯ uλ (p) =

pμ γ μ + m , 2m

(11.18)

is the opposite of the choice of the basis (11.2), where 4-spinors are the eigenstates of the parity operator. 3 Of course, there are no mathematical diﬃculties to change the normalization to ±m, which may be more convenient for the study of the massless limit.

140

Valeri V. Dvoeglazov and J.L. Quintanar Gonz´ alez P− = −

vλ (p)¯ vλ (p) =

λ

m − pμ γ μ , 2m

(11.19)

with the properties P+ + P− = 1 and P±2 = P± . This permits us to expand the 4-spinors deﬁned in the basis (11.2) in linear superpositions of the helicity basis 4-spinors and to ﬁnd corresponding coeﬃcients of the expansion: uσ (p) = Aσλ uλ (p) + Bσλ vλ (p) , vσ (p) = Cσλ uλ (p) + Dσλ vλ (p) .

(11.20) (11.21)

Multiplying the above equations by u ¯λ , v¯λ and using the normalization conditions, we ¯λ uσ , Bσλ = Cσλ = −¯ vλ uσ . Thus, the transformation matrix from obtain Aσλ = Dσλ = u the commonly used basis to the helicity basis is uσ uλ AB =U , U= (11.22) vσ vλ BA Neither A nor B are unitary: A = (a++ + a+− )(σμ aμ ) + (−a−+ + a−− )(σμ aμ )σ3 , B = (−a++ + a+− )(σμ aμ ) + (a−+ + a−− )(σμ aμ )σ3 ,

(11.23) (11.24)

where a0 = −i cos(θ/2) sin(φ/2) ∈ *m , 2

a = sin(θ/2) sin(φ/2) ∈ +e ,

a1 = sin(θ/2) cos(φ/2) ∈ +e ,

3

a = cos(θ/2) cos(φ/2) ∈ +e ,

(11.25) (11.26)

and

a++ a−+

(E + m)(E + p) √ , = 2 2m (E − m)(E + p) √ , = 2 2m

(E + m)(E − p) √ , 2 2m (E − m)(E − p) √ . = 2 2m

a+− =

(11.27)

a−−

(11.28)

However, A† A + B † B = 11, so the matrix U is unitary. Please note that the 4 × 4 matrix acts on the spin indices (σ,λ), and does not act on the spinorial indices. Alternatively, the transformation can be written: β 5 uα σ = [Aσλ ⊗ Iαβ + Bσλ ⊗ γαβ ]uλ ,

vσα

= [Aσλ ⊗ Iαβ + Bσλ ⊗

5 γαβ ]vλβ

.

(11.29) (11.30)

We now investigate the properties of the helicity-basis 4-spinors with respect to the discrete symmetry operations P and C. It is expected that λ → −λ under parity, as Berestetski˘ı, Lifshitz, and Pitaevski˘ı claimed [10].4 With respect to p → −p (i.e., the spherical system angles θ → π − θ, φ → π + φ) the helicity 2-eigenspinors transform as follows: φ↑↓ ⇒ −iφ↓↑ , ref. [15]. Hence, P u↑ (−p) = −iu↓ (p) , P v↑ (−p) = +iv↓ (p) , P u↓ (−p) = −iu↑ (p) , P v↓ (−p) = +iv↑ (p) .

(11.31) (11.32)

4 Indeed, if x → −x, then the vector p → −p, but the axial vector S → S, that implies the above statement.

Helicity basis for spin 1/2 and 1

141

Thus, on the level of classical ﬁelds, we observe that the helicity 4-spinors transform to the 4-spinors of the opposite helicity. The charge conjugation operation is deﬁned as 0 Θ C= K. (11.33) −Θ 0 Hence, we observe Cu↑ (p) = −v↓ (p) , Cv↑ (p) = +u↓ (p) , Cu↓ (p) = +v↑ (p) , Cv↓ (p) = −u↑ (p) .

(11.34) (11.35)

due to the properties of the Wigner operator Θφ∗↑ = −φ↓ and Θφ∗↓ = +φ↑ . For the CP (and P C) operation we get CP u↑ (−p) = −P Cu↑ (−p) = +iv↑ (p) ,

(11.36)

CP u↓ (−p) = −P Cu↓ (−p) = −iv↓ (p) , CP v↑ (−p) = −P Cv↑ (−p) = +iu↑ (p) ,

(11.37) (11.38)

CP v↓ (−p) = −P Cv↓ (−p) = −iu↓ (p) ,

(11.39)

which are diﬀerent from the Dirac commonly used case. Similar conclusions can be drawn in the Fock space.

11.3

The (1, 0) ⊕ (0, 1) Case

In this section we are going to investigate the behavior of the ﬁeld functions of the (1, 0) ⊕ (0, 1) representation in the helicity basis with respect to P , C, and CP operations. Let us start from the Klein-Gordon equation written for the 3-component function ( = c = 1): (11.40) (E 2 − p2 )ψ(3) = m2 ψ(3) . The function ψ describes the particles, usually referred to as spin-1; we refer to it as a “3-spinor.” On choosing the basis where Sijk = −iijk one can derive the following property for any 3-vector a: (11.41) (S · a)2ij = a2 δij − ai aj . Then Eq. (11.40) can be rewritten in the form: (E − S · p)(E + S · p)ij ψ j − pi pj ψ j = m2 ψ i .

(11.42)

In the coordinate space it is of the second order in the time derivative, but as in the spin-1/2 case [17] we can reduce it to the set of the 3-“spinor” equations of the ﬁrst orders. The procedure permits us to consider the Hamiltonian-like form i ∂ψ ∂t = Hψ and make it easier to ﬁnd the energy eigenstates of the problem. We can denote (E + S · p)ψ = mξ

(11.43)

p p ψ = p (p · ψ) = mp ϕ.

(11.44)

i j

j

Hence Eq. (11.42) is written as m(E − S · p)ξ − mp ϕ = m2 ψ.

(11.45)

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Valeri V. Dvoeglazov and J.L. Quintanar Gonz´ alez

Now, we insert the properties (S · p)ij ψ j = (∇ × ψ)i ,

pi pj ψ j = −[∇ (∇ · ψ)]i ,

(11.46)

and deﬁne ψ = E − iB. We can obtain (cf. with ref. [4]) ∇×B−

∂E = −m · Im(ξ), ∂t

∂B = m · Re(ξ), ∂t

(11.47)

∇ · E = −m · Im(ϕ) + constx ,

(11.48)

∇×E+

and ∇ · B = −m · Re(ϕ) + constx ,

respectively, by means of separation of Eqs. (11.43) and (11.44) into the real and imaginary parts. Next, we ﬁx ϕ = imφ and ξ = imA, with φ and A being the electromagnetic-like potentials. The well-known Proca equation follows: ∂μ F μν + m2 Aν = 0.

(11.49)

For the sake of completeness let us substitute ϕ and ξ in Eq.(11.45). The result is − ∂A ∂t − ∇φ = E y ∇ × A = B that is equivalent to the second Proca equation: F μν = ∂ μ Aν − ∂ ν Aμ .

(11.50)

Let us take the complex conjugates of Eqs.(11.43), (11.44), and (11.45) and now deﬁne χ = E + iB. As a result we have (E − S · p)χ = −mξ

or

(E − S · p)(E + iB) = −im2 A,

(11.51)

pi pj χj = p(p · χ) = −mpϕ (E + S · p)ξ − pϕ = −mχ

or or

p [p · (E + iB)] = −im2 pφ, (E + S · p)A − pφ = i(E + iB),

(11.52) (11.53)

It is possible to rewrite the above equations (and their complex conjugates) in the 10 × 10 matrix equation (with appropriate choice of matrices) for spin-1 particles (cf. [22]): ⎛ ⎞ ⎛ ⎞⎛ ⎞ χ1 χ1 0 0 0 0 0 0 −E ipz −ipy px ⎜ χ2 ⎟ ⎜ 0 ⎟ ⎜ χ2 ⎟ 0 0 0 0 0 −ip −E ip p z x y ⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ χ3 ⎟ ⎜ 0 0 0 0 0 0 ipy −ipx −E pz ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ χ3 ⎟ ⎜ ⎜ψ1 ⎟ ⎜ 0 ⎟ ⎟ 0 0 0 0 0 E ipz −ipy −px ⎟ ⎜ψ1 ⎟ ⎜ ⎟ ⎜ ⎜ ⎜ψ2 ⎟ ⎜ 0 ⎟ ⎟ 0 0 0 0 0 −ipz E ipx −py ⎟ ⎜ψ2 ⎟ ⎜ ⎟ = m⎜ (11.54) ⎜ ⎜ψ3 ⎟ , ⎜ 0 ⎟ ⎟ 0 0 0 0 0 ipy −ipx E −pz ⎟ ⎜ψ3 ⎟ ⎜ ⎟ ⎜ ⎜ ⎟ ⎜ ξ1 ⎟ ⎜ −E −ipz ipy 0 0 0 0 0 0 0 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ξ1 ⎟ ⎜ ⎜ ξ2 ⎟ ⎜ ipz −E −ipx 0 0 0 0 ⎟ ⎟ 0 0 0 ⎟ ⎜ ξ2 ⎟ ⎜ ⎟ ⎜ ⎝ ⎝ ξ3 ⎠ ⎝−ipy ipx −E 0 0 0 0 ⎠ ⎠ 0 0 0 ξ3 0 0 0 ϕ ϕ −px −py −pz 0 0 0 0 which is in the symbolic form: ⎛ 03×3 ⎜ 0 3×3 ⎜ ⎝ −(E − S · p)3×3 −p1×3

⎞ 03×3 −(E + S · p)3×3 p3×1 03×3 (E − S · p)3×3 −p3×1 ⎟ ⎟ Ξ = mΞ 03×3 03×3 03×1 ⎠ 01×3 01×3 0

(11.55)

ξ, ϕ). This ﬁrst-order equation is for the 10-component ﬁeld function Ξ = column( χ, ψ, known as the Duﬃn-Kemmer-Petiau (DKP) equation [22]. It contains the part corresponding to the 4-vector potential. 5 At ﬁrst sight, for the construction of (11.54) we have used 5 It would be of interest to research the helicity basis for the DKP equation, as we did for the Dirac equation. However, we leave this task for future works. Instead, we are going to consider the helicity basis of the solutions of the Weinberg-Tucker-Hammer second-order equations that follow.

Helicity basis for spin 1/2 and 1

143

Eqs. (11.45) and (11.51–11.53), and omitted Eqs. (11.43) and (11.44). However, one can show that our DKP equation contains that information. If we write (11.43)–(11.45) and (11.51) in the matrix form, we can also write ⎞⎛ ⎞ ⎛ 03×3 03×1 −m3×3 ψ (E + S · p)3×3 ⎟ ⎜ ⎟ ⎜ 0 (E − S · p) 0 m χ 3×3 3×3 3×1 3×3 ⎟⎜ ⎟ = 0, ⎜ (11.56) ⎠ ⎝ϕ⎠ ⎝ p1×3 01×3 −m1×1 01×3 −m3×3 03×3 −p3×1 (E − S · p)3×3 ξ which is related to (11.55). It is more convenient to write this equation in terms of E, B, φ and A. We use the unitary transformation with ⎞ ⎛ 13×3 13×3 03×1 03×3 1 ⎜ i3×3 −i3×3 03×1 03×3 ⎟ ⎟. V=√ ⎜ (11.57) 2 ⎝01×3 01×3 21×1 01×3 ⎠ 03×3 03×3 03×1 23×3 As a result we have ⎛ ⎞⎛ ⎞ E3×3 −i(S · p)3×3 03×1 03×3 E ⎜i(S · p)3×3 ⎜ ⎟ E3×3 03×1 −2im3×3 ⎟ ⎜ ⎟⎜ B ⎟, ⎝ p1×3 ⎠ ⎝ imφ ⎠ −ip1×3 −2m1×1 01×3 −m3×3 im3×3 −2p3×1 2(E − S · p)3×3 imA

(11.58)

where p1×3 = (px , py , pz ) is the row and p3×1 is the column. It is equivalent to the Proca set. Taking into account the Proca Eqs. (11.49) and (11.50), the deﬁnitions of Ei = F i0 , i B = − 12 ijk F jk and the deﬁnition of the Levi-Civita tensor, we can obtain the TuckerHammer equation [18]: E 2 − p2 + 2E(S · p) + 2(S · p)2 χ E 2 − p2 − 2m2 = 0. (11.59) E 2 − p2 − 2E(S · p) + 2(S · p)2 E 2 − p2 − 2m2 ψ In the covariant form Eq. (11.59) is written as μν γ pμ pν + pμ pμ − 2m2 Ψ(6) (pμ ) = 0. with the 6 × 6 matrices [19]), 0 13×3 0 −S i γ 00 = , γ i0 = γ 0i = , 13×3 0 Si 0 0 −δij + Si Sj + Sj Si ij γ = . −δij + Si Sj + Sj Si 0 In the coordinate space we have μν γ ∂μ ∂ν + ∂ μ ∂μ + 2m2 Ψ(xμ ) = 0 .

(11.60)

(11.61)

(11.62)

If we set the condition ∂μ ∂μ → −m2 we can recover the Weinberg equation, ref. [20]6 : m2 + 2E(S · p) + 2(S · p)2 χ −m2 χ Γ = = 0 , (11.63) m2 − 2E(S · p) + 2(S · p)2 −m2 ψ ψ 6 We should mention that this procedure is not quite clear, because the dispersion relations of the Weinberg equation and the Tucker-Hammer equation may be diﬀerent (see [21]). The Weinberg equation permits, in general, the tachyonic solutions, E 2 − p2 = −m2 .

144

Valeri V. Dvoeglazov and J.L. Quintanar Gonz´ alez

which is in the covariant form (γ μν ∂μ ∂ν + m2 )Ψ(xμ ) = 0 .

(11.64)

Thus, from what we have seen above, we can conclude that the Duﬃn-Kemmer-Petiau, Proca, Weinberg, and Tucker-Hammer equations are all related to one another. They can be obtained by various transformations from the relativistic dispersion relation, E 2 − p2 = m2 . Let us consider Eq. (11.59) as a set of equations for the bivector components in the helicity basis. Then, we have (p =| p |): (E 2 − p2 + 2Ep + 2p2 )ψ↑ = (2m2 − (E 2 − p2 ))χ↑ , (E 2 − p2 − 2Ep + 2p2 )χ↑ = (2m2 − (E 2 − p2 ))ψ↑ , (E 2 − p2 − 2Ep + 2p2 )ψ↓ = (2m2 − (E 2 − p2 ))χ↓ , (E 2 − p2 + 2Ep + 2p2 )χ↓ = (2m2 − (E 2 − p2 ))ψ↓ , 2

2

2

2

(h = 1)

(11.65)

(h = −1)

(11.66)

(h = 0).

(11.67)

2

(E − p )ψ→ = (2m − (E − p ))χ→ , (E 2 − p2 )χ→ = (2m2 − (E 2 − p2 ))ψ→ ,

where the 3-“spinors” are in the helicity basis (see [14, p. 192]): ⎞ ⎛ ⎛ 1+cos θ −iφ ⎞ ⎛ 1−cos θ −iφ ⎞ sin √ θ e−iφ − e e 2 2 2 ⎟ ⎜ sin sin ⎠, ⎝ ⎠ √θ √θ − cos θ χ = = , χ χ↑ = ⎝ ⎠ ⎝ → ↓ 2 2 sin θ iφ 1−cos θ iφ 1+cos θ iφ √ e e e . 2 2 2 The normalization condition is chosen χ† χ = 1.

Taking into account (11.65)–(11.67) we can write the bivectors u↑,↓,→ = following way: u1,↑ = N↑

χ↑

2m2 −(E 2 −p2 ) E 2 −p2 +2Ep+2p2 χ↑

,

u1,↓ =N↓

u1,→ = N→

χ↓

χ→

2m2 −(E 2 −p2 ) χ→ E 2 −p2

2m2 −(E 2 −p2 ) E 2 −p2 −2Ep+2p2 χ↓

χ↑,↓,→ ψ↑,↓,→

(11.68)

in the

, (11.69)

.

13×3 03×3 ). After the 03×3 −13×3 normalization to the unit and imposing m2 = E 2 − p2 , our bivectors are then E+p m χ↓ 1 1 1 χ↑ χ→ E+p m , (11.70) , u1,→ = √ , u1,↓ = √ u1,↑ = √ m E+p 2 E+p χ↑ 2 χ→ 2 m χ↓ E+p m 1 1 1 χ→ E+p χ↓ m χ↑ √ √ . (11.71) v1,↑ = √ = = , v , v 1,→ 1,↓ m E+p 2 − E+p χ↑ 2 −χ→ 2 − m χ↓ Let us now introduce uλ = u† γ 00 , vλ = γ 5 uλ (where γ 5 =

Now we study the discrete symmetry operations for the spin-1 case (as we did for the spin-1/2 case in the previous section). The bivectors have the following properties: 1. The parity (p → −p, θ → π − θ, φ → π + φ). We note that the 3-“spinors” are transformed as χh → −χ−h ; the parity operator is P = γ 00 (it is analogous to that which was used for spin-1/2 (see (11.16)). Therefore, P u1,↑ (−p) = −u1,↓ (p), P u1,→ (−p) = −u1,→ (p), P u1,↓ (−p) = −u1,↑ (p) . (11.72) And, P v1,↑ (−p) = +v1,↓ (p), P v1,→ (−p) = +v1,→ (p), P v1,↓ (−p) = +v1,↑ (p) . (11.73)

Helicity basis for spin 1/2 and 1

145

2. The charge conjugation is deﬁned as C=e

iα

⎛

(as in (11.33)) with Θ[1]

0 0 = ⎝ 0 −1 1 0

0 Θ K −Θ 0

(11.74)

⎞ 1 0 ⎠. Hence, Θχ∗↑ = χ↓ , Θχ∗↓ = χ↑ , Θχ∗→ = −χ→ . 0

Finally, we have Cu1,↑ (p) = +eiα v1,↓ (p),

Cu1,→ (p) = −eiα v1,→ (p),

Cu1,↓ (p) = + eiα v1,↑ (p) ,

(11.75)

and Cv1,↑ (p) = −eiα u1,↓ (p),

Cv1,→ (p) = +eiα u1,→ (p),

Cv1,↓ (p) = − eiα u1,↑ (p) .

(11.76)

3. The CP and P C operations: CP u1,↑ (−p) = −P Cu1,↑ (−p) = −eiα v1,↑ (p),

(11.77)

CP v1,↑ (−p) = −P Cv1,↑ (−p) = −eiα u1,↑ (p),

(11.78)

CP u1,↓ (−p) = −P Cu1,↓ (−p) = −e v1,↓ (p),

(11.79)

CP v1,↓ (−p) = −P Cv1,↓ (−p) = −eiα u1,↓ (p),

(11.80)

CP u1,→ (−p) = −P Cu1,→ (−p) = +e v1,→ (p),

(11.81)

CP v1,→ (−p) = −P Cv1,→ (−p) = +e u1,→ (p).

(11.82)

iα

iα

iα

We found within the classical ﬁeld theory that the properties of particle and antiparticle of spin-1 are diﬀerent compared with the known cases (when the basis is chosen in such a way that the solutions are the eigenstates of the parity).

11.4

Conclusions

• As in the ( 12 , 12 ) representation [12], the ( 12 , 0)⊕(0, 12 ) and (1, 0)⊕(0, 1) ﬁeld functions in the helicity basis are not eigenstates of the commonly used parity operator; |p, λ >⇒ | − p, −λ > on the classical level. This is in accordance with the earlier consideration of Berestetski˘ı, Lifshitz, and Pitaevski˘ı. • Helicity ﬁeld functions may satisfy the ordinary Dirac equation with γ’s to be in the spinorial representation. • Helicity ﬁeld functions can be expanded in the set of the Dirac 4-spinors by means of the matrix U −1 given in this chapter. • P and C operations anticommute in this framework on the classical level. • The diﬀerent formulations of the spin-1 particles are all connected by algebraic transformations.

146

Valeri V. Dvoeglazov and J.L. Quintanar Gonz´ alez • The properties of spin-1 solutions in the helicity basis with respect to P , C, CP are similar to those in the spin-1/2 case.

In order to make the above conclusions more ﬁrm, one should repeat the calculations in the Fock space within the “secondary quantization” framework (see [17] for the spin-1/2 case).

References [1] G. Ziino, Ann. Fond. Broglie 14 (1989) 427; ibid 16 (1991) 343; A. Barut and G. Ziino, Mod. Phys. Lett. A8 (1993) 1011; G. Ziino, Int. J. Mod. Phys. A11 (1996) 2081. [2] N. D. S. Gupta, Nucl. Phys. B4 (1967) 147; D. V. Ahluwalia, Int. J. Mod. Phys. A11 (1996) 1855; V. Dvoeglazov, Hadronic J. 20 (1997) 435. [3] V. V. Dvoeglazov, Mod. Phys. Lett. A12 (1997) 2741. [4] V. V. Dvoeglazov, Spacetime and Substance 3(12) (2002) 28; Rev. Mex. Fis., Supl. 1, 49 (2003) 99. [5] V. V. Dvoeglazov, Physica Scripta 64 (2001) 201. [6] V. V. Dvoeglazov, Hadronic J. 25 (2002) 137. [7] V. V. Dvoeglazov, Hadronic J. 26 (2003) 299. [8] A. Sankaranarayanan and R. H. Good, jr., Nuovo Cim. 36 (1965) 1303. [9] D. V. Ahluwalia, M. B. Johnson and T. Goldman, Phys. Lett. B316 (1993) 102; V. V. Dvoeglazov, Int. J. Theor. Phys. 37, (1998) 1915, and references therein. [10] V. B. Berestetski˘ı, E. M. Lifshitz and L. P. Pitaevski˘ı, Quantum Electrodynamics (Pergamon Press, 1982), §16. [11] Yu. V. Novozhilov, Introduction to Elementary Particle Physics (Pergamon Press, 1975), §4.3, 6.2. [12] H. M. Ruck y W. Greiner, J. Phys. G: Nucl. Phys. 3 (1977) 657. [13] M. Jackob and G. C. Wick, Ann. Phys. 7 (1959) 404. [14] D. A. Varshalovich, A. N. Moskalev and V. K. Khersonski˘ı, Quantum Theory of Angular Momentum (World Scientiﬁc, 1988), §6.2.5. [15] V. V. Dvoeglazov, Fizika B6 (1997) 111. [16] J. J. Sakurai, Advanced Quantum Mechanics. (Addison-Wesley, 1967), §3.2; A. Gersten, Found. Phys. Lett. 12 (1999) 291; ibid. 13 (2000) 185; V. V. Dvoeglazov, J. Phys. A: Math. Gen. 33 (2000) 5011. [17] V. V. Dvoeglazov, In Memorias de la 8a Reuni´ on Nacional Acad´emica de F´ısica y

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Matem´ aticas, 12-16 de Mayo de 2003, ESFM-IPN, M´exico, D.F., pp. 45-54. [18] R. H. Tucker y C. L. Hammer, Phys. Rev. D3 (1971) 2448. [19] A. O. Barut, I. Muzinich and D. Williams, Phys. Rev. 130 (1963) 442. [20] S. Weinberg, Phys. Rev. 133B (1964) 1318. [21] V. V. Dvoeglazov, Helv. Phys. Acta, 70 (1997) 677. [22] W. Greiner, Relativistic Quantum Mechanics. The 1st English Ed. (Springer, 1990).

Chapter 12 A new look at Freudenthal’s Magic Square Alberto Elduque Departamento de Matem´aticas, Universidad de Zaragoza, Spain

12.1 12.2 12.3 12.4 12.5 12.6

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Symmetric composition algebras and Freudenthal’s magic square . . . . . . . . . . . . . . . . . . . . . Okubo algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Invariant form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Real forms of the simple exceptional Lie algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

149 150 154 156 163 164 164

Abstract A recent construction of Freudenthal’s Magic Square by means of symmetric composition algebras will be reviewed. In dimension 8 there are two classes of such algebras: para-Hurwitz and Okubo, and the relationship between the exceptional simple Lie algebras obtained from these two classes will be given. A natural invariant bilinear form will be deﬁned and used to show that all the real central simple exceptional Lie algebras of type F and E do appear in the construction. Key words: Freudenthal Magic Square, symmetric composition algebra, triality, exceptional Lie algebra 2000 MSC: Primary 17B25; Secondary 17A75

12.1

Introduction

In 1966 Tits gave a uniﬁed construction of the exceptional simple Lie algebras that uses a couple of ingredients: a unital composition algebra C and a simple Jordan algebra J of degree 3 [Tit66], thus obtaining Freudenthal’s Magic Square (see [Sch95, Fre64]): C \J

H3 (F )

H3 (K)

H3 (Q)

H3 (C)

F

A1

A2

C3

F4

K

A2

A2 ⊕ A2

A5

E6

Q

C3

A5

D6

E7

C

F4

E6

E7

E8

Here, F denotes the ground ﬁeld, of characteristic = 2, 3, K is a quadratic ´etale algebra, Q a generalized quaternion algebra, and C a Cayley algebra.

149

150

Alberto Elduque

At least in the split cases, this is a construction that depends on two unital composition algebras, since the Jordan algebra involved consists of the 3 × 3-hermitian matrices over a unital composition algebra. Even though the construction is not symmetric in the two composition algebras that are being used, the outcome (the Magic Square) is indeed symmetric. Over the years, more symmetric constructions have been given, starting with a construction by Vinberg in 1966 [OV94]. Later on, a quite general construction was given by Allison and Faulkner [AF93] of Lie algebras out of structurable ones. In the particular case of the tensor product of two unital composition algebras, this construction provides another symmetric construction of Freudenthal’s Magic Square. Quite recently, Barton and Sudbery [BS00, BS03] (see also Landsberg and Manivel [LM02, LM04]) gave a simple recipe to obtain the Magic Square in terms of two unital composition algebras and their triality Lie algebras which, in perspective, is subsumed in Allison-Faulkner’s construction. However, as shown in [KMRT98], the triality phenomenon is better dealt with by means of the so-called symmetric composition algebras, instead of the classical unital composition algebras. This led the author [Eld04] to reinterpret Barton-Sudbery’s construction in terms of two symmetric composition algebras. With a few exceptions in dimension 2, any symmetric composition algebra of dimension 1, 2, or 4 is a para-Hurwitz algebra, while in dimension 8, besides the para-Hurwitz algebras, there appears a new family of symmetric composition algebras, the Okubo algebras. The construction in [Eld04] using para-Hurwitz algebras reduces naturally to the construction by Barton and Sudbery (although with slightly simpler formulas). Okubo algebras provide new constructions that highlight diﬀerent order 3 automorphisms and diﬀerent subalgebras of the exceptional simple Lie algebras. The purpose of this chapter is twofold. First, it will be shown how the Lie algebras built out of Okubo algebras are related to the ones deﬁned in terms of para-Hurwitz algebras, at least in the presence of an idempotent in the Okubo algebras. This will make strong use of the results in [EP96] and will be the subject of Section 12.3. Second, a natural invariant bilinear form will be considered for these Lie algebras in Section 12.4. This invariant form will allow us, in the real case, to check which exceptional simple real Lie algebras appear under the construction. Although this bilinear form and the real forms have been studied in [LM02, Section 2] and [BS03, Section 7], it is worth considering them again in terms of symmetric composition algebras. It will be shown in Section 12.5 that, suitably varying the symmetric composition algebras involved, plus some parameters, all the central simple exceptional real Lie algebras of type F and E appear in our construction. But before proceeding any further, a review of the symmetric composition algebras, their triality Lie algebras, and the corresponding construction of Freudenthal’s Magic Square is in order. This will be the subject of Section 12.2.

12.2

Symmetric Composition Algebras and Freudenthal’s Magic Square

A composition algebra is a triple (S, ∗, q), where (S, ∗) is a (nonassociative) algebra over a ﬁeld F with multiplication denoted by x ∗ y for x, y ∈ S, and where q : S → F is a regular quadratic form (the norm) satisfying for any x, y ∈ S: q(x ∗ y) = q(x)q(y). Usually, ∗ and q will be omitted.

(12.1)

A new look at Freudenthal’s magic square

151

In what follows, the ground ﬁeld F will always be assumed to be of characteristic = 2. Unital composition algebras (or Hurwitz algebras) form a well-known class of algebras. Any Hurwitz algebra has a ﬁnite dimension equal to either 1, 2, 4, or 8. The two-dimensional Hurwitz algebras are the quadratic ´etale algebras over the ground ﬁeld F , the four dimensional ones are the generalized quaternion algebras, and the eight dimensional Hurwitz algebras are called Cayley algebras, and are analog to the classical algebra of octonions (for a good survey of the latter, see [Bae02]). A composition algebra (S, ∗, q) is said to be symmetric if it satisﬁes q(x ∗ y, z) = q(x, y ∗ z),

(12.2)

where q(x, y) = q(x + y) − q(x) − q(y) is the polar of q. In what follows, any quadratic form and its polar will always be denoted by the same letter. Equation (12.2) is equivalent to (x ∗ y) ∗ x = x ∗ (y ∗ x) = q(x)y

(12.3)

for any x, y ∈ S. (See [KMRT98, Ch. VIII] for the basic facts and notations.) The classiﬁcation of the symmetric composition algebras was obtained in [EM93] for characteristic = 3 and in [Eld97] for characteristic 3. Given any Hurwitz algebra C with norm q, standard involution x → x ¯ = q(x, 1)1 − x, and multiplication denoted by juxtaposition, the new algebra deﬁned on C but with multiplication x∗y =x ¯y¯, is a symmetric composition algebra, called the associated para-Hurwitz algebra. In dimension 1, 2, or 4, any symmetric composition algebra is a para-Hurwitz algebra, with a few exceptions in dimension 2, which are, nevertheless, forms of para-Hurwitz algebras; while in dimension 8, apart from the para-Hurwitz algebras, there is a new family of symmetric composition algebras termed Okubo algebras. If (S, ∗, q) is any symmetric composition algebra, consider the corresponding orthogonal Lie algebra: o(S, q) = {d ∈ EndF (S) : q d(x), y + q x, d(y) = 0 ∀x, y ∈ S}. and the subalgebra of o(S, q)3 deﬁned by tri(S, ∗, q) = {(d0 , d1 , d2 ) ∈ o(S, q)3 : d0 (x ∗ y) = d1 (x) ∗ y + x ∗ d2 (y) ∀x, y ∈ S}.

(12.4)

The map θ : tri(S, ∗, q) → tri(S, ∗, q),

(d0 , d1 , d2 ) → (d2 , d0 , d1 ),

is an automorphism of (S, ∗, q) of order 3, the triality automorphism. Its ﬁxed subalgebra is (isomorphic to) the derivation algebra of (S, ∗) which, if the dimension is 8 and the characteristic of the ground ﬁeld is = 2, 3, is a simple Lie algebra of type G2 in the paraHurwitz case and a simple Lie algebra of type A2 (a form of sl3 ) in the Okubo case. For any x, y ∈ S, the triple 1 1 tx,y = σx,y , q(x, y)id − rx ly , q(x, y)id − lx ry (12.5) 2 2 is in tri(S, ∗, q), where σx,y (z) = q(x, z)y − q(y, z)x, rx (z) = z ∗ x, and lx (z) = x ∗ z for any x, y, z ∈ S. Let (S, ∗, q) and (S , ∗, q ) be two symmetric composition algebras and deﬁne g = g(S, S ) to be the Z2 ×Z2 -graded anticommutative algebra such that g(¯0,¯0) = tri(S, ∗, q)⊕tri(S , ∗, q ),

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g(¯1,¯0) = g(¯0,¯1) = g(¯1,¯1) = S ⊗S . (Unadorned tensor products are considered over the ground ﬁeld F .) For any a ∈ S and x ∈ S , denote by ιi (a⊗x) the element a⊗x in g(¯1,¯0) (respectively g(¯0,¯1) , g(¯1,¯1) ) if i = 0 (respectively, i = 1, 2). The anticommutative multiplication on g is deﬁned by means of: • g(¯0,¯0) is a Lie subalgebra of g. • [(d0 , d1 , d2 ), ιi (a ⊗ x)] = ιi di (a) ⊗ x , [(d0 , d1 , d2 ), ιi (a ⊗ x)] = ιi a ⊗ di (x) , for any (d0 , d1 , d2 ) ∈ tri(S, ∗, q), (d0 , d1 , d2 ) ∈ tri(S , ∗, q ), a ∈ S and x ∈ S . • [ιi (a ⊗ x), ιi+1 (b ⊗ y)] = ιi+2 (a ∗ b) ⊗ (x ∗ y) (indices modulo 3), for any a, b ∈ S, x, y ∈ S . • [ιi (a ⊗ x), ιi (b ⊗ y)] = q (x, y)θi (ta,b ) + q(a, b)θi (tx,y ), for any i = 0, 1, 2, a, b ∈ S and x, y ∈ S , where ta,b ∈ tri(S, ∗, q) (respectively tx,y ∈ tri(S , ∗, q )) is the element in (12.5) for a, b ∈ S (resp. x, y ∈ S ) and θ (resp. θ ) is the triality automorphism of tri(S, ∗, q) (resp. tri(S , ∗, q )). The main result in [Eld04] asserts that, with this multiplication, g(S, S ) is a Lie algebra and, if the characteristic of the ground ﬁeld is = 2, 3, Freudenthal’s Magic Square is recovered:

1

dim S

dim S 2 4

1

A1

A2

2

A2

4

C3

8

F4

8

C3

F4

A2 ⊕ A2

A5

E6

A5

D6

E7

E6

E7

E8

Let us ﬁnish this section with a useful conclusion: THEOREM 12.1 Let S1 , S2 , and S be symmetric composition algebras, such that S1 is a subalgebra of S2 . Then, in a natural way, g(S1 , S ) is isomorphic to a subalgebra of g(S2 , S ). PROOF First note that (12.3) implies that the norm of S1 is the restriction of the norm of S2 . Denote the latter one by q. ˜ of g(S2 , S ) generated by ⊕2i=0 ιi (S1 ⊗ S ) (considered as a Consider the subalgebra g 2 subspace of ⊕i=0 ιi (S2 ⊗ S ). Thus 2

i ˜= θ (tS1 ,S1 ) ⊕ tri(S , ∗, q ) ⊕ ⊕2i=0 ιi (S1 ⊗ S ) . g i=0

Let tri(S2 /S1 , ∗, q) = {(d0 , d1 , d2 ) ∈ tri(S2 , ∗, q) : di (S1 ) ⊆ S1 , i = 0, 1, 2}, which is a subalgebra of tri(S2 , ∗, q), and consider the “restriction” map: ρ : tri(S2 /S1 , ∗, q) −→ tri(S1 , ∗, q) (d0 , d1 , d2 ) → d0 |S1 , d1 |S1 , d2 |S1 .

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Note that ρ θi (tS1 ,S1 ) = θi (tS1 ,S1 ), where there is a slight abuse of notation: on the left ta,b ∈ tri(S2 , ∗, q), while on the right ta,b ∈ tri(S1 , ∗, q) for any a, b ∈ S1 . 2 The map ρ is surjective, since tri(S1 , ∗, q) = i=0 θi (tS1 ,S1 ) (see [Eld04, Section 3]). To 2 prove the theorem it is enough to prove that the restriction of ρ to i=0 θi (tS1 ,S1 ) is an isomorphism. This is trivial if dim S1 = 1, since then tS1 ,S1 = tri(S1 , ∗, q) = 0. On the other hand we can assume that the ground ﬁeld F is algebraically closed. Assume then that dim S1 = 2. Then S1 is the para-Hurwitz algebra of the unique twodimensional Hurwitz algebra K = F × F , so S1 = F e1 ⊕ F e2 , with e1 ∗ e1 = e2 ,

e2 ∗ e2 = e1 ,

and

e1 ∗ e2 = e2 ∗ e1 = 0.

Also, q(e1 ) = 0 = q(e2 ) and q(e1 , e2 ) = 1. Then e = e1 + e2 is an idempotent of S1 , and hence of S2 , too. Therefore, C = S2 is a Hurwitz algebra with the same norm q, but with the new multiplication given by ab = (e ∗ a) ∗ (b ∗ e). Moreover, the unity of C is e and the map τ : a → q(a, e)e − a ∗ e is an automorphism of a)τ 2 (¯b) for any a, b ∈ C = S2 , where C (see [EP96, (2.7]) such that τ 3 = id and a ∗ b = τ (¯ a ¯ = q(a, e)e − a for any a ∈ C. This also implies that τ is an automorphism, too, of the symmetric composition algebra S2 . Now, tS1 ,S1 = F te1 ,e2 and it is enough to prove that te1 ,e2 + θ(te1 ,e2 ) + θ2 (te1 ,e2 ) = 0 2 in tri(S2 , ∗, q). (In this way, dim i=0 θi (tS1 ,S1 ) ≤ 2 = dim tri(S1 , ∗, q) and, since ρ is surjective, it must be one-to-one too.) By its own deﬁnition (12.5) te1 ,e2 + θ(te1 ,e2 ) + θ2 (te1 ,e2 ) = (d, d, d) with d = id + σe1 ,e2 − re1 le2 + le1 re2 = σe1 ,e2 + [re2 , le1 ] since re1 le2 + re2 le1 = q(e1 , e2 ) id = id = σe1 ,e2 + [re , le1 ] since [re1 , le1 ] = 0 by (12.3),

by (12.3),

and this is 0 on S1 , while on S1⊥ = {a ∈ S2 : q(a, S1 ) = 0} it acts as [re2 , le1 ] = −[τ, le1 ], which is 0, because τ ∈ Aut(S2 , ∗) and τ (e1 ) = e1 . Finally, let us assume that dim S1 = 4 and dim S2 = 8. In this case, the decomposition 2 S2 = S1 ⊕ S1⊥ is a Z2 -grading. It is enough to prove that i=0 θi (tS1 ,S1 ) is a subspace of tri(S2 , ∗, q) of dimension 9. Since dim S2 = 8, tri(S2 , ∗, q) = tS2 ,S2 , which is isomorphic to the orthogonal Lie algebra o(S2 , q) by means of the projection π0 : tri(S2 , ∗, q) → o(2 , q), (d0 , d1 , d2 ) → d0 [Eld04, Lemma 2.1]. Also, because of the Z2 -grading above, tri(S2 /S1 , ∗, q) = tS1 ,S1 ⊕ tS1⊥ ,S1⊥ (a direct sum of ideals by [Eld04, Lemma 2.1]), and π0 restricts to isomorphisms tS1 ,S1 ∼ = o(S1 , q) and tS1⊥ ,S1⊥ ∼ = o(S1⊥ , q). Thus π0 restricts to an isomorphism tri(S2 /S1 , ∗, q) → o(S1 , q) ⊕ o(S1⊥ , q), which is a direct sum of four simple three-dimensional ideals. By symmetry, the same happens with the other projections π1 and π2 , so θi (tS1 ,S1 ) ∼ = o(S1 , q) and θi (tS1⊥ ,S1⊥ ) ∼ = o(S1⊥ , q). But S1 is a para-Hurwitz algebra. Let e be its para-unit (the unity of the associated Hurwitz algebra, which spans the commutative center). This means that there is a multiplication ab on S1 with unity e such that S1 is a Hurwitz algebra for the new multiplication with the same norm, and such that a ∗ b = a ¯¯b for any a, b, where a ¯ = q(a, e)e − a. Since dim S1 = 4, the Hurwitz algebra is a generalized quaternion algebra, so it is associative. Now, for any a, b ∈ S1 , 1 π0 θ(ta,b ) |S1 = q(a, b)id − ra lb |S1 , 2

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c¯b) = a∗(bc) = a ¯(bc) = (¯ ab)c and q(a, b) = a ¯b+¯ba. but for any c ∈ S1, la rb c = a∗(c∗b) = a∗(¯ 1 Hence π0 ρ θ(ta,b ) = 2 L¯ba−¯ ab , whereLc denotes the left multiplication by c in the Hurwitz algebra. In particular, π0 ρ θ(tS1 ,S1 ) = L(F e)⊥ and, in the same vein, π0 ρ θ2 (tS1 ,S1 ) = R(F e)⊥ (right multiplications in the Hurwitz algebra) and π0 ρ(tS1 ,S1 ) = σS1 ,S1 = o(S1 , q), which is known to be equal to L(F e)⊥ ⊕R(F e)⊥ (a direct sum of two simple three-dimensional ideals). By dimension count, π0 ρ|tS1 ,S1 is an isomorphism. Moreover, π0 ρ|tS⊥ ,S⊥ = 0. 1 1 Hence,

tS1 ,S1 = tS1 ,S1 ∩ θ(tS1 ,S1 ) ⊕ tS1 ,S1 ∩ θ2 (tS1 ,S1 )

(a direct sum of two simple three-dimensional ideals) and, by symmetry,

θi (tS1 ,S1 ) = θi (tS1 ,S1 ) ∩ θi+1 (tS1 ,S1 ) ⊕ θi (tS1 ,S1 ) ∩ θi+2 (tS1 ,S1 ) (indices modulo 3), and 2

θi (tS1 ,S1 ) = ⊕2i=0 θi (tS1 ,S1 ) ∩ θi+1 (tS1 ,S1 )

i=0

has dimension 9, as required.

12.3

Okubo Algebras

Symmetric composition algebras are interesting mainly because of the existence of Okubo algebras. Otherwise, with a few exceptions in dimension 2 over some ﬁelds, the theory will reduce to that of the classical Hurwitz algebras, since any para-Hurwitz algebra is completely determined by its Hurwitz counterpart. It is shown in [Eld04], that the construction g(S, S ) for para-Hurwitz algebras S and S coincides, in natural ways, with the constructions by Barton and Sudbery, Landsberg and Manivel, and Allison and Faulkner. Therefore our construction, besides its simplicity, due to the simpler formulas for triality that appear with the use of symmetric composition algebras, has the extra interest of the constructions obtained by means of the Okubo algebras. In particular, over an algebraically closed ﬁeld, we have a priori three diﬀerent constructions of the simple split Lie algebra E8 , namely g(pH8 , pH8 ), g(pH8 , Ok) and g(Ok, Ok), where pH8 denotes the unique eight-dimensional para-Hurwitz algebra and Ok the unique Okubo algebra over such a ﬁeld. Each one of these three constructions highlights a diﬀerent order 3 automorphism of E8 [Eld04, Section 4]. The aim of this section is to show the relationship between the Lie algebras built out from para-Hurwitz algebras and those constructed from Okubo algebras. Let (S, ∗, q) be a symmetric composition algebra endowed with an automorphism ϕ ∈ Aut S of order 3 (ϕ3 = id). Then (12.3) shows that ϕ is an isometry of the norm q, and also that if we deﬁne a new multiplication on S by means of x y = ϕ(x) ∗ ϕ2 (y)

(12.6)

for any x, y ∈ S, then (S, , q) is again a symmetric composition algebra. Let us write S to refer to this new algebra.

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155

THEOREM 12.2 Let (S, ∗, q) and (S , ∗, q ) be two symmetric composition algebras, and let ϕ be an order 3 automorphism of (S, ∗, q). Then the linear map Φ : g(S, S ) → g(S , S ) determined by • Φ (d0 , d1 , d2 ) = (d0 , ϕ2 d1 ϕ, ϕd2 ϕ2 ) for any (d0 , d1 , d2 ) ∈ tri(S, ∗, q), • Φ (d0 , d1 , d2 ) = (d0 , d1 , d2 ) for any (d0 , d1 , d2 ) ∈ tri(S , ∗, q ), • Φ ιi (a ⊗ x) = ιi (ϕ2i (a) ⊗ x), for any i = 0, 1, 2, a ∈ S and x ∈ S , is a Lie algebra isomorphism. PROOF First of all, note that Φ is well deﬁned, since for any (d0 , d1 , d2 ) ∈ tri(S, ∗, q), Φ (d0 , d1 , d2 ) = (d0 , ϕ2 d1 ϕ, ϕd2 ϕ2 ) ∈ tri(S , , q) because, for any a, b ∈ S, d0 (a b) = d0 ϕ(a) ∗ ϕ2 (b) = d1 ϕ(a) ∗ ϕ2 (b) + ϕ(a) ∗ d2 ϕ2 (b) = ϕ(ϕ2 d1 ϕ)(a) ∗ ϕ2 (b) + ϕ(a) ∗ ϕ2 (ϕd2 ϕ2 )(b) = (ϕ2 d1 ϕ)(a) b + a (ϕd2 ϕ2 )(b). Let us denote by la and ra the left and right multiplications by the element a in S , and for any a, b ∈ S, let ta,b be the element of tri(S , , q) deﬁned by (12.5); that is, ta,b = σa,b , 12 q(a, b)id − ra lb , 12 q(a, b)id − la rb . For any a, b ∈ S, a b = ϕ(a) ∗ ϕ2 (b) = ϕ a ∗ ϕ(b) = ϕ2 ϕ2 (a) ∗ b , so that la = ϕla ϕ, rb = ϕ2 rb ϕ2 , ra lb = ϕ2 ra lb ϕ, la rb = ϕla rb ϕ2 . Therefore, for any a, b ∈ S, Φ(ta,b ) = ta,b . 2i i Moreover, ϕ is clearly an automorphism, too, of S , so ϕ2i la ϕi = lϕ = 2i (a) and ϕ ra ϕ 2i i rϕ2i (a) for any a, b ∈ S and i = 0, 1, 2. Since ϕ is an isometry, also ϕ σa,b ϕ = σϕ2i (a),ϕ2i (b) for any a, b ∈ S. As a consequence,

Φ(θi (ta,b ) = θi (tϕ2i (a),ϕ2i (b) ) for any a, b ∈ S and i = 0, 1, 2. Now, using the deﬁnition of the multiplication in the Lie algebras g(S, S ) and g(S , S ), it is easy to check that Φ is an isomorphism. Let (S, , q) be any symmetric composition algebra over F . Then either it contains a nonzero idempotent or there exists a cubic ﬁeld extension L/F such that L ⊗ S does contain a nonzero idempotent ([KMRT98, (34.10)]). Assuming that e is a nonzero idempotent of S, S becomes a Hurwitz algebra with the same norm and the new multiplication given by ab = (e a) (b e), for any a, b, the map ϕ : S → S, a → e (e a) is an automorphism for both multiplications on S, with ϕ3 = id and for any a, b ∈ S, the original multiplication ¯ = q(e, a)e − a and is given by a b = ϕ(a) ϕ2 (b) = ϕ(a) ∗ ϕ2 (b) for any a, b ∈ S, where a

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a∗b = a ¯¯b (see [KMRT98, (34.9)]). Therefore, (S, ∗, q) is a para-Hurwitz algebra and (S, , q) is obtained from it by means of (12.6). Then, as a simple consequence of our Theorem, one obtains the following. COROLLARY 12.1 Let S be an Okubo algebra with a nonzero idempotent and let S be any symmetric composition algebra. Then there is a para-Hurwitz algebra Sˆ such that ˆ S ). g(S, S ) is isomorphic to g(S,

In particular, over ﬁelds with no cubic ﬁeld extensions (for instance, over algebraically closed ﬁelds or the real ﬁeld), the previous corollary shows how the Lie algebras g(S, S ), with S or S an Okubo algebra, are related to those Lie algebras constructed from paraHurwitz algebras. Nevertheless, as shown in [Eld04], it is nice to deal with Okubo algebras, too, in order to highlight some interesting automorphisms and subalgebras of the exceptional simple Lie algebras.

12.4

Invariant Form

The purpose of this section is to consider a natural invariant form on the Lie algebras g(S, S ), whose restriction to the three copies of S ⊗ S is the natural bilinear form induced by the norms on S and S : (q⊗q )(a⊗x, b⊗y) = q(a, b)q (x, y) for any a, b ∈ S and x, y ∈ S . This has also been dealt with in [LM02, Lemma 2.2], in terms of Hurwitz algebras. THEOREM 12.3 Let (S, ∗, q) be a symmetric composition algebra of dimension ≥ 2 over a ﬁeld F of characteristic = 2, 3. Then there exists a unique quadratic form Q : tri(S, ∗, q) → F such that

Q (d0 , d1 , d2 ), θi (tx,y ) = q di (x), y (12.7) for any (d0 , d1 , d2 ) ∈ tri(S, ∗, q) and any x, y ∈ S. PROOF The uniqueness follows from (12.7) since tri(S, ∗, q) = tS,S + θ(tS,S ) + θ2 (tS,S ) ([Eld04, Section 3]). This also shows that for any such Q, θ is an isometry of Q. Let us ﬁrst prove the existence in the case dim S = 8. In this case tri(S, ∗, q) = tS,S and π0 : tS,S → o(S, q), tx,y → σx,y is an isomorphism (principle of local triality, see [Eld04, Lemma 2.1]). Deﬁne the quadratic form Q by

1 Q (d0 , d1 , d2 ) = − trace(d20 ). 4

(12.8)

Note that σa,b σx,y =q(a, y)q(x, −)b − q(b, y)q(x, −)a − q(a, x)q(y, −)b + q(b, x)q(y, −)a so

% % %q(a, x) q(a, y)% %. trace(σa,b σx,y ) = −2 %% q(b, x) q(b, y) %

(12.9)

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Since o(S, q) is simple, the bilinear form (f, g) → trace(f g) is, up to scalars, the unique ˆ x,y ) = 1 q(x, y)id− invariant bilinear form on o(S, q). Let θˆ be the triality automorphism: θ(σ 2 ˆ )θ(g) ˆ rx ly for any x, y ∈ S, and consider the bilinear form f, g = trace θ(f . Then there is a nonzero scalar α ∈ f such that f, g = α trace(f g) for any f, g ∈ o(S, q). But for any

2 1 1 a, b ∈ S, σa,b , σa,b = trace . Now, trace 2 q(a, b)id − ra lb = 0, as 2 q(a, b)id − ra lb this endomorphism belongs to o(S, q), so trace(ra lb ) = 4q(a, b). Thus, using (12.3),

1 2 q(a, b)id − ra lb trace 2 1 = trace ra lb ra lb − q(a, b)ra lb 2 1 = trace ra (lb ra + la rb )lb − ra la rb lb − q(a, b)ra lb 2 1 = trace q(a, b)ra lb − q(a)q(b)id 2 = 2q(a, b)2 − 8q(a)q(b)

= −2 q(a, a)q(b, b) − q(a, b)2

2 . = trace σa,b Hence α = 1 and θˆ is an isometry of the quadratic form f → trace(f 2 ). (This argument is valid in characteristic 3, too. Actually, by using that θˆ is an automorphism of the Lie algebra o(S, q), and hence an isometry of its Killing form, and that this is 6 times the trace form (see Remark 12.2), a trivial proof comes out immediately for characteristic = 2, 3.) Therefore, it is enough to check (12.7) for i = 0 and (d0 , d1 , d0 ) = ta,b for a, b ∈ S. However, in this case, % % %q(a, x) q(a, y)% 1 % % Q(ta,b , tx,y ) = − trace σa,b σx,y = % q(b, x) q(b, y) % 2 % % %q(a, x) q(a, y)% %, % q σa,b (x), y = q q(a, x)b − q(b, x)a, y = % q(b, x) q(b, y) % as required. Now, if the dimension of S is 2, let {u, v} be a basis of S. Then by [Eld04, Corollary 3.4], with d = σu,v , tri(S, ∗, q) = {(α0 d, α1 d, α2 d) : αi ∈ F (i = 0, 1, 2), α0 + α1 + α2 = 0}. Let us consider the quadratic form Q deﬁned by % % %q(u, u) q(u, v)% 1 2 %. (12.10) Q (α0 d, α1 d, α2 d) = α0 + α12 + α22 %% q(u, v) q(v, v)% 3 % % %q(u, u) q(u, v)% %. To prove the validity of (12.7), we must check that % For simplicity, let Δ = % q(u, v) q(v, v)% Q (α0 d, α1 d, α2 d), θi (tu,v ) = αi q d(u), v for any scalars α0 , α1 , α2 whose sum is 0. But

1 1 1 1 tu,v = σu,v , q(u, v)id − ru lv , q(u, v)id − lu rv = d, − d, − d , 2 2 2 2

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since (S, ∗) is commutative, so the second and third components of tu,v coincide. Thus, 2 Q (α0 d, α1 d, α2 d), θi (tu,v ) = 3 while

αi −

αi+1 + αi+2 2

Δ = αi Δ ,

q αi d(u), v = αi q σu,v (u), v = αi Δ .

Finally, assume that dim S = 4. Then S is a para-Hurwitz algebra, so if e is its para-unit and xy = (e∗x)∗(y ∗e), then S is a Hurwitz algebra with this new multiplication, with unity 1 = e and the same norm q. Let us denote by Lx and Rx the left and right multiplications by x in this Hurwitz algebra. Here S can be embedded in an eight-dimensional para-Hurwitz ˜ ∗, q) and, as in Theorem 12.1, we may identify tri(S, ∗, q) with tS,S + θ(tS,S ) + algebra ( S, ˜ ∗, q) . In this way, the existence of Q is guaranteed as the restriction of θ2 (tS,S ) ≤ tri(S, ˜ ∗, q). the corresponding quadratic form on tri(S, REMARK 12.1 As shown in the proof above, the restriction on the characteristic to be = 3 is only necessary for the case of two-dimensional S. See also the last paragraph in [Eld04, Section 3] COROLLARY 12.2 Let (S, ∗, q) be a symmetric composition algebra of dimension ≥ 2 over a ﬁeld F of characteristic = 2, 3 and let Q be the unique quadratic form on tri(S, ∗, q) satisfying (12.7). Then, for any (d0 , d1 , d2 ) ∈ tri(S, ∗, q): • If dim S = 8, Q (d0 , d1 , d2 ) = − 41 trace(d2i ), for any i = 0, 1, 2. 2 • If dim S = 4, Q (d0 , d1 , d2 ) = − 81 i=0 trace(d2i ). 2 • If dim S = 2, Q (d0 , d1 , d2 ) = − 61 i=0 trace(d2i ). In particular, Q is an invariant quadratic form. PROOF In case dim S = 8, the assertion already appears in (12.8). If dim S = 2, by (12.10) above, with d = σu,v , 1 Q (α0 d, α1 d, α2 d) = (α02 + α12 + α22 )Δ , 3 2 = −2Δ, so but trace(d2 ) = trace σu,v 2 1 1 Q (α0 d, α1 d, α2 d) = − (α02 + α12 + α22 ) trace(d2 ) = − trace (αi d)2 . 6 6 i=0

Finally, if dim S = 4, Q is given by the restriction to tS,S +θ(tS,S )+θ2 (tS,S ) of an invariant ˜ ∗, q), where S˜ is an eight-dimensional para-Hurwitz algebra, so Q quadratic form on tri(S, is invariant. Moreover, by [Eld04, Corollary 3.4], tri(S, ∗, q) = ⊕2i=0 ker πi , a direct sum of three simple three-dimensional ideals so, by invariance, Q(ker πi , ker πj ) = 0 for any

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i = j. Consider the Hurwitz algebra deﬁned on S in the proof of 12.3. Then, for any ¯ = −u, v¯ = −v), u, v ∈ [S, S] = (F 1)⊥ (so u σu,v (x) = q(u, x)v − q(v, x)u = (u¯ x + x¯ u)v − u(¯ xv + v¯x) = −u¯ v x + x¯ uv 1 = [uv, x] = [[u, v], x] as uv + vu ∈ F 1, 2 while

Therefore, tu,v

1 1 q(u, v)id − ru lv (x) = −x(uv + vu) − (v ∗ x) ∗ u 2 2 1 = − x(uv + vu) − xv ∗ u 2 1 u = − x(uv + vu) − (xv)¯ 2 1 = − R[u,v] (x) . 2 1 1 q(u, v)id − lu rv (x) = L[u,v] (x) . 2 2 = 12 ad[u,v] , −R[u,v] , L[u,v] , and tu,v − θ(tu,v ) − θ2 (tu,v ) = (0, −L[u,v] , R[u,v] ) ∈ ker π0 .

Now, if τ denotes the standard involution of the Hurwitz algebra (τ (x) = x ¯ = q(x, 1)1 − x for any x), then ker π0 = {(0, la τ, −ra τ ) : a ∈ [S, S]} (see [Eld04, Corollary 3.4]), and for any a, u, v ∈ [S, S]:

Q (0, la τ, −ra τ ), (0, −L[u,v] , R[u,v] ) = Q (0, −La , Ra ), (tu,v − θ(tu,v ) − θ2 (tu,v ) = q(0, v) + q(au, v) − q(ua, v) = q(a, v¯ u−u ¯v) = q(a, [u, v]) . Therefore,

Q (0, la τ, −ra τ ), (0, lb τ, −rb τ ) = q(a, b)

(12.11)

for any a, b ∈ [S, S]. Besides, trace(la τ lb τ ) = trace(La Lb ) = trace Lab = 2q(ab, 1) = −2q(a, b) and also trace(ra τ rb τ ) = −2q(a, b) for any a, b ∈ [S, S]. Hence, in ker π0 , (12.11) gives 2 1 Q (d0 , d1 , d2 ) = − trace (di )2 , 8 i=0 and, by symmetry, the same happens in ker πi , i = 1, 2. Now, we are ready to deﬁne a natural invariant bilinear form on g(S, S ): COROLLARY 12.3 Let (S, ∗, q) and (S , ∗, q ) be two symmetric composition algebras over a ﬁeld F of characteristic = 2, 3, and let Q and Q be the forms deﬁned on tri(S, ∗, q) and tri(S , ∗, q ) by means of (12.7). Then on g(S, S ) = tri(S, ∗, q) ⊕ tri(S , ∗, q ) ⊕ ι0 (S ⊗ S ) ⊕ ι1 (S ⊗ S ) ⊕ ι2 (S ⊗ S ),

(12.12)

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consider the bilinear form B = BS,S determined by the following conditions: • The direct summands in (12.12) are orthogonal relative to B, • The restriction of B to tri(S, ∗, q) (respectively, tri(S , ∗, q )) is Q (respectively Q ). • B ιi (a ⊗ x), ιi (b ⊗ y) = q(a, b)q (x, y), for any i = 0, 1, 2, a, b ∈ S and x, y ∈ S . Then B is invariant. PROOF It is enough to take into account that: 1. For any a, b, c ∈ S, x, y, z ∈ S and i = 0, 1, 2 (indices modulo 3), due to (12.2):

B [ιi (a ⊗ x), ιi+1 (b ⊗ y)], ιi+2 (c ⊗ z) = q(a ∗ b, c)q (x ∗ y, z) = q(a, b ∗ c)q (x, y ∗ z) = B (ιi (a ⊗ x), [ιi+1 (b ⊗ y), ιi+2 (c ⊗ z)]) . 2.

For any (d0 , d1 , d2 ) ∈ tri(S, ∗, q), a, b ∈ S, x, y ∈ S , and i = 0, 1, 2,

B [(d0 , d1 , d2 ), ιi (a ⊗ x)], ιi (b ⊗ y) + B ιi (a ⊗ x), [(d0 , d1 , d2 ), ιi (b ⊗ y)] = q di (a), b q (x, y) + q a, di (b) q (x, y)

= q di (a), b + q a, di (b) q (x, y) = 0 ,

because di ∈ o(S, q), and similarly for elements in tri(S , ∗, q ). 3. For any (d0 , d1 , d2 ) ∈ tri(S, ∗, q), a, b ∈ S, x, y ∈ S and i = 0, 1, 2,

B [(d0 , d1 , d2 ), ιi (a ⊗ x)], ιi (b ⊗ y) = q di (a), b)q (x, y) = Q (d0 , d1 , d2 ), θi (ta,b ) q (x, y) = B (d0 , d1 , d2 ), [ιi (a ⊗ x), ιi (b ⊗ y)] stemming from (12.7), and similarly for elements in tri(S , ∗, q ).

The relationship between the Killing form and the invariant form deﬁned in the previous corollary depends on the dimension of S and S . PROPOSITION 12.1 Let (S, ∗, q) and (S , ∗, q ) be two symmetric composition algebras over a ﬁeld F of characteristic = 2, 3, Let B = BS,S be the invariant bilinear form deﬁned on g(S, S ) in Corollary 12.3, and let κ = κS,S be the Killing form of g(S, S ). Then κ=−

1

dim S dim S + 4(dim S + dim S ) − 8 B . 2

(12.13)

PROOF Since κ is invariant, all thedirect summands in the Z2 × Z2 -graduation of 2 g(S, S ) are orthogonal. Moreover, since i=0 ιi (S ⊗ S ) generates g(S, S ), it is enough to check (12.13) for ιi (S ⊗ S ) and, by symmetry, the case i = 0 suﬃces.

A new look at Freudenthal’s magic square

161

From [Eld04, Lemma 3.3], we know that tri(S, ∗, q) = tS,S ⊕ ker π0 and for any a, b ∈ S and x, y ∈ S , adι0 (a⊗x) adι0 (b⊗y) ker π0 = 0, while for any u, v ∈ S: adι0 (a⊗x) adι0 (b⊗y) (tu,v ) = − adι0 (a⊗x) ι0 (σu,v (b) ⊗ y) = −q (x, y)ta,σu,v (b) − q a, σu,v (b) tx,y . Identifying tS,S with the second exterior power 42 42 linear map S→ S given by

42

S, we must consider the trace of the

u ∧ v → a ∧ σu,v (b) = q(u, b)a ∧ v + q(v, b)u ∧ a . The trace of this map is (dim S − 1)q(a, b) (just extend scalars and work with a suit able orthogonal basis). Thus, the contribution to κ ι0 (a ⊗ x), ι0 (b ⊗ y) of tri(S, ∗, q) is −q(a, b)q (x, y)(dim S − 1) and, in the same vein, the contribution of tri(S , ∗, q ) is −q(a, b)q (x, y)(dim S − 1). Now, for any c ∈ S and z ∈ S : adι0 (a⊗x) adι0 (b⊗y) ι0 (c ⊗ z) = ι0 (a ⊗ x), q (y, z)ty,z + q(b, c)ty,z = − (q (y, z)σb,c (a) ⊗ x + q(b, c)a ⊗ σy,z (x)) , so, identifying ι0 (S ⊗ S ) with S ⊗ S , the restriction of adι0 (a⊗x) adι0 (b⊗y) to ι0 (S ⊗ S ) is the linear map − q(a, b)id ⊗ q (y, −)x + q(a, −)b ⊗ q (y, −)x − q(b, −)a ⊗ q (x, y)id + q(b, −)a ⊗ q (x, −)y , whose trace is −q(a, b)q (x, y) dim S + q(a, b)q (x, y) − q(a, b)q (x, y) dim S + q(a, b)q (x, y) = −q(a, b)q (x, y) dim S + dim S − 2 . Finally, the action of adι0 (a⊗x) adι0 (b⊗y) on ι1 (S ⊗ S ) is given by adι0 (a⊗x) adι0 (b⊗y) ι1 (c ⊗ z) = −ι0 (b ∗ c) ∗ a ⊗ (y ∗ z) ∗ x . But 12 q(a, b)id − ra lb has zero trace, since it belongs to o(S,q), so trace(ra lb ) = and the contribution of ι1 (S ⊗ S ) to κ ι0 (a ⊗ x), ι0 (b ⊗ y) is

dim S 2 q(a, b),

dim S dim S q(a, b)q (x, y) , 4 and exactly the same contribution is provided by ι2 (S ⊗ S ). Therefore, κ ι0 (a ⊗ x), ι0 (b ⊗ y) = trace adι0 (a⊗x) adι0 (b⊗y)

= −q(a, b)q (x, y) (dim S − 1) + (dim S − 1) dim S dim S + dim S + dim S + 2 + 2 q(a, b)q (x, y) =− (dim S dim S + 4(dim S + dim S ) − 8) , 2 as required.

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REMARK 12.2 The result above is valid in characteristic 3 provided dim S = 2 = dim S . Even for dim S = 2 or dim S = 2, the proof above shows that the restriction of the Killing form to ιi (S ⊗ S ) is (with obvious notation): −

1

dim S dim S + 4(dim S + dim S ) − 8 ιi (q ⊗ q ) . 2

The restriction of the Killing form to tri(S, ∗, q) is easily computed in any characteristic = 2, since for any (d0 , d1 , d2 ), (e0 , e1 , e2 ) ∈ tri(S, ∗, q), a ∈ S and x ∈ S : ad(d0 ,d1 ,d2 ) ad(e0 ,e1 ,e2 ) |tri(S ,∗,q ) = 0 ad(d0 ,d1 ,d2 ) ad(e0 ,e1 ,e2 ) ιi (a ⊗ x) = ιi (di ei (a) ⊗ x , so the contribution to κ (d0 , d1 , d2 ), (e0 , e1 , e2 ) of the sum tri(S , ∗, q) ⊕ ⊕2i=0 ιi (S ⊗ S ) 2 is (dim S ) i=0 trace(di ei ). Now, if dim S = 2, tri(S, ∗, q) is abelian and there is no further contribution. If dim S = 8, the relationship between the Killing form and the natural trace form on o(S, q) is given by trace adσa,b adσu,v = 6 trace σa,b σu,v for any a, b, u, v ∈ S (just extend scalars and work with a suitable orthogonal basis), so κ (d0 , d1 , d2 ), (e0 , e1 , e2 ) = 2 (dim S + 2) i=0 trace(di ei ), since trace(d0 e0 ) = trace(d1 e1 ) = trace(d2 e2 ). Note that 2 −6Q (d0 , d1 , d2 ), (e0 , e1 , e2 ) by Corollary 12.2, and that − 12 8 dim S + i=0 trace(di ei ) = 4(8 + dim S ) − 8 = −6(dim S + 2), so all this agrees with Proposition 12.1. Finally, if dim S = 4, for any a ∈ [S, S], ad2(0,la τ,−ra τ ) = ad2(0,−La ,Ra ) (with the notation in 12.2). But [La , Lb ] = L[a,b] and [Ra , Rb ] = −R[a,b] , hence

the proof of Corollary 2 trace ad(0,la τ,−ra τ ) |ker π0 = trace ad2a |[S,S] , and for any b ∈ [S, S], [a, [a, b]] = a2 b + ba2 − 2aba = 2(a2 b + ba2 ) − a(ab + ba) − (ab + ba)a = 2q(a, b)a − 4q(a)b = 2 q(a, b)a − q(a, a)b , so trace ad2a |[S,S] = 2q(a, a) − 6q(a, a) = −4q(a, a). By (12.11), Corollary 12.2, and since tri(S, ∗, q) is the orthogonal sum of the ideals ker πi (i = 0, 1, 2), it follows that 2

trace ad2(d0 ,d1 ,d2 ) |tri(S,∗,q) = trace(di ei ) i=0

for any (d0 , d1 , d2 ) ∈ tri(S, ∗, q), so 2 κ (d0 , d1 , d2 ), (e0 , e1 , e2 ) = (dim S + 1) trace(di ei ) . i=0

Note that in this case, 2

trace(di ei ) = −4Q ((d0 , d1 , d2 ), (e0 , e1 , e2 ))

i=0

and − 12 4 dim S +4(4+dim S )−8 = −4(dim S +1), so again this ﬁts well into Proposition 12.1.

A new look at Freudenthal’s magic square

12.5

163

Real Forms of the Simple Exceptional Lie Algebras

Given a ﬁnite dimensional composition algebra C with norm q and multiplication denoted 1 a2 satisﬁes by juxtaposition, and an arbitrary element a with q(a) = 0, the element e = q(a) −1 −1 q(e) = 1, and the new multiplication deﬁned on C by means of x · y = Re (x) Le (y) is a Hurwitz algebra with the same norm q and unity 1 = e2 . Hence q is the norm of a Hurwitz algebra (a Pﬁster form). Over R and for each dimension 2, 4, or 8, there are only two possibilities for q: either it has a maximal Witt index (the split case) or it is positive deﬁnite. It is well known that the real forms of the simple exceptional Lie algebras are determined by the signature of their Killing forms (see, for instance, [Hel78]), the compact ones having negative deﬁnite Killing forms. For a real quadratic form q, let δq be its signature, deﬁned as the diﬀerence between the number of positive and negative entries in any diagonal coordinate matrix of q. PROPOSITION 12.2 algebras. Then:

Let (S, ∗, q) and (S , ∗, q ) be two real symmetric composition

(i) If q and q are positive deﬁnite, then g(S, S ) is compact. (ii) If q is split, then the restriction of the Killing form of g(S, S ) to ⊕2i=0 ιi (S ⊗ S ) has maximal Witt index, and the signature of its restriction to tri(S, ∗, q) is 2, 3, or 4, according to dim S being 2, 4, or 8. PROOF For (i) it is enough to check that the invariant form B = BS,S deﬁned in Corollary 12.3 is positive deﬁnite. But this is clear for dim S = dim S = 8 (Corollaries 12.2 and 12.3), and for the other cases it is enough to use Theorem 12.1, because by Theorem 12.3, if S1 is a subalgebra of S2 , then BS1 ,S is the restriction of BS2 ,S . Moreover, over R any symmetric composition algebra of dimension 2 or 4 is para-Hurwitz (see [?, Theorem 4.3]) and hence can be embedded in a para-Hurwitz algebra of dimension 8. For (ii) note ﬁrst that if q has a maximal Witt index, so does q ⊗ q on S ⊗ S , thus only the last part needs to be veriﬁed. If dim S = 2, Δ < 0 in (12.10), and hence Q is negative deﬁnite, and the restriction of the Killing form (which is a negative scalar multiple of B by Proposition 12.1) is positive deﬁnite. If dim S = 4, by (12.11), Q has signature −1 on the three-dimensional ideal ker πi (i = 0, 1, 2), so the Killing form has signature 1 on each ker πi and hence 3 on tri(S, ∗, q). Finally, if dim S = 8, the result follows from (12.8), which implies that the signature of Q is −4. COROLLARY 12.4 Let (S, ∗, q) and (S , ∗, q ) be two real symmetric composition algebras with dim S = 8, then the Lie algebra g(S, S ) is, up to isomorphism, the Lie algebra given by the following tables (notation as in [Hel78]): dim S:

1

2

4

8

q deﬁnite q deﬁnite

q split q deﬁnite

f4,−52

e6,−26 e6,−78

e7,−25 e7,−133

e8,−24 e8,−248

q split q split

q split q deﬁnite

f4,4

e6,6 e6,2

e7,7 e7,−5

e8,8 e8,−24

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Alberto Elduque

Therefore, the only missing real forms of the exceptional simple Lie algebras f4 , e6 , e7 and e8 are f4,−20 and e6,−14 . (See also [BS03, Section 7], but note that it is incorrectly asserted there that the missing real forms are f4,−20 and e6,2 , which contradicts the second table in [BS03, p. 607], where this latter Lie algebra appears.) However, there is a minor variation of the construction of g(S, S ) pointed out in [Eld04, Remark 3.2]. Take nonzero scalars α0 , α1 α2 and consider α = (α0 , α1 , α2 ) and the Lie algebra gα (S, S ), which is deﬁned over the same vector space as g(S, S ), but with new multiplication given by [ιi (a ⊗ x), ιi+1 (b ⊗ y)] = αi+2 ιi+2 ((a ∗ b) ⊗ (x ∗ y)), [ιi (a ⊗ x), ιi (b ⊗ y)] = αi+1 αi+2 q (x, y)θi (ta,b ) + q(a, b)θi (tx,y ) , where the indices are taken modulo 3, other products as for g(S, S ). By [Eld04, Remark 3.2], over R, any gα (S, S ) is isomorphic either to g(S, S ) or to g(1,−1,−1) (S, S ). Then, if κα denotes the Killing form of gα (S, S ) and κ the one of g(S, S ), it follows that κα |tri(S,∗,q)⊕tri(S ,∗,q ) = κ|tri(S,∗,q)⊕tri(S ,∗,q ) , κα |ιi (S⊗S ) = αi+1 αi+2 κ|ιi (S⊗S ) . Thus, if q or q is split, the signature of κα coincides with the signature of κ, while if both q and q are positive deﬁnite and α = (1, −1, −1), then the Killing form changes signs on ι1 (S ⊗ S ) ⊕ ι2 (S ⊗ S ); hence the following Lie algebras are obtained (q and q deﬁnite, α = (1, −1, −1), and dim S = 8): dim S:

1

2

4

8

f4,−20

e6,−14

e7,−5

e8,8

thus obtaining, in particular, the two missing exceptional real Lie algebras.

12.6

Acknowledgments

This work was supported by the Spanish Ministerio de Ciencia y Tecnolog´ıa and FEDER (BFM 2001-3239-C03-03) and by the Diputaci´ on General de Arag´ on (Grupo de Investigaci´ on ´ de Algebra).

References [AF93] B. N. Allison and J. R. Faulkner, Nonassociative coeﬃcient algebras for Steinberg unitary Lie algebras, J. Algebra 161 (1993), no. 1, 1–19. [Bae02] John C. Baez, The octonions, Bull. Amer. Math. Soc. (N.S.) 39 (2002), no. 2, 145–205 (electronic).

A new look at Freudenthal’s magic square [BS00] C. H. Barton and A. arXiv:math.RA/0001083

Sudbery,

Magic

Squares

165 of

Lie

Algebras.

[BS03] C. H. Barton and A. Sudbery, Magic squares and matrix models of Lie algebras, Adv. Math. 180 (2003), no. 2, 596–647. [Eld97] Alberto Elduque, Symmetric composition algebras, J. Algebra 196 (1997), no. 1, 282–300. [Eld04] Alberto Elduque, The magic square and symmetric compositions, Rev. Mat. Iberoamericana 20 (2004), 477–493. [EM91] Alberto Elduque and Hyo Chul Myung, Flexible composition algebras and Okubo algebras, Comm. Algebra 19 (1991), no. 4, 1197–1227. [EM93] Alberto Elduque and Hyo Chul Myung, On ﬂexible composition algebras, Comm. Algebra 21 (1993), no. 7, 2481–2505. [EP96] Alberto Elduque and Jos´e Mar´ıa P´erez, Composition algebras with associative bilinear form, Comm. Algebra 24 (1996), no. 3, 1091–1116. [Fre64] Hans Freudenthal, Lie groups in the foundations of geometry, Advances in Math. 1 (1964), no. fasc. 2, 145–190 (1964). [Hel78] Sigurdur Helgason, Diﬀerential geometry, Lie groups, and symmetric spaces, Pure and Applied Mathematics, vol. 80, Academic Press Inc. [Harcourt Brace Jovanovich Publishers], New York, 1978. [KMRT98] Max-Albert Knus, Alexander Merkurjev, Markus Rost, and Jean-Pierre Tignol, The book of involutions, American Mathematical Society Colloquium Publications, vol. 44, American Mathematical Society, Providence, RI, 1998. [LM02] J. M. Landsberg and L. Manivel, Triality, exceptional Lie algebras and Deligne dimension formulas, Adv. Math. 171 (2002), no. 1, 59–85. [LM04] J. M. Landsberg and L. Manivel, Representation theory and projective geometry. In Algebraic transformation groups and algebraic varieties, Encyclopedia of Mathematical Sciences, vol. 132, 71-122, Springer, Berlin, 2004. [OV94] Lie groups and Lie algebras, III, Encyclopaedia of Mathematical Sciences, vol. 41, Springer-Verlag, Berlin, 1994. [Sch95] Richard D. Schafer, An introduction to nonassociative algebras, Dover Publications Inc., New York, 1995. [Tit66] J. Tits, Alg`ebres alternatives, alg`ebres de Jordan et alg`ebres de Lie exceptionnelles. I. Construction, Nederl. Akad. Wetensch. Proc. Ser. A 69 = Indag. Math. 28 (1966), 223–237.

Chapter 13 A Survey on Approximation of Locally Compact Groups by Finite Groups, Semigroups and Quasigroups L.Yu. Glebsky, E.I. Gordon, and C.J. Rubio Universidad de San Luis Potosi, M´exico

13.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.2 Approximation of locally compact groups by ﬁnite groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.2.1 Positive results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.2.2 Negative results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.3 Approximation of locally compact groups by ﬁnite semigroups and quasigroups . . . . . . 13.3.1 Approximation by ﬁnite semigroups and its application . . . . . . . . . . . . . . . . . . . . . . . . 13.3.2 Approximation of locally compact groups by ﬁnite loops . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

167 169 169 175 177 177 179 181

Abstract In this survey we discuss the approximations of topological groups by ﬁnite algebraic systems: groups, semigroups, quasigroups and loops. Key words: Approximation, group, quasigroup 2000 MSC: 26E35, 03H05

13.1

Introduction

In this chapter we discuss the approximations of topological groups by ﬁnite algebraic systems. The deﬁnition of approximation that we explore was introduced in [9] for investigation of the approximation of locally compact abelian groups by ﬁnite abelian groups. According to this deﬁnition (Deﬁnition 13.1) the approximating ﬁnite groups are embedded into the locally compact group G that they approximate only as sets and their group laws converge to the group law of G in the topology of the uniform convergence on compacts. This deﬁnition arose from the consideration of representation of numbers in the computer, see Examples 13.2 and 13.3 in 13.3.1. It was shown that harmonic analysis on locally compact abelian groups can be developed using this deﬁnition on the base of harmonic analysis on ﬁnite abelian groups. Some of these results are presented in Section 13.2.1 below. In [1] this notion of approximation was used for construction of ﬁnite dimensional approximations of pseudodiﬀerential operators. It was shown that if we manage to approximate operators together with some algebraic structures connected with them (e.g., their symmetry groups), then we obtain the better properties of approximation. Thus, it is interesting to investigate the approximation of nonabelian groups also. Lie groups are of special interest.

167

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Unfortunately, the positive results here were obtained only for some nilpotent Lie groups [10, 18]. On the other hand it was proved that the most important Lie groups such as the group SO(3) are not approximable by ﬁnite groups. This fact stimulated investigation of the approximation of groups by some more general algebraic systems. The most popular algebraic systems close to groups are semigroups and quasigroups. We found out [7] that the approximation by ﬁnite semigroups yieds nothing new: any group approximable by ﬁnite semigroups is approximable by ﬁnite groups. Nevertheless, this result allowed us to obtain an interesting negative result [6]: the ﬁeld R cannot be approximated by ﬁnite associative rings. Finite algebraic systems that approximate the ﬁeld R can be considered as numerical systems implemented in computers for the simulation of the ﬁeld R. Thus, this result shows that computer arithmetic cannot be constructed as associative rings. It was proved [8] that a locally compact group is approximable by ﬁnite quasigroups if and only if it is unimodular. This result answers the old open question about characterization of unimodular groups in algebraic and topological terms. For the case of discrete groups our deﬁnition is equivalent to the well-known notion of local embedding in the model theory, which derives from A.I. Mal’tsev [14] (see also [4]). The class of groups that are locally embeddable in the class of ﬁnite groups was studied in [18]. It was shown that it has some important applications in the ergodic theory of group actions. One interesting application of this class in symbolic dynamics is contained in [11]. We start with a rigorous deﬁnition of approximation. Let G be a locally compact group. We denote by · the multiplication in G and use the usual notations: XY = {x · y | x ∈ X, y ∈ Y } X −1 = {x−1 | x ∈ X} gX = {g · x | x ∈ X} for X, Y ⊂ G, g ∈ G. DEFINITION 13.1 Let C ⊂ G be a compact set, U be a relatively compact neighborhood of the unity in G, and (H, .) be a ﬁnite universal algebra with one binary operation. 1. We say that a set M ⊂ G is a U -grid of C if and only if C ⊂ M U . 2. A map j : H → G is called a (C, U )-homomorphism if ∀x, y ∈ H ((j(x), j(y), j(x) · j(y) ∈ C) ⇒ (j(x . y) ∈ j(x)j(y)U )) 3. We say that the pair < H, j > is a (C, U )-approximation of G if j(H) is a U -grid of C and j : H → G is a (C, U )-homomorphism. 4. Let K be a class of ﬁnite algebras. We say that G is approximable by the systems of the class K if for any compact C ⊂ G and for any neighborhood U of the unity there exists a (C, U )-approximation < H, j > of G such that H ∈ K and j is an injection. The main question under consideration here is Given class K and group G if G is approximable by the systems of the class K? REMARK 13.1 Since in item 2 elements j(x . y) and j(x) · j(y) are U -close in the left uniformity on G it may seem that the deﬁnition of approximability of G by systems of K depends on which of two uniformities we consider. However, this is not so. Indeed it is clear from the deﬁnition that we deal only with the restrictions of the uniformities on compacts.

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But it is well known that the restrictions of the left uniformity and of the right uniformity on any compact are equivalent. REMARK 13.2 It is easy to see that a similar deﬁnition can be formulated for any topological universal algebra. Even a group one can consider as an algebra with one constant and two operation: the unity, the multiplication, and the inversion. Such a consideration leads to Deﬁnition 13.3 of the strong approximability. REMARK 13.3 We believe that any reasonable computer representation of a topological algebraic system should be a (C, U )-approximation for a large enough C and a small enough U . From this point of view it is interesting to know how eﬀective an approximation may be. We will not consider such questions. (An analogy with analysis: one can prove convergence without knowing how fast it is.) REMARK 13.4 It is not necessary to assume that approximated algebras are ﬁnite. For example, the approximations of discrete groups by amenable groups have been introduced in [2]. We recall also the deﬁnitions of semigroups and quasigroups. DEFINITION 13.2 1. We say that an algebra (A, ◦) is a right quasigroup (left quasigroup) if and only if for every a, b ∈ A the equation a ◦ x = b (x ◦ a = b) has the unique solution x = /(b, a) (x = \(b, a)). 2. An algebra (A, ◦) is a quasigroup if and only if it is a right quasigroup and left quasigroup. A quasigroup A with a unity (the element e ∈ A such that ∀ a ∈ A a ◦ e = e ◦ a = a) is called a loop. 3. We say that an algebra (A, ◦) is a semigroup if the operation ◦ satisﬁes the law of associativity.

13.2

Approximation of Locally Compact Groups by Finite Groups

13.2.1 13.2.1.1

Positive results Approximation of locally compact abelian (LCA) groups by ﬁnite abelian groups

The results of this section are contained in [1] and [9]. Here we need a stronger deﬁnition of approximability. For the sake of simplicity we restrict ourselves by the case of separable locally compact groups, which is most important for applications. The results formulated here hold for nonseparable groups after an obvious slight modiﬁcation. If a locally compact group G is separable, then it is metrizable. Fix any metric ρ on G that deﬁnes its topology.

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DEFINITION 13.3 The group G is strongly approximable by ﬁnite groups if and only if there exists a sequence (Gn , jn ) of ﬁnite groups Gn and maps jn : Gn → G that satisﬁes the following condition. For any ε > 0 and for any compact K ⊆ G there exists N > 0 such that for all n > N : 1. jn (Gn ) is an ε-grid for K 2. ∀g1 , g2 ∈ jn−1 (K) ρ(jn (g1 ◦n g2±1 ), jn (g1 )·jn (g2 )±1 ) < ε, where ◦n is the multiplication in Gn 3. jn (en ) = e, where en and e are units in the groups Gn and G respectively A sequence (Gn , jn ) that satisﬁes the conditions of Deﬁnition 13.3 is called an approximating sequence (a.s.) for G. Obviously, if a separable group G is strongly approximable by ﬁnite groups, then it is approximable by ﬁnite groups in the sense of Deﬁnition 13.1. The inverse is known to be true for discrete and compact groups. It is an open question in general. It is easy to see that the following proposition holds. PROPOSITION 13.1 If a locally compact separable group G is approximable by ﬁnite algebraic systems of a class K in the sense of Deﬁnition 13.1, then there exists a sequence (Gn , jn ) of ﬁnite algebraic systems Gn ∈ K and maps jn : Gn → G that satisﬁes the following condition. For any ε > 0 and for any compact K ⊆ G there exists N > 0 such that for all n > N : 1. jn (Gn ) is an ε-grid for K 2. ∀g1 , g2 ∈ jn−1 (K) ρ(jn (g1 ◦n g2 ), jn (g1 ) · jn (g2 )) < ε, where ◦n is the operation in Gn THEOREM 13.1

Any LCA group G is strongly approximable by ﬁnite abelian groups.

Indeed, all basic constructions of the harmonic analysis on LCA groups can be approximated by respective constructions of harmonic analysis on ﬁnite groups. Here we introduce the main theorems from [9] about these approximations. THEOREM 13.2 Let G be an LCA group, μ, Haar measure on G, U , a relatively compact neighborhood of the unit in G, K, the family of all compact subsets of G. Then if f : G → C is bounded, almost μ-everywhere continuous and satisﬁes the following condition: ∀ε > 0 ∃K ∈ K∃n0 ∀n > n0 ∀B ⊆ jn−1 (G \ K) μ(U ) (13.1) |f (jn (g))| < ε, where Δn = −1 Δn · |jn (U )| g∈B then f ∈ L1 (μ) and

f dμ = lim Δn G

n→∞

f (jn (g)).

(13.2)

g∈Gn

Obviously condition (13.1) holds for any bounded a.e. continuous function f with the compact support on a locally compact group G. Notation. For each p ≥ 1 we denote by Lp (G) the space of functions f , such that |f |p satisﬁes the conditions of Theorem 13.2.

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The Δn , deﬁned in (1) will be called the normalizing multiplier (n.m.) of a.s. (Gn , jn ). We can take any sequence Δn , equivalent to Δn for a n.m. of all continuous homomorphisms γ : G → Let G be an LCA group. Recall that a set G T = {z ∈ C | |z| = 1} endowed with the operation of pointwise multiplication and the topology of the uniform convergence on compact sets is an LCA group, which is called the dual group to G. be the base of neighborhoods of the unit in G, Introduce the following notation. Let Γ which consists of the neighborhoods of the form U (K, ε), where K ⊆ G is compact, and | ∀g ∈ K|γ(g) − 1| < ε}. U (K, ε) = {γ ∈ G n , n is the dual group for Gn and n → G jn ), where G jn : G Fix an a.s. (Gn , jn ) for G. Fix (G n we say that χ belongs are more or less arbitrary (not related with jn ) now. For χ ∈ G to U (K, ε) strongly (χ ∈s U (K, ε)) if jn (χ) ∈ U (K, ε), and χ belongs to U (K, ε) weakly (χ ∈w U (K, ε)) if ∀g ∈ jn−1 (K) |χ(g) − 1| < ε n , DEFINITION 13.4 Let (Gn , jn ) be an a.s. for LCA group G and (G jn ) be an a.s. for its dual group G. We call this pair of a.s. mutually dual (the ﬁrst is dual to the second and vice versa) if the following two conditions are fulﬁlled: 0 ∀n ≥ n0 ∀χ ∈ G n (χ ∈w U −→ χ ∈s U ) ∈ Γ∃n (i) ∀U ∈ Γ∃U n , lim jn (gn ) = ξ, lim (ii). If gn ∈ Gn , χn ∈ G jn (χn ) = χ, then χn (gn ) → χ(ξ) when n→∞ n→∞ n→∞ THEOREM 13.3 For every LCA group G there exists a pair of dual approximations respectively. for G and G, The dual approximations are necessary for approximating the Fourier transform on L2 (G). deﬁned by the formula The linear operator F : L2 (G) → L2 (G) F (f )(γ) = f (g)γ(g)dμ(g) = f(γ) (13.3) G

is called the Fourier transform on G. Strictly speaking the Fourier transform is deﬁned by formula (13.3) only for f ∈ L1 (G) ∩ L2 (G). However, since F is bounded in L2 -norm on L1 (G) ∩ L2 (G) it can be extended on the whole L2 (G). such that the Fourier transform is a norm There exists the dual Haar measure μ on G . The inverse operator preserving operator with respect to L2 -norms deﬁned by μ and μ → L2 (G) is given by the formula F −1 : L2 (G) −1 f (g) = F (f )(g) = f(γ)γ(g)d μ(γ). (13.4) G

n , jn ) is a pair of dual a.s. for G and Δn is PROPOSITION 13.2 If (Gn , jn ) and (G a normalizing multiplier for μ then Δn = (|Gn | · Δn )−1 is a normalizing multiplier for the Haar measure μ . n is isomorphic to Gn , thus Note that for a ﬁnite abelian group Gn its dual group G |Gn | = |Gn |.

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n ) is deﬁned by formula The Fourier transform Fn : L2 (Gn ) → L2 (G Fn (ϕ)(χ) = Δn · ϕ(g)χ(g) g∈Gn

and the inverse Fourier transform Fn−1 by the formula n · Fn−1 (ψ)(g) = Δ ψ(χ)χ(g) χ∈Gn

Now we are able to formulate the approximation theorem for the Fourier transform. , THEOREM 13.4 If (Gn , jn ) and (G jn ) is a pair of dual a.s. for G, f ∈ L2 (G) and (see formula (3)). Then f ∈ L2 (G) n · lim Δ |(f ◦ j)(χ) − Fn (f ◦ j)(χ)|2 = 0. (13.5) n→∞

χ∈Gn

is isomorphic to R. One Example 13.1 Let G be the additive group R. Then R of the possible isomorphisms Λ : R → R is given by the formula Λ(t)(x) = exp(2πixt), we obtain the canonical formula for the Fourier where x, t ∈ R. After identifying R and R transform: +∞ f (x) exp(−2πixt)dx. (13.6) f (t) = −∞

The Haar measure is in this case the usual Lebesgue measure dx. The dual Haar measure 1 and the formula (13.4) has the following form: is 2π 1 f (x) = 2π

+∞ f(t) exp(2πixt)dt. −∞

To construct the approximating sequence + , for R choose two sequences of positive reals an and Nn = 2Mn + 1. Here [α] is the integral an → +∞ and Δn → 0. Put Mn = Δ n

part of a real α. Now the group Gn = {−Mn , . . . , Mn } is the additive group of residuals modulo Nn and jn : Gn → R is deﬁned by the formula: jn (k) = k · Δn , k ∈ Gn . It is easy to see that jn is a ([−an , an ], ε)-approximation of R. Indeed, obviously jn (Gn ) is an ε-grid on [−an , an ] and for k, m ∈ Gn holds k · Δn + m · Δn ∈ [−an , an ] if and only if k + m ∈ Gn , here we denote by + the operation of addition in R and in Z. But k + m ∈ Gn if and only if k ⊕n m = k + m, where ⊕n is the addition modulo Nn , i.e., the operation in Gn . We have proved that if jn (k), jn (m), jn (k) + jn (m) ∈ [−an , an ] then jn (k) + jn (m) = j(k ⊕n m), i.e. jn is a ([−an , an ], ε)-approximation. One can easily see that the conditions of Deﬁnition 13.3 are also satisﬁed. n deﬁned by the formula Λn (k)(m) = It is easy to see that the map Λn : Gn → G 2πikm n . Let exp( Nn ), k, m ∈ Gn is an isomorphism between Gn and G n = (Nn Δn )−1 . Δ

(13.7)

n . Then identifying Gn with G n by the Deﬁne jn : Gn → R by the formula jn (k) = k · Δ jn ) is isomorphism Λn and R with R by the isomorphism Λ one can easily see that (Gn , the dual approximating sequence to the approximating sequence (Gn , jn ).

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The formula (13.5) for approximation of the Fourier transform has the following form in this case. lim

Nn →∞, Nn Δn →0

n Δ

Mn

Mn

n ) − Δn |f(k Δ

m=−Mn

k=−Mn

2πikm f (mΔn ) exp − |2 = 0, (13.8) N n

n is deﬁned by the formula (13.7), f is the Fourier transform of f deﬁned by the where Δ formula (13.6) and f, f ∈ L2 (R). By Theorem 13.2 the pointwise approximation f(t) =

lim

Nn →∞, Nn Δn →0

Mn

Δn

f (mΔn ) exp(−2πitmΔn ) = 0.

(13.9)

m=−Mn

holds for any bounded a.e. continuous function f that decreases fast enough on inﬁnity and for any t ∈ R, in particular t = k Δ. The convergence (13.8) is stronger then (13.9) one. It is natural to ask, whether n Δ

Mn k=−Mn

n ) − Δn |f(k Δ

Mn

n )|2 → 0 f (mΔn ) exp(−2πikmΔn Δ

(13.10)

m=−Mn

n → ∞ as n → ∞. n such that Δn , Δ n → 0, Nn Δn , Nn Δ for an arbitrary Nn , Δn , Δ It was shown by M.Yu. Zdorovenko in [19] that the convergence (13.10) holds only if n → 1 as n → ∞. This result shows that if we discretize the Fourier transform, Nn Δn Δ we obtain the best approximating results if we keep its group theoretic properties in our discretization. The same is true for many other approximations. Unfortunately, it is possible to keep algebraic properties under approximation not so often. In the monograph [9] a lot of examples of dual approximating sequences for various concrete LCA groups were constructed. In [1] the approximation of Fourier transform was used for construction of ﬁnite dimensional approximations of pseudodiﬀerential operators. 13.2.1.2

Approximation of nilpotent Lie groups

The following theorem was obtained in [18]. THEOREM 13.5 If G is a simply connected nilpotent Lie group and its Lie algebra G has a basis with rational structural constants, then G is approximable by ﬁnite groups. Consider as an example a topological approximation of nilpotent group U T (m, R). Recall that if K is a commutative associative ring with identity then the group of all m × m matrices over K with zeros under the principal diagonal and units on it is denoted as U T (m, K). We shall write the elements of this group in the form x = (xij )1≤i<j≤m . In addition, if y = (yij ) ∈ U T (m, K), then it is convenient to write the elements of the product z = (zij ) in the form zij = xij + yij +

j−i−1

xi,i+k yi+k,j .

(13.11)

k=1

So, let G = U T (m, R). Since this group is separable it is possible to construct a sequence of ﬁnite groups Gn , which satisﬁes the conditions of Proposition 13.1. Let Gn = U T (m, ZN ),

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where N = 2nm +1, and ZN is the ring of residues mod N , which is convenient to represent here as the ring of least absolute value residues, i.e., ZN = {−nm , . . . , nm }. Now deﬁne an injective map jn : Gn −→ G in the following way. If k = (kij ) ∈ Gn , i.e., kij ∈ ZN , then jn (k) = x = (xij ), where xij =

kij . nj−i

To check the conditions of Deﬁnition 13.3 we shall use the metric ρ(x, y) = max |xij − yij |. Verify the ﬁrst condition of Deﬁnition 13.3. Let ξ = (ξij ) ∈ U T (m, R), ε > 0. Put n0 = [max{ξij , ε−1 }] and deﬁne for any n > n0 the element k = kij ∈ Gn , which satisﬁes the condition kij kij + 1 ≤ ξij < . nj−i nj−i By the choice of n it follows immediately that |kij | < nm , i.e., indeed k ∈ Gn , and also that ρ(ξ, jn (k)) < ε. To verify the second condition of Deﬁnition 13.3 note that it is enough to consider the compact sets of the form Δa = {k ∈ G | |kij | ≤ a}, a ∈ R+ . Let 1

1

n2 m+2

> a, k = (kij ) ∈ Gn .

nj−i+ 2 m+2

Then if jn (k) ∈ Δa , it is easy to see that |kij | < . As the calculations by formula (13.11) of matrices, which satisfy the latter inequality, in Z and in ZN are the same it is easy to see that ∀k, s ∈ Gn (jn (k), jn (s) ∈ Δa −→ jn (k · s) = jn (k) · jn (s)). The third condition of Deﬁnition 13.3 is obvious. The following theorem is due to A. Gorodnik ([10]). THEOREM 13.6 Let G be a simply connected nilpotent Lie group of degree 2. Then G is approximable by ﬁnite groups. The idea of the proof of Theorem 13.6 that is contained in [10] follows. Denote by NN (S) the Lie algebra of upper triangular matrices of degree N over a ring S. It is well-known that any nilpotent Lie algebra over R can be embedded in NN (R) for some positive integer N . PROPOSITION 13.3 Let G0 be a nilpotent Lie algebra over R of degree 2. Then there exist a natural number N and an embedding δ0 : G0 → NN (R) such that for any > 0 and a basis {yi : i = 1, . . . n} of G0 one can choose a Lie subalgebra G1 in NN (Q) with a basis {zi : i = 1, . . . n} such that δ0 (yi ) − zi < for i = 1, . . . n. Let G2 = G1 ⊗Q R. Then G = exp(G2 ) satisﬁes the condition of Theorem 13.5 and, thus, approximable by ﬁnite groups. Then it can be proved that if the Lie algebra of a nilpotent Lie group G is approximable in the sense of Proposition 13.3 by the Lie algebras of nilpotent Lie groups that are approximable by ﬁnite groups, then G is approximable by ﬁnite groups itself. Whether the ideas of this proof can be extended on the case of arbitrary nilpotent Lie groups, is an open question.

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Discrete groups

The results of this section are contained in [18] Applying Deﬁnition 13.1 to the particular case of a group G with the discrete topology one obtains the following. DEFINITION 13.5 A discrete group G is approximable by ﬁnite groups (locally embedded in the class of ﬁnite groups [14, 18]) if for any ﬁnite subset K ⊆ G there exists a ﬁnite group H and a map j : H → G such that 1. j(H) ⊇ K 2. if j(h1 ), j(h2 ), j(h1 ) · j(h2 ) ∈ K, then j(h1 ◦ h2 ) = j(h1 ) · j(h2 ), where ◦ is the group operation in H, and · is the group operation in G Obviously the approximability by ﬁnite groups is a local property, i.e., if any ﬁnitely generated subgroup of a group G is approximable by ﬁnite groups, then G itself is also approximated by ﬁnite groups. Recall that a group G is residually ﬁnite if for any ﬁnite subset K there exists a ﬁnite group H and a surjective homomorphism ϕ : G → H such that ϕ|K is injective. It is easy to see that any residually ﬁnite group G is approximable by ﬁnite groups. Indeed, let j : H → G be any right inverse to ϕ such that K ⊆ j(H). Then obviously the pair (H, j) satisﬁes Deﬁnition 13.5. By the locality of approximability mentioned above we obtain the following. THEOREM 13.7

Any locally residually ﬁnite group is approximable by ﬁnite groups.

Since any matrix group over a ﬁeld is locally residually ﬁnite [15] we obtain the following. COROLLARY 13.1 imable by ﬁnite groups.

Any matrix group (with discrete topology) over a ﬁeld is approx-

There exist groups approximable by ﬁnite ones that are not locally residually ﬁnite. The following example is due to A.M. Vershik. Let Sf (Z) be a group of all permutations of Z that move only ﬁnitely many elements. Then Z acts on Sf (Z) by shifts. Let G be the semidirect product Sf (Z) ∝ Z. It is easy to see that this group has two generators and since it contains a simple subgroup of even permutations, G is not locally residually ﬁnite. On the other hand it is easy to see that G is approximable by ﬁnite groups. Indeed, let Z(m) = {−m, . . . , m} be the additive group of residues modulo 2m + 1, S(Z(m) ), the group of permutations of the set Z(m) . Then Z(m) acts on S(Z(m) ) by shifts, and it is easy to see that G is approximable by ﬁnite groups S(Z(m) ) ∝ Z(m) .

13.2.2

Negative results

1. A lot of examples of nonapproximable locally compact groups are provided by the following. THEOREM 13.8 If a locally compact group G is approximable by ﬁnite quasigroups then it is unimodular.

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The idea of the proof of this theorem is following. Let (Gn , jn ) is a sequence of ﬁnite quasigroups Gn and injective mappings jn : Gn → G that satisﬁes the conditions of Proposition 13.1 and Δn , a sequence of positive reals deﬁned by formula (13.1) (see Theorem 13.2). Then it can be proved that for any f ∈ C0 (G), the space of continuous functions with the compact support, I(f ) = lim Δn f (jn (h)), n→∞

h∈Gn

exists and is nontrivial: if f > 0, then I(f ) > 0. The positive linear functional I(f ) on C0 (G) is left and right invariant. It follows from the fact that for any quasigroup Q and an element q ∈ Q the left and right multiplications by q are permutations of Q. Thus, for any function ϕ on Gn and for any g ∈ Gn the following equalities hold: ϕ(h) = ϕ(gh) = ϕ(hg). h∈Gn

h∈Gn

h∈Gn

By the deﬁnition, the existence of left and right positive nontrivial functional on C0 (G) means the unimodularity of G. It is equivalent to the existence of left and right invariant Borel measures on G. COROLLARY 13.2 ﬁnite groups.

Any nonunimodular locally compact group is nonapproximable by

The simplest example of a nonunimodular group is the matrix-group

5 ab G= | a = 0, b ∈ R . 01

(13.12)

By Corollary 13.2 this group is non-approximable as a topological group. Notice that by Corollary 13.1 the group G is approximabale by ﬁnite groups as a discrete group. 2. The nonapproximability of simple Lie groups follows from the Kazhdan’s well-known theorem on -representations [12]. Here we formulate an almost obvious modiﬁcation of this theorem that was ﬁrst used in [2]. DEFINITION 13.6 Let H be a group and G be a compact topological group with invariant metric d. A map ρ : H → G is called an -homomorphism if ∀ x, y ∈ H (d(ρ(xy), ρ(x)ρ(y)) < ). THEOREM 13.9 For any ﬁnite-dimensional compact Lie group G there exists an > 0 such that for any 0 < < and any -homomorphism ρ : H → G of a compact (in particular, ﬁnite) group H there exists a homomorphism φ : H → G such that d(φ(h), ρ(h)) < 2 for all h ∈ H. The following proposition is a straightforward consequence of the deﬁnitions of an homomorphism and of the approximability of a compact group by ﬁnite ones. PROPOSITION 13.4 A compact group G is approximable by ﬁnite groups if and only if for every > 0 there exists a ﬁnite group H and an -homomorphism ρ such that ρ(H) is an -grid in G.

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The following theorem, which follows immediately from Proposition 13.4 and Theorem 13.9, gives a necessary and suﬃcient condition for approximability of compact Lie groups. THEOREM 13.10 A compact ﬁnite dimensional Lie group G is approximable by ﬁnite groups if and only if for any > 0 there exists a ﬁnite subgroup H ⊂ G that is an -grid in G. The next statement is a consequence of this theorem and the fact that there are only a ﬁnite number of subgroups of SO(3) that are not reduced to rotations around the same axis. COROLLARY 13.3

The group SO(3) is not approximable by ﬁnite ones.

3. All known examples of non-approximable discrete groups are based on the following THEOREM 13.11 A ﬁnitely presented group G is approximable by ﬁnite groups if and only if it is residually ﬁnite. This theorem is contained in [18] (see also [4]). It was shown by A.I. Mal’tsev [16] that the word problem for ﬁnitely presented residually ﬁnite groups is decidable (see also [5]). Thus, we obtain the following COROLLARY 13.4 Finitely presented groups with the undecidable word problem are non-approximable by ﬁnite groups. Recall that a group G is called hopﬁan if there does not exist a noninjective surjective homomorphism of G onto itself. Otherwise G is called non-hopﬁan. It was shown in [15] that any ﬁnitely generated residually ﬁnite groups are hopﬁan. The simplest example of a nonhopﬁan group is given by the Baumslag-Solitar group, G = b, t; t−1 b2 t = b3 . Since G is ﬁnitely presented by Theorem 13.11 it is not approximable by ﬁnite groups.

13.3 13.3.1

Approximation of Locally Compact Groups by Finite Semigroups and Quasigroups Approximation by ﬁnite semigroups and its application

The following theorem holds (see [6] and [7]). THEOREM 13.12 A locally compact group is approximable by ﬁnite semigroups if and only if it is approximable by ﬁnite groups.

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The proof of Theorem 13.12 is based on some results about the structure of ﬁnite semigroups from [17]. Theorem 13.12 has an interesting corollary about the approximability of the ﬁeld R by ﬁnite rings. THEOREM 13.13

The ﬁeld R is not approximable by ﬁnite associative rings.

Let us sketch the proof of this theorem. Consider the nonunimodular group G deﬁned by formula (13.12). As it was discussed above, this group is not approximable by ﬁnite groups. By Theorem 13.12 G is not approximable by ﬁnite semigroups. On the other hand if we could approximate R by ﬁnite associative rings then G would be approximable by semigroups of matrices of the same type with the elements of ﬁnite rings that approximate R, contradiction. Finite algebraic systems in the signature σ =< +, · > that approximate the ﬁeld R can be considered as the numerical systems implemented in computers for the simulation of the ﬁeld R. To justify this statement let us discuss the following two examples. Since any compact C ⊂ R is contained in the interval [−a, a] for an appropriate a and the sets Uε = {x ∈ R | |x| < ε}, ε > 0 form a base of the neighborhoods of zero in R, it is enough to consider only the ([−a, a], Uε )-approximations of R. We will call these approximations the (a, ε)-approximations. Example 13.2

Recall that the normal (computer) form of a real α is its representation: α = ±10p · 0.a1 a2 . . . ,

(13.13)

where p ∈ Z, and a1 a2 . . . is a ﬁnite or inﬁnite sequence of decimal digits 0 ≤ an ≤ 9, and a1 = 0. The integer p is called the exponent of α and a1 a2 . . . , its mantissa. Fix two natural numbers P > Q and consider the ﬁnite set AP Q of reals in the form (13.13) such that the exponent p of α satisﬁes the inequality |p| ≤ P and its mantissa contains no more than Q decimal digits. Deﬁne the two binary operations ⊕ and . on AP Q . Let α, β ∈ AP Q and the normal form of α × β, where × is either + or ·, is α × β = ±10r · 0.c1 c2 . . . . Notice that the mantissa of α × β may contain more than Q digits. Now ⎧ ±10r · 0.c1 c2 . . . cQ , if |r| ≤ P ⎪ ⎪ ⎨ ±10P · 0. 99 . . . 9 , if r > P. 6 78 9 α⊗β = ⎪ Q digits ⎪ ⎩ 0, if r < −P In the case when the mantissa of α × β contains fewer than Q digits we complete it to a Q-digits mantissa by zeros. We will denote the universal algebra < AP Q , σ >, such that the interpretations of the functional symbols + and · are the functions ⊕ and ., respectively, by AP Q . It is easy to see that for any positive a and ε there exist natural numbers P and Q such that the universal algebra AP Q is an (a, ε)-approximation of R. The described systems AP Q are implemented in working computers. What properties of addition and multiplication of reals hold for ⊕ and .? It is easy to see that the operations ⊕ and . are commutative, ξ ⊕ (−ξ) = 0 and ξ + 0 = ξ for any ξ ∈ AP Q .

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. . . 59 Then α ⊕ β = α ⊕ γ, so the cancellation law fails Let α = β = 0. 60 . . 069, γ = 0. 60 6 78 6 .78 Q digits Q digits for ⊕ and thus, the law of associativity for ⊕ fails. It is easy to construct examples that show that the laws of associativity for . and distributivity in AP Q also fail. Example 13.3 Fix a natural number M and a positive ε. Put AM ε = {kε | k = −M . . . M }. Let N = 2M + 1. For any n ∈ Z we will denote by n ( mod N ) the element of the set {−M, . . . , M }, congruent to n modulo N . The operations ⊕ and . on AM,ε are deﬁned as follows: kε ⊕ mε = (k + m) ( mod N )ε (13.14) kε . mε = [kmε] ( mod N )ε.

(13.15)

AM,ε

We will denote by the universal algebra in the signature σ with the underlying set AM,ε and the interpretation of the functional symbols deﬁned by formulas (13.14) and (13.15). It is easy to see that AM ε is an (M ε, ε)-approximation of R. It is obvious that AM,ε is an abelian group with respect to ⊕ (see (3)). However one can easily construct examples that show that for any big enough M and small enough ε the multiplication . satisﬁes neither the law of associativity, nor the law of distributivity. This example shows that it is possible to implement in computers a numerical system that simulate reals, which is an abelian group with respect to addition, while by Theorem 13.13 it is impossible to implement such a system that would be an associative ring (even noncommutative). It is an interesting question: Is it possible to approximate R by any ﬁnite non-associative rings?

13.3.2

Approximation of locally compact groups by ﬁnite loops

The following theorem was introduced in [6]. See [8] for detailed proofs. THEOREM 13.14 A locally compact group G is unimodular if and only if it is approximable by ﬁnite quasigroups. The suﬃciency was discussed above (see Theorem 13.8). The necessity can be strengthened a little. Recall that an element e of a quasigroup (Q, ◦) is called the unity if ∀a ∈ Q a◦e = e◦a = a. A quasigroup with unity is called a loop. THEOREM 13.15 imable by ﬁnite loops.

Any nondiscrete locally compact unimodular group G is approx-

The proof of this theorem can be obtained by a slight modiﬁcation of the proof of Theorem 13.14, contained in [8]. For discrete groups the following stronger result due to M. Ziman [20] holds. DEFINITION 13.7

We will say that a loop L has two-sided inverse if and only if ∀x ∈ L∃x−1 xx−1 = x−1 x = 1.

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A loop L with (two-sided) inverses has the inverse antiautomorphism property if the mapping x → x−1 is an antiautomorphism of (L, ·), i.e., (xy)−1 = y −1 x−1 , for every x, y ∈ L. A loop with the inverse antiautomorphism property is brieﬂy called an IAA loop. Obviously, every group is an IAA loop. On the other hand, an IAA loop does not necessarily satisfy the conditions x−1 (xy) = y and (xy)y −1 = x. THEOREM 13.16

Every discrete group is approximable by ﬁnite IAA loops.

We believe that the result holds for any locally compact group. In fact a more general assertion than Theorem 13.16 is proved: Every discrete IAA loop is approximable by ﬁnite IAA loops. Let us ﬁnish the chapter with a sketch of the proof of Theorem 13.16. The proof utilizes the well-known relation between quasigroups and Latin squares (a multiplication table of a quasigroup is a Latin square). Its key ingredient is a kind of embedding theorem for Latin subsquares (Theorem 13.17). It gives some suﬃcient conditions guaranteeing the extendability of a Latin subsquare, symmetric with respect to some involutive permutation of the set of its elements and with constant diagonal, to a Latin square with the same property. A p × q matrix R = (rij ) with elements from a set A is called a Latin rectangle of size p × q over A if every element of A occurs at most once in each row as well as in each column. If p = q then the Latin rectangle is called a Latin subsquare of order p. If p = q equals the number n of elements of the ﬁnite set A, then the Latin rectangle is called a Latin square of order n over A. DEFINITION 13.8 Let α : A → A be an involutive permutation of the set A, i.e., α2 = id. A Latin (sub)square R = (rij ) over A is called α-symmetric if α(rij ) = rji for all i, j. Obviously, if (rij ) is an α-symmetric Latin (sub)square then α(rii ) = rii ; in other words, all the diagonal elements are ﬁxed by α. The next theorem is a partial case, for α = id, of the Cruse theorem on extensions of commutative Latin squares (cf. [3], Theorem 1 or [13], Theorem 4.1). On the the other hand, it can be regarded as a generalization of the special case (rii = 1) of the quoted result from commutative Latin squares to the α-symmetric ones. The number of occurrences of an element a ∈ A in a Latin rectangle R will be denoted by NR (a). THEOREM 13.17 Let n be even, α be an involutive permutation of the set A = {1, . . . , n} with α(1) = 1, and R = (rij ) be an α-symmetric Latin subsquare over A of order m < n such that rii = 1 and r1i = i for all i ≤ m. Then R can be extended to an α-symmetric Latin square S = (sij ) over A satisfying sii = 1 and s1i = i for all i ≤ n if and only if NR (k) ≥ 2m − n for all k ∈ A. Theorem 13.17 is proved in [20]. Theorem 13.17 implies Theorem 13.16 because there exists a one-to-one correspondence between α-symmetric Latin squares and IIA loops. Let

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us explain it in more detail. Let Q = {1, 2, ..., n}, α : Q → Q be an involution such that α(1) = 1. Let rij be an α-symmetric Latin square on Q with rii = 1 and r1j = j. Latin square rij is a multiplication table of the quasigroup (Q, ·), x · y = rxy . For all x, y ∈ Q hold α(x · y) = y · x and x2 = 1. Introduce operation ∗, x ∗ y = α(x) · y. It is a straightforward calculation to check that (Q, ∗) is a IAA loop with inverse x−1 = α(x) and the unity 1. On the other hand, if (Q, ∗) is an IAA loop with the unity 1, we may introduce operation x · y = x−1 ∗ y. It is easy to check that rij = i · j is an α-symmetric Latin square on Q with rii = 1 and r1j = j, where α(x) = x−1 . Now, under proper deﬁnition of partial IIA loop, Theorem 13.17 implies that any ﬁnite partial IIA loop may be extended to a ﬁnite IIA loop, that implies Theorem 13.16.

References [1] S. Albeverio, E. Gordon, A. Khrennikov. Finite dimensional approximations of operators in the spaces of functions on locally compact abelian groups. Acta Applicandae Mathematicae 64(1) pp. 33-73, October 2000. [2] M.A. Alekseev, L.Yu. Glebskii, E.I. Gordon. On approximations of groups, group actions and Hopf algebras. Representation Theory, Dynamical Systems, Combinatorial and Algebraic Methods. III, A.M. Vershik, editor, Russian Academy of Science. St. Petersburg Branch of V.A. Steklov’s Mathematical Institute. Zapiski nauchnih seminarov POMI 256 (1999), 224-262. (in Russian; Engl. Transl. in Journal of Mathematical Sciences, 107, No.5 (2001), pp. 4305-4332. [3] A.B.Cruse. On embedding incomplete symmetric Latin squares. J. of Combinatorial Theory. (A) 16 (1974), 18-22. [4] T. Evans. Some connection between residual ﬁniteness, ﬁnite embeddability and the word problem. J. Lond. Math. Soc., (2),1 (1969), pp. 399-403 [5] T. Evans. Word problems, Bull. of American Math. Soc., 84, No. 5 (1978), pp. 789802. [6] L.Yu. Glebsky, E.I. Gordon. On approximation of topological groups by ﬁnite algebraic systems, Illinois Math. J. V.49 N1, 1-16, 2005. [7] L.Yu. Glebsky, E.I. Gordon. On approximation of locally compact groups by ﬁnite algebraic systems, Electroni research ann. of AMS V 10, pp. 21-28, 2004. [8] L.Yu. Glebsky. E.I. Gordon, C.J. Rubio. On approximation of topological groups by ﬁnite algebraic systems. II, Illinois Math. J. V.49, N1, 17-31, 2005. [9] E. Gordon. Nonstandard Methods in Commutative Harmonic Analysis. AMS, Providence, RI, 1997. [10] A. Gorodnik. On approximation of nilpotent Lie groups. Unpublished. [11] M. Gromov. Endomorphisms of symbolic algebraic varieties. J. Eur. Math. Soc. 1 (1999), pp.109–197.

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[12] D. Kazhdan. On -representations. Israel J. Math. 43, No. 4, 315-323 (1982). [13] C.C. Lindner. Embeddings theorem for partial Latin squares. Annals of Discrete Mathematics 46, Latin squares ed. by J. Denes and A.D. Keedwell, North-Holland, Amsterdam, 1991, 217-265. [14] A.I. Mal’tsev. Algebraic Systems, (Russian), Nauka, Moscow, 1967. [15] A.I. Mal’tsev. On an isomorphic representation of inﬁnite groups by matrices. In: A.I. Mal’tsev. Selected works, Volume I. Classical algebra. Nauka, Moscow, 1976, pp. 58–73 (in Russian). [16] A.I. Mal’tsev. On homomorphisms onto ﬁnite groups. In: A.I. Mal’tsev. Selected works, Volume I. Classical algebra. Nauka, Moscow, 1976, pp. 540–462 (in Russian). [17] J. Rhodes, B. Tilson. Theorems on local structure of ﬁnite semigroups. In: Algebraic theory of machines, languages and semigroups, ed. M.A. Arbib, Academic Press, New York and London, 1968. [18] A.M. Vershik, E.I. Gordon. Groups locally embedded into the class of ﬁnite groups. (Russian) Algebra i Analiz 9 (1997), no. 1, pp. 71–97; translation in St. Petersburg Math. J. 9 (1998), no. 1, 49-67. [19] M.Yu. Zdorovenko. Hyperﬁnite approximation of the Fourier transform (Russian). Izv. Vyssh. Uchebn. Zaved. Mat. 1999, no. 4, pp. 68–72; translation in Russian Math. (Iz. VUZ) 43 (1999), no. 4, pp. 67–71. [20] Milos Ziman. Extensions of Latin subsquares and local embeddability of groups and group algebras. Quasigroups and Related Systems. 11 (2004), pp. 115-125.

Chapter 14 The Transformation Algebras of Bernstein Graph Algebras H. Guzzo Jr. Universidade de S˜ ao Paulo, Instituto de Matem´ atica e Estat´ıstica, S˜ ao Paulo, Brazil P. Vicente Universidad de Le´ on, Campus de Vegazana, Departamento de Matem´ aticas, Le´on, Spain 14.1 14.2 14.3 14.4

The Bernstein graph algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The associative algebra Γ(G) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The associative algebra T (G) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Abstract In this chapter we present a decomposition of the subalgebra T (G) of Hom(A(G)) generated by {idA(G) } ∪ {Lt |Lt : A(G) → A(G), t ∈ A(G)}, where G is a graph, A(G) is the Bernstein graph algebra of G, idA(G) is the identity function on A(G) and Lt is the left (= right) multiplication by t. If G is a simple connected graph, without loop and |V (G)| > 2, then we present a characterization of T (G). Key words: Baric algebras y Bernstein algebras, Bernstein graph algebra, transformation algebras 2000 MSC: 17D92

14.1

The Bernstein Graph Algebra

Let G = (V (G), E(G)) be a ﬁnite graph, where V (G) is the set of vertices and E(G) the set of edges of G. The edges will be denoted by, say, α = ab, where a and b are the vertices linked by α. Thus we have ab = ba, and aa is a loop in the vertex a. Suppose now F is a ﬁeld of characteristic not 2, G has no isolated vertices and let N = N (G) be the vector space that is the direct sum of U and Z, where U is the F -vector space freely generated by V (G) and Z is the vector space freely generated by E(G). We introduce in N the following commutative multiplication (on the basis of U and Z) aab = b;

bab = a;

other products are zero.

(14.1)

For some x ∈ N , we have (x2 )2 = 0. In fact, we see that if x1 , x2 , x3 , x4 ∈ N then (x1 x2 )(x3 x4 ) = 0 because both x1 x2 and x3 x4 are in U . So N is solvable of index 3. By applying [4, Prop. 7], we can embed N in an exceptional Bernstein algebra, by taking the linear operator τ : N −→ N deﬁned by τ (u) = 12 u, u ∈ U , and τ (z) = 0, z ∈ Z, and using [1, (1)] as the product in A = A(G) = F ⊕ N (G). The Bernstein algebra just deﬁned

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will be referred to as the Bernstein algebra associated to the graph G or Bernstein graph algebra of G. For e = (1, 0) the elements of U are proper vectors of the linear operator Le : N (G) −→ N (G) deﬁned by x −→ ex corresponding to the proper value 12 . The elements of Z are the proper vectors of the proper value 0 of Le . We have the following relations: U 2 = 0, U Z ⊆ U, Z2 = 0 as special cases of (19.1). Moreover the type of A(G) is (1 + |V (G)|, |E(G)|). Let G = (V (G), E(G)) be a graph, we will denote by T (G) the subalgebra of Hom(A(G)) generated by {idA(G) } ∪ {Lt | Lt : A(G)→A(G), t ∈ A(G)}, where idA(G) is the identity function on A(G) and Lt is the left (= right) multiplication by t. We will use (A, ω)∼ =b (A , ω ) or A∼ =b A , to denote the existence of a baric isomorphism from (A, ω) to (A , ω ). PROPOSITION 14.1 If G is a graph, then there exists a simple graph G such that A(G)/Ann(A(G))∼ =b A(G ). PROOF Consider the graph G = (V (G ), E(G )) such that V (G ) = {a | a ∈ V (G)} and E(G ) = {α | α ∈ E(G)}, where x = x + Ann(A(G)), x ∈ A(G). Then the following conditions hold: (i) For all a, b ∈ V (G), a = b⇐⇒a = b (ii) For all α, β ∈ E(G) α = β⇐⇒Lα = Lβ ⇐⇒α and β have the same vertices; (iii) If a and b are the vertices linked by α ∈ E(G), then α a = b and α b = a So A(G)/Ann(A(G))∼ =b A(G ). PROPOSITION 14.2 A(G1 )∼ =b A(G2 ). PROOF

If G1 and G2 are graphs such that A(G1 )∼ =b A(G2 ), then

If ϕ : A(G1 )→A(G2 ) is a baric isomorphism, then ϕ (Ann(A(G1 ))) = Ann(A(G2 ))

and

A(G1 )/Ann(A(G1 ))∼ =b A(G2 )/Ann(A(G2 )),

so by Proposition 14.1, A(G1 )∼ =b A(G2 ).

14.2

The Associative Algebra Γ(G)

We denote by Γ(G) the subalgebra of T (G) generated by {Lα |α ∈ E(G)}. PROPOSITION 14.3 tions hold:

Let G be a graph and α, β, γ, δ ∈ E(G). Then following condi-

The transformation algebras of Bernstein graph algebras

x, if x ∈ F a ⊕ F b 2 (a) If α = ab, then Lα (x) = 0, if x ∈ F e ⊕ V (G) \ {a, b} ⊕ E(G)

185

2 2n+1 = Lα , for all n≥1 (b) L2n α = Lα and Lα

(c) Lα ◦ Lβ = 0 if and only if the vertices of α and β are distinct (d) Lα = Lβ if and only if α and β have the same vertices (e) Lα ◦ Lβ = Lγ ◦ Lδ if and only if Lβ ◦ Lα = Lδ ◦ Lγ PROOF (a), (b), (c), and (d) are trivial. (e) By (c) we can suppose that Lα ◦Lβ =0. If α = ab and β = ac, we have (Lγ ◦Lδ )(c) = b, so γ = bd and δ = cd. Hence Lβ ◦ Lα = Lδ ◦ Lγ . DEFINITION 14.1 γab : A(G)→A(G) by

Let G be a graph and a, b ∈ V (G). We deﬁne the linear operator

γab (x) =

a, if x = b; 0, if x ∈ {e} ∪ (V (G) \ {b}) ∪ E(G).

When a = b we denote by γa = γaa . Clearly γab ◦ γbc = γac and γab ◦ γdc = 0 if b =d. For simple proof we have Propositions 14.4 and 14.5. PROPOSITION 14.4 Let G be a graph, α = ab, β = ac ∈ E(G) such that a, b, c are distinct. Then the following conditions hold: (a) Lα ◦ Lβ = γbc , Lβ ◦ Lα = γcb (b) L2α ◦ Lβ = γac , Lβ ◦ L2α = γca (c) Lα ◦ L2β = γba , L2β ◦ Lα = γab (d) L2α ◦ L2β = L2β ◦ L2α = γa ; L2α − L2α ◦ L2β = γb ; L2β − L2β ◦ L2α = γc (e) Lα = γab + γba and Lβ = γac + γca PROPOSITION 14.5 conditions hold:

Let G be a graph, ab, aa ∈ E(G) with a =b. Then the following

(a) γa = Laa (d) γba = γb ◦ Lab

(b) γab = Laa ◦ Lab (e) Lab = γab + γba

(c) γb = L2ab − Laa

THEOREM 14.1 Let G be a connected graph such that |V (G)|≤2 and there exists aa ∈ E(G) for some a ∈ V (G) or |V (G)| > 2. The following conditions hold: (a) Γ(G) = γab | a, b ∈ V (G) (b) 11G = γa is the identity element of Γ(G) a∈V (G)

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(c) dim Γ(G) = |V (G)|2 PROOF (a) If V (G) = {a}, then by Proposition 14.5, γa = Lα for all α ∈ E(G), so Γ(G) = F γa . If V (G) = {a, b} with a =b, then we can suppose that {ab, aa} ⊆ E(G), so by Propositions 14.3 and 14.5, we have Γ(G) = γa , γb , γab , γba . If |V (G)| > 2, given J = γab | a, b ∈ V (G). Clearly J is a subalgebra of T (G). Given a, a1 , a2 , a3 ∈ V (G) such that a1 , a2 , a3 are distinct, a ∈ {a1 , a2 , a3 } and there exist a1 a2 and a2 a3 . Then by Proposition 14.4, γa ∈ Γ(G). Given a, b ∈ V (G) such that a =b. If there exists ab ∈ E(G), let c ∈ V (G) \ {a, b} such that there exists ac or bc, so by Proposition 14.4, γab ∈ Γ(G). Otherwise there exist aa1 , a1 a2 , . . . , an b ∈ E(G) such that a, a1 , . . . , an , b all distinct, because G is connected. So γab = Laa1 ◦ . . . ◦ Lan b ∈ Γ(G) and J ⊆ Γ(G). Given ab ∈ E(G). If a = b, then Laa = γa ∈ J. If a =b, then there exists ac or bc for some c ∈ V (G), so by Proposition 14.4 or 14.5, Lab ∈ J. Hence J = Γ(G). (b) Given γcd ∈ Γ(G), then ⎞ ⎛ ⎝ γa ⎠ ◦ γcd = γa ◦ γcd = γc ◦ γcd = γcd . a∈V (G)

⎛ Similarly γcd ◦ ⎝

a∈V (G)

⎞

γa ⎠ = γcd .

a∈V (G)

(c) Given λa,b ∈ F such that

λa,b γab = 0.

a,b∈V (G)

For all c ∈ V (G), we have

0=

a,b∈V (G)

λa,b γab (c) =

λa,c a,

a∈V (G)

so λa,c = 0, for all a ∈ V (G).

COROLLARY 14.1 Let G be a connected graph such that |V (G)|≤2 and there exists aa ∈ E(G) for some a ∈ V (G) or |V (G)| > 2. Then Γ(G) ∼ = M|V (G)| (F ). COROLLARY 14.2 Let G be a connected graph such that |V (G)|≤2 and there exists aa ∈ E(G) for some a ∈ V (G) or |V (G)| > 2. Then Γ(G) has no nonzero proper two-sided ideals. COROLLARY 14.3 Let G1 and G2 be two connected graphs with |V (Gi )|≤2 and there exists aa ∈ E(Gi ) for some a ∈ V (Gi ) or |V (Gi )| > 2. Then Γ(G1 ) ∼ = Γ(G2 ) if and only if |V (G1 )| = |V (G2 )|.

The transformation algebras of Bernstein graph algebras COROLLARY 14.4 with identity element.

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If G is a connected graph, then Γ(G) is an associative algebra

PROOF From Theorem 14.1 and Proposition 14.3 we need to study the following case: V (G) = {a, b} and E(G) = {ab} with a =b. In this case, by Proposition 14.3, we have Γ(G) = Lab , L2ab and L2ab is the identity element of Γ(G). : Note that if G is a graph, then for all edge ab ∈ E(G) with a =b, we have Lab , L2ab is a

cyclic group of order 2 and F (Lab +L2ab ) and F (Lab −L2ab ) are two-sided ideals of Lab , L2ab . PROPOSITION 14.6 Let G be a graph and G1 , . . . , Gn its connected components. The following conditions hold: (a) Γ(G) = Γ(G1 ) ⊕ . . . ⊕ Γ(Gn ) (b) Γ(Gi ) is a two-sided ideal of Γ(G) (c) 11G = 11G1 + . . . + 11Gn is identity element of Γ(G) PROOF (a) For all α ∈ E(Gi ) and β ∈ E(Gj ) with i =j, we have Lα ◦ Lβ = 0, so Γ(G) = Γ(G1 ) + . . . + Γ(Gn ) Given gi ∈ Γ(Gi ) (i = 1, . . . , n) such that g1 + . . . + gn = 0, then for all u ∈ Uj = V (Gj ), 0 = (g1 + . . . + gn )(u) = gj (u), so gj = 0 (b) As f ◦ g = g ◦ f = 0, for all f ∈ Γ(Gi ) and g ∈ Γ(Gj ) with i =j, so we obtain (b) (c) Given gi ∈ Γ(Gi ) (i = 1, . . . , n), then ⎛ ⎞ ⎞⎛ ⎞ ⎛ n n n n ⎝ 11Gj ⎠ (gi ) = gi , so ⎝ 11Gj ⎠ ⎝ gi ⎠ = gi . j=1

j=1

j=1

i=1

COROLLARY 14.5 Let G be a graph and G1 , . . . , Gn its connected components. If |V (Gi )|≤2 and there exists aa ∈ E(Gi ) for some a ∈ V (Gi ) or |V (Gi )| > 2, then Γ(G) ∼ = M|V (G1 )| (F ) ⊕ . . . ⊕ M|V (Gn )| (F ).

14.3

The Associative Algebra T (G)

The following proposition has an easy proof. PROPOSITION 14.7

Let G be a graph. Then the following conditions hold:

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(a) L3e = 32 L2e − 12 Le ; {idA(G) , Le , L2e } is free (b) Lu ◦ Lz = 0, for all u ∈ U, z ∈ Z (c) Lu ◦ Lv = 0, for all u, v ∈ U (d) Le ◦ Lz = 12 Lz = Lz ◦ Le , for all z ∈ Z (e) Le ◦ Lu = 12 Lu , for all u ∈ U (f ) Lu ◦ Le = Lu ◦ L2e , for all u ∈ U If G is a graph, we denote by Δ(G) = gLc , gLc Le |g ∈ Γ(G), c ∈ V (G). PROPOSITION 14.8

If G is a graph, then we have the following conditions:

(a) L2e = Le − 14 11G (b) (Δ(G))2 = 0,

Δ(G)Γ(G) = 0,

Γ(G)Δ(G) ⊆ Δ(G)

PROOF (a) As L2e (e) = Le (e) = e, 11G (e) = 0 and L2e (z) = Le (z) = 11G (z) = 0, for all z ∈ Z, we have 1 1 1 1 (Le − 11G )(λe + u + z) = λe + u − u = λe + u = L2e (λe + u + z) 4 2 4 4 (b) If g1 , g2 ∈ Γ(G) and a, b ∈ V (G), then by Proposition 14.7, (g1 La )(g2 Lb ) = g1 (La g2 )Lb = 0 (g1 La )g2 = g1 (La g2 ) = 0

(g1 La Le )(g2 Lb ) = (g1 La Le )g2 =

1 g1 La g2 Lb = 0 2

1 g1 La g2 = 0 2

g1 (g2 La ) = (g1 g2 )La ∈ Δ(G)

THEOREM 14.2

If G is a graph, then T (G) = F idA(G) ⊕ F Le ⊕ Γ(G) ⊕ Δ(G).

PROOF Given λ1 , λ2 ∈ F, γ ∈ Γ(G) and δ ∈ Δ(G) such that λ1 idA(G) +λ2 Le +γ +δ = 0. By Proposition 14.7, we have 2λ1 Le + 2λ2 L2e + γ + δ = 0, then λ1 idA(G) + (λ2 − 2λ1 )Le − 2λ2 L2e = 0, so λ1 = λ2 = 0. For all u ∈ U , we have δ(u) = 0, then γ(u) = 0, so γ = 0 and δ = 0. Given J = F idA(G) ⊕ F Le ⊕ Γ(G) ⊕ Δ(G). We will prove that J is a subalgebra of T (G). By Proposition 14.8, we have L2e = Le − 14 11G ∈ F Le ⊕ Γ(G). By Proposition 14.7, for all γ ∈ Γ(G), Le γ = γLe = 12 γ and for all δ ∈ Δ(G), Le δ = 12 δ and δLe ∈ Δ(G). As Γ(G) is a subalgebra of T (G), then (Γ(G))2 ⊆ Γ(G), besides by Proposition 14.8, Δ(G)Γ(G) = 0, Γ(G)Δ(G) ⊆ Δ(G) and (Δ(G))2 = 0. Hence J is a subalgebra of T (G).

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Thus in J, we have the following multiplication: (λ1 idA(G) + λ2 Le + γ + δ)(λ1 idA(G) + λ2 Le + γ + δ ) = λ1 λ1 idA(G) + (λ1 λ2 + λ2 λ1 + λ2 λ2 )Le 1 1 1 − λ2 λ2 11G + (λ1 + λ2 )γ + (λ1 + λ2 )γ + γγ 4 2 2 1 + (λ1 + λ2 )δ + λ1 δ + γδ + λ2 δLe . 2

(14.2)

For α ∈ E(G), we have Lα ∈ Γ(G), then Lα ∈ J. Given a ∈ V (G), then La = 11G La ∈ Δ(G), because La (λe + u + z) =

1 1 λa + az = 11G ( λa + az) = (11G La )(λe + u + z). 2 2

Therefore J = T (G). COROLLARY 14.6 is a left ideal of T (G).

If G is a graph, then Δ(G) is a two-sided ideal of T (G) and Γ(G)

Note that if G is a graph, then 11G is a left identity element of Γ(G) ⊕ Δ(G), but is not a right identity element, because gLa Le 11G = 0. PROPOSITION 14.9 Let G be a connected graph such that |V (G)|≤2 and there exist aa ∈ E(G) for some a ∈ V (G) or |V (G)| > 2. If there is no dd for some d ∈ V (G), then Δ(G) = γab Lc | a, b, c ∈ V (G) PROOF

By Theorem 14.1, we have Γ(G) = γab | a, b ∈ V (G), then Δ(G) = γab Lc , γab Lc Le | a, b, c ∈ V (G).

Given γab Lc Le ∈ δ(G). We will prove that

0 if b =c , γab Lc Le = γad Ld if b = c where there is no dd ∈ E(G). If b =c, 1 1 (γab Lc Le )(λe + u + z) = (γab Lc )(λe + u) = λγab (c) = 0. 2 2 If b = c, 1 1 1 (γab Lb Le )(λe + u + z) = (γab Lb )(λe + u) = λγab (b) = λa, 2 2 2 1 1 1 (γad Ld )(λe + u + z) = γad ( λd + zd) = λγad (d) + γad (zd) = λa, 2 2 2 because d does not appear in zd, for all z ∈ Z.

Note that if there exist bc ∈ E(G), then (γab Lc )(bc) = γab (b) = a =0, then Δ(G) =0.

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A graph G is simple if the number of edges joining any two points is at most 1. PROPOSITION 14.10 following conditions hold:

If G is a simple graph, without loop and |V (G)|≥2, then the

(a) For all a, b ∈ V (G) γab Lb (x) =

1

2a

if x = e 0 if x ∈ V (G) ∪ E(G)

(b) For all a, b, c ∈ V (G), γab Lb = γac Lc (c) For all a ∈ V (G) and α = bc ∈ E(G),

a if x = α γac Lb (x) = γab Lc (x) = 0 if x ∈ {e} ∪ V (G) ∪ (E(G) \ {α}) PROOF (a) (γab Lb )(e) = 12 γab (b) = 12 a. As G has no loop, then xb =b for all x ∈ E(G), so (γab Lb )(x) = γab (xb) = 0. (b) By (a), we obtain (b). (c) (γab Lc )(α) = γab (b) = a. For all γ ∈ E(G) with γ =α, we have γc =b, because G is simple, so (γab Lc )(γ) = 0.

DEFINITION 14.2 Let G be a simple graph, without loop and |V (G)|≥2. Then we deﬁne by γae = 2γab Lb and γaα = γab Lc , where a, b, c ∈ V (G) and α = bc. Clearly,

γae (x) =

a if x = e 0 if x ∈ V (G) ∪ E(G)

a if x = α 0 if x ∈ {e} ∪ V (G) ∪ (E(G) \ {α})

γre if s = a γae ◦ γrs = 0 γae ◦ Le = γae γrs ◦ γae = 0 if s =a

γrα if s = a γrs ◦ γaα = γaα ◦ γrs = 0 0 if s =a γaα (x) =

where r, s, a ∈ V (G) and α ∈ E(G). PROPOSITION 14.11 Let G be a simple connected graph, without loop and |V (G)| > 2. Then the following conditions hold: (a) Δ(G) = γae , γbα | a, b ∈ V (G), α ∈ E(G) (b) dim Δ(G) = |V (G)|(1 + |E(G)|) PROOF

By Proposition 14.9, we have Δ(G) = γab Lc | a, b, c ∈ V (G).

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(a) Given γab Lc ∈ Δ(G), then we have the following cases: (i) If b = c, then γab Lb = 12 γae (ii) If b =c and there exists α = bc ∈ E(G), then γab Lc = γaα (iii) If b =c and there is not edges with vertices b and c, then for all γ ∈ E(G), we have γc =b, so (γab Lc )(γ) = 0, therefore γab Lc = 0 ) is free. (b) We will prove that (γae , γbαa,b∈V (G) α∈E(G)

Given λa , λb,α ∈ F (a, b ∈ V (G), α ∈ E(G)) such that μ= λa γae + λb,α γbα = 0, a∈V (G)

then μ(e) =

a,b∈V (G) α∈E(G)

λa a = 0, so λa = 0, for all a ∈ V (G).

a∈V (G)

Given γ ∈ E(G), then 0 =

λb,α γbα (γ) =

a,b∈V (G) α∈E(G)

λb,γ b, so λb,γ = 0, for all

b∈V (G)

b ∈ V (G).

Let G be a graph such that V (G) = {a, b} with a =b and

PROPOSITION 14.12 E(G) = {ab}. Then (a) Γ(G) = Lab , L2ab

(b) Δ(G) = γae , γbe , γaα , γbα , where α = ab (c) dim Δ(G) = 4 PROOF (a) By proof of Corollary 14.4, we have (a). (b) As 11G = L2ab , then L2ab La = La and L2ab Lb = Lb , so Δ(G) = Lab La , Lab Lb , Lab La Le , Lab Lb Le , La , Lb , La Le , Lb Le . We observe that (i) γae = 2Lab Lb Le , (ii) γaα = Lab La −

γbe = 2Lab La Le

1 2 γbe ,

(iii) La = 12 γae + γbα ,

γbα = Lab Lb − 12 γae

Lb = 12 γbe + γaα

(iv) γae = 2La Le , γbe = 2Lb Le Hence Δ(G) = γae , γbe , γaα , γbα .

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(c) The proof of (c) is the same proof as Proposition 14.11.

From Propositions 14.11 and 14.12, we have PROPOSITION 14.13 If G is a simple connected graph, without loop and |V (G)| ≥2, then Δ(G) = γae , γbα | a, b ∈ V (G), α ∈ E(G) and dim Δ(G) = |V (G)|(1 + |E(G)|). By Theorems 14.1 and 14.2, Proposition 14.13, and γe = Le − 12 11G , we have the next theorem. THEOREM 14.3 If G is a simple connected graph, without loop and |V (G)| > 2, then the following conditions hold: (a) T (G) = idA(G) , γe , γab , γce , γdα | a, b, c, d ∈ V (G), α ∈ E(G) (b) dim T (G) = 2 + |V (G)|(|V (G) + |E(G)| + 1) By Corollary 14.3 and Proposition 14.13, we have the following proposition. PROPOSITION 14.14 If G1 and G2 are simple connected graphs, without loop and |V (Gi )| > 2 (i = 1, 2). Then Γ(G1 ) ∼ = Γ(G2 ) and Δ(G1 ) ∼ = Δ(G2 ) if and only if |V (G1 )| = |V (G2 )| and |E(G1 )| = |E(G2 )|.

14.4

Acknowledgments

The ﬁrst author is partially supported by FAPESP proc. 03/02162-6. The second author is partially supported by DGICYT, PB94-1311-C03-01.

References [1] Costa, R. and Guzzo Jr, H.: Indecomposable baric algebras, Linear Algebra Appl. 183, 223-236 (1993). [2] Costa, R. and Guzzo Jr, H.: Indecomposable baric algebras, II, Linear Algebra Appl. 196, 233-242 (1994). [3] Costa, R. and Guzzo Jr, H.: Graphs and Bernstein algebras, Comm. Algebra 25, no. 7, 2129-2139 (1997). [4] Costa, R., Neuburg, M. and Suazo, A.: Some results in the theory of Bernstein algebras, submitted to Comm. Algebra.

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[5] Grishkov, A., Costa, R.: Graphs and non-associative algebras, Algebra, 11. J. Math. Sci. (New York) 93, no. 6, 877–882 (1999). [6] Lyubich, Yu. I.: Mathematical Structures in Population Genetics (Biomathematics 22, Springer, Berlin-Heidelberg-New York, 1992). [7] W¨ orz, A.: Algebras in Genetics (Lecture Notes in Biomathematics 36, Springer, Berlin-Heidelberg-New York, 1980).

Chapter 15 Central Elements of Minimal Degree in the Free Alternative Algebra Irvin Roy Hentzel Department of Mathematics, Iowa State University, Ames Luiz Antonio Peresi Departamento de Matem´atica, Universidade de S˜ ao Paulo, Brazil 15.1 15.2 15.3 15.4 15.5 15.6 15.7

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Representation techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Proof of Theorem 15.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Annihilators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Further work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

195 197 199 199 201 203 203 203

Abstract Filippov (1999) conjectured that 7 is the minimal degree of nonzero elements in the center of the free alternative algebra. We prove that this conjecture is true. We give all the central elements of degree 7 in the free alternative algebra over the ﬁeld Z103 . We give some results that indicate that these elements are also central elements in characteristic zero. Key words: free alternative algebra, central elements, free Malcev algebra, annihilators 2000 MSC: 17D05, 17D10, 20C30, 17D08

15.1

Introduction

In any nonassociative algebra we deﬁne the associator (a, b, c), the commutator [a, b], and the symmetric product a ◦ b by (a, b, c) := (ab)c − a(bc), [a, b] := ab − ba and a ◦ b := ab + ba. An alternative algebra is a nonassociative algebra satisfying the identities (a, b, b) = 0 and (a, a, b) = 0. Alternative algebras may be described as nonassociative algebras where any subalgebra generated by two elements is associative. Under the skew product [a, b], an alternative algebra becomes an anticommutative algebra satisfying the identity malcev(a, b, c, d) := (ac)(bd) − ((ab)c)d − ((bc)d)a − ((cd)a)b − ((da)b)c = 0. This algebra is called a special Malcev algebra. A central element in an alternative algebra (A, +, .) becomes an annihilator in the special Malcev algebra (A, +, [, ]). Any anticommutative algebra satisfying malcev(a, b, c, d) = 0 is called a Malcev algebra.

195

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Until now, it is unknown whether the center of the free alternative algebra has a ﬁnite number of generators as a T-subalgebra. Therefore, it is of interest to ﬁnd nonzero elements of minimal degree in the center of the free alternative algebra. This problem is related to the problem of ﬁnding nonzero elements in the annihilator of the free Malcev algebra. It is known that there are nonzero central elements in the free alternative algebra. Dorofeev [1] and Shelipov [9] found (independently) an element of degree 32. Shestakov [10] found an element of degree 12. In the free alternative algebra with more than ﬁve generators, Filippov found an element of degree 8 in [3] and an element of degree 7 in [4]. We presented in [7] a new element of degree 7. In [4] Filippov conjectured that the minimal degree of nonzero central elements in the free alternative algebra is 7. In this chapter we prove that this conjecture is true. We give all of the degree 7 central elements in the free alternative algebra over the ﬁeld Z103 . To state our results we introduce the concept of alternating sum. To compute the alternating sum alt{x1 ,x2 ,...,xk } f (x1 , x2 , ..., xk , xk+1 , ..., xn ) for a function of n arguments, take each permutation π of the letters x1 , x2 , ..., xk , aﬃx the sgn(π) to f (xπ(1) , xπ(2) , ..., xπ(k) , xk+1 , ..., xn ) and then sum these k! terms. THEOREM 15.1 In the free alternative algebra over Z103 on generators {a, b, c, d, e, f, g} we have the following: (i) There are no nonzero central elements of degree < 7. (ii) The following are nonzero central elements of degree 7: { 3 [e, [e, (a, b, (e, c, d))]] + 4 [e, [e, (e, a, (b, c, d))]] alt{a,b,c,d}

+6 [e, [a, (e, b, (e, c, d))]] + 3 [e, a] ◦ (e, b, (e, c, d)) −6 (e, a, (e, b, (e, c, d))) },

(15.1)

{ (d, e, (d, a, (e, b, c))) − (d, e, (e, a, (d, b, c)))

alt{a,b,c}

−(d, e, (a, b, (d, e, c))) − (d, a, (d, e, (e, b, c))) +(e, a, (d, e, (d, b, c))) + (a, b, (d, e, (d, e, c))) },

(15.2)

{ 4 (f, a, (f, b, (c, d, e))) + 3 (f, a, (b, c, (f, d, e)))

alt{a,b,c,d,e}

−3 (a, b, (f, c, (f, d, e))) + 5 ((f, a, b), f, (c, d, e)) },

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

(15.3) (15.4)

alt{a,b,c,d,e,f,g}

(iii) All the central elements of degree 7 are consequences of the alternative identities of degree 7 and central elements (15.1)–(15.4). Filippov’s central element is described by a series of constructions. Let J(x, y, z) = (xy)z + (yz)x + (zx)y, {x, y, z} = J(x, y, z) + 3x(yz),

Central elements of minimal degree in the free alternative algebra

197

s(z, y, t, a, b) = −{J(t, a, b), z, y} − {J(y, a, b), t, z} + {J(z, a, b), y, t}, f (z, y, t, a, b, x) = s(z, y, t, a, b)x − s(zx, y, t, a, b). Let fˆ(z, y, t, a, b, x) be the function obtained from f (z, y, t, a, b, x) by replacing the products by skew products. Finally, Filippov’s central element is [fˆ(z, y, t, a, b, x), z]. We prove that Filippov’s central element is equivalent to our central element (15.3). The central element we presented in [7] is (15.4). We prove that the central elements (15.2), (15.3), and (15.4) can be expressed in terms of the skew product. Furthermore, we prove that each one of these expansions of (15.2), (15.3), and (15.4) is an annihilator in the free Malcev algebra. We prove that (15.1), (15.2), (15.3), and (15.4) are in the radical of the free alternative algebra.

15.2

Representation Techniques

Let Sn be the symmetric group and F Sn be the group ring of Sn over a ﬁeld F . We assume that the characteristic of F is zero or greater than n. We denote by I the identity element of F Sn . To ﬁnd nonzero central elements of degree n, we look for a multilinear polynomial p(x1 , x2 , . . . , xn ) in the free alternative algebra such that [p(x1 , x2 , . . . , xn ), xn+1 ] = 0 and p(x1 , x2 , . . . , xn ) = 0. We start with the alternative identities (x, y, z) + (y, x, z) = 0 and (x, y, z) + (x, z, y) = 0. Suppose we are looking for central elements of degree n. We ﬁrst construct all multilinear identities of degree n + 1 in the T-ideal generated by the alternative identities. For our purposes, we only need to store a (ﬁnite) representative collection of these identities. We indicate this collection as I = {fi (x1 , x2 , x3 , . . . , xn+1 ) | i ∈ Λ}, where Λ is just some index set. It is necessary that every degree n + 1 multilinear element f in the T-ideal generated by the alternative identities is expressible as f=

λπ fi (xπ(1) , xπ(2) , . . . , xπ(n+1) ).

π∈Sn+1 , λπ ∈F, i∈Λ

The set I is obtained by “lifting” the alternative identities to identities of degree four, then to identities of degree ﬁve, and continuing until we reach degree n + 1. In the process of lifting, an identity f (x1 , x2 , x3 , . . . , xk ) with k arguments generates k +2 identities. Each of these lifted identities have k + 1 arguments. They are xk+1 f (x1 , x2 , x3 , . . . , xk ), f (xk+1 x1 , x2 , x3 , . . . , xk ), f (x1 , xk+1 x2 , x3 , . . . , xk ), f (x1 , x2 , xk+1 x3 , . . . , xk ), . . . , f (x1 , x2 , x3 , . . . , xk+1 xk ), f (x1 , x2 , x3 , . . . , xk )xk+1 . If we augment the set I by including as more identities, fi (xπ(1) , xπ(2) , . . . , xπ(n+1) ) for all (n + 1)! permutations π of 1, 2, . . . , n + 1, and i ∈ Λ, we get a spanning set of the T-ideal of multilinear identities of degree n + 1 for the free alternative algebra. We perform this extension to all substitutions in two steps. The ﬁrst step is an actual substitution. The second step is a virtual substitution accomplished by using the representations of the group ring F Sn .

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The ﬁrst step creates the set I # , which has n + 1 times as many elements as I. For each element fi (x1 , x2 , . . . , xn+1 ) in I, we interchange xj and xn+1 . The set I # contains the elements fi (xn+1 , x2 , x3 , . . . , xj , . . . , x1 ), fi (x1 , xn+1 , x3 , . . . , xj , . . . , x2 ) fi (x1 , x2 , xn+1 , . . . , xj , . . . , x3 ), . . . , fi (x1 , x2 , x3 , . . . , xn+1 , . . . , xj ), . . . fi (x1 , x2 , x3 , . . . , xj , . . . , xn+1 ) for each i in Λ. This set I # has the property that we can get a spanning set of the multilinear elements of the T-ideal of degree n + 1 by substituting the permutations of x1 , x2 , . . . , xn . Suppose that I = {p1 , p2 , . . . , pq }. Each pi is composed of terms of degree n + 1 with xn+1 occupying one of the positions. We sort the terms of pi into classes based on their association type and the position of xn+1 . There are (n + 1)c(n + 1) of these types, where is the Catalan number. We label these types T1 , T2 , ..., T(n+1)c(n+1) . All c(n) = n1 2n−2 n−1 the terms of the same type can be thought of as an element of the group ring F Sn . The identity pi can be expressed as gi1 ⊕ gi2 ⊕ · · · ⊕ gi(n+1)c(n+1) . This huge matrix is given in Table 15.1. Table 15.1. Central elements T1 g11 g21 . . . gq1 h11 h21 . . . hc(n)1

... ... ... . . . ... ... ... . . . ...

T(n+1)c(n+1) g1(n+1)c(n+1) g2(n+1)c(n+1) . . . gq(n+1)c(n+1) h1(n+1)c(n+1) h2(n+1)c(n+1) . . . hc(n)(n+1)c(n+1)

[T1 , xn+1 ] 0 0 . . . 0 I 0 . . . 0

[T2 , xn+1 ] 0 0 . . . 0 0 I . . . 0

... ... ... . . . ... ... ... . . . ...

[Tc(n) , xn+1 ] 0 0 . . . 0 0 0 . . . I

The types on the right are the c(n) association types of degree n, T1 , . . . , Tc(n) , inside the commutator. If we let wi be x1 x2 . . . xn associated in type Ti , then the hij represent the expansion of [wi , xn+1 ] in terms of the (n + 1)c(n + 1) types Tj . We now evaluate each of these group ring elements in each of the matrix representations of Sn . In each representation we get a huge matrix that we reduce to row canonical form. The portion of the row canonical form with leading ones under the [Tj , xn+1 ] portion are central elements in the free alternative algebra. We compare these central elements with the identities of the free alternative algebra and look for the new leading ones. These are the nonzero central elements in the free alternative algebra of degree n.

Central elements of minimal degree in the free alternative algebra

15.3

199

Proof of Theorem 15.1

To prove Theorem 15.1 we use the representation techniques described in Section 15.2. We perform row reductions modulo characteristic 103. Therefore we assume that F = Z103 . There are 15 representations of S7 . They are listed in Table 15.2. There are c(7) = 132 association types of degree 7. We lift the alternative identities to degree 7 and then compute the ranks of these alternative identities in all 15 representations. Since any alternative identity of degree 7 is a central identity, we compare ranks to tell if there are any nonzero central elements. Where the ranks diﬀer, we look for the additional leading one in the right half of the row canonical form of the matrix given in Table 15.1. The row with this additional leading one is a nonzero central element. There is an additional leading one in each of representations 11, 13, 14, 15. The ranks are given in Table 15.2. The elements (15.1), (15.2), (15.3), (15.4) are those nonzero central elements. Table 15.2. Rank of the matrices Representation 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

15.4

Partition 7 61 52 511 43 421 4111 331 322 3211 31111 2221 22111 211111 1111111

Alternative identities 131 786 1834 1960 1834 4579 2606 2748 2745 4566 1946 1822 1815 768 123

Central elements 131 786 1834 1960 1834 4579 2606 2748 2745 4566 1947 1822 1816 769 124

Annihilators

The central elements (15.2), (15.3), and (15.4) are Malcev admissible. That means that they can be expressed in terms of the skew product. This is immediate from the identity 6(a, b, c) = [[a, b], c] + [[b, c], a] + [[c, a], b] that holds in any alternative algebra (see [7, Lemma 1]). We denote by m2 , m3 , and m4 the expansions of the central elements (15.2), (15.3), and (15.4) in terms of the skew product. When written in terms of the skew product the number of terms in the central elements increases by a factor of 27. Although the central elements become unmanageable written

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in the skew product, it is necessary to do so if we wish to check whether m2 , m3 , and m4 are annihilators in the free Malcev algebra. We already know they are annihilators in any special Malcev algebra.

THEOREM 15.2 Malcev algebra.

The elements m2 , m3 , and m4 are nonzero annihilators in the free

PROOF We lift anticommutativity and the Malcev identity m(a, b, c, d) = 0 to degree 8. We obtain the rank in each of the 22 representations. We use only 23 association types this time using anticommutativity to reduce the number from c(8) = 429. Then we add additional identities m2 x8 = 0, m3 x8 = 0, and m4 x8 = 0. We check that the ranks do not increase with the addition of these identities. The ranks are given in Table 15.3. This shows that these three identities are consequences of the anticommutative and Malcev identities. Therefore the elements m2 , m3 , and m4 are annihilators in the free Malcev algebra. Since (15.2), (15.3), and (15.4) are nonzero in the free alternative algebra, and they are Malcev admissible, then m2 , m3 , and m4 are nonzero in the free special Malcev algebra. Therefore m2 , m3 , and m4 are nonzero in the free Malcev algebra.

Table 15.3. Anticommutative and Malcev identities: degree 8 Representation 1 2 3 4 5 6 7 8 9 10 11

Partition 8 71 62 611 53 521 5111 44 431 422 4211

Rank 23 160 458 479 640 1463 799 321 1600 1281 2054

Representation 12 13 14 15 16 17 18 19 20 21 22

Partition 41111 332 3311 3221 32111 311111 2222 22211 221111 2111111 11111111

Rank 800 958 1280 1596 1461 478 320 637 457 157 22

The central element (15.1) is not Malcev admissible. We prove this by using types T1 , S1 , S2 , . . . , S11 , where Ti are the association types of degree 7 and Si are the skew types of degree 7. We set up a matrix with the degree 7 consequences of the alternative . We add to the matrix the expansion of the monomial identities under types T1 , T2 , . . . , Tc(7) x1 x2 x3 x4 x5 x6 x7 bracketed as type Si . We reduce to row canonical form and obtain the rank in each of the 15 representations. Then we add (15.1) to the list of identities and look where the additional stair step one of the row canonical form appears. It appears in representation 11 under a type Ti . This means that (15.1) is not expressible in terms of the skew product. , T2 , . . . , Tc(7)

Central elements of minimal degree in the free alternative algebra

15.5

201

Discussion

The Filippov central element is equivalent to our central element (15.3) modulo alternativity. When either of them is added to the alternative identities, the rank increases only in representation 14. There, it increases by exactly one and the new row from Filippov’s central element and the new row from (15.3) are the same. An alternative way to search for central identities was used in [1]. They found identities satisﬁed by the Cayley-Dickson algebra using a probabilistic technique. Then from the identities expressible using only associators, they looked for linear combinations that vanished inside a commutator in the free alternative algebra. Because they only looked for central elements expressible as associators, they limited themselves from ﬁnding all central elements of degree 7 (see [1], p. 272, Concluding Remarks). Shestakov [11] found an inﬁnite family of elements, which are annihilators in the free Malcev algebra. These elements, interpreted in terms of the skew product, give an inﬁnite family of elements in the center of the free alternative algebra. The ﬁrst element of this family is our element (15.4). Shestakov and Zhukavets [12] found a basis of the space of skew-symmetric annihilators of the free Malcev algebra. A semisimple alternative algebra is a subdirect sum of associative and Cayley-Dickson algebras. Thus any element of a free alternative algebra, which is zero in every associative and Cayley-Dickson algebra, must be in the radical (see [198], p. 271, Corollary 1). The span of the associators and commutators do not contain the identity element. Therefore (15.2), (15.3), (15.4) have to evaluate to zero in the Cayley-Dickson algebra. Therefore (15.2), (15.3), (15.4) are in the radical of the free alternative algebra. Because (15.1) contains a symmetric product, it is not as easy to establish that it vanishes in the Cayley-Dickson algebra. We need three results to establish this. In any alternative algebra we have [a ◦ b, c] = a ◦ [b, c] + b ◦ [a, c] and [a, b] ◦ (a, b, c) = 0. In a Cayley-Dickson algebra [a, (b, c, d) ◦ (e, f, g)] = 0. Since element (15.1) evaluates to an element of the center in any alternative algebra, (15.1) has to evaluate to a scalar multiple of the identity element of the Cayley-Dickson algebra. The only terms that produces a nonzero coeﬃcient for the identity element are the terms in alt{a,b,c,d} [e, a] ◦ (e, b, (e, c, d)). In any alternative algebra [(e, x, y) ◦ (e, z, w), e] = (e, x, y) ◦ [(e, z, w), e] + [(e, x, y), e] ◦ (e, z, w) = −[e, (e, z, w)] ◦ (e, x, y) − [e, (e, x, y)] ◦ (e, z, w) = [e, x] ◦ (e, (e, z, w), y) + [e, z] ◦ (e, (e, x, y), w) = −[e, x] ◦ (e, y, (e, z, w)) − [e, z] ◦ (e, w, (e, x, y)). In a Cayley-Dickson algebra, the left-hand side is zero. Therefore, in a Cayley-Dickson algebra, [e, x]◦(e, y, (e, z, w))+[e, z]◦(e, w, (e, x, y)) = 0. In the alternating sum on {a, b, c, d} the terms [e, a] ◦ (e, b, (e, c, d)) and [e, c] ◦ (e, d, (e, a, b)) have the same sign. Their sum is zero since in a Cayley-Dickson algebra [e, x] ◦ (e, y, (e, z, w)) + [e, z] ◦ (e, w, (e, x, y)) = 0. Similarly, the rest of the 24 terms pair oﬀ and cancel leaving the alternating sum equal to zero. The central element (15.1) evaluates to zero in the Cayley-Dickson algebra. It also evaluates to zero in any associative algebra. Therefore (15.1) is in the radical of the free alternative algebra. If there were a central element over the rationals, then it would generate a central element modulo 103 so we know there are no central elements of degree less than 7 over the rationals.

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The hard part is to construct the central elements using the representation techniques described in Section 15.2. Constructing the central elements requires working with 8∗c(8)+ c(7) = 3564 types. Once the central elements are constructed we can perform further checking. There are three things to check. We have to establish that each element is in the center, that each element is not zero, and that the four elements are independent. To show that the elements are in the center, we lift the alternative identities to degree 8. There are c(8) = 429 association types in degree 8. We obtain the rank of the alternative identities in each of the 22 representations of degree 8. The ranks are listed in Table 15.4. Then we add the central elements (15.1), (15.2), (15.3), (15.4) as though they are the identities [(15.1), x8 ] = 0, [(15.2), x8 ] = 0, [(15.3), x8 ] = 0, [(15.4), x8 ] = 0. We obtain the ranks in each representation and verify that none of them increases. This means that [(15.1), x8 ] = 0, [(15.2), x8 ] = 0, [(15.3), x8 ] = 0, and [(15.4), x8 ] = 0 are consequence of the alternative identities. In other words, (15.1), (15.2), (15.3), and (15.4) are in the center of the free alternative algebra. Table 15.4. Alternative identities Representation 1 2 3 4 5 6 7 8 9 10 11

Partition 8 71 62 611 53 521 5111 44 431 422 4211

Rank 428 2996 8560 8982 11984 27384 14962 5992 29954 23959 38491

Representation 12 13 14 15 16 17 18 19 20 21 22

Partition 41111 332 3311 3221 32111 311111 2222 22211 221111 2111111 11111111

Rank 14954 17970 23954 29933 27353 8958 5981 11962 8532 2970 418

To show that the central elements (15.1), (15.2), (15.3), and (15.4) are not zero and independent, we lift the alternative identities to degree 7. There are c(7) = 132 association types. We obtain the ranks in each of the 15 representations. They are listed in Table 15.2. Then we add the elements separately to the list of alternative identities of degree 7. Each central element increases the rank by one in a diﬀerent representation. This shows that none of the elements are consequences of the alternative identities, and that no two of them imply the third in the presence of alternativity. Analogously, we show that expansions m2 , m3 , and m4 are not zero and independent. We lift the anticommutative and Malcev identities to degree 7. We obtain the ranks in each of the 15 representations. They are listed in Table 15.5. Then we add the expansions separately to the list of identities of degree 7. Each expansion increases the rank by one. The expansion m2 gives rank 150 in representation 13. The expansion m3 gives rank 63 in representation 14. The expansion m4 gives rank 10 in representation 15. Therefore m2 , m3 , and m4 are not consequences of the anticommutative and Malcev identities, and no two of them imply the third in the presence of these identities.

Central elements of minimal degree in the free alternative algebra

15.6

203

Further Work

We would like to see a direct proof that the identities (15.1), (15.2), and (15.3) are central identities. We would like to see a direct proof that identity (15.3) is equivalent to Filippov’s central element. Since the identities m2 x8 = 0, m3 x8 = 0, and m4 x8 = 0 hold in the free Malcev algebra, these identities are not special Malcev identities. The element (15.1) is not Malcev admissible. We wonder if there might be some way to use (15.1) to construct a special Malcev identity. The representation techniques of Section 15.2 can be generalized to ﬁnd elements in the nucleus of the free alternative algebra, and also to look for special Malcev identities. Table 15.5. Anticommutative and Malcev identities: degree 7 Representation 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

15.7

Partition 7 61 52 511 43 421 4111 331 322 3211 31111 2221 22111 211111 1111111

Anticommutative and Malcev identities 11 65 152 162 152 379 215 227 226 377 161 149 149 62 9

Acknowledgments

This chapter was written while the second author held grants from CNPq of Brazil and FAPESP. Part of the research was done when this author was visiting Iowa State University on a grant from FAPESP.

References [1] Bremner, M., Hentzel, I. R. (2002). Identities for the associator in alternative algebras. J. Symbolic Computation 33:255-273.

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[2] Dorofeev, G. V. (1973). Centers of nonassociative rings. Algebra and Logic 12:297-309. [3] Filippov, V. T. (1982). Free Mal’tsev algebras and alternative algebras. Algebra and Logic 21:58-73. [4] Filippov, V. T. (1999). Centers of Mal’tsev and alternative algebras. Algebra and Logic 38:335-350. [5] Hentzel, I. R. (1977). Processing identities by group representation. In Beck, R. E., Kolman, B., eds. Computers in Nonassociative Rings and Algebras. New York: Academic Press, pp. 13-40. [6] Hentzel, I. R. (1978). Alternators of a right alternative algebra. Trans. Amer. Math. Soc. 242:141-156. [7] Hentzel, I. R., Peresi, L. A. (2003). A nonzero element of degree 7 in the center of the free alternative algebra. Comm. Alg. 31:1279-1299. [8] Myung, H. C. (1986). Malcev admissible algebras. Progress in Mathematics Vol. 64. Boston: Birkhauser Boston Inc. [9] Shelipov, A. N. (1973). Some properties of kernel of an alternative ring. Mat. Issled. 8:183-187. [10] Shestakov, I. P. (1976). Centers of alternative algebras. Algebra and Logic 15:214-226. [11] Shestakov, I.P. (2003). Free Malcev superalgebra on one odd generator. J. Algebra Appl. 2:451-461. [12] Shestakov, I.P., Zhukavets, N. (2004). Universal multiplicative envelope of free Malcev superalgebra on one odd generator, preprint. [13] Zhevlakov, K. A., Slin’ko, A. M., Shestakov, I. P., Shirshov, A. I. (1982). Rings That Are Nearly Associative. New York: Academic Press.

Chapter 16 Composition, Quadratic, and Some Triple Systems Noriaki Kamiya Center for Mathematical Sciences, University of Aizu, Japan Susumu Okubo Department of Physics and Astronomy, University of Rochester, New York 16.1 16.2 16.3 16.4 16.5

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Relationship between triple systems and bilinear algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Composition triple systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Final comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

205 206 211 224 228 228

Abstract We will discuss the mutual relationships between triple and bilinear products for many triple systems. Especially, the octonionic triple product can be shown to be realizable in terms of the bilinear Hurwitz product. We also study an analog of the Hurwitz theorem for composition triple systems. As an application, the existence of quintuple octonionic identities is demonstrated. Keywords: composition triple systems, nonassociative algebras, Yang-Baxter equations 2000 MSC: 17B25; 17A75

16.1

Introduction

In the authors’ judgment, it seems that composition algebra was ﬁrst introduced by Hurwitz at the end of the 19th century. The composition algebra is characterized by quaternion and octonion algebras permitting composition N (xy) = N (x)N (y) called quadratic norm N (the square of the absolute value), that is, describing for the inner product < a|b > of a vector space, this norm is denoted by < xy|xy >=< x|x >< y|y > . Then Zorn studied them in the framework of alternative algebras, subsequently many mathematicians and physicists are investigating these subjects called nonassociative algebras (e.g., quadratic, alternative, ﬂexible, involutive and Jordan algebras). For these nonassociative algebras, we will mainly refer to the books [J], [O.5], [Sch], and [Z-S-S-S]. From an other viewpoint, for the construction of simple Lie algebras except G2 , F4 , and E8 types, it is well known that Jordan algebras are a useful concept as worked by the study 205

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of Koecher, Tits, and Kantor. Jordan algebras can be characterized in terms of the triple product xyz := x(yz) + (xy)z − y(xz). This triple product has the properties as follows; xyz = zyx ab(xyz) = (abx)yz − x(bay)z + xy(abz). A triple system with above properties is said to be a Jordan triple system. Thereafter, in order to construct all classical, exceptional simple Lie algebras, Lie superalgebras, and Jordan superalgebras, the notation of several triple systems were produced from a generalization of Jordan triple systems (e.g., generalized Jordan triple systems of second order, (ε, δ)-Freudenthal-Kantor triple systems, and δ-Jordan-Lie triple systems). For these constructions, we would like to refer the articles of authors and earlier references quoted therein as follows; [E-K-O],[K.1],[K.2],[K.3],[K-K], [K-O.1],[K-O.2],[K-O.3],[O-K.1], and [O-K.2]. The aim of our study is to give a uniﬁed explicit framework for the relationships between triple and bilinear products for many triple systems, which play an important role in mathematical physics and geometrical phenomena. In this chapter, the following are main Propositions. PROPOSITION 16.1 identities

If a vector space V with a triple product xyz satisﬁes the

xy(yxz) =< x|x >< y|y > z

and

< u|xyv >=< v|yxu >,

then this triple system is a composition triple system, i.e., < xyz|xyz >=< x|x >< y|y >< z|z > . In particular, if we put x · y := xey, then we have < x · z|x · z >=< x|x >< z|z > (this means the deﬁnition of composition algebra), where e is any ﬁxed element of V such that < e|e >= 1. PROPOSITION 16.2 composition triple system.

A quadratic weakly alternative triple system is an involutive

Throughout this article, we shall consider only algebras and triple systems that are ﬁnite dimensional over a ﬁeld F of characteristic = 2, unless otherwise is speciﬁed.

16.2

Relationship between Triple Systems and Bilinear Algebras

In order to render this chapter as self-contained as possible, we will ﬁrst recall the deﬁnition of a Jordan triple system.

Composition, quadratic, and some triple systems

207

Let V be a vector space over a ﬁled F of characteristic = 2. A triple (or ternary) linear product speciﬁed by juxtaposition xyz for x, y, z ∈ V is a linear map xyz : V × V × V → V in contrast to the familiar bilinear product xy : V × V → V. In general, there exists some relationship between a triple and bilinear product. A familiar example is that of the Jordan triple system [J], where the triple product satisﬁes the conditions: xyz = zyx uv(xyz) = (uvx)yz − x(vuy)z + xy(uvz)

(i) (ii)

(16.1a) (16.1b)

for u, v, x, y, z ∈ V. Suppose that xy is a bilinear Jordan product satisfying the Jordan relations xy = yx

(16.2a)

{(xx)y}x = (xx)(yx).

(16.2b)

xyz := (xy)z + x(yz) − y(xz)

(16.3)

(i) (ii) Then, the triple product given by

deﬁnes a Jordan triple system [J]. But the converse is not necessarily true. We, however, note that if a Jordan algebra has the unit element e, then Eq. (16.3) implies the validity of eye = y. The converse statement is also correct as in the following proposition. PROPOSITION 16.3 Let V be a Jordan triple system with the triple product xyz. Suppose that there exists an element e ∈ V satisfying exe = x

(16.4)

for all x ∈ V . When we introduce a bilinear product x ∗ y in V by x ∗ y := xey,

(16.5)

then the resulting bilinear algebra V ∗ is a unital Jordan algebra with e being the unit element, i.e., we have (i) (ii) (iii)

e ∗ x = x ∗ e = x,

(16.6a)

x ∗ y = y ∗ x, {(x ∗ x) ∗ y} ∗ x = (x ∗ x) ∗ (y ∗ x),

(16.6b) (16.6c)

provided that the underlying ﬁeld F is of characteristic neither 2 nor 3. Moreover, the triple product xyz is realizable as xyz = (x ∗ y) ∗ z + x ∗ (y ∗ z) − y ∗ (x ∗ z)

(16.7)

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in terms of the bilinear product x ∗ y deﬁned by Eq. (16.5). Note that Eq. (16.7) has the same form as Eq. (16.3). PROOF Equation (16.6b) follows immediately from Eqs. (16.1a) and (16.5). We next set v = y = e in Eq. (16.1b) to obtain ue(xez) = (uex)ez − x(eue)z + xe(uez). Setting u = y, and noting Eq. (16.5), this gives x(eye)z = (x ∗ y) ∗ z + x ∗ (y ∗ z) − y ∗ (x ∗ z), which leads to the validity of Eq. (16.7) because of Eq. (16.4). We now note that eee = e.

(16.8)

(16.9)

If we set x = e in Eq. (16.4). We then calculate ee(exe) = (eee)xe − e(eex)e + ex(eee) by using Eq. (16.1b). Together with Eqs. (16.5) and (16.9), this gives eex = exe = x, which proves e ∗ x = x as in Eq. (16.6a). When we set x = z = e in Eq. (16.1b), we also obtain uv(eye) = (uve)ye − e(vuy)e + ey(uve), which is rewritten as uvy + vuy = 2ey(uve).

(16.10)

Furthermore, by setting v = e this relation yields u ∗ y = 2eyu − euy. However, since the left side of this equation is symmetric for u ↔ y, this requires the validity of u ∗ y = eyu = euy. (16.11) Here, the assumption of the underlying ﬁeld F to be of characteristic = 3 is crucial. Equation (16.10) is then rewritten as uvy + vuy = 2y ∗ (u ∗ v)

(16.12)

since uve = evu = v ∗ u = u ∗ v. Especially, this gives xxy = y ∗ (x ∗ x).

(16.12 )

With the choices of x = y and v = e, Eq. (16.1b) also leads to ue(xxz) = (uex)xz − x(eux)z + xx(uez) or equivalently to u ∗ {z ∗ (x ∗ x)} = (u ∗ x)xz − x(u ∗ x)z + (u ∗ z) ∗ (x ∗ x)

(16.13)

Composition, quadratic, and some triple systems

209

because of Eqs. (16.5) and (16.12 ). Furthermore, we set u = x in Eq. (16.13) and note the validity of (x ∗ x)xz = x(x ∗ x)z (16.14) as we will prove shortly. Then, Eq. (16.13) for u = x gives Eq. (16.6c) when we replace z by y. In order to prove Eq. (16.14), we let x ↔ u and y ↔ v in Eq. (16.1b) to obtain uv(xyz) − xy(uvz) = (uvx)yz − x(vuy)z = −(xyu)vz + u(yxv)z. By setting x = y = u and v = e, the 2nd relation of the above equation now yields (xex)xz − x(exx)z = −(xxx)ez + x(xxe)z or equivalently, (x ∗ x)xz − 2x(x ∗ x)z = −{(x ∗ x) ∗ x} ∗ z.

(16.15)

However, we also have x(x ∗ x)z + (x ∗ x)xz = 2{(x ∗ x) ∗ x} ∗ z by using Eq. (16.12). From these two relations, we obtain 3x(x ∗ x)z = 3{(x ∗ x) ∗ x} ∗ z and 3(x ∗ x)xz = 3{(x ∗ x) ∗ x}ez so that these lead to x(x ∗ x)z = (x ∗ x)xz = {(x ∗ x) ∗ x} ∗ z and hence to the validity of Eq. (16.14). Again the condition of F being of characteristic = 3 is important for obtaining the desired result. This completes the proof of Proposition 16.3 Proposition 16.3 implies that there is a one-to-one correspodence between any unital Jordan algebra and a Jordan triple system satisfying the condition Eq. (16.4), if the underlying ﬁeld F is of characteristic = 2 and = 3. REMARK 16.1 We can slightly relax the condition Eq. (16.4) as follows. Instead, we suppose only that there exists e ∈ V satisfying eV e = V.

(16.16)

In that case, for any x ∈ V , there exists x ¯ ∈ V satisfying e¯ xe = x.

(16.17)

xyz = (x ∗ y¯) ∗ z + x ∗ (¯ y ∗ z) − y¯ ∗ (x ∗ z).

(16.18)

Then, Eq. (16.8) gives

If we replace y by y¯. Moreover, if V is ﬁnite dimensional, then x ¯ is uniquely determined by x. Furthermore, we could obtain ¯ = x, x

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provided that we assume an additional ansatz of e(exe)e = x. The purpose of this chapter is to oﬀer similar examples, where we can express both triple and bilinear products in terms of each other. For this purpose, we ﬁrst introduce the notion of a (ε, δ)-Freudenthal-Kantor triple system (hereafter referred to as (ε, δ)FKTS) (see [Y-O], [K.1], [K.2], [K-O.1], [K-O.2]). Let ε = ±1 and δ = ±1. For any triple system product xyz, two multiplication operators L(x, y) and K(x, y) are deﬁned by L(x, y)z : = xyz, K(x, y)z : = xzy − δyzx.

(16.19a) (16.19b)

[L(u, v), L(x, y)] = L(uvx, y) + εL(x, vuy),

(16.20a)

K(xyu, v) + K(u, xyv) + δK(x, K(u, v)y) = 0,

(16.20b)

If they satisfy

and we call the system a (ε, δ)FKTS. The simplest case is when we have K(x, y) = 0 identically. In that case, Eqs. (16.20a), (16.20b) can be equivalently described by

(i) (ii)

xyz = δzyx

(16.21a)

uv(xyz) = (uvx)yz + εx(vuy)z + xy(uvz).

(16.21b)

Any system satisfying Eqs. (16.21a), and (16.21b) is called a (ε, δ) Jordan triple system. The standard Jordan triple system speciﬁed by Eqs. (16.1a), (16.1b) correspond to the special case of ε = −1 and δ = 1, i.e., (−1, 1) Jordan triple system. REMARK 16.2 The case of (±1, −1) Jordan triple system is of some interest. Deﬁning x ∗ y again by using Eq. (16.5), i.e., x ∗ y := xey, we now obtain (i) (ii)

x ∗ y = −y ∗ x

(16.22a)

x ∗ (y ∗ z) + y ∗ (z ∗ x) + z ∗ (x ∗ y) = 0.

(16.22b)

In other words, it gives a Lie algebra. To demonstrate it, Eq. (16.21a) for δ = −1 immediately gives Eq. (16.22a) by setting y = e and changing x by y. Similarly, if we set v = y = e in Eq. (16.21b), we have ue(xez) = (uex)ez + εx(eue)z + xe(uez).

(16.23)

However, we obtain eue = 0 by using Eq. (16.21a) for δ = −1, so that Eq. (16.23) leads to Eq. (16.22b) after suitably changing variables. For this case, it is not possible to express xyz in terms of x ∗ y. Returning to the original discussion, the second interesting special case of (ε, δ)FKTS is obtained as follows. We suppose that V possesses a bilinear form < .|. > satisfying < y|x >= −ε < x|y >,

(16.24)

Composition, quadratic, and some triple systems

211

which is symmetric for ε = −1 but antisymmetric for ε = +1. We now assume the validity K(x, y) = 2 < x|y > Id,

(16.25)

where Id stands for the identity map in V . If < .|. > is nonzero, Eqs. (16.20a), (16.20b) can be shown (e.g., [K-O.1]) to be equivalent to the validity of (i) (ii) (iii) (iv) (v)

ε = δ, xyz − εyxz = −2ε < x|y > z,

(16.26a) (16.26b)

xyz − εzyx = 2 < x|z > y, uv(xyz) = (uvx)yz + εx(vuy)z + xy(uvz),

(16.26c) (16.26d)

< y|x >= −ε < x|y > .

(16.26e)

Any triple system satisfying Eqs. (16.26a)–(16.26e) is called a (ε, ε) balanced FreudenthalKantor triple system (hereafter, referred to as (ε, ε)BFKTS). We have added here the factor of 2 in the right side of Eqs. (16.26b) and (16.26c) for later conveniences in contrast to the usual convention. This can be readily rectiﬁed, if we so wish, by replacing xyz → 2xyz. Then by straightforward calculations, we have the following proposition. PROPOSITION 16.4

For any (ε, ε)BFKTS, we have

< x|uvy >=< y|vux >=< u|xyv >=< v|yxu > .

(16.27)

In our forthcoming work, we will discuss the relation between the triple product of (−1, −1)-BFKTS and the bilinear product of same underlying vector space.

16.3

Composition Triple Systems

The idea presented in the previous section can be readily extented to many other cases. For example, we may call a system satisfying uv(xyz) = (uvx)yz

(16.28)

to be an associative triple system by the following reason. For any ﬁxed element e ∈ V, we introduce a bilinear product by xy := xey. (16.29) We have here and hereafter changed the notation from x ∗ y in Eq. (16.5) simply to xy. Setting v = y = e in Eq. (16.28), we ﬁnd the associative law u(xz) = (ux)z. Conversely, given an associative algebra with associative product xy, a triple product xyz deﬁned by xyz := (xy)z = x(yz) satisﬁes Eq. (16.28). We note that such a notion has already been given in [B-G]. We can now similarly introduce the notion of a weakly alternative triple system as follows.

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DEFINITION 16.1

Suppose that a triple product xyz in V satisﬁes (i) (ii)

xy(xyz) = (xyx)yz zy(xyx) = (zyx)yx.

(16.30a) (16.30b)

We call V to be a weakly alternative triple system. Similarly, if we have (xyz)yx = xy(zyx),

(16.31)

then the system is called a weakly ﬂexible triple system. REMARK 16.3 If we set y = e in Eqs. (16.30a), (16.30b) then the bilinear product xy deﬁned by Eq. (16.29) will obey the alternative law [Sch], x(xz) = (xx)z z(xx) = (zx)z. Similarly, Eq. (16.31) leads to the ﬂexible law [Sch], (xz)x = x(zx). However, the reason why we called the triple system to be a weakly alternative, since the terminology of an alternative triple system has been used in [C-M] and [Lo] for stronger relations. We then note that a weakly alternative triple system automatically implies a weakly ﬂexible triple system, following the same argument used in the bilinear algebra. Hereafter in this article, we assume that the vector space V possesses a nonzero bilinear symmetric form < .|. > so that < x|y >=< y|x > . (16.32a) We assume, moreover, the existence of an element e ∈ V satisfying < e|e >= 1,

(16.32b)

and introduce the bilinear product xy by using Eq. (16.29), i.e., xy := xey unless otherwise is stated. DEFINITION 16.2

A triple system is called a composition triple system, if we have < xyz|xyz >=< x|x >< y|y >< z|z > .

REMARK 16.4

(16.33)

If we set y = e in Eq. (16.33), we obtain < xz|xz >=< x|x >< z|z >

so that xz deﬁnes a composition algebra. Especially, if < .|. > is nondegenerate, the dimension of V must be restricted to either of 1,2,4, or 8, unless it is inﬁnite dimensional. Note that for a nonunital composition algebra, the dimension can be inﬁnite [U-W] and [E-M]. For the study of composition algebras, we refer the works as follows; for example, [E.1], [E.2], [E-O.1], [E-O.2], [Kap], and [O-O]. DEFINITION 16.3

A triple product is called quadratic, if it satisﬁes xxy = yxx =< x|x > y.

(16.34)

Composition, quadratic, and some triple systems

213

The terminology will be justiﬁed soon below. LEMMA 16.1 (i) (ii) (iii) PROOF

Let V be a quadratic triple system. We then have xzy + zxy = yzx + yxz = 2 < x|z > y, xyz − zxy = 2 < y|z > x − 2 < x|z > y,

(16.35a) (16.35b)

xyz + zyx = 2 < y|z > x − 2 < x|z > y + 2 < x|y > z.

(16.35c)

First, Eq. (16.35a) is a simple linearization of Eq. (16.34). Then, we calculate xyz = 2 < y|z >x − xzy = 2 < y|z > x − {2 < x|z > y − zxy} = zxy + 2 < y|z > x − 2 < x|z > y

which gives Eq. (16.35b). Then Eq. (16.35c) follows immediately from xzy = 2 < y|z > x − xyz. LEMMA 16.2 Let V be a quadratic triple system. Then, for any ﬁxed element e ∈ V satisfying < e|e >= 1, the bilinear product deﬁned by Eq. (16.29) satisﬁes (i) (ii)

xe = ex = x,

(16.36a)

xx − 2 < x|e > x+ < x|x > e = 0.

(16.36b)

In other words, the resulting algebra is quadratic with e being the unit element. PROOF If we set y = e and x = z in Eq. (16.35c), it will immediately give Eq. (16.36b). On the other hand, when we choose x = z = e in Eq. (16.35a), we ﬁnd eey = yee =< e|e > y = y, which is essentially Eq. (16.36a) by letting y → x. LEMMA 16.3

Let V be a quadratic triple system, and set x ¯ := 2 < e|x > e − x,

(16.37)

which evidently satisﬁes ¯ = x, x

(16.38a)

(ii) (iii)

e¯ = e, <x ¯|¯ y >=< x|y > .

(16.38b) (16.38c)

(iv)

x¯ x=x ¯x =< x|x > e,

(16.39)

(i)

We have moreover (v) PROOF calculate

exy = x ¯y,

xye = x¯ y.

(16.40)

Equation (16.39) is a simpliﬁed rewriting form of Eq. (16.36b). Next, we exy = 2 < e|x > y − xey = 2 < e|x > y − xy = x ¯y

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and xye = 2 < y|e > x − xey = 2 < e|y > x − xy = x¯ y. This proves Eq. (16.40). If V is a quadratic composition triple system, then the algebra deﬁned by xy is a unital composition algebra. However, an interesting fact is that the original triple product xyz can then be realizable in terms of the bilinear product as has been noted by several authors [B-G], [Sh.1], and [E.1], as shown in the following lemma. LEMMA 16.4 Let V be a quadratic composition triple system with < .|. > being nondegenerate. Then, the triple product xyz can be realized in terms of the Hurwitz product xy by either (i)

xyz = (x¯ y )z

(16.41a)

(ii)

xyz = x(¯ y z).

(16.41b)

or

Since the Hurwitz algebra (i.e., the unital composition algebra) is quadratic and alternative, we will consider whether our previous notion of weakly alternative triple systems may have some relevance on Lemma 16.4; the algebra satisfying Eqs.16.43 and 16.44 below has been called 3C algebra by Shaw [Sh.1]. We ﬁrst note: PROPOSITION 16.5

Suppose that a triple product xyz satisﬁes (i)

xy(yxz) =< x|x >< y|y > z,

(16.42)

and (ii)

< u|xyv >=< v|yxu >,

(16.43)

then V is a composition triple system, i.e., we have < xyz|xyz >=< x|x >< y|y >< z|z > .

(16.44)

Also, the validity of Eqs. (16.43) and (16.44) implies that of Eq. (16.42), provided that < .|. > is nondegenerate. PROOF

For simplicity, we set u = xyz and calculate < xyz|xyv >=< u|xyv >=< v|yxu > .

(16.45)

However, we obtain yxu = yx(xyz) =< y|y >< x|x > z by using Eq. (16.42), so that Eq. (16.45) becomes < xyz|xyv >=< x|x >< y|y >< z|v > .

(16.46)

Especially, if we set z = v, this gives Eq. (16.44). Conversely, if we have Eq. (16.44) and Eq. (16.43), we can reverse the direction of the argument to ﬁnd the validity of Eq. (16.42), provided that < .|. > is nondegenerate.

Composition, quadratic, and some triple systems

215

This completes the proof of Proposition 16.5. We note that the resulting bilinear algebra, satisfying the conditions of Proposition 16.5 is in general nonunital. Moreover, Eq. (16.43) is one of the relations given in Eq. (16.27). This suggests the following deﬁnition. If the triple product xyz satisﬁes

DEFINITION 16.4

< x|uvy >=< y|vux >=< u|xyv >=< v|yxu >,

(16.47)

then it is called involutive. PROPOSITION 16.6

Let V be quadratic and involutive. Then, we ﬁnd (i)

(ii)

xy = y¯x ¯

(16.48a)

< z¯|xy >=< x ¯|yz >=< y¯|xz > .

(16.48b)

Moreover, V is weakly ﬂexible. PROOF Equations (16.48a) and (16.48b) can be derived by straightforward calculations. In order to prove V to be weakly ﬂexible, we ﬁrst calculate xy(xyz) + (xyz)yx = 2 < y|xyz > x − 2 < x|xyz > y + 2 < x|y > xyz

(16.49)

by replacing z → xyz in Eq. (16.35c). Moreover, by Eq. (16.35c), we know that xyz + zyx = 2 < y|z > x − 2 < x|z > y + 2 < x|y > z so that Eq. (16.49) is rewritten as (xyz)yx − xy(zyx) + 2 < y|z > xyx − 2 < x|z > xyy + 2 < x|y > xyz = 2 < y|xyz > x − 2 < x|xyz > y + 2 < x|y > xyz. Furthermore, Eqs. (16.34) and (16.35c) give xyy =< y|y > x

xyx = 2 < x|y > x− < x|x > y

(16.50)

(xyz)yx − xy(zyx) = 2{< y|xyz > + < x|z >< y|y > −2 < y|z >< x|y >}x − 2{< x|xyz > − < x|x >< y|z >}y.

(16.51)

and

so that the above relation becomes

But the right side of Eq. (16.51) vanishes identically, since we calculate < x|xyz >=< y|zxx >=< y| < x|x > z >=< x|x >< y|z > and

< y|xyz > =< x|yzy >=< x|2 < y|z > y− < y|y > z > = 2 < y|z >< x|y > − < y|y >< x|z >

in view of Eqs. (16.47) and (16.50).

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This proves the validity of the weak ﬂexibility law (xyz)yx = xy(zyx).

LEMMA 16.5 Let V be a quadratic triple system. Then, the weak alternative law is equivalent to the validity of xy(yxz) = (zxy)yx =< x|x >< y|y > z. PROOF

(16.52)

Suppose that V is quadratic. Then, we have xyz + yxz = 2 < x|y > z

so that Eq. (16.30a) is rewritten as xy(−yxz + 2 < x|y > z) = 2(< x|y > x− < x|x > y)yz or xy(yxz) =< x|x > yyz =< x|x >< y|y > z. Similarly, Eq. (16.30b) is equivalent to have zy(2 < x|y > x− < x|x > y) = (−zxy + 2 < x|y > z)yx, which gives (zxy)yx =< x|x > zyy =< x|x >< y|y > z. Therefore, we ﬁnd Eq. (16.52). Conversely, the validity of Eq. (16.52) leads to the weak alternative law, proving the lemma. We note also that one relation of Eq. (16.52) is the same as Eq. (16.42) in Proposition 16.5. REMARK 16.5

We can also rewrite Eq. (16.52) as xy(yxz) = (xyy)xz = x(yyx)z, (zxy)yx = zx(yyx) = z(xyy)x.

(16.53a) (16.53b)

We may call the combined relations of Eqs. (16.30a), (16.30b) and (16.53a), (16.53b) to be a strong alternative law. PROPOSITION 16.7 Let V be a quadratic composition triple system with < .|. > being nondegenerate. Then V is also involutive and weakly alternative. PROOF Because of Lemma 16.4, it suﬃces to verify that the triple product xyz deﬁned by either Eq. (16.41a) or Eq. (16.41b) satisﬁes the results stated in Proposition 16.7. Let us assume here Eq. (16.41a), i.e., xyz = (x¯ y )z. Then, we calculate < x|uvy >=< x|(u¯ v )y >=< x¯ y |u¯ v>

Composition, quadratic, and some triple systems

217

and < y|vux >=< y|(v u ¯)x >=< y¯ x|v u ¯ >=< x¯ y |u¯ v >, which proves < x|uvy >=< y|uvx >. Similarly, the rest of the relation in Eq. (16.47) can be veriﬁed. This proves that it is involutive. We next verify the validity of Eqs. (16.52) as follows x){(y¯ x)z} xy(yxz) = (x¯ y ){(y¯ x)z} = (y¯ =< y¯ x|y¯ x > z =< y|y >< x ¯|¯ x > z =< y|y >< x|x > z and (zxy)yx = [{(z x ¯)y}¯ y ]x =< y|y > (z x ¯)x =< y|y >< x|x > z. Thus, V is also weakly alternative by Lemma 16.5. The second case of Eq. (16.41b), that is, xyz = x(¯ y z) can be similarly veriﬁed. We can also prove the converse of Proposition 16.7. PROPOSITION 16.8 A quadratic weakly alternative triple system is an involutive composition triple system. Here, we need not assume the nondegeneracy of < .|. >. In order to prove this, we ﬁrst need the following lemma. LEMMA 16.6 A necessary and suﬃcient condition for the triple product xyz being quadratic and involutive is that the 2nd triple product [x, y, z] given by [x, y, z] + xyz :=< y|z > x− < x|z > y+ < x|y > z, satisﬁes the following two conditions: (i) [x, y, z] is totally antisymmetric in x, y, and z (ii) < w|[x, y, z] > is totally antisymmetric in w, x, y, and z. PROOF

If we set x = y in Eq. (16.55), it leads to [x, x, z] = −xxz+ < x|x > z = 0

so that [x, y, z] is antisymmetric in x and y. Similarly, with y = z, we ﬁnd that [x, y, y] = −xyy+ < y|y > x = 0, which shows the antisymmetricity of [x, y, z] for y ↔ z. We next calculate < w|[x, y, z] >+ < w|xyz > =< y|z >< w|x > − < x|z >< w|y > + < x|y >< w|z > . Setting w = y, and noting < y|xyz > =< x|yzy >=< x|2 < y|z > y− < y|y > z > = 2 < y|z >< x|y > − < y|y >< x|z >, this gives < y|[x, y, z] >= 0

(16.54)

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Noriaki Kamiya and Susumu Okubo

so that < w|[x, y, z] > is antisymmetric in y ↔ w. The converse statement can be similarly veriﬁed, and this completes the proof of Lemma 16.6. We will now give the proof of Proposition 16.8. PROOF

We ﬁrst replace z in Eq. (16.35b) by yxz to ﬁnd xy(yzx) − (yzx)xy = 2 < y|yzx > x − 2 < x|yzx > y.

(16.55)

Moreover, we have yzx = −yxz + 2 < x|z > y = −zyx + 2 < y|z > x so that Eq. (16.55) is rewritten as −xy(yxz) + 2 < x|z > xyy + (zyx)xy − 2 < y|z > xxy = 2 < y|yzx > x − 2 < x|yzx > y. Therefore, if Eq. (16.52) is valid, this leads to 2{< y|yzx > − < x|z >< y|y >}x − 2{< x|yzx > − < y|z >< x|x >}y = 0, and hence < y|yzx >=< x|z >< y|y > < x|yzx >=< y|z >< x|x > . In terms of [x, y, z] given by Eq. (16.55), these are equivalent to < y|[y, z, x] >=< x|[y, z, x] >= 0. Since [x, y, z] is totally antisymmetric in x, y, and z in view of the quadratic property of xyz, this proves that < w|[x, y, z] > is also totally antisymmetric in w, x, y and z. Therefore by Lemma 16.6, xyz is involutive. Then, V is also a composition triple system by virtue of Proposition 16.5. This completes the proof of Proposition 16.8. REMARK 16.6 Propositions 16.7 and 16.8 may be regarded as triple analogs of similar theorems in the familiar Hurwitz algebra [Sch]. PROPOSITION 16.9 A necessary and suﬃcient condition that V is a quadratic weakly alternative triple system is to have [x, [x, y, z], z] = Bx (y, z)z + Bz (x, y)x − Bz (x, x)y

(16.56)

for the completely antisymmetric triple product [x, y, z] as in Lemma 16.6, where the function Bz (x, y) over V is given by Bz (x, y) :=< x|y >< z|z > − < x|z >< y|z > .

(16.57)

PROOF It is a simple matter of straightforward computation as in Lemma 16.6 so that it is omitted here.

Composition, quadratic, and some triple systems

219

REMARK 16.7 The relation Eq. (16.56) without the assumption of < w|[x, y, z] > being completely antisymmetric has been given in [O.1] and it has been used to construct the Cayley-Dickson process and Zorn’s vector matrix approach out of the triple system [O.2]. However, in these works, the system has been named either alternative or psedoalternative. Because we use the terminology diﬀerently, we may call the system satisfying Eq. (16.56) to be a Hurwitz triple system. We note that the quadratic weakly alternative triple system is a special example of the Hurwitz systems in which we must have < w|[x, y, z] > to be completely antisymmetric in w, x, y, and z with Eq. (16.57) for Bz (x, y). Consider now the octonionic triple system [O.3] satisfying [u, v, [x, y, z]] = {< y|v >< z|u > − < y|u >< z|v > −β < u|[v, y, z] >}x + {< z|v >< x|u > − < z|u >< x|v > −β < u|[v, z, x] >}y + {< x|v >< y|u > − < x|u >< y|v > −β < u|[v, x, y] >}z − β{< x|v > [u, y, z]+ < y|v >}[u, z, x]+ < z|v > [u, x, y]

(16.58)

− < x|u > [v, y, z]− < y|u > [v, z, x]− < z|u > [v, x, y]}, where we have normalized the value of α in [O.3] to be α = 1. We will assume hereafter in this section that < .|. > is nondegenerate. Then, as has been noted in [O.3] and [E.1], the consistency requires that the allowed values of β are β = ±1 for the 8-dimensional case and β = 0 for the 4-dimensional case. At any rate, we can easily see the validity of the following lemma: LEMMA 16.7

Equation (16.56) with Eq. (16.57) is satisﬁed by Eq. (16.58).

Actually, it is often more convenient to use triple product xyz instead. Then, we have the following proposition as a generalization of Lemma 16.4. PROPOSITION 16.10 Suppose that the triple product xyz in V is quadratic and satisﬁes a triple product equation: 1 1 (1 + β)Q1 + (1 − β)Q2 , 2 2 Q1 = < u|v > xyz+ < x|y > uvz+ < y|z > uvx− < x|z > uvy

uv(xyz) =

(16.59a)

+ < x|v > uyz+ < y|v > xuz+ < z|v > xyu − < x|u > vyz− < y|u > xvz− < z|u > xyv − < u|vzy > x+ < u|vzx > y− < u|vyx > z − < v|zyx > u+ < u|zyx > v,

(16.59b)

Q2 = < u|v > xyz+ < x|y > uvz+ < y|z > uvx− < x|z > uvy − < x|v > uyz− < y|v > xuz− < z|v > xyu + < x|u > vyz+ < y|u > xvz+ < z|u > xyv − < u|yzv > x+ < u|xzv > y− < u|xyv > z + < v|xyz > u− < u|xyz > v,

(16.59c)

for β = 0 and β = ±1. Then, V is a weakly alternative composition triple system for β = ±1. For β = 0, we must assume, in addition, V to be involutive in order to obtain the same conclusion. We also have the following relations:

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Noriaki Kamiya and Susumu Okubo

(a) For β = −1, uv(xyz) + yx(vuz) = 2 < v|xyu > z,

(16.60a)

(ii)

uv(xyz) = (uvx)yz + x(vuy)z − xy(uvz).

(16.60b)

(i) (ii)

(zuv)xy + (zyx)vu = 2 < v|uyx > z (zyx)vu = zy(xvu) + z(yuv)x − (zvu)yx.

(16.61a) (16.61b)

(i)

(b) For β = +1,

(c) For β = 0, both Eqs. (16.60a) and (16.60b) and (16.61a) and (16.60b) are valid simultaneously. Finally, in terms of the bilinear product xy := xey, the triple product is realized as (a) (b)

For β = −1, xyz = (x¯ y )z For β = +1, xyz = x(¯ y z)

(16.62a) (16.62b)

(c)

For β = 0, xyz = (x¯ y )z = x(¯ y z)

(16.62c)

in conformity with the results of Lemma 16.4. Note that Eq. (16.62a) reproduces the result of [O.3] and [E.1]. PROOF

We will present the proof in the following steps.

1. We ﬁrst establish that V is involutive for β = ±1. We will verify the consistency of Eqs. (16.59a)–(16.59c) with the quadratic properties of V . If x = y or y = z in Eqs. (16.59a)– (16.59c), we can easily ﬁnd uv(xxz) =< x|x > uvz, and uv(xyy) =< y|y > uvx as is expected. However, if we set u = v in Eqs. (16.59a)–(16.59c) for the case of β = −1, for example, we ﬁnd uu(xyz) =< u|u > xyz − {< u|yzu > − < u|u >< y|z >}x + {< u|xzu > − < u|u >< x|z >}y − {< u|xyu > − < u|u >< x|y >}z. Since the left side must be equal to < u|u > xyz, this requires the validity of < u|xyu > − < u|u >< x|y >= 0.

(16.63)

Introducing [x, y, z] by Eq. (16.55), it must be totally antisymmetric in x, y, and z because of the quadratic property of xyz. Then, Eq. (16.63) is rewritten as < u|[x, y, z] >= 0, which shows that < u|[x, y, z] > is totally antisymmetric in u, v, x, and y. Then, V is involutive by Lemma 16.6. We can prove the same for the case of β = ±1. However, for β = 0, Eqs. (16.59a)–(16.59c) simply leads to uu(xyz) =< u|u > xyz

Composition, quadratic, and some triple systems

221

without giving the desired involutive property of V . For this case, we have to assume V to be involutive in what follows. 2. When we rewrite Eqs. (16.59a)–(16.59c) in terms of [x, y, z] given by Eq. (16.55), then it becomes Eq. (16.58), and [x, y, z] now satisﬁes the conditions of Lemma 16.6 by virtue of the involutivity of the triple product xyz. Then, by Proposition 16.9 and Lemma 16.7, we conclude that V is a weakly alternative composition triple system. 3. The validity of Eqs. (16.60a) and (16.60b) for β = −1 can be established after somewhat long calculations. In order to show the validity of Eqs. (16.61a) and (16.61b) for β = +1, we observe the following fact. Returning to Eq. (16.58), the case of β = +1 is obtained from the one for β = −1, if we reverse the sigh of [x, y, z]. However, this is equivalent to make the following replacement: xyz → zyx in view of Eq. (16.55). Then, Eqs. (16.61a) and (16.61b) follow immediately from Eqs. (16.60a) and (16.60b). Finally, the case of β = 0 requires some explanations. As we stated in [O.3], it corresponds to the 4-dimensional quaternionic space, where there are many identities. Especially, the ﬁrst relation in Eq. (4.4) of [O.3] implies the validity of the identity: < z|y > [u, v, x] =< z|x > [y, u, v]+ < z|u > [x, y, v] + < z|v > [y, x, u]− < u|[v, x, y] > z. Interchanging z ↔ v, this relation is rewritten as < x|v > uyz+ < y|v > xuz+ < z|v > xyu− < u|v > xyz = {< y|v >< z|u > + < y|u >< z|v > − < u|v >< y|z >}x − {< x|v >< z|u > + < x|u >< z|v > − < u|v >< x|z >}y + {< y|v >< x|u > + < y|u >< x|v > − < u|v >< x|y >}z + {< x|v >< y|z > + < z|v >< x|y > − < y|v >< x|z >}u

(16.64)

+ < u|[x, y, z] > v. Interchanging u ↔ v and subtracting both, it leads to < x|v > uyz+ < y|v > xuz+ < z|v > xyu− < x|u > vyz − < y|u > xvz− < z|u > xyv

(16.65)

=< v|xyz > u− < u|xyz > v. We have moreover the following identity (see Eq. (4.3) in [O.3]) for β = 0: < y|[z, u, v] > x+ < z|[u, v, x] > y+ < u|[v, x, y, ] > z + < v|[x, y, z] > u+ < x|[y, z, u] > v = 0 since the left side of this equation is totally antisymmetric in 5 variables x, y, z, u, and v for the 4-dimensional quaternionic space. Combining these two relations, we ﬁnd that a presence or an absence of the term proportional to β in Eqs. (16.59a)– (16.59c) is completely immaterial for β = 0. Especially, this is the reason why we could have both Eqs. (16.60a)– (16.60b) and Eqs. (16.61a)– (16.61b) for the case of β = 0. 4. Consider ﬁrst the case of β = −1. If we set v = y = e in Eq. (16.60b), it gives ue(xez) = (uex)ez + x(eue)z − xe(uez).

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However, eue = e¯ u=u ¯ by Eq. (16.40) so that it is rewritten as u(xz) = (ux)z + x¯ uz − x(uz). Replacing u by y¯, we obtain xyz = y¯(xz) − (¯ y x)z + x(¯ y z) = −(¯ y , x, z) + x(¯ y z),

(16.66)

(¯ y , x, z) := (¯ y x)z − y¯(xz)

(16.67)

where is the associator of the Hurwitz algebra. Since the latter is alternative [Sch], we have (¯ y , x, z) = −(x, y¯, z) and hence we calculate y z) + x(¯ y z) = (x¯ y )z. xyz = (x, y¯, z) + x(¯ y z) = (x¯ y )z − x(¯ This proves Eq. (16.62a). We next consider the case of β = +1, and set y = v = e in Eq. (16.61b) to ﬁnd (zex)eu = ze(xeu) + z(eue)x − (zeu)ex, which gives zu ¯x = (zx)u − z(xu) + (zu)x = (z, x, u) + (zu)x = −(z, u, x) + (zu)x = −(zu)x + z(ux) + (zu)x = z(ux). Replacing x ↔ z and setting u = y¯, this gives xyz = x(¯ y z), which is Eq. (16.62b). For the last case of β = 0, we then evidently have Eq. (16.62c). Especially, the Hurwirz product xy is associative in conformity with the fact that it must be a 4-dimensional quaternion algebra. This completes the proof of Proposition 16.10. REMARK 16.8 Discussions of Eq. (16.58) for β = −1 appeared in physics problems in [D-G-T] and [deW-N], although they are not written in the language of the triple product but in terms of their structure constants with the standard octonionic basis. Its realization for β = −1 is given in [O.4] where [x, y, z] is expressed in terms of 8-dimensional spinors in 7-dimensional Cliﬀord algebra. It has also been used to ﬁnd some solutions of the YangBaxter equation in [O.3]. Automorphism group of the triple product xyz = (x¯ y )z has been extensively studied in [Sh.2], [Sh.3], and [E.1],. Especially, if the underlying ﬁeld F is the complex ﬁeld, then its derivation Lie algebra is D4 , which is large than G2 of the corresponding octonion algebra. Also, the triality principle of the octonion algebra is related to the corresponding triple system [E.2]. Returning to the consequences of Proposition 16.10, we have already seen in the proof of Proposition 16.7 that the weak alternative law is identically satisﬁed by the realization of Eqs. (16.62a)–(16.62c). We can verify the same for Eqs. (16.60a) and (16.60b) and for Eqs. (16.61a) and (16.61b). Here, we consider the case of Eqs. (16.60a) and (16.60b) for β = −1 with xyz = (x¯ y )z. For example, we then calculate uv(xyz) + yx(vuz) = (u¯ v {(x¯ y )z} + (y¯ x){(v u ¯)z}.

(16.68)

Composition, quadratic, and some triple systems

223

If we set for simplicity t = v u ¯ and s = x¯ y , then the right side of Eq. (16.68) is computed to be (u¯ v ){(x¯ y )z} + (y¯ x){(v u ¯)z} ¯ = t(sz) + s¯(tz) = 2 < t|s > z = 2 < v u ¯|x¯ y>z = 2 < v|(x¯ y )u > z = 2 < v|xyu > z, which reproduces Eq. (16.60a). Similarly, we can verify the validity of Eq. (16.60b). However, Eq. (16.59c) for β = −1 is rewritten as y )z (u¯ v ){(x¯ y )z} =< u|v > (x¯ + < x|y > (u¯ v )z+ < y|z > (u¯ v )x− < x|z > (u¯ v )y − < z|v > (x¯ y )u− < x|v > (u¯ y )z− < y|v > (x¯ u)z + < z|u > (x¯ y )v+ < x|u > (v y¯)z+ < y|u > (x¯ v )z

(16.69)

+ < u¯ v |x¯ z > y− < u¯ v |y¯ z > x− < u¯ v |x¯ y>z + < v¯ z |x¯ y > u− < u¯ z |x¯ y > v. But the existence of such an octonionic quintuple identity is not really surprising, since any product involving more than 3 elements in the Hurwitz algebra can always be expressed as a linear sum of terms involving products of at most 3 elements [O.6]. Actually, if we set any one of u, v, x, y, and z in Eq. (16.69) to be the unit element e, then Eq. (16.69) gives the quartic identity in [O.6]. However, we will not go into detail. Similarly, Eq. (16.59b) for the case of β = +1 yields another quintuple relation: u{¯ v {x(¯ y z)}} =< u|v > x(¯ y z)+ < x|y > u(¯ v z)+ < y|z > u(¯ v x)− < x|z > u(¯ v y) + < x|v > u(¯ y z)+ < y|v > x(¯ uz)+ < z|v > x(¯ y u) − < x|u > v(¯ y z)− < y|u > x(¯ v z)− < z|u > x(¯ y v)

(16.69 )

− x+ < u ¯v|¯ xz > y− < u ¯v|¯ xy > z − < v¯z|¯ xy > u+ < u ¯z|¯ xy > v, when we note xyz = x(¯ y z) now by Eq. (16.62b). In ending this section, we note a connection between the triple product given here and the (−1, −1)-BFKTS discussed in Section 16.2. We note that Eq. (16.60b) diﬀers in signs in some places in contrast to Eq. (16.15) for ε = −1. It is also a variation of the generalized Jordan-Lie triple system [O-K.1]. In [O.3], we already noted that Eq. (16.58) is intimately related to another triple system that we called an orthogonal triple system. However, there exists (see [K-O.1]) a simple relationship between the orthogonal triple system and (−1, −1)BFKTS. Consider the case of β = −1, and introduce a new triple product by x∗y∗z =

2 1 xyz + {< x|y > z + 2 < x|z > y − 2 < y|z > x}, 3 3

(16.70)

assuming that the underlying ﬁeld F is not of characteristic 3. Then, this deﬁnes a 8dimensional octonionic (−1, −1)BFKTS. However, the triple product x ∗ y ∗ z is not a composition triple system. As a matter of fact, we calculate < x ∗ y ∗ z|x ∗ y ∗ z >=< x|x >< y|y >< z|z > −

8 < x × y × z|x × y × z >, 9

(16.71)

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Noriaki Kamiya and Susumu Okubo

where x × y × z is the Freudenthal’s cross-product deﬁned by 1 {x(¯ y z) − z(¯ y x)} 2 1 = {(x, y, z)− < e|x > [y, z]− < e|y >}[z, x] 2 − < e|z > [x, y]+ < z|[x, y] > e}

x×y×z :=

(16.72)

= (x, y, z) − [x, y, z] in terms of the octonionic product xy with [x, y] := xy − yx for the case of β = −1. REMARK 16.9 If the ﬁeld F is of characteristic 3, we can then show that [x, y, z] deﬁnes both a Lie triple system [Li] and a (−1, −1)Jordan triple system [K-O.1]. Especially [x, y] := −2[x, e, y] deﬁnes a Lie algebra by Remark 16.2. However, we will not go into detail. REMARK 16.10 We can also construct a 7-dimensional (−1, −1)BFKTS out of octonion as in [O.3] and [K-O.2].

16.4

Final Comments

1. We can relax the notion of the composition triple system as follows. For a nonzero symmetric bilinear form < .|. >, which is not necessarily nondegenerate, we assume the validity of (i) (ii)

xyz = zyx

(16.73a)

xy(xyx) =< x|x >< y|y > x.

(16.73b)

If V possesses an element e ∈ V satisfying < e|e >= 1, we introduce a bilinear product xy in V by xy := xey (16.74) as described before. Then, Eqs. (16.73a) and (16.73b) for y = e become (i) (ii)

xz = zx x(xx) =< x|x > x.

(16.75a) (16.75b)

The bilinear algebra satisfying Eqs. (16.75a) and (16.75b) has been called a pseudocomposition algebra in [M-O], since we will then have the pseudocomposition law < xx|xx >=< x|x >< x|x > .

(16.76)

If the bilinear form < .|. > is nondegenerate, this algebra allows dimensions 5, 8, 14, and 26 in addition to some other cases. We shall now prove the following proposition. PROPOSITION 16.11 Let V be the triple system satisfying Eqs. (16.73a) and (16.73b). We then have the validity of

Composition, quadratic, and some triple systems

(i) (ii)

< xwy|z >=< x|ywz > < xyx|xyx >=< x|x >< x|x >< y|y > .

225

(16.77a) (16.77b)

PROOF Let us consider the corresponding pseudocomposition algebras satisfying Eqs. (16.75a) and (16.75b). We have shown elsewhere [E-O.2] that the bilinear form < .|. > must be automatically associative, i.e., < xy|z >=< x|yz > .

(16.78)

In terms of the triple product, this is rewritten as < xey|z >=< x|yez >,

(16.79)

which holds for any e ∈ V satisfying < e|e >= 1. Extending the underlying ﬁeld F , if necessary, we can relax the condition < e|e >= 1 to < e|e > = 0, since for any such e, 1 we can choose e0 = e(< e|e >)− 2 instead of e. We now consider Zarisky topology [Mc]. Then, the set consisting of e ∈ V satisfying < e|e > = 0, is dense in this topology, so that Eq. (16.79) should be valid for any e. Rewriting e as w, this gives Eq. (16.77a). Without using the Zarisky topology, we can prove the same also as follows. Since < .|. > is assumed to be nonzero, there exists an element v ∈ V such that < v|v > = 0 so that we have < xvy|z >=< x|yvz >

(16.80)

for any x, y, z ∈ V . Suppose that e ∈ V satisﬁes < e|e >= 0. For a constant variable λ ∈ F, we set wλ := e + λv. We then calculate < wλ |wλ >=< e|e > +2λ < e|v > +λ2 < v|v >= λ{2 < e|v > +λ < v|v >}. Choose λ ∈ F to be λ = 0 and λ = − 2<e|v> . This implies < wλ |wλ > = 0, so that we also have < xwλ y|z >=< x|ywλ z > . Comparing this with Eq. (16.80), we ﬁnd the validity of Eq. (16.79) even for the case of < e|e >= 0. Therefore, Eq. (16.77a) is valid for all x, y, z ∈ V. We now set z = xwy in Eq. (16.77a) to ﬁnd < xwy|xwy >=< x|yw(xwy) > . If we choose y = x, it gives < xwx|xwx > =< x|xw(xwx) > =< x| < x|x >< w|w > x >=< x|x >< x|x >< w|w > . Changing w → y, this proves Eq. (16.77b). We note that if we set y = e in Eq. (16.77b) it will reproduce the pseudocomposition law Eq. (16.76). 2. We can also discuss some nonunital composition algebras in triple forms. We have shown the following elsewhere [O.5]:

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Noriaki Kamiya and Susumu Okubo

Let < .|. > be a nondegenerate symmetric bilinear form. Then, a necessary and suﬃcient condition that we have (i) (ii)

< xy|xy >=< x|x >< y|y > < x|yz >=< xy|z >

(16.81a) (16.81b)

is that the algebra satisﬁes x(yx) = (xy)x =< x|x > y.

(16.82)

This relation can be immediately translated as follows in the language of the triple system. PROPOSITION 16.12 A necessary and suﬃcient condition that a composition triple system with < .|. > being nondegenerate satisﬁes < x|ywz >=< z|xwy >

(16.83)

xy(zyx) = (xyz)yx =< x|x >< y|y > z.

(16.84)

is to have Especially, it is weakly ﬂexible. PROOF

This can be readily proved as in Proposition 29.13, and will not be repeated.

Note that the algebras satisfying Eqs. (16.81a) and (16.81b) are known to be either paraHurwitz or 8-dimensional pseudo-octonion algebra [O-O], [O.5], and [My]. The triple system corresponding to the para-Hurwitz algebra is easily characterized by the para-quadratic law: xxy = yxx = 2 < x|y > x− < x|x > y,

(16.85)

instead of the quadratic law Eq. (16.34) for the Hurwitz case. However, for the triple system corresponding to the pseudo-octonion algebra, we do not know yet an analogous characterization. 3. Let us now introduce a notion of noncommutative quadratic Jordan triple system where we have (i) (ii)

xxy = yxx =< x|x > y, uv(xyz) = (uvx)yz − x(vuy)z + xy(uvz).

(16.86a) (16.86b)

Note that we replaced the condition Eq. (16.1a) by Eq. (16.86a). Equation (16.86b) are evidently a variation of Eq. (16.60b), as well as (−1, −1)BFKTS. Deﬁning the bilinear product xy by xy := xey for any ﬁxed element e satisfying < e|e >= 1 as described before, it gives a quadratic algebra with Eqs. (16.36a), and (16.36b). However xy is not necessarily commutative. Then, the triple product xyz is realized by xyz = x(¯ y z) + (¯ y x)z − y¯(xz) just as in Eq. (16.18).

(16.87)

Composition, quadratic, and some triple systems

227

4. For the relation between the quadratic triple system , the Jordan triple system, and the Freudenthal-Kantor triple system, we have the following. PROPOSITION 16.13

Let U be a Jordan triple system equipped with the product

xyz :=< x|y > z+ < y|z > x− < z|x > y, where < | >is a symmetric bilinear form. Then (U, xyz) is a quadratic triple system. PROPOSITION 16.14 Let (U, (xyz)) be a quadratic triple system. Then (U, {xyz}) is a Jordan triple system with respect to a new product, {xyz} := 2(xyz) + (yxz) − (zxy). PROPOSITION 16.15 Let (U, (xyz)) be a quadratic triple system. Then (U, {xyz}) is a (−1, δ) Freudenthal-Kantor triple system with respect to new product {xyz} := (yzx) + (zyx). About the proof of the above three propositions, we omit it, but we will discuss the detail in a forthcoming work. DEFINITION 16.5 satisﬁes

([K-K]) A triple product is called weakly commutative, if it xy(xxx) = (xxx)yx.

PROPOSITION 16.16 Let (U, (xyz))be a quadratic or Jordan triple system. Then (U, (xyz)) is a weakly commutative triple system. Similarly we have the following. PROPOSITION 16.17 Let (U, {xyz}) be a (−1, −1) balanced Freudenthal-Kantor triple system. Then (U, {xyz}) is a weakly commutative triple system. PROPOSITION 16.18 ([K.3]) Let (U, {xyz}) be a simple (1, 1) balanced FreudenthalKantor triple system with weakly commutative property. Then the product of (U, {xyz}) is given by {xyz} =< y|x > z+ < y|z > x+ < x|z > y, where < | > is an antisymmetric bilinear form. PROPOSITION 16.19 Let V be a triple system with < xyz|w >=< y|zwx > and < | > be a nondegenerate symmetric bilinear form. Then this triple system is weakly commutative. PROOF

From < xyz|w >=< y|zwx >, it follows that < xy(xxx)|y >=< y|(xxx)yx >,

228

Noriaki Kamiya and Susumu Okubo

(if we put z = (xxx) and w = y ). By the assumption of nondegenerate bilinear form < | >, we have xy(xxx) = (xxx)yx, and this means a weakly commutative. This completes the proof. COROLLARY 16.1 Let V be an involutive triple system with nondegenerate < | >. Then V is a weakly commutative. PROPOSITION 16.20 involutive triple system.

Let U be a (−1, −1) BFKTS. Then this triple system U is an

REMARK 16.11 ([O-K.2]) We note that any involutive triple system is a quasiclassical triple system with respect to the symmetric bilinear form <, >. PROPOSITION 16.21 Let (U, {xyz}) be a (−1, −1) BFKTS. Then this triple system (U, (xyz))is a quadratic triple system with respect to new product deﬁned by (xyz) = {yxz}. In our ﬁnal comments, it seems that composition triple systems or quadratic triple systems and other triple systems are useful concepts as that of mathematical physics, diﬀerential geometry, quantum groups and integrable systems, as well as a reformulation of Yang-Baxter equation in terms of triple product system. (see [K-O.3] and [O-K.2]), and a construction of Lie algebras and superalgebras (for example, [K-O.1], [K-O.2].)

16.5

Acknowledgments

One of the authors, S. Okubo, is supported in part by the U.S. Department of Energy Grant No. DE-FG02-91ER40685.

References [B-G] R.B. Brown and A. Gray, Vector Cross Products, Comment. Math. Helvetici 42 (1967) 222-236. [C-M] A. Castellon and C. Martin, Prime Alternative Triple Systems, in Proceedings of Non-associative Algebras and Its Applications, ed. by Santos Gonzales, Kluwer Acad. Pub., Math and Its Applications 303, 73-79 (Dordrecht, 1994). [deW-N] B. de Wit and H. Nicolai, The Paralleling S 7 Torsion in Gauged N = 8 Supergravity, Nucl. Phys. B231 (1984) 506-532.

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[D-G-T] R. D¨ undarer, F. G¨ usey, and C-H. Tze, Generalized Vector Products, Duality, and Octonionic Identities in D = 8 Geometry, Jour. Math. Phys. 25 (1984) 1496-1506. [E.1] A. Elduque, On a Class of Ternary Composition Algebras, Jour. Korean Math. Soc. 33(1996) 183-203. [E.2] A. Elduque, On Triality and Automorphisms and Derivations of Composition Algebras, Linear Algebra Appl. 314 (2000), 49-74. [E-K-O] A. Elduque, N. Kamiya, and S. Okubo, Simple (−1, −1) Balanced FreudenthalKantor Triple Systems, Glasgow Math. J. 45 (2003) 353-372 [E-M] A. Elduque and H.C. Myung, On Flexible Composition Algebras, Comm. Alg. 21(7) (1993) 2481-2505. [E-O.1] A. Elduque and S. Okubo, On Algebras Satisfying x2 x2 = N (x)x, Math. Zeit. 235 (2000) 275-314. [E-O.2] A. Elduque and S. Okubo, Pseudo-composition Super-Algebras, Jour. Alg. 277 (2000) 1-25. [J] N. Jacobson, Structures and Representations of Jordan Algebras, Amer. Math. Soc. Colloq. Pub. Vol. 39, Amer. Math. Soc. Providence, RI (1968). [K.1] N. Kamiya, A Structure Theory of Freudenthal-Kantor Triple Systems, J. Alg. 110, (1987) 108-123. [K.2] N. Kamiya, On Freudenthal-Kantor Triple Systems and Generalized Structure Algebras, in Proceedings of Non-Associative Algebras and Its Application, ed. by Santos Gonzales, Kluwer Acad. Pub. Math. and Its Applications 303, 198-203 (Dordrecht 1994) and earlier references quoted therein. [K.3] N.Kamiya, A Structure Theory of Freudenthal-Kantor Triple Systems. II, Comment. Math. Univ. Sancti Pauli, 23 (1989) 33-51. [K-K] I. Kantor and N. Kamiya, A Peirce decomposition for Generalized Jordan Triple Systems of Second Order, Comm Alg. 31, no 12 (2003) 5875-5913. [Kap] I. Kaplansky, Inﬁnite-dimensional Quadratic Forms Permitting Composition, Proc. Amer. Math. Soc. 4 (1953) 956-960. [K-O.1] N. Kamiya and S. Okubo, On δ-Lie Supertriple Systems Associated with (ε, δ) Freudenthal-Kantor Supertriple Systems, Proceedings of Edinburugh Math. Soc. 43, (2000) 243-260. [K-O.2] N. Kamiya and S. Okubo, Construction of Lie Superalgebras D(2, 1; α), G(3) and F (4) from Some Triple Systems, Proceeding of Edinburugh Math. Soc. 46, (2003) 87-98. [K-O.3] N. Kamiya and S. Okubo, On Generalized Freudenthal-Kantor Triple Systems and Yang-Baxter Equations, Proc. XXIV International Coll. Group Theoretical Methods in Physics, IPCS, vol 173 (2003) 815-818. [Li] W.G. Lister, A Structure Theory of Lie Triple System, Trans. Amer. Math. Soc.

230

Noriaki Kamiya and Susumu Okubo 72 (1952) 217-242. [Lo] O. Loos, Alternative Triple System, Math. Ann. 198 (1972) 205-238. [Mc] K. McCrimmon, Generalized Algebraic Algebras, Trans. Amer. Math. Soc. 127 (1967) 527-551. [My] H.C. Myung, Malcev admissible Algebras, Birkhauser (Boston) 1986 [M-O] K. Meyberg and S. Osborn, Pseudo-composition Algebras, Maht. Z.214 (1993) 67-77 and earlier references quoted therein. [O.1] S. Okubo, Deformation of the Lie-admissible Pseudo-Octonion Algebra into the Octonion Algebra, Hadronic Jour. 1 (1978) 1383-1431. [O.2] S. Okubo, Octonion, Pseudo-Alternative and Lie-Admissible Algebras, Hadronic Jour. 2 (1979) 39-66. [O.3] S. Okubo, Triple Products and Yang-Baxter Equation I. Octonionic and Quaternionic Triple Systems, Jour. Math. Phys. 34 (1993) 3273-3291. [O.4] S. Okubo, Representation of Cliﬀord Algebras and its Applications, Mathematica Japonica 41 (1995) 59-79. [O.5] S. Okubo, Introduction to Octonion and Other Non-associative Algebras in Physics, Cambridge Univ. Press (1995). [O.6] S. Okubo, Eigenvalue Problems for Symmetric 3 × 3 Octonionic Matrix, Advances in Applied Cliﬀord Algebras, 9 (1999) 131-176.

[O-K.1] S. Okubo and Kamiya, Jordan-Lie Super Algebras and Jordan-Lie Triple System, Jour. Alg. 198 (1997) 388-411. [O-K.2] S. Okubo and N. Kamiya, Quasi-Classical Lie superalgebras and Lie supertriple Systems, Comm. in Alg. 30, No. 8 (2002) 3825-3850. [O-O] S. Okubo and S. Osborn, Algebras with Non-degenerate Associative Symmetric Bilinear Forms Admitting Composition, Comm. Alg. 9 (1981) 1233-1261 and 2015-2073. [Sch] R.D. Schaﬀer, An Introduction to Non-associative Algebras, Academic Press, New York, 1966. [Sh.1] R. Shaw, Ternary Composition Algebras and Hurwitz’s Theorem, Jour. Math. Phys. 29 (1988) 2329-2333. [Sh.2] R. Shaw, Ternary Composition Algebras I. Structure Theorems: Deﬁnite and Neutral Signatures, Proc. R. Soc. London, A431 (1990) 1-19. [Sh.3] R. Shaw, Ternary Composition Algebras II. Automorphism Groups and Subgroups, Proc. R. Soc. London A431 (1990) 21-36. [U-W] K. Urbanik and F.B. Wright, Absolute Valued Algebras, Proc. Amer. Math. Soc. 11 (1960) 861-866. [Y-O] K. Yamaguchi and A. Ono, On Representation of Freudenthal-Kantor Triple

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System U (ε, δ), Bull. Fac. School Ed. Hiroshima Univ. Part II, 7 (1984) 43-51 and earlier references quoted therein. [Z-S-S-S] Zhevlakov, Slinko, Shestakov, and Shirshov, Rings that are Nearly Associative, Academic Press, New York-London, 1982.

Chapter 17 An Introduction to Associator Quantization Jaak L˜ ohmus Institute of Physics, University of Tartu, Estonia Leo Sorgsepp Tartu Observatory, T˜ oravere, Estonia 17.1 Nonassociativity and associator quantization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.2 Octonion algebra and its regular bimodule representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.2.1 Regular birepresentation of octonions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.3 The Dirac equation and its “octonionization”: Introduction of color charges . . . . . . . . . 17.3.1 The Dirac equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.3.2 “Octonionization” of the Dirac equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.4 The spectrum of fundamental fermions from double octonion structure . . . . . . . . . . . . . . . 17.5 Evolution of observables in Hamiltonian quantum mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . 17.6 Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

234 235 235 236 236 237 238 242 243 243

Abstract A review is presented about the work of the authors on the meaning of nonassociativity and application of the nonassociative algebra of octonions to some problems of elementary particle physics. We discuss the following topics: 1. Nonassociativity and associator quantization. 2. Octonion algebra and its regular bimodule representation. 3. The Dirac equation and its “octonionization.” Introduction of color charges. 4. The spectrum of fundamental fermions: families of leptons and quarks from double octonion structure. 5. Evolution of observables in the Hamiltonian quantum mechanics. In Sections 17.1 and 17.2 we present a brief description of nonassociativity, its representation and the most unique nonassociative algebra of octonions. The central part of the report consists of the construction of the spectrum of leptons and quarks (Sections 17.3 and 17.4) from octonionized version of the Dirac gamma-structure. Section 17.5 introduces the octonion bimodule formalism into Hamiltonian quantum mechanics. Key words: nonassociative algebra, octonions, bimodule representation, applications in physics, associator quantization 2000 MSC: 17A35, 81V22

233

234

17.1

Jaak L˜ ohmus and Leo Sorgsepp

Nonassociativity and Associator Quantization

A naive and speculative original thought about nonassociativity and its meaning in physics was the following. All of us know the applicability of real and complex numbers in physics and elsewhere. There is no comment about reals, but it seems that just the imaginary unit of complex numbers is responsible for the space-time metric and the velocity of light can be supposedly related to it. The next plausible hypercomplex number system is the noncommutative associative quaternion algebra, which can be represented by Pauli matrices and can be intuitively related (via spin concept) to the fundamental constant of quantum mechanics, the Planck constant. In mathematics there are two classical theorems (generalized theorems of Frobenius and Hurwitz) restricting the alternative division algebras to the set of four, real and complex numbers, quaternions, and octonions. Then, what is the role of octonions and the meaning of nonassociativity in physics? In the text that follows we discuss this problem in more deﬁnitive terms. We shall call associator quantization the introduction of nonassociativity for some elements (observables or something like that) of the deep structure of matter. This property is measured by associator (a, b, c) = (ab)c − a(bc) ;

a, b, c ∈ A ,

(17.1)

where A is some nonassociative algebra of generalized observables with a binary operation a, b → ab. The precedent of quantum mechanics where noncommutativity (measured by a commutator [a, b] = ab − ba) plays a fundamental role, suggests that nonassociativity, too, may have some good sense in physics. If some algebraic structures, such as groups and algebras, have to be used in physics, some apparatus is needed to establish a connection with physical measurements representable by numbers or arrays of numbers. For this representation theories have been elaborated, for associative systems representation meaning a homomorphic map into the matrix algebra. For nonassociative systems such representation is not possible, nevertheless there remains the possibility of “two-sided” bimodule representations [1], [9] (see also [4], [10]). The most simple version of such representation is the regular bimodule representation (regbirep) by left- and right-multiplication operators (L-,R-operators), we also call this associative projection: a → La , Ra :

La x = ax ,

Ra x = xa ,

∀a, x ∈ A .

(17.2)

Nonassociativity violates the homomorphism property: (x, a, b) = (Rb Ra − Rab )x = 0 ⇒ Rab = Rb Ra , (a, b, x) = (Lab − La Lb )x = 0 ⇒ Lab = La Lb , and is measured by the commutators of L-,R-matrices: (a, x, b) = (ax)b − a(xb) = (Rb La − La Rb )x = [Rb , La ]x .

(17.3)

We speak about associator quantization because in the associative projection of an nonassociative theory we have a commutator measure for associators, but it is a special kind of commutator, originating from nonassociativity.

An Introduction to associator quantization

235

In the interpretations of associator quantization the L- and R-operators are treated as representing internal and external degrees of freedom, respectively. For example, constructing the spacetime part of the Dirac equation with γ-matrices expressed through R-matrices, the nonobservability of colored particles appears as a new kind of strict indeterminacy between the spacetime and color properties, represented by [R, L] = 0 but having deeper origin in nonassociativity as mentioned above.

17.2

Octonion Algebra and Its Regular Bimodule Representation

Octonion algebra O is an 8-dimensional algebra with the general elements X = x0 e0 + xi ei ;

x0 , xi ∈ R ,

i = 1, 2, . . . , 7

(17.4)

with multiplication rules for the octonion units e0 , e1 , . . . , e7 : ei ej = −δij e0 + εijk ek ,

(17.5)

where εijk is a completely antisymmetric tensor = +1 for the values of the indices ijk = 123, 145, 176, 246, 257, 347, 365, and ei e0 = e0 ei = ei , e0 e0 = e0 .

(17.6)

The triples ijk above (called cycles) give the Cayley original multiplication table (A. Cayley, 1845) for octonion units; about octonions see review [13] and the book [4]. Octonion units are anticommutative and antiassociative, the last property meaning that antiassociator {ei , ej , ek } = (ei ej )ek + ei (ej ek ) = 0 for cycles. Octonion algebra is an alternative composition division algebra, alternativity meaning the alternation of associator under the permutations of arguments, composition meaning ¯ = XX ¯ = x2 + xi xi (i = the property N (XY ) = N (X)N (Y ) for the norm N (X) = XX 0 1, 2, . . . , 7), and the division algebra property meaning that division is possible by every element X = 0. Octonion algebra is an exceptional algebraic system, it is the last alternative division (and composition) algebra over R (real numbers), terminating the quadruple of most interesting and outstanding number systems R (real numbers), C (complex numbers), H (quaternions), and O (octonions). Octonion algebra is related with numerous interesting exceptional structures in group theory (exceptional groups), geometry (exceptional geometries of the 2dimensional octonion plane), and many other parts of mathematics such as number theory and coding.

17.2.1

Regular birepresentation of octonions

We have already introduced the notion of bimodule representation for a general nonassociative algebra. Now we represent it in more particular form for octonion algebra. This regbirep (we shall use this short argot mode) may be constructed from Li - and Ri -matrices corresponding to the octonion units ei → Li , Ri :

Li x = ei x,

Ri x = xei .

(17.7)

Due to linearity regbirep matrices can be constructed for every general element 17.4 of the octonion algebra.

236

Jaak L˜ ohmus and Leo Sorgsepp

According to Eq. (17.3) nonassociativity of octonions is measured by the nonzero commutators [Li , Rj ] = 0. From the alternativity property it follows that ei ej → Lij = Li Lj + [Li , Rj ] := Li ∗ Lj

(17.8)

ei ej → Rij = Rj Ri + [Ri , Lj ] := Ri ∗ Rj ,

(17.9)

where commutator terms appear due to alternativity (otherwise there would be the common homomorphic representation of an associative algebra!), and by ∗ a new “regbirep operation” is denoted. The two halves of the regbirep, L- and R-matrices, behave quite symmetrically and we shall proceed meanwhile with R-matrices only, because they appear to be more suitable for some physical formulations (e.g., for the Dirac equation [14]). Ri -matrices (i=0,1,. . . ,7) are anticommuting antisymmetric 8 × 8-matrices forming the 64-dimensional Cliﬀord algebra C6 with 6 generic elements (e.g., R1 , R2 , R3 , R4 , R5 , R6 ). Some very simple properties are R0 = R1 R2 R3 R4 R5 R6 R7 = I, R1 R2 R3 R4 R5 R6 = R7 . For the cycles (ijk) from (17.5) the regbirep operation ∗ (17.9) for Ri -matrices may be written in a projective form: ei ej → Rij := Ri ∗ Rj = Γ(ijk) Rj Ri ,

(17.10)

where Γ(ijk) = Ri Rj Rk are “reﬂection-type” diagonal matrices with ±1 on the main diagonal: Γ(123) = diag (1, 1, 1, 1, −1, −1, −1, −1) Γ(145) = diag (1, 1, −1, −1, 1, 1, −1, −1) Γ(167) = diag (1, 1, −1, −1, −1, −1, 1, 1) Γ(246) = diag (1, −1, 1, −1, 1, −1, 1, −1) Γ(257) = diag (1, −1, 1, −1, −1, 1, −1, 1)

(17.11)

Γ(347) = diag (1, −1, −1, 1, 1, −1, −1, 1) Γ(365) = diag (1, −1, −1, 1, −1, 1, 1, −1) . Any three can be regarded as generic ones, e.g., Γ(123) (≡ Γ1 ), Γ(145) (≡ Γ2 ), Γ(246) (≡ Γ3 ), the other being expressed as their products: Γ(167) = Γ1 Γ2 , Γ(572) = Γ1 Γ3 , Γ(347) = Γ2 Γ3 , Γ(653) = Γ1 Γ2 Γ3 .

17.3 17.3.1

The Dirac Equation and Its “Octonionization”: Introduction of Color Charges The Dirac equation

Our starting point is the Dirac equation with the electromagnetic term and γ-matrices in the representation where the “helicity operator” γ 5 is diagonal (Pauli representation, see e.g., [8]):

An Introduction to associator quantization

−iγ μ

∂ψ e + γ μ Aμ ψ − imcψ = 0 , ∂xμ c

237

(17.12)

where x4 = ix0 = ict, iA0 = A4 ; the connection with α, β-matrices of [8] is γ 4 = β, γ k = −iβαk , αk = iγ 4 γ k , k = 1, 2, 3. Dirac matrices γ μ satisfy the relations, {γ μ , γ ν } = γ μ γ ν + γ ν γ μ = 2δμν I ;

μ, ν = 1, 2, 3, 4.

(17.13)

A suitable choice [5], [14] of triple products of Ri -matrices gives us 8 × 8 real γ-matrices, which reproduce exactly real equation obtainable from Pauli matrices by the 8-dimensional 10 0 −1 replacements 1 → , i→ . In the present context it is convenient to use the 01 1 0 μ representation γ = Ri Rj Rk , where μ = 1, 2, 3, 4 ⇔ ijk = 654, 745, 647, 756 .

17.3.2

“Octonionization” of the Dirac equation

As we have said already, the Dirac equation can be written using the products of three Rmatrices, Ri Rj Rk with certain values of indices i, j, k (cycles!) [5], [14]. From such a Dirac equation in the “R-form” we can deduce some interesting consequences. With every such “triple” product of R-matrices one can associate the two that are obtained from Ri Rj Rk by cyclic permutation of indices; these are Rk Ri Rj and Rj Rk Ri . If we want the Dirac equation to be invariant, we must write 13 before these terms, so we can write in general Ri Rj Rk =

1 1 1 Ri Rj Rk + Rk Ri Rj + Rj Rk Ri . 3 3 3

(17.14)

We can arrange the following correspondence: ((ψei )ej )ek ↔ Rk Rj Ri ψ ,

(17.15)

where ψ in the right-hand side is an 8-column consisting of “coeﬃcient functions” ψ0 , ψ1 , . . . , ψ7 of octonionic wave function ψ = ψ0 e0 + ψ1 e1 + . . . + ψ7 e7 on the left-hand side. Now the following situation can be analyzed. We can transform the Dirac equation in R-form adding equal terms obtained by cyclization (17.14), and then, passing over to nonassociative entities (octonion units) through the correspondence (17.15), perform all the possible rearrangements of parentheses, i.e., perform all the possible associations: ((ψek )ej )ei ↔ Ri Rj Rk ψ , (ψek )(ej ei ) ↔ Rji Rk ψ , (ψ(ek ej ))ei ↔ Ri Rkj ψ , ψ((ek ej )ei ) ↔ R(kj)i ψ ,

(17.16)

ψ(ek (ej ei )) ↔ Rk(ji) ψ , where, for example, R(kj)i represents multiplication from the right by (ek ej )ei . It is easy to see that here the last two terms mutually cancel! At ﬁrst glance it may seem that performing these transformations (i.e., symmetrization (17.14), transfer from R-matrices to octonion units according to (17.15), performing all bracketings (17.16) and returning again to R-matrices, we call all this together octonionization), the initial Dirac equation undergoes serious changes. Actually in the Dirac equation the electromagnetic (interaction) term remains invariant, and after returning to the “Rform” the mass term disappears [5], i.e., we are dealing with particles of zero rest mass.

238

Jaak L˜ ohmus and Leo Sorgsepp

For every γ μ -matrix in the Dirac equation we can now write the following expression (here for μ = 1, 2, 3, 4 ijk = 654, 745, 647, 756 respectively): γ μ = Ri Rj Rk =

1 1 1 Ri Rj Rk + Rk Ri Rj + Rj Rk Ri ⇒ 3 3 3

1 + (Ri Rj Rk + Rji Rk − Rkj Ri ) 3 1 + (Rk Ri Rj + Rik Rj − Rji Rk ) 3 1 + (Rj Rk Ri + Rkj Ri − Rik Rj ) 3 1 ¯ + 1 (Rk Ri Rj + G) ¯ + 1 (Rj Rk Ri + B) ¯ , = (Ri Rj Rk + R) 3 3 3

(17.17)

it means that for each of index combinations ijk, kij, jki, we have after octonionization two terms (color charges of [2]) representing a new entity that may be interpreted as color. If all terms in the sum (17.14) or (17.17) are present, i.e., the electromagnetic term sums up to form charge Q = ±e, color charges cancel. But taking one or two terms, i.e., using the Dirac equation in R-form to describe quarks with Q = ± 31 , ± 32 , color charges remain uncancelled. It is remarkable that this scheme can be carried through exactly in the same way for every γ μ , provided that we are dealing with helicity eigenstates, i.e., our fundamental fermions must have zero mass. We summarize the situation in Table 17.1 Table 17.1. Spectrum and structure of fundamental fermions from the octonization procedure of the Dirac equation ¯ ijk + R

¯ kij + G

¯ jki + B

−(ijk + R)

−(kij + G)

−(jki + B)

+

e d¯R d¯G d¯B

1 1 0 0

1 0 1 0

1 0 0 1

−1 −1 0 0

−1 0 −1 0

−1 0 0 −1

e− dR dG dB

νe uR uG uB

0 0 1 1

0 1 0 1

0 1 1 0

0 0 −1 −1

0 −1 0 −1

0 −1 −1 0

ν¯e u ¯R u ¯G u ¯B

As we see, our quantum numbers reproduce the subquark models of Harari [3] and Shupe [12], if we take 1 for “tohu” and 0 for “vohu” in terms of [3].

17.4

The Spectrum of Fundamental Fermions from Double Octonion Structure

To grasp all the essence of the regbirep of the octonion algebra as of a nonassociative algebra we must pass over to the full (128-dim) Cliﬀord algebra C7 formed from both Rand L-matrices.

An Introduction to associator quantization

239

For the classiﬁcation of fundamental fermions we shall form 4 projection-reﬂection operators as diagonal 16 × 16-matrices with 0, ±1 on the main diagonal, ˆ1 ∓ Γ ˆ (123) , Γ

ˆ1 ∓ Γ ˆ (145) , Γ

ˆ1 ∓ Γ ˆ (246) , Γ

ˆ1 ∓ Γ ˆ (347) , Γ

(17.18)

where ˆ1 = Γ

I 0 0 −I

ˆ (ijk) = Γ

,

(R)

Γ(ijk) 0

0 (L)

Γ(ijk)

,

(17.19)

(L)

(R)

Γ(ijk) being reﬂection-type matrices formed from Li -matrices of (17.7), and Γijk = Γijk from (17.11). We call the ﬁrst operator in (17.18) the helicity operator, the other three color-type helicity operators. From diagonal operators (17.18) we can now read out quantum numbers of fundamental fermions (Table 17.2). Table 17.2. Spectrum of fundamental fermions with particle codons ˆ1 − Γ ˆ (ijk) ) Particles 12 (Γ

ˆ1 + Γ ˆ (ijk) ) Antiparticles 12 (Γ

ijk

145 k1

246 k2

347 k3

123 k4

145 k1

246 k2

347 k3

123 k4

ijk

(νR )

0 0 1 1

0 1 0 1

0 1 1 0

0 0 0 0

1 1 0 0

1 0 1 0

1 0 0 1

1 1 1 1

e+ R

0 0 1 1

0 1 0 1

0 1 1 0

1 1 1 1

1 1 0 0

1 0 1 0

1 0 0 1

0 0 0 0

e+ L

−1 −1 0 0

−1 0 −1 0

−1 0 0 −1

−1 −1 −1 −1

0 0 −1 −1

0 −1 0 −1

0 −1 −1 0

0 0 0 0

(¯ νL )

−1 −1 0 0

−1 0 −1 0

−1 0 0 −1

0 0 0 0

0 0 −1 −1

0 −1 0 −1

0 −1 −1 0

−1 −1 −1 −1

ν¯R

uR νL uL e− L dL e− R dR

d¯R

d¯L

u ¯L

u ¯L

Here we have “infovectors” (k1 , k2 , k3 , k4 ) consisting of particle codons k1 , k2 , k3 , k4 . The ﬁrst 3 determine electric charge, Q=

1 (k1 + k2 + k3 ) , 3

(17.20)

240

Jaak L˜ ohmus and Leo Sorgsepp

and colors R (red), G (green), and B (blue). The color of a quark is speciﬁed by the quantum number ki diﬀering from the other two (e.g., the quark is green if k1 , k2 , k3 = 1, 0, 1 (or −1, 0 − 1), the two diﬀerent possibilities related to the fact that the quantum numbers k1 , k2 , k3 also determine the electric charge). For the observable (nonconﬁned) fundamental fermions (leptons) k1 = k2 = k3 .

(17.21)

The fourth codon determines the third component of the weak isospin: 1 k4 . (17.22) 2 In [6] we introduced the second (open) octonionization to get a weak charge and family codons. The resulting quantum number system is quite symmetric with respect to color and ﬂavor parts. Without digging into details we summarize: we have three color codons k1 , k2 , k3 (which can be written directly from Table 17.1, and which are the ﬁrst three codons from the k-system of the previous section). These codons determine color and electric charge. Flavor codons l1 , l2 , l3 determine the ﬂavor of the family (if one of the ﬂavor codons is diﬀerent from two other, then its position determines the ﬂavor of the family, its value and sign giving the weak charge). There also appear three common families (generations) and one nonconventional hypothetical one, the ﬂavorless family. The latter may represent a new kind of matter with very massive particles and/or not interacting with ordinary matter. The scheme is presented in Tables 17.3 – 17.6 T3w =

Table 17.3. Fundamental fermions: I (electron) family Particles

Antiparticles

k1

k2

k3

l1

l2

l3

k1

k2

k3

l1

l2

l3

e (νR )

0 0 1 1

0 1 0 1

0 1 1 0

0 0 0 0

1 1 1 1

1 1 1 1

1 1 0 0

1 0 1 0

1 0 0 1

1 1 1 1

0 0 0 0

0 0 0 0

e+ R

0 0 1 1

0 1 0 1

0 1 1 0

1 1 1 1

0 0 0 0

0 0 0 0

1 1 0 0

1 0 1 0

1 0 0 1

0 0 0 0

1 1 1 1

1 1 1 1

e+ L

−1 −1 0 0

−1 0 −1 0

−1 0 0 −1

−1 −1 −1 −1

0 0 0 0

0 0 0 0

0 0 −1 −1

0 −1 0 −1

0 −1 −1 0

0 0 0 0

−1 −1 −1 −1

−1 −1 −1 −1

(¯ νLe )

−1 −1 0 0

−1 0 −1 0

−1 0 0 −1

0 0 0 0

−1 −1 −1 −1

−1 −1 −1 −1

0 0 −1 −1

0 −1 0 −1

0 −1 −1 0

−1 −1 −1 −1

0 0 0 0

0 0 0 0

e ν¯R

uR νLe uL e− L dL e− R dR

d¯R

d¯L

u ¯L

u ¯R

An Introduction to associator quantization

241

Table 17.4. Fundamental fermions: II (muon) family Particles μ (νR ) cR νLμ cL μ− L sL μ− R sR

k1 0 0 1 1 0 0 1 1 −1 −1 0 0 −1 −1 0 0

k2 0 1 0 1 0 1 0 1 −1 0 −1 0 −1 0 −1 0

k3 0 1 1 0 0 1 1 0 −1 0 0 −1 −1 0 0 −1

Antiparticles l1 1 1 1 1 0 0 0 0 0 0 0 0 −1 −1 −1 −1

l2 0 0 0 0 1 1 1 1 −1 −1 −1 −1 0 0 0 0

l3 1 1 1 1 0 0 0 0 0 0 0 0 −1 −1 −1 −1

k1 1 1 0 0 1 1 0 0 0 0 −1 −1 0 0 −1 −1

k2 1 0 1 0 1 0 1 0 0 −1 0 −1 0 −1 0 −1

k3 1 0 0 1 1 0 0 1 0 −1 −1 0 0 −1 −1 0

l1 0 0 0 0 1 1 1 1 −1 −1 −1 −1 0 0 0 0

l2 1 1 1 1 0 0 0 0 0 0 0 0 −1 −1 −1 −1

l3 0 0 0 0 1 1 1 1 −1 −1 −1 −1 0 0 0 0

μ+ R s¯R μ+ L s¯L (¯ νLμ ) c¯L μ ν¯R

c¯R

Table 17.5. Fundamental fermions: III (tauon) family Particles τ (νR ) tR νLτ tL τL− bL τR− bR

k1 0 0 1 1 0 0 1 1 −1 −1 0 0 −1 −1 0 0

k2 0 1 0 1 0 1 0 1 −1 0 −1 0 −1 0 −1 0

k3 0 1 1 0 0 1 1 0 −1 0 0 −1 −1 0 0 −1

Antiparticles l1 1 1 1 1 0 0 0 0 0 0 0 0 −1 −1 −1 −1

l2 1 1 1 1 0 0 0 0 0 0 0 0 −1 −1 −1 −1

l3 0 0 0 0 1 1 1 1 −1 −1 −1 −1 0 0 0 0

k1 1 1 0 0 1 1 0 0 0 0 −1 −1 0 0 −1 −1

k2 1 0 1 0 1 0 1 0 0 −1 0 −1 0 −1 0 −1

k3 1 0 0 1 1 0 0 1 0 −1 −1 0 0 −1 −1 0

l1 0 0 0 0 1 1 1 1 −1 −1 −1 −1 0 0 −0 0

l2 0 0 0 0 1 1 1 1 −1 −1 −1 −1 0 0 −0 0

l3 1 1 1 1 0 0 0 0 0 0 0 0 −1 −1 −1 −1

τR+ ¯bR τL+ ¯bL (¯ νLτ ) t¯L τ ν¯R

t¯R

242

Jaak L˜ ohmus and Leo Sorgsepp Table 17.6. Fundamental fermions: IV (ﬂavorless) family Particles

Antiparticles

k1

k2

k3

l1

l2

l3

k1

k2

k3

l1

l2

l3

z (νR )

0 0 1 1

0 1 0 1

0 1 1 0

0 0 0 0

0 0 0 0

0 0 0 0

1 1 0 0

1 0 1 0

1 0 0 1

1 1 1 1

1 1 1 1

1 1 1 1

0 0 1 1

0 1 0 1

0 1 1 0

1 1 1 1

1 1 1 1

1 1 1 1

1 1 0 0

1 0 1 0

1 0 0 1

0 0 0 0

0 0 0 0

0 0 0 0

−1 −1 0 0

−1 0 −1 0

−1 0 0 −1

−1 −1 −1 −1

−1 −1 −1 −1

−1 −1 −1 −1

0 0 −1 −1

0 −1 0 −1

0 −1 −1 0

0 0 0 0

0 0 0 0

0 0 0 0

(¯ νLz )

−1 −1 0 0

−1 0 −1 0

−1 0 0 −1

0 0 0 0

0 0 0 0

0 0 0 0

0 0 −1 −1

0 −1 0 −1

0 −1 −1 0

−1 −1 −1 −1

−1 −1 −1 −1

−1 −1 −1 −1

z ν¯R

xR νLz xL − zL

yL − zR

yR

17.5

+ zR

y¯R + zL

y¯L

x ¯L

x ¯R

Evolution of Observables in Hamiltonian Quantum Mechanics

According to the general mathematical principles of QM [7], [11] the time evolution of the observable (operator) F belonging to the algebra of observables F is determined by a 1dimensional subgroup of the automorphism group Aut(F) of the algebra of observables. So we can take any derivation D from the derivation algebra D(F) of the algebra of observables. If we take the algebra F alternative, we may use the derivation formula (from R. Schafer [10]): Dx,z = R[x,z] − L[x,z] − 3[Lx , Rz ] ,

(17.23)

expressing derivations by L, R-operators of the regular birepresentation. If we take F the algebra of octonions, the formula (17.23) represents an element of the Lie algebra G2 (the Lie algebra of the automorphism group of octonions) by means of octonions themselves. After some identiﬁcations, redenotings, dimensional considerations and adding the partial derivative term we have dF ∂F 1 1 3m [H1 , H2 ], F − = + (H1 , F, H2 ) , dt12 ∂t12 i i mP

(17.24)

where H1 , H2 are “factor-Hamiltonians” acting in the deepest (third) level of nonassociative 2 entities, and mP is Planck mass (= (c/κ)1/2 ≈ 2, 2 · 10−5 g ≈ 1019 GeV/c ). In the associative case this formula reduces to the common Heisenberg formula. So our time evolution equation (17.24) has three terms:

An Introduction to associator quantization

243

1) The partial derivative term describing the explicit change in time, but instead of proper time t we have “deep level” time t12 . 2) The modiﬁed quantum mechanical commutator term containing factor-Hamiltonians. 3) The associator term describing the Planck region behavior of particles with mass.

17.6

Acknowledgments

This work was supported by the Estonian Science Foundation Grant No. 4510.

References [1] Eilenberg, S. Extension of general algebras. Ann. Soc. Polon. Math., 21(1):125134, 1948. [2] Georgi, H. A uniﬁed theory of elementary particles and forces. Sci. Amer., 244(4):48-63, 1981. [3] Harari, H. A schematic model of quarks and leptons. Phys. Lett., 86B(1):83-86, 1979. [4] L˜ ohmus, J.; E. Paal; L. Sorgsepp. Nonassociative algebras in physics. Hadronic Press, Palm Harbor, Florida, 1994. [5] L˜ ohmus, J.; L. Sorgsepp. About nonassociative extensions of matrix structure of Dirac equation. In: Group-theoretical methods in physics (Proc. 3rd Internat. Sem. Yurmala (Riga, Latvia), May 22-24, 1985), vol. 2, pp. 603-608. Nauka, Moscow, 1986. [6] L˜ ohmus, J.; L. Sorgsepp. About charge-algebraic structure of matter. Annual of the Estonian Phys. Soc., 1996, pp. 28-41 (In Estonian). [7] Mackey, G.W. The mathematical foundations of quantum mechanics. W.A. Benjamin, Inc., New York, Amsterdam, 1963. [8] Pauli, W. Allgemeine Prinziepen der Quantenmechanik. Springer, Berlin, 1933. [9] Schafer, R.D. Representations of alternative algebras. Trans. Amer. Math. Soc., 72(1):1-17, 1952. [10] Schafer, R.D. Introduction to Nonassociative Algebras. Academic Press, New York, London, 1966. 2nd ed.: Dover, New York, 1995. [11] Segal, I.E. Mathematical problems of relativistic physics. Amer. Math. Soc., Providence, RI, 1963. [12] Shupe, M.A. A composite model of leptons and quarks. Phys. Lett., 86B(1):87-92, 1979.

244

Jaak L˜ ohmus and Leo Sorgsepp [13] Sorgsepp, L.; J. L˜ ohmus. About nonassociativity in physics and Cayley-Graves’ octonions. Hadronic J. (USA), 2(6):1388-1459, 1979. [14] Sorgsepp, L.; J. L˜ ohmus. Dirac equation in the regular bimodule representation of octonions. In: Fundamental interactions, vol. 62 of Eesti NSV. Tead. Akad. F¨ uu ¨s. Inst. Uurim. (Trans. Inst. Phys. Acad. Sci. Estonian SSR), pp. 159-173. Acad. Sci. Estonian SSR, Tartu, 1987.

Chapter 18 A Theorem on Bernstein Quadratic Integral Operators Yu. Lyubich Department of Mathematics, Technion, Israel Institute of Technology

Abstract In his seminal paper [1] S. N. Bernstein formulated a theorem on the stationary (“Bernstein” in the modern terminology, cf. [2]) quadratic integral operators with positive kernels. There is no proof of that in [1] and no matter where until last time. At the end of our paper [5] the Bernstein theorem was placed with an indication that it can be proved combining the theory created in [5] with a geometrical construction from [3]. In the present paper we realize this plan even for a more general result. The chapter is dedicated to the 80th anniversary of Bernstein’s work. Key words: Bernstein algebras, quadratic integral operators, evolutionary operator of a population 2000 MSC: 17D92, 45G10 Let X be a compact topological space with a regular Borel measure μ such that supp μ = X, i.e., μ(Y ) > 0 for every nonempty open subset Y ⊂X. We consider the Banach space C(X) of real-valued continuous function on X provided with the standard norm f = max |f (x)|. x∈X

The measure μ determines a continuous linear functional on C(X), f dμ, ω(f ) =

(18.1)

(18.2)

X

the norm of which is ω = μ(X).

(18.3)

Let K(x, y, z) be a continuous real-valued function of x, y, z ∈ X such that K(x, y, z) = K(y, x, z).

(18.4)

One can use it as the kernel of a continuous quadratic integral operator Q in C(X), K(x, y, z)f (x)f (y)dμ(x)dμ(y). (18.5) (Qf )(z) = X

X

It is easy to prove that the operator Q is compact.

245

246

Yu. Lyubich

It follows from (18.2) and (18.5) that ω(Qf ) = ω(K(x, y, .))f (x)f (y)dμ(x)dμ(y) X

(18.6)

X

If we assume that ω(K(x, y, .)) ≡ 1, i.e., K(x, y, z)dμ(z) = 1 (x, y ∈ X),

(18.7)

X

then (18.6) reduces to ω(Qf ) = ω 2 (f ),

(18.8)

which geometrically means that the aﬃne hyperplane Ω = {f ∈ C(X) : ω(f ) = 1}

(18.9)

is Q-invariant. Setting V = Q|Ω we can iterate this as a mapping Ω → Ω. The simplest situation V2 =V

(18.10)

is just that of a Bernstein operator V , cf. [2]. This terminology is equally applicable to Q and to the kernel K satisfying (18.4) and (18.7), a priori. THEOREM 18.1 (Main Theorem) K(x, y, z) ≥ 0,

Let K be a Bernstein kernel such that K(x, x, z) > 0

(18.11)

for all x, y, z ∈ X. Then K is a function of z only. The original Bernstein formulation relates to the case X = [0, 1], dμ(x) = dx. The functional space is not mentioned in [1] but C[0, 1] seems to be the most relevant. The positivity assumption in [1] is stronger than (18.11), namely, it is K(x, y, z) > 0 for all x, y, z. Under the condition K(x, y, z) ≥ 0 the operator Q preserves the cone C+ (X) of pointwise nonnegative functions from C(X). So, the operator V preserves the inﬁnite-dimensional simplex Δ = C+ (X) ∩ Ω = f ∈ C(X) : f ≥ 0, ω(f ) = 1 .

(18.12)

In this sense the quadratic operator V is stochastic. One can interpret it as the evolutionary operator of a population the states of which are the probability densities f ∈ Δ. Taking X = {1, 2, . . . , n} and the uniform measure μ we return to the standard model of a selection free population with n hereditary types, see [2] or [4, Section 1.2]. In this case the main theorem was proved in [1] by a complicated inductive argument. The latter fails if X is inﬁnite, for example, X = C[0, 1]. In contrast, the apparent geometric approach of [3] works in our general proof below.

A theorem on Bernstein quadratic integral operators

247

PROOF (Proof of the main theorem) The operator Q can be represented as Qf = f 2 in terms of the Banach algebra with underlying space C(X) where the multiplication is K(x, y, z)f (x)g(y)dμ(x)dμ(y)

(f ◦ g)(z) = X

(18.13)

X

instead of the standard (pointwise) one. This algebra AK is nonassociative but commutative because of (18.4). The property (18.8) in the form ω(f 2 ) = ω 2 (f ) yields ω(f ◦ g) = ω(f )ω(g),

(18.14)

so (AK , ω) is a baric algebra. Moreover, this is a Bernstein algebra since (18.10) means (f 2 )2 = f 2 for ω(f ) = 1. On the other hand, AK is a Banach algebra because of the inequality f ◦ g ≤ M f · g,

K(x, y, z)dμ(x)dμ(y),

M = max z

X

(18.15)

X

which immediately follows from (18.13). In addition, AK is compact in the sense of [5], i.e., the operator Q is compact (≡ the Q-image of every ball f ≤ r is relatively compact). This property easily follows from the Arzela–Askoli theorem (cf. [5, Lemma 4]). As a result, the subspace L = A2K is ﬁnite-dimensional according to [5, Theorem 1]. Note that L is nothing but the linear span of ImQ. Now we consider the set Γ = QΔ (the Q-image of the simplex Δ) and show that Γ = Fix Q ∩ Δ.

(18.16)

We already know that Γ⊂Δ. On the other hand, Γ⊂Fix (Q) since QΩ⊂Fix Q by (18.10). Thus, Γ⊂Fix Q ∩ Δ. Conversely, if Qf = f ∈ Δ then, obviously, f ∈ QΔ = Γ. The set Fix Q is closed since the operator Q is continuous. The simplex Δ is also closed since so are the cone C+ (X) and the hyperplane Ω, see (18.9). As a result, Γ is closed. In addition, Γ is bounded (though Fix A and Δ are unbounded, in general). Indeed, if f ∈ Γ, then f ∈ Fix Q, so K(x, y, z)f (x)f (y)dμ(x)dμ(y).

f (z) = X

(18.17)

X

On the other hand, f ∈ Δ, so f dμ = 1.

f ≥ 0,

(18.18)

X

It follows from (18.17) and (18.18) that f = max f (z) ≤ max K(x, y, z). z

x,y,z

(18.19)

Since the closed bounded set Γ lies in the ﬁnite-dimensional space L, it turns out to be compact.

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Yu. Lyubich

The inclusion Γ⊂Δ can be sharpened, namely, Γ⊂Δ0 , where Δ0 = f ∈ C(X) : f > 0, ω(f ) = 1 ⊂Δ

(18.20)

and the inequality f > 0 means f (x) > 0 for all x. The latter means that we need to prove for f ∈ Γ. To this end we ﬁnd a point x0 such that f (x0 ) > 0 and turn to (18.17). By (18.11) the integrand in (18.17) is positive on a neighborhood W ⊂X × X of (x0 , x0 ). On the other hand, mes(W ) > 0 by the assumption supp μ = X that we introduced in the beginning of the chapter. Therefore, f (z) > 0. Note that Δ0 is open in Δ, since inf f > 0 for f ∈ Δ0 and X is compact. Fix a point z ∈ X and consider the continuous quadratic functional ζ(f ) = (Qf )(z). If f ∈ Δ, then ζ(Qf ) = (Q2 f )(z) = (Qf )(z) = ζ(f ). Therefore, sup(ζ|Δ) = sup(ζ|Γ) and inf(ζ|Δ) = inf(ζ|Γ). Since Γ is compact, there are f0 , f1 ∈ Γ such that ζ(f0 ) = sup(ζ|Γ), ζ(f1 ) = inf(ζ|Γ). Finally, we have ζ(f0 ) = max(ζ|Δ),

ζ(f1 ) = min(ζ|Δ).

(18.21)

S = fτ = (1 − τ )f0 + τ f1 : 0 ≤ τ ≤ 1 .

(18.22)

Now we consider the rectilinear segment

Its ends lie in Δ0 since Γ⊂Δ0 . Hence, S⊂Δ0 by convexity of Δ0 . Since the latter is open in Δ, there exists ε > 0 such that fτ ∈ Δ for −ε ≤ τ ≤ 1 + ε. The function 0 , f1 ), ζ(fτ ) = (Qfτ )(z) = (1 − τ )2 ζ(f0 ) + τ 2 ζ(f1 ) + 2(1 − τ )τ Q(f 0 , f1 ) = f0 ◦ f1 , is a quadratic polynomial of τ ∈ [−ε, 1 + ε] but its maxiwhere Q(f mum and minimum on this segment are attained at the interior points τ = 0 and τ = 1 by (18.21). Hence, ζ(fτ ) = const, ζ(f0 ) = ζ(f1 ). Turning to (18.21) again, we conclude that max(ξ|Δ) = min(ξ|Δ), hence ζ|Δ = const. In fact, this constant depends on z, so that e(z) = (Qf )(z),

z ∈ X,

(18.23)

for all f ∈ Δ. We see that e ∈ Γ, so e ∈ Fix Q ∩ Δ+ . In particular, e = Qe. Now we extend (18.23) to f ∈ Ω, i.e., to f with ω(f ) = 1 but not necessarily f ≥ 0. In this case we consider the segment S = f τ = (1 − τ )e + τ f : 0 ≤ τ ≤ 1

(18.24)

in the hyperplane Ω and the quadratic polynomial ξ(f τ ) = (Qf τ )(z), where z ∈ X and z is ﬁxed again. An initial piece of this segment belongs to Δ, hence this polynomial is a constant there, by (18.23). Then it is constant everywhere, in particular, Q(f ) = Q(f 1 ) = Q(e) = e. Finally, if f ∈ C(X), ω(f ) = 0, then f /ω(f ) ∈ Ω, hence (Qf )(z) = e(z)ω 2 (f ).

(18.25)

The constraint ω(f ) = 0 does not matter because of continuity of the functionals in (18.25).

A theorem on Bernstein quadratic integral operators

249

By polarization, we obtain (f ◦ g)(z) = e(z)ω(f )ω(g) for arbitrary f, g ∈ C(X), i.e., K(x, y, z) − e(z) f (x)g(y)dμ(x)dμ(y) = 0. X

X

It is well known that the family of all functions (f ⊗ g)(x, y) = f (x)g(y) is complete in the space of continuous functions on X × X. Hence K(x, y, z) − e(z) h(x, y)dμ(x)dμ(y) = 0 X

X

for all h ∈ C(X × X) and for all z ∈ X. This implies that K(x, y, z) = e(z) (x, y, z ∈ X) as required. REMARK 18.1

Condition (18.10) can be weakened to V 2 f = V f,

f ≥ 0,

(18.26)

since it is suﬃcient to conclude that the algebra (AK , ω) is Bernstein. Indeed, let f > 0 and ω(f ) = 1. Then the function h = f 2 is a positive idempotent by (18.26) and (18.11). The same is true for all functions hε = (f + εg)2 where g is such that ω(g) = 0 and ε is a small parameter. Thus, 2 (f + εg)2 = (f + εg)2

(18.27)

which yields the system of identities just characterizing the Bernstein algebras. (These identities follow when comparing the coeﬃcients of εν , 1 ≤ ν ≤ 4, in (18.27).) Note that the constraint f ≥ 0 is included into Bernstein’s original formulation. Our main theorem, jointly with the previous remark, covers this formally stronger result. COROLLARY 18.1

Let K(x, y) be a continuous positive function of x, z ∈ X and let K(x, z)dμ(z) = 1.

(18.28)

X

If the linear integral operator (P f )(z) =

K(x, z)f (x)dμ(x)

(18.29)

X

in C(X) is such that P 2 = P,

(18.30)

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Yu. Lyubich

then K is a function of z only. PROOF

We introduce

K(x, y, z) =

K(x, z) + K(y, z) 2

(18.31)

in order to meet (18.4), and then (18.28) implies (18.7). Moreover, the quadratic operator Q determined by (18.5) with the kernel (18.31) coincides with P on the hyperplane Ω. Indeed, if f dμ = 1 X

then K(x, y, z)f (x)f (y)dμ(x)dμ(y)

(Qf )(z) = X

1

= 2

X

K(x, z)f (x)dμ(x) +

X

K(y, z)f (y)dμ(y) = (P f )(y).

X

Finally, the conditions (18.11) are fulﬁlled since K(x, z) > 0 for all x, z. The main theorem turns out to be applicable, so K(x, z) + K(y, z) = e(z) 2 for all x, y. Setting y = x we obtain K(x, z) = e(z). The corollary we have proved is formulated in [1] for X = [0, 1], dμ(x) = dx, as a theorem preceding the theorem on quadratic operators, also with no proof. For X = {1, . . . , n} a proof is given in [1]. It is simple, unlike the quadratic situation with the same X. Below we give a short direct proof for the linear situation in full. Note that (18.30) means that P is a projection in the space C(X). By (18.29), it is a compact operator. Hence, the image ImP is ﬁnite-dimensional, so, rankP ≡ dim(ImP ) = r < ∞. We show that r = 1 using positivity f ≥ 0 ⇒ P f > 0 or f = 0.

(18.32)

To this end we use the easily veriﬁed equality C(X)+ ∩ ImP = P (C(X)+ ).

(18.33)

If r > 1 then the cone CP ⊂ImP determined by (18.33) contains two linearly independent functions, say, e0 and e1 . Indeed, any function g ∈ C(X) is a diﬀerence of two functions g1 , g2 ∈ C(X)+ . (For instance, g(x) = max(g(x), 0), g2 (x) = − min(g(x), 0).) Hence, if P g1 and P g2 are always linearly dependent, then r = 1. Now we consider the straight line L = f = (1 − τ )e0 + τ e1 : τ ∈ R ,

A theorem on Bernstein quadratic integral operators

251

the segment of which, corresponding to τ ∈ [0, 1], belongs to the cone CP . On the other hand, there exists f ∈ L\CP ; otherwise, (1 − τ )e0 + τ e1 ∈ CP |τ | for all τ = 0, and sending τ to ∞, we obtain ±(e1 − e0 ) ∈ CP , hence e1 − e0 = 0. Let τ0 be the inﬁmum of those τ > 1 such that / CP (1 − τ )e0 + τ e1 ∈

(18.34)

if such τ exists (otherwise, τ0 would be taken as the supremum of those τ < 0 such that (18.34) is valid). Then (1 − τ0 )e0 + τ0 e1 ≥ 0. Hence, (1 − τ0 )e0 + τ0 e1 > 0 because of (18.32) and linear independence of e0 , e1 . Therefore, (1 − τ )e0 + τ e1 > 0 if τ > τ0 and τ − τ0 is small enough. This contradiction proves that r = 1. (The case r = 0 is also impossible because of (18.31) and (18.32).) The cone CP turns out to be a ray, CP = {f : f = αe, α ≥ 0}, where e is any nonzero vector from CP ; actually, e > 0. If e is chosen under condition ω(e) = 1 then α = ω(f ). Hence, P f = ω(P f )e

(18.35)

for all f ≥ 0 and then for all diﬀerences f1 − f2 with f1 , f2 ∈ C(X). Therefore, (18.35) is valid for all f ∈ C(X). By (18.2), (18.28), and (18.29), ω(P f ) = f (x)dμ(x) K(x, z)dμ(z) = f (x)dμ(x) = ω(f ). X

X

X

As a result, P f (z) = ω(f )e(z) =

e(z)f (x)dμ(x)

(18.36)

X

which yields what we need, by comparison with the kernels in (18.29) and (18.36).

References [1] S. N. Bernstein, Solution to a mathematical problem related to the theory of inheritance, Uch. Zap. N.-i. Kafedr Ukrainy 1 (1924), 83–115 (in Russian). [2] Yu. I. Lyubich, Basic concepts and theorems of evolutionary genetics for free populations, Russian Math. Surveys 26, No. 5 (1971), 51–123. [3] Yu. I. Lyubich, About a theorem of S. N. Bernstein, Sib. Math. J. 14 (1973), 474. [4] Yu. I. Lyubich, Mathematical Structures in Population Genetics, Springer–Verlag, 1992. [5] Yu. I. Lyubich, Banach–Bernstein algebras and their applications, In: Nonass. Algebra and its Applications, edited by R. Costa et.al., Marcel Dekker Inc., No. 1 (2000), 205– 210.

Chapter 19 The Real Prosymmetric Spaces O.A. Matveyev Moscow State Regional University, Russia E.L. Nesterenko Friendship of Nations University, Russia

Abstract The algebraic description of the aﬃnely connected spaces is closely related to symmetric manifolds once the quasigroup language is developed. The invariant almost reductive and reductive prosymmetric spaces are discussed. Keywords: aﬃnely connected spaces, quasigroup, loop, odule, prosymmetric spaces, space with geodesics 2000 MSC: 20XX, 53XX The notions of the quasigroup and the loop are universally recognized. Contemporary results show that the notion of odule is very important in the applications to the geometry of aﬃnely connected manifolds. The real odule is natural generalization of the well-known notion of module or vector space. DEFINITION 19.1 An algebra Me =< M , · ,{te }t∈R , e >=< M, Le , {te }t∈R , e > is e called a real odule, if the following conditions are fulﬁlled: a) < M, · , e >=< M, Le , e > is a left loop with neutral e, if x, y ∈ M then x· y = Lex y is e e a binary multiplication. The neutral e has the properties: x· e = e· x = x e

e

(19.1)

for any x from M . If a, b ∈ M then the equation a· x = b e

(19.2)

has the unique solution: x = a \e b. So the left division is deﬁned. We can also write x = (Lea )−1 b. This means that the map Lea : M → M is a bijection. The following identities are fulﬁlled:

253

254

O.A. Matveyev and E.L. Nesterenko x· (x \e y) = y <=> Lex (Lex )−1 y = y

(19.3)

x \e (x· y) = y <=> (Lex )−1 Lex y = y.

(19.4)

e

e

b) The multiplication of any element from M by any real number t is deﬁned as (x, t) → te x ∈ M. Sometimes the following notation is used: xte = te x. The following identities are fulﬁlled: (te x)· (ue x) = (t + u)e x ,

(19.5)

te (ue x) = (t · u)e x, 1e x = x,

(19.6) (19.7)

e

monoassociative identities,

where x ∈ M, t, u ∈ R and 1e is the unity of the real numbers. REMARK 19.1 If the multiplication is associative and commutative, i.e., the loop is an abelian group, and the following identity is fulﬁlled: te (x· y) = te x· te y, e

e

(19.8)

where x, y ∈ M, t ∈ R, then we come to the notion of real vector space. DEFINITION 19.2 satisﬁed: where

The odule Me is called geometric, if the following identity is le (x, te , y)ue y = ue le (x, te y)y,

(19.9)

le (x, y) = (LeLex y )−1 ◦ Lex ◦ Ley .

(19.10)

DEFINITION 19.3 An algebra P =< M, (ωt )t∈R >, where ωt (x, y) = tx y, x, y ∈ ∈ M, t ∈ R is called a space with geodesics, if the following identities are fulﬁlled: tx (ux y) = (t · u)x y, tx y = (1 − t)y x, 1x y = y.

(19.11) (19.12) (19.13)

From an algebraic viewpoint a space with geodesics is a 1-parametric family of two-sided quasigroups. PROPOSITION 19.1 ([5]) There exists one-to-one correspondence between local diﬀerential manifolds with geodesics and aﬃnely connected manifolds with zero torsion tensor ﬁeld.

The real prosymmetric spaces PROPOSITION 19.2 space with geodesics.

([9], [5])

255

An odule is geometric if and only if it deﬁnes the

Indeed, if Me =< M, Le , {te }t∈R , e > is a geometric odule, then we can deﬁne the operations on the space with geodesics P =< M, (ωt )t∈R > by the formulas: ωt (x, y) = tx y = Lex te (Lex )−1 y.

(19.14)

PROPOSITION 19.3 ([9]) A local diﬀerential odule is geometric if and only if it is a geodesic odule of some aﬃnely connected manifold. DEFINITION 19.4 The space with geodesics is called symmetric if it corresponds to a symmetric aﬃnely connected manifold. ((M, ∇) is symmetric if and only if T = 0, ∇R = 0, where T, R are the torsion and the curvature tensors ﬁelds). PROPOSITION 19.4 ([5]) following identities are fulﬁlled:

The space with geodesics is symmetric if and only if the

(−1)x ty z = t(−1)x y (−1)x z, (−1)x (−1)tx y z = (−1)ty x (−1)y z,

(19.15) (19.16)

where x, y, z ∈ M, t ∈ R. REMARK 19.2

The identity (19.16) is equivalent to the following (see (19.15)): (−1)(u+t)x y ◦ (−1)(2u)x y = (−1)tx y ◦ (−1)ux y ,

(19.17)

where u, t ∈ R, x, y ∈ M . The identity (19.17) is equivalent to the following identity: (−1)(t+1)x y ◦ (−1)tx y = (−1)y ◦ (−1)x ,

(19.18)

where x, y ∈ M, t ∈ R. DEFINITION 19.5 An odule Me =< M, Le , e > is called prosymmetric if it is geometric and it deﬁnes symmetric space with geodesics P =< M, (ωt )t∈R > by the formulas (19.14). An odule Me =< M, S e , e > is called symmetric if it

DEFINITION 19.6 ([10]) satisﬁes the following identities:

e , Stee x ◦ Sue e x = S(t+u) ex

SSe xe Sye x

=

Sxe

◦

Sye

◦

(19.19) Sxe ,

e ◦ (−1)e , (−1)e ◦ Sxe = S(−1) ex e e ¯ ¯ l (x, y) ◦ te = te ◦ l (x, y).

where l¯e (x, y) = (SSe xe y )−1 ◦ Sxe ◦ Sye .

(19.20) (19.21) (19.22)

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O.A. Matveyev and E.L. Nesterenko

PROPOSITION 19.5 An odule Me =< M, S e , e > is symmetric if and only if it can be realized as a geodesic odule of a symmetric space. COROLLARY 19.1

Symmetric odule is prosymmetric (but not converse).

PROPOSITION 19.6 An odule Me =< M, Le , {te }t∈R , e > is prosymmetric if and only if Me =< M, S e , {te }t∈R , e > is a symmetric odule, where Sxe = (−1)( 12 )e x ◦ (−1)e

(19.23)

= Le( 1 )e x ◦ (−1)e ◦ (Le( 1 )e x )−1 ◦ (−1)e . 2

2

PROPOSITION 19.7 Let Me =< M, S e , {te }t∈R , e > be a symmetric odule. Let the map ψ e : M × M → M, ψ e (x, y) = ψxe (y) be a biection such that for any x from M and any real number t the following conditions are fulﬁlled: ψtee x (x) = x, ψxe

◦ te = te ◦

(19.24) ψxe .

(19.25)

Then an algebra Me =< M, Le , {te }t∈R , e > is a prosymmetric odule and the multiplication is deﬁned by the formula: Lex = Sxe ◦ ψxe . (19.26) For the theory of prosymmetric spaces the map ψxe = (Sxe )−1 ◦ Lex = (−1)e ◦ (−1)( 12 )e x ◦ Lex = (−1)e ◦ Le( 1 )e x ◦ (−1)e ◦ [Le( 1 )e x ]−1 ◦ Lex 2

2

plays the deﬁning role and may be called the deformation map at the point e or the torsion map at the point e. PROPOSITION 19.8 Let Me =< M, Le , {te }t∈R , e > be a geometric odule. Let us deﬁne ψ e : M × M by the formula ψxe = (Sxe )−1 ◦ Lex = (−1)e ◦ (−1)( 12 )e x ◦ Lex = (−1)e ◦ Le( 1 )e x ◦ (−1)e ◦ [Le( 1 )e x ]−1 ◦ Lex 2

(19.27)

2

Then Me is prosymmetric if and only if the conditions (19.24), (19.25) and e Lete x ◦ (ψtee x ) ◦ Leue x ◦ (ψue e x )−1 = Le(t+u)e x ◦ (ψ(t+u) )−1 . ex

(19.28)

are fulﬁlled. PROPOSITION 19.9 deﬁne the map

Let Me =< M, Le , {te }t∈R , e > be a geometric odule. Let us

g e (x, y) = (Ley )−1 ◦ (−1)( 12 )x y ◦ Lex ◦ (−1)e = (Ley )−1 ◦ LeLe ( 1 )

e −1 x 2 e (Lx )y

for any x, y from M .

◦ (−1)e ◦ (LeLe ( 1 )

e −1 x 2 e (Lx )y

)−1 ◦ Lex ◦ (−1)e ,

(19.29)

The real prosymmetric spaces

257

Then Me is prosymmetric if and only if the following identities are fulﬁlled g e (Lex ue y, Lex te y) ◦ g e (x, Lex ue y) = g e (x, Lex te y),

(19.30)

g (x, y) ◦ te = te ◦ g (x, y),

(19.31)

e

e

where x, y ∈ M, t, u ∈ R. PROPOSITION 19.10 A geometric odule Me =< M, Le , {te }t∈R , e > is prosymmetric if and only if the following identities are fulﬁlled: [LeLex (−1)e y ]−1 ◦ Lex ◦ (−1)e ◦ (Lex )−1 ◦ LeLex y ◦ te = te ◦ [LeLex (−1)e y ]−1 ◦ Lex ◦ (−1)e ◦ (Lex )−1 ◦ LeLex y , LeLex (t+1)e y ◦ (−1)e ◦ [LeLex (t+1)e y ]−1 ◦ LeLex te y ◦ (−1)e ◦ [LeLex te y ]−1 = LeLex y ◦ (−1)e ◦ (LeLex y )−1 ◦ Lex ◦ (−1)e ◦ (Lex )−1 .

(19.32)

(19.33)

DEFINITION 19.7 A geometric odule Me =< M, Le , {te }t∈R , e > is called invariant, if its left translations Lex : M → M are the automorphisms of its manifold with geodesics, i.e., the following identity is fulﬁlled: Lex ◦ ty = tLex y ◦ Lex .

(19.34)

PROPOSITION 19.11 A geometric odule Me =< M, Le , {te }t∈R , e > is invariant, if and only if it satisﬁes the following identity: le (x, y) ◦ te = te ◦ le (x, y). PROOF

(19.35)

Indeed, using the formula (19.14), we obtain

analogically,

Lex ◦ ty = Lex ◦ Ley ◦ te ◦ (Ley )−1 ,

(19.36)

tLex y ◦ Lex = LeLex y ◦ te ◦ (LeLex y )−1 ◦ Lex .

(19.37)

Using (19.36) and (19.37) we obtain (19.35), from the identity (19.35), using (19.14), we obtain (19.34). REMARK 19.3

The identity (19.9) follows the identity (19.35).

PROPOSITION 19.12 A geometric odule Me =< M, Le , {te }t∈R , e > is invariant prosymmetric if and only if the identity (19.35) and the following identities (19.38), (19.39) are fulﬁlled: Lex ◦ (−1)e ◦ Lex ◦ te = te ◦ Lex ◦ (−1)e ◦ Lex , Lete x

◦

Lex

◦ (−1)e ◦

(Lex )−1

◦ (−1)e =

Lex

◦ (−1)e ◦

(Lex )−1

(19.38) ◦ (−1)e ◦

Lete x .

(19.39)

PROOF It is very easy to see that the identities (19.38) and (19.31) are equivalent, if we have the identity (19.35). Missing the symbol “e” we can obtain, using (19.29) and

258

O.A. Matveyev and E.L. Nesterenko

(19.10), g(x, y) = L−1 y ◦ (−1)( 12 )x y ◦ Lx ◦ (−1) = L−1 ]−1 ◦ Lx ◦ (−1) y ◦ L( 12 )x y ◦ (−1) ◦ [LLx ( 1 )L−1 x y 2

= L−1 ]−1 ◦ Lx ◦ L( 1 )L−1 ◦ L−1 ◦ (−1) y ◦ L( 12 )x y ◦ (−1) ◦ [LLx ( 1 )L−1 x y x y ( 1 )L−1 y 2

2

2

x

1 = ◦ L( 12 )x y ◦ (−1) ◦ l(x, L−1 y) ◦ (−1) ◦ L−1 ◦ (−1) ( 12 )L−1 x y 2 x = L−1 ]−1 ◦ Lx ◦ L( 1 )L−1 ◦ (−1) ◦ L−1 ◦ (−1) y ◦ L( 12 )x y ◦ [LLx ( 1 )L−1 x y x y ( 1 )L−1 y L−1 y

2

2

2

x

= L−1 ◦ (−1) ◦ L−1 ◦ (−1). y ◦ Lx ◦ L( 1 )L−1 x y ( 1 )L−1 y 2

2

x

Let L−1 x y = z, then y = Lx z, and −1 g(x, Lx z) = L−1 Lx z ◦ Lx ◦ L( 12 )z ◦ (−1) ◦ L( 1 )z ◦ (−1) 2

−1 −1 = L−1 Lx z ◦ Lx ◦ Lz ◦ Lz ◦ L( 12 )z ◦ (−1) ◦ L( 1 )z ◦ (−1) 2

= l(x, z) ◦

L−1 z

L−1 ( 12 )z

◦ (−1) ◦ ◦ (−1) 1 = l(x, z) ◦ l z, z ◦ L−1 ◦ (−1) ◦ L−1 ◦ (−1). ( 12 )z ( 12 )z 2 2

1

◦L

( 12 )z

Now, using (19.31) and (19.38) we have L−1 ◦ (−1) ◦ L−1 ◦ (−1) ◦ t = t ◦ L−1 ◦ (−1) ◦ L−1 ◦ (−1), ( 1 )z ( 1 )z ( 1 )z ( 1 )z 2

2

2

2

it is easy to see that (−1) ◦ t = t ◦ (−1), so

1 2

Putting +

◦ (−1) ◦ L−1 ◦ t = t ◦ L−1 ◦ (−1) ◦ L−1 . L−1 ( 1 )z ( 1 )z ( 1 )z ( 1 )z 2

z = a, V =

2

1 t,

2

2

we obtain

L−1 ◦ (−1) ◦ L−1 ◦t ( 1 )z ( 1 )z 2

2

,−1

=

1 ◦ La ◦ (−1) ◦ La = V ◦ La ◦ (−1) ◦ La . t

Analogically, ,−1 + 1 −1 ◦ (−1) ◦ L = La ◦ (−1) ◦ La ◦ = La ◦ (−1) ◦ La ◦ V. t ◦ L−1 1 1 ( 2 )z ( 2 )z t If we denote x = a, V = t, then we come to the identity (19.38). Straightforward calculation shows that identity (19.31) implies (19.38). Let us deduce the identity (19.39). In the identity (19.18) we put x = e. (−1)(t+1)y ◦ (−1)ty = (−1)y ◦ (−1).

(19.40)

It is easy to see that (−1)(t+1)y = (−1)y ◦ (−1) ◦ (−1)ty , −1 −1 L(t+1)y ◦ (−1) ◦ L−1 (t+1)y = Ly ◦ (−1) ◦ Ly ◦ (−1) ◦ Lty ◦ (−1) ◦ Lty , −1 L(t+1)y ◦ (−1) ◦ L−1 (t+1)y ◦ Lty ◦ Ly = Ly ◦ (−1) ◦ Ly ◦ (−1) ◦ Lty ◦ (−1) ◦ Ly .

The real prosymmetric spaces Since

259

L−1 (t+1)y ◦ Lty ◦ Ly = l(ty, y)

and (−1) ◦ l(ty, y) = l(ty, y) ◦ (−1), we have Lty ◦ Ly ◦ (−1) = Ly ◦ (−1) ◦ L−1 y ◦ (−1) ◦ Lty ◦ (−1) ◦ Ly , −1 Lty ◦ Ly ◦ (−1) ◦ L−1 y ◦ (−1) = Ly (−1) ◦ Ly ◦ (−1) ◦ Lty .

Changing y and x we get the identity (19.39). PROPOSITION 19.13 Let an odule Me =< M, Le , {te }t∈R , e > be left monoassociative and invariant prosymmetric. Then it is characterized by the identities (19.38) and

PROOF

Lete x ◦ Lex = Le(t+1)e x ,

(19.41)

Lete x ◦ (−1)e ◦ Lex ◦ (−1)e = (−1)e ◦ Lex ◦ (−1)e ◦ Lete x .

(19.42)

Using (19.41), we have (Lex )−1 ◦ Lete x ◦ Lex = (Lex )−1 ◦ Le(t+1)e x = Le(−1)e x ◦ Le(t+1)e x = Lete x

and from (19.38) we come to (19.42). COROLLARY 19.2 An odule Me =< M, Le , {te }t∈R , e > is a reductive and prosymmetric if and only if the identities (19.41), (19.42), (19.38) and le (x, y) ◦ Lez = Lele (x,y)z ◦ le (x, y)

(19.43)

(19.43) are fulﬁlled. REMARK 19.4 then

Let Me =< M, Le , {te }t∈R , e > is a reductive prosymmetric odule, Lete x ◦ Sxe = Sxe ◦ Lete x .

(19.44)

References [1] Belousov V.D. Foundations of quasigroups and loops theory. Nauka, Moscow, 1967. [2] Gromoll D., Klingenbery W., Meyer W. Riemannische geometry in grossen, Berlin New York, 1968. [3] Loos O. Symmetric spaces. 1-2. New York, Benjamin, 1969. [4] Malcev A.I. Analytic loops. Math. Collection, 1955, 36 (78), 3, 569-573. [5] Matveyev O.A. On manifolds with geodesics. Webs and quasigroups. Kalinin, 1986, 44-49.

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[6] Matveyev O.A., Nesterenko E.L. To the theory prosymmetric reductive spaces. Vestnik of Freindship of Nations University of Russia, 2000, 7(1), 114-126. [7] Micheev P.O. Idempotent quasigroups and manifolds with geodesics. Webs and quasigroups. Kalinin, 1988, pp. 41-46. [8] Sabinin L.V. Algebraic structures of Nonlinear Geometric Algebra. Quasigroups and systems of quasigroups. Mathematical Explorations 113 (1990); Stiinca Press, Kishinev, pp. 83-88 (Russian). [9] Sabinin L.V. Analytic Quasigroups and Geometry (Monograph) Friendship of Nations University Press, Moscow, 1991 (Russian). [10] Sabinin L.V. Supplement to Russian translation of S.K. Kobayasy and K. Nomizu, Foundations of Diﬀerential Geometry, 1, Moscow, Nauka Press, 1981, p. 295. [11] Loos O. Spiegelungsraume and homogene Symmetrische Mannigfaltigkeiten. Dissertation. Munchen, 1966. [12] Hitotsuyanagi N. Manifolds with a triple multiplication. Math. Japonica. 1997. V. 45. N 2. pp. 345-353.

Chapter 20 Deep Matrices and their Frankenstein Actions Kevin McCrimmon Department of Mathematics, University of Virginia, Charlottesville

20.1 20.2 20.3 20.4 20.5 20.6

Prolegomenon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Heads and Bodies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Deep Matrix Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Scalar Multiple Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Frankenstein Actions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Irreducible Actions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

261 263 265 267 269 271 273

Abstract The algebra of deep matrices E(X, A) is spanned over a coordinate algebra A by “deep matrix units” Ehk parameterized, not by single natural numbers like the standard matrix units Eij , but by all “deep indices” or “heads” h,k (ﬁnite strings of natural numbers or some other inﬁnite set X). This algebra has a natural Frankenstein action on the free right A-module V (X, A) with basis of all “bodies” b (inﬁnite sequences or strings), where Ehk chops oﬀ head k from the body b and sews on a new head h (replaces an initial string k of b by h): Ehk (kd) = hd, Ehk (b) = 0 if b does not begin with string k. As with ordinary matrix algebras, the center and the ideals of the deep matrix algebra are just those of the coordinate algebra, because each nonzero element A is only “distance 1” away from a scalar: there exist a coordinate a and deep matrix units E, F such that EAF = a1. In particular, over a simple coordinate algebra A the deep matrices form a simple unital algebra which acts irreducibly on each tail subspace of V (X, A), spanned by all b having the same “tail,” where two strings b,b have the same tail if they become the same after chopping oﬀ suitable heads (of perhaps diﬀerent sizes). Key words: simple unital algebra, irreducible actions 2000 MSC: Primary 16D30, Secondary 16D60

20.1

Prolegomenon

Deep matrices were born of musings on the diﬃculty of creating ideals in quadratic Jordan algebras, where the ideal generated by an element a consists of all ﬁnite sums of ﬁnite quadratic products of a by elements of the algebra. The number of summands and factors in such an expression could be considered a measure of its complexity. This is much clearer in an associative algebra: we can deﬁne the algebraic distance from a to b to be n the length of the shortest expression b = i=1 xi ayi for b in terms of a (or ∞, if no such expression exists). The diameter of an algebra would be the supremum of all distances

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between nonzero elements. An algebra is simple precisely when every two nonzero elements are a ﬁnite distance apart, and P.M. Cohn showed that an algebra has ﬁnite diameter precisely when it is simple and all its ultrapowers remain simple. We write dA (a, b) if there is any ambiguity about the algebra in which we are computing distance. Distance increases (generating power decreases) under multiplication of a and decreases (reachability increases) under multiplication of b by elements x ˆ, yˆ of the unital hull; distances shrink in homomorphic images and grow in subalgebras (but remain the same in Peirce subalgebras): d(ˆ xaˆ y , b) ≥ d(a, b) ≥ d(a, x ˆbˆ y ), d(a, b + c) ≤ d(a, b) + d(a, c), dA (¯ a, ¯b) ≤ dA (a, b), dB (a, b) = dA (a, b),

d(a, c) ≤ d(a, b)d(b, c),

d(A) ≤ d(A),

dB (a, b) ≥ dA (a, b),

d(B) ≤ d(A) for B = eAe for an idempotent e ∈ A.

For the subalgebra B = ΦE11 + ΦE12 of the algebra A of 2 × 2 matrices over Φ, a = E12 , b = E11 have b = E11 aE21 in A, so dA (a, b) = 1 but dB (a, c) = ∞ for all c = 0 in B. If A is a dense algebra of linear transformations on an inﬁnite-dimensional right vector space V over a division algebra Δ, A can still retain ﬁnite diameter, but only with diﬃculty: by Litoﬀ’s theorem, A contains for each ﬁnite n a subalgebra An having quotient An isomorphic to M (n, Δ) and hence d(An ) ≥ d(An ) = n. Despite having these subalgebras of large diameter, A itself may have ﬁnite diameter (even diameter 1, as in the case of Deep Matrices), since the diameter of subalgebras not of the form eAe may exceed the diameter of A. of distance rapidly loses signiﬁcance in commutative algebras: then b = The notion xi ayi = ( xi yi )a1, so d(a, b) is either ∞ or 1, and A has ﬁnite diameter (= 1) iﬀ A is simple (= a ﬁeld). But for noncommutative algebras, distance and diameter do give an algebraic notion of “size.” It is easy to see that the algebra M(n, Δ) of n × n matrices over a division ring Δ (equivalently, the algebra End(VΔ ) of linear transformations on an n-dimensional right vector space V over Δ) has diameter n. In particular, every division algebra has diameter 1. But the converse turns out to be false: just because any two nonzero elements are a distance 1 apart (each a = 0 has two friends x, y such that xay = 1) does not imply the algebra is a division algebra. Algebras of diameter 1 have been constructed by L.A. Bokut [1], using transﬁnite induction and free algebras to show that every simple algebra without zero divisors imbeds in an algebra of diameter 1 (indeed, in an algebra A with the property that for every a = 0, b, c, d, e, f, g ∈ A, α, β ∈ Φ, one can solve the equation xay + ybx + αxy + βyx + cx + xd + ey + yf + g = 0 for x, y, not merely the equation xay = 1). Prof. Ken Goodearl suggests the following quick argument that every algebra B over a ﬁeld Φ imbeds in one of diameter 1. We may assume the algebra B is unital, ; and let E = EndΦ (V ) be the ring of Φ-linear transformations of a free B-module V = Ba over an index set of inﬁnite cardinality ℵ ≥ dimΦ (B), so that dimΦ (V ) = ℵdimΦ (B) = ℵ. B imbeds via the left regular action in A := E/M for the maximal ideal M = {x ∈ E | rank(x) ; < ℵ}, since each ≥ bB left multiplication Lb ∈ M if b = 0. [Note that it has rank dimΦ a a a1 = ℵ since bBa = 0 for b = 0 and B unital]. A has diameter 1 since for any endomorphism a ∈ E \ M we have V = ker(a) ⊕ W = U ⊕ im(a) with a an isomorphism of W on im (a), thus dim (W ) = dim (im (a)) = rank (a) = ℵ = dim (V ) gives rise to a Φ-isomorphism y : V → W, hence xay = 1V for proj

a−1

y −1

x : V −→ im(a) −→ W −→ V. Thus in A = E/M we have x ¯a ¯ y¯ = ¯1, and A has diameter 1.

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This example is fairly universal. Whenever 1 is a ﬁnite distance n < ∞ away from an r = ℵ. Indeed, if element a ∈ EndΔ (V ) with ℵ = dimΔ (V ) inﬁnite, then a must have rank n strictly smaller, the dimension of V = 1(V ) = r= dimΔ (a(V )) < ℵ were i=1 xi ayi (V ) ⊆ n n n x a(V ) would be ≤ dim (x (a(V ))) ≤ dim (a(V )) (transformations cani Δ i Δ i=1 i=1 i=1 not increase dimension) = nr < nℵ = ℵ, a contradiction. Thus all nonzero elements a in a diameter 1 algebra must be “within striking distance of invertibility.” In particular, algebras of diameter 1 containing matrices of ﬁnite rank must already be division algebras. Algebras of diameter 1 have been called “purely inﬁnite” and studied intensively in the setting of C ∗ -algebras.1 In particular, J. Cuntz [2] introduced an algebra O∞ which is the C ∗ -closure of the algebra of deep matrices with complex coordinates over a countable index set, and established the basic diameter 1 property making heavy use of the complete norm topology. We will develop a purely algebraic theory of deep matrices over arbitrary coordinate rings. We will work thoughout with unital associative algebras over an irrelevant (unital, associative, commutative) ring of scalars Φ. Andy Warhol used to say that each algebra (he meant, of course, only associative algebras) deserves to be famous for 10 minutes. We want to give the algebra of deep matrices a few pages in the limelight, in the hope that it may ﬁnd useful employment in the algebraic community.

20.2

Heads and Bodies

We want to create an algebra of square matrices A = h,k ah,k Ehk whose entries ah,k come from some unital associative Φ-algebra A, and whose deep matrix units Ehk have “deep” row- and column-indices h, k from a set H(X) of “heads” based on some underlying nonempty index set X. The set of all “deep X-indices” or “X-heads” ∞ <

H(X) =

Xn

n=0

consists of all ﬁnite strings (n-tuples) h = (x1 , . . . , xn ) of arbitrary depth |h| = n ≥ 0 whose individual indices xi come from X. The number of heads is inﬁnite. Notice that we include one important head, the empty head ∅ of depth 0. The reader may for concreteness think of X as the natural numbers N = {1, 2, . . .}, though neither countability nor ordering of the indices is relevant. Also, we are primarily interested in the case when the coordinate algebra A is a division algebra, or at least simple. Our matrix units act in a gruesome way on a free right A-module bA V (X, A) := b∈B

with basis vectors b from the set of all “X-bodies” B(X) =

∞ =

X

1

consisting of all inﬁnite strings (sequences) b = (y1 , y2 , . . .) of indices from X. The number of bodies is uncountable if |X| ≥ 2. When A is commutative we can ignore the distinction between right and left modules. 1 I would like to thank Prof. Goodearl for directing me to the C ∗ literature. See [3] for several equivalent versions of the purely-inﬁnite condition.

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We cannot sew bodies together, but we can sew heads onto bodies: we can concatenate ﬁnite tuples with inﬁnite sequences, hb := (x1 , . . . , xn , y1 , y2 , . . .). In addition to sewing heads on, we can also cut them oﬀ. The N th head and tail operations ηN : B(X) → H(X), τN : B(X) → B(X) for ﬁnite N = 0, 1, . . . are deﬁned by ηN (b) := (y1 , . . . , yN ), τN (b) = (yN +1 , yN +2 , . . .) (b = (y1 , y2 , . . .)). Thus the head operation decapitates the N th head (the ﬁrst N indices) ηN (b) from the body and carries it away, leaving behind the N th tail τN (b) (all but the ﬁrst N indices). We agree that η0 (b) = ∅ is the empty head (no decapitation), and τ0 (b) = b is the identity map. The humpty-dumpty concatenation restores the original body by sewing its N th head back on to its N th tail: b = ηN (b)τN (b). If we are careful we can even cut heads oﬀ heads, forming ηN (h) as long as N ≤ |h|. We say that a ﬁnite or inﬁnite string d has head or begins with h, or that h heads d (written h 2 d) if h = ηN (d) is an initial segment of d for some N, i.e., d results from concatenation with h. We say h is a proper head or properly heads or properly begins d (written h < d) if it is a proper initial segment: h 2 d ∈ H (resp. B) iﬀ d = hd for some d ∈ H (resp. B), h < d ∈ H iﬀ h 2 d = h

(i.e., d = hd for d = ∅)

Note that always ∅ 2 h. The relation of heading is a partial ordering of heads: it is reﬂexive, h 2 h, transitive, j 2 h 2 k =⇒ j 2 k, and is antisymmetric, j 2 h 2 j =⇒ j = h. Two heads h,k are related under this partial order (written h ∼ k) if one is a head of the other, h 2 k or k 2 h, otherwise they are unrelated (written h ∼ k). The direction of a relation is determined by depth: if |h| = |k| then h ∼ k ⇐⇒ h = k, if |h| < |k| then h ∼ k ⇐⇒ h < k, if |h| > |k| then h ∼ k ⇐⇒ k < h. Note that each of our creatures is polycephalic, having lots of diﬀerent heads (including an empty head), though fortunately all its heads are related. The key anatomical result follows. THEOREM 20.1 (Heads) (i) [Relatedness] Let h, k, h ∈ H(X) be heads, d, d ∈ H(X) ∪ B(X) be heads or bodies. Then h heads kd only if h,k are related; more precisely, h heads kd iﬀ either h heads " k, or k properly heads h and the remainder of h heads d: (i) h 2 k or h 2 kd ⇐⇒ (ii) k < h = kh and ∅ = h 2 d = h d . If h,kd are related then so are h,k: h ∼ kd =⇒ h ∼ k. (ii) [Unrelatedness] If h = k are distinct heads in H(X) and y, z ∈ X are indices not appearing in either head (y = z allowed), then hy, kz are unrelated: hy ∼ kz

(y, z ∈ h, k).

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(iii) [Head Separation] For any ﬁnite collection b1 , . . . , bn of distinct bodies in B(X), there exists a head k such that k 2 b1 ,

but

k 2 bi

for

i = 2, . . . , n.

Indeed, there is a natural number N so that all the heads ηN (bi ) of depth N are already distinct. PROOF (1) Suppose h = (x1 , . . . , xr ) heads kd = (y1 , . . . , ys , z1 , z2 , . . .) for k = (y1 , . . . , ys ) (xi , yj , zk indices in X). When r ≤ s (so k lasts as long as h), we need x1 = y1 , x2 = y2 , ..., xr = yr , i.e., that h 2 k, as in (i). When r > s, so k stops before h does, we must have y1 = x1 , . . . . , ys = xs (i.e., k < h = kh for h = (w1 , . . . , wr−s ) of length r − s > 0) and w1 = xs+1 = z1 , w2 = xs+2 = z2 , . . . , wr−s = xr = zr−s . i.e., h 2 d, as in (ii). (2) follows since kd 2 h =⇒ k 2 kd 2 h. (3) If hy 2 kz then h 2 hy 2 kz =⇒ h 2 k (since h does not involve z) =⇒ k = hh for h = ∅ (since k = h). But then hy 2 kz = hh z =⇒ y 2 h z (cancelling h), whereas y does not appear in the nonempty part h of k. Analogously kz 2 hy. (4) Since the bodies are all distinct, for any two labels i = j the bodies bi , bj are distinct, and if Nij is the ﬁrst place they diﬀer then their heads ηN (bi ) = ηN (bj ) of length N already diﬀer for any N ≥ Nij . If we take N = maxi =j Nij to be the largest of these “diﬀerentiating places,” any two bodies will already be diﬀerent by their N th place: ηN (bi ) = ηN (bj ) if i = j. In particular, if we take k := ηN (b1 ) we have k 2 b1 but k 2 bi for all other i (since their initial segment of depth N is ηN (bi ) = ηN (b1 ) = k).

20.3

The Deep Matrix Algebra

Here we put our heads together to construct an algebra of “matrices” spanned by formal “matrix units” Ehk labelled by “deep” row and column indices h,k. THEOREM 20.2 (Deep Matrix Algebra Construction) The deep matrix algebra E(X, A) based on X over A consists of the free left A-module with the basis of all deep matrix units Ehk for ﬁnite strings h, k ∈ H(X), together with the Deep Multiplication Rules for the products aEhi · bEjk (a, b ∈ A) : k (DMI) (aEhi )(bEjk ) = (aEhi )(bEijk ) = abEhj if i 2 j = ij ,

(DMII) (aEhi )(bEjk ) = (aEhji )(bEjk ) = abEhki if j 2 i = ji ,

(DMIII) (aEhi )(bEjk ) = 0 if i ∼ j are unrelated

(i 2 j and j 2 i).

This is an associative algebra with unit 1deep = E∅∅ . The construction is an increasing function of both variables, and the construction for general A is just the usual scalar extension by A of the construction for the ground ring Φ: E(X, A) ⊆ E(X, B), E(X, A) ⊆ E(Y, A), E(X, A) ∼ = A ⊗Φ E(X, Φ)

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Kevin McCrimmon

under the natural inclusions for unital subalgebras A ⊆ B and subsets X ⊆ Y, and the natural isomorphism a ⊗ Ehk → aEhk . In particular, E(·, X) is a functor on unital associative algebras. We also have a functor E(X, ·) on unital associative ∗-algebras: if A carries an involution a→a ¯ (e.g. if A is commutative, a ¯ = a), then E(X, A) carries a natural conjugate transpose involution uniquely determined by k ∗ ¯Ekh . aEh := a In particular, we always have a transpose involution on the subalgebra E(X, Φ). The deep matrix algebra is generated by the “forward and backward shifts” determined by elements of X: Ehk = Eh∅ E∅k , where for heads ∅ E(x = Ex∅1 · · · Ex∅n , 1 ,...,xn )

h = (x1 , . . . , xn ), k = (y1 , . . . , ym ) (y ,...,ym ) E∅ 1 = E∅ym · · · E∅y1 .

It can be characterized as the free algebra generated over A by “orthogonal shifts” Sx , Sx∗ satisfying the deﬁning relations Sx∗ Sy = δx,y 1 under the correspondence Sx1 · · · Sxn Sy∗m · · · Sy∗1 = hk ∗ → Ehk . PROOF The Deep Multiplication Rules (1) for products of basis elements uniquely determine an algebra structure E(X, A); it is associative by a tedious direct calculation (superseded by the Deep Frankenstein Isomorphism 20.7 (vii)). E∅∅ acts as unit from the left on the basis elements Ejk by the Deep Multiplication Rule (DMI) with i = h = ∅, and from the right on Ehi by (DMII) with j = k = ∅ (note that always ∅ 2 j, k with trivial concatenations ∅m = m = m∅). The natural inclusions (2) follow immediately from the Deep Multiplication Rules. Since E(X, Φ) is free as Φ-module with basis Ehk , the tensor product A ⊗Φ E(X, Φ) as well as E(X, A) are free as left A-modules with bases 1 ⊗ Ehk and Ehk , and in view of the Deep Multiplication Rules the natural Φ-linear isomorphism a ⊗ Ehk → aEhk is an algebra isoφ

morphism. Tensoring A → A ⊗Φ E(X, Φ) is always a functor (or, directly, note A −→ A E(φ)

extends to E(X, A) −→ E(X, A ) via E(φ)(aEhk ) = φ(a)Ehk ). (3) The conjugate transpose involution ∗ certainly deﬁnes a linear transformation on E of period 2, A∗∗ = A, which is an algebra anti-homomorphism (AB)∗ = B ∗ A∗ on the basis elements A, B by straightforward veriﬁcation: when the middle deep indices ∗ i,j are ∗ unre∗ aEih ) = bEjk aEhi , while lated we have by (DMIII) (aEhi )(bEjk ) = 0∗ = 0 = (¯bEkj )(¯ ∗ k ∗ when i 2 j = ij we have by (DMI) (aEhi )(bEjk ) = abEhj = ¯b a ¯Ekhj = (¯bEkj )(¯ aEih ) = k ∗ i ∗ ∗ ∗ bEj aEh , and ﬁnally when j 2 i = ji we have by (DMII) (aEhi )(bEjk ) = abEhki = k ∗ i ∗ j h h ¯ ¯b a aEh . Thus ∗ is an algebra involution. aEi ) = bEj ¯Eki = (bEk )(¯ (4) These generation formulas follow immediately from the Deep Multiplication Rules. (5) Since the shallow matrix units satisfy E∅x Ey∅ = δx,y E∅∅ , the map Sy → Ey∅ , Sx∗ → E∅x induces an epimorphism S → E of the free algebra. The deﬁning relations (5) (shortening any product with an Sx∗ to the left of an Sy ) show that S is spanned over A by elements hk ∗ , and these spanning elements are sent to the basis elements Ehk ∈ E by (4). But then the hk ∗ must be A-independent too, and the map is an isomorphism sending the natural A-basis of S to that of E. This establishes the description (5). Let us also comment on the “deepness” of these matrix units. Inside E∅ := E∅∅ EE∅∅ = E of depth 0 we have “shallow” matrix units Exy (x, y ∈ X), which by the Deep Multiplication

Deep Matrices and their Frankenstein actions

267

Rules form an inﬁnite family of ordinary matrix units Exy Ezw = δyz Exw (δyz = 0 if y = z, δyz = 1 if y = z), and thus span an inﬁnite matrix subalgebra M∞ ∼ = span{Exy | x, y ∈ m m mk ) of arbitrary X}. B ut inside any diagonal subalgebra Em := Em EEm (the span of all Emh my my mw mw , depth |m| we have another family of matrix units Emx (x, y ∈ X) with Emx Emz = δyz Emx ∼ which again spans an inﬁnite matrix subalgebra = M∞ . Thus no matter how deeply we descend, we still ﬁnd copies of M∞ stretching below us. Indeed, E has a “fractal” nature: by the Deep Multiplication Rules every diagonal subalgebra Em is a clone (isomorphic copy) mh mi mk , in view of the rules (i) Emh Emij = of the entire algebra E under the map Ekh −→ Emk

mji mk mk mki mi mk Emhj if j 2 i = ji , (iii) Emh Emj = 0 if i, j are if i 2 j = ij , (ii) Emh Emj = Emh not related (i 2 j and j 2 i).

20.4

The Scalar Multiple Theorem

THROUGHOUT THE REST OF THIS CHAPTER WE WILL ASSUME THAT X IS AN INFINITE INDEX SET. J. Cuntz showed that the complex C ∗ -algebra O∞ of operators on a separable Hilbert space generated by a countable family of orthogonal isometries Si , Si∗ satisﬁed the Deep Multiplication Rules [2, 1.2, p. 175] and, making heavy use of the norm topology, had diameter 1 [2, 3.4, p. 184].2 We now turn to a direct computational proof that, for an arbitrary inﬁnite index set X and arbitrary coordinate algebra A, every nonzero deep matrix has a two-sided multiple which is a “scalar”; over a division algebra A this implies E(X, A) has diameter 1. THEOREM 20.3 (Scalar Multiple) Every nonzero element of the deep matrix algebra E(X, A) for an inﬁnite set X has a “scalar multiple”: if 0 = A ∈ E(X, A) there exist a nonzero coordinate 0 = a ∈ A and deep matrix units E, F (backward and forward shifts) with EAF = a1deep and EF = δ1deep (δ = 1 or 0). More speciﬁcally, let A = h,k ah,k Ehk = 0 be a ﬁnite sum with coeﬃcients ah,k ∈ A, and let ah0 ,k0 be a nonzero coeﬃcient which is minimal in the sense that ah,k = 0 when h < h0 is a proper initial segment, and also when h = h0 but k < k0 is a proper initial segment. Then EAF = ah0 ,k0 1deep

and

EF = δh0 ,k0 1deep .

for the backward shift E = E∅h0 y and forward shift F = Ek∅0 y for any index y ∈ X which does not appear in any of the ﬁnite number of deep indices h,k with ah,k = 0. PROOF It suﬃces to ﬁnd shifts E = E∅h0 y , F = Ek∅0 y which isolate the minimal matrix unit in the sense that EEhk00 F = E∅∅ but EEhk F = 0 for all other pairs of deep indices appearing in A. Then multiplication by E, F will pick out exactly the given minimal term of A and turn it into ah0 ,k0 1, and automatically EF = E∅h0 y Ek∅0 y = δh0 ,k0 E∅∅ by Heads Unrelatedness 20.1 (iii). 2 In [2, 1.13, p. 179] Cuntz established similar results for C ∗ -algebra O of operators on a separable n Hilbert space generated by a ﬁnite family of n orthogonal isometries Si , Si∗ subject to the additional conn ∗ dition i=1 Si Si = 1. In general, for ﬁnite |X| = n < ∞, the “correct” deep matrices require this extra ∗ condition n i=1 xi xi = 1, and require a slightly diﬀerent treatment.

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Kevin McCrimmon

Certainly we have E∅h0 y Ehk00 Ek∅0 y = E∅k0 y Ek∅0 y = E∅∅ = 1 from the Deep Multiplication Rules for any index y ∈ X. We claim that E Ehk F = 0 for all other terms as long as we choose y distinct from all indices x which appear in h,k (and we can do this for all the nonzero terms in a simultaneously since there are only ﬁnitely many of them and there is by hypothesis an inﬁnite set X of indices y to choose from). So assume ah,k is some other nonzero coeﬃcient. By the Deep Multiplication Rule (DMIII) we already have E∅h0 y Ehk = 0 unless h, h0 y are related. Since y was chosen not to appear in h, h0 y cannot be part of h, so we must have h 2 h0 y, and again since h does not contain y we must have h 2 h0 . By ah,k = 0 and minimality of h0 we cannot have h < h0 a proper initial segment, so we must have h = h0 . Since we are working with diﬀerent coeﬃcients, we have h = h0 , Ehk

But then by (DMII) E F plication Rule (DMIII) since

k = k0 .

= E∅h0 y Ehk0 Ek∅0 y = E∅ky Ek∅0 y = 0 vanishes by the Deep Multiky, k0 y are not related by Heads Unrelatedness 20.1 (iii).

Note that is crucial to isolate a minimal matrix unit, and it is crucial for isolation that X be an inﬁnite set. Also, we must use deep matrix units; for ordinary n × n matrix units there is no trouble isolating a single matrix unit by multiplication since all coeﬃcients are automatically minimal, but because ∅ is not allowed as an index we must create the identity n matrix as a ﬁnite sum j=1 Ejj of n matrix units rather than as a single matrix unit E∅∅ . The Scalar Multiple Theorem has important consequences for ideals and centers. First, it guarantees that ideals of E correspond to ideals of A, just as for ﬁnite matrix algebras. THEOREM 20.4 (Ideal Lattice) The lattice of ideals K of E(X, A) is isomorphic to the lattice of ideals I of A, since the ideals of E are precisely all IEhk for I deﬁned by I1deep := K ∩ A1deep . E(X, I) := h,k∈B(X)

If A has an involution then the lattice of ∗-ideals of E(X, A) under the conjugate transpose involution is isomorphic to the lattice of ∗-ideals of A. PROOF Let K be an ideal of E, and deﬁne I as above. Clearly I is an ideal of A, and by deﬁnition K ⊇ (I1deep )E ⊇ IEhk = E(X, I). We must establish the reverse inclusion, and it suﬃces by surgery to prove K = ¯0 in the quotient algebra E(X, A)/E(X, I) ∼ = E(X, A) for A := A/I the quotient coordinate algebra. But by the Scalar Multiple Theorem 20.3 ¯ A¯ F¯ ∈ 0 = A¯ ∈ K there is a nonzero scalar ¯0 = a ¯1deep = E (applied to A), as soon as ¯ K ∩ A1deep . Since the kernel E(X, I) is contained in K, taking preimages gives a1deep ∈ ¯ = ¯0, a contradiction. Thus K must be ¯0, as claimed. K ∩ A1deep so by deﬁnition a ∈ I and a In particular, all ideals are invariant under the transpose map, and K is invariant under the conjugate transpose involution iﬀ I is invariant under the conjugation of A. THEOREM 20.5 (Simplicity) The deep matrix algebra E(X, A) is simple iﬀ the coordinate algebra A is simple, and is ∗-simple iﬀ the coordinate algebra is ∗-simple. Secondly, the center of the deep matrix algebra consists of the scalar matrices coming from the center of the coordinate algebra, just as with ﬁnite matrix algebras.

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269

THEOREM 20.6 (Center) The centralizer in the deep matrix algebra E(X, A) of the deep matrix units (even just the shallow backward or forward shifts) consists of the scalar multiples of the identity, and the center of E(X, A) corresponds to the central multiples of the identity: ∅ ) = A1deep , CentralizerE (E∅X ) = CentralizerE (EX

Center(E) = Center(A)1deep . PROOF It suﬃces to show a centralizer is a scalar. By the Scalar Multiple Theorem 20.3, if C = 0 we have 0 = a1deep = ECF for matrix units E = E∅k , F = Eh∅ with EF = δ1deep . If C commutes with the shallow backward shifts E∅x it commutes with all (y ,...,y )

n backward shifts E∅k = E∅ 1 = E∅yn · · · E∅y1 , so 0 = a1deep = ECF = CEF = δC. Then δ = 0 forces δ = 1 and C = a1deep is a scalar. Similarly, if C commutes with all backward shifts then a1deep = ECF = EF C = δC = C.

20.5

Frankenstein Actions

We can realize the abstract algebra of deep matrices as operators on the space spanned by all “bodies.” Because we are dealing with an inﬁnite index set X, the set B(X) of bodies is always uncountable. The standard matrix units Eij (i, j ∈ N) have a natural representation as A-linear transformations on a free right A-module with basis {vj } via Eij (vk ) = vi δjk , so Eij replaces vj by vi and kills all other vk . In a similar way, the deep matrix units Ehk have a natural representation as A-linear operators Fhk on the Frankenstein module, the free right A-module with basis of all bodies b, V (X, A) =

; b∈B

bA,

where Fhk transforms basic bodies beginning with k into ones beginning with h according to the basic Frankenstein Action Rules (FAR)

Fhk (ba) = 0 if k 2 b,

Fhk (kb a) = hb a if k 2 b = kb

(a ∈ A).

Thus for heads h,k the hth “insertion” or “forward shift” or “sewing operator” (sewer, but watch the pronunciation!) Fh∅ sews a new head onto the body (in front of its old one), the kth “deletion” or “backward shift” or “chopping operator” (chopper) F∅k removes the head k (so the operation is not a success, killing the patient, if it has a diﬀerent |k|-th head), and the hkth “chop-and-sewer” or general Frankenstein operator Fhk removes the head k and sews on the head h in its place. The Frankenstein projection Fkk kills all bodies not having k as head, but leaves bodies with head k alone (actually, it removes the head and then quickly sews it back on). In particular, F∅∅ is the identity operator. We can take linear combinations of Frankenstein operators to form an algebra. THEOREM 20.7 (Frankenstein Algebra) As A-linear transformations on V (X, A)A , the Frankenstein operators have (for b ∈ B(X), a ∈ A) the actions

270

Kevin McCrimmon (i) F∅∅ is the identity operator F∅∅ (ba) = ba, (ii) Fh∅ is the hth “insertion” or “forward shift” or “sewer” Fh∅ (ba) = hba, (iii) F∅k is the kth “deletion” or “backward shift” or “chopper” F∅k (ba) = 0 if k 2 b,

F∅k (kda) = da if k 2 b = kd

(iv) Fhk = Fh∅ F∅k is the hkth “chop-and-sewer” Fhk (ba) = 0 if k 2 b,

Fhk (kba) = hda if k 2 b = kd,

(v) the Frankenstein projection Fkk is the projection onto the subspace of V spanned by all b beginning with k Fkk (ba) = 0 if k 2 b,

Fkk (kda) = kda if k 2 b = kd.

The Frankenstein operators have the following multiplication table as linear transformations on the Frankenstein module V (X, A)A : (i 2 j = ij )

k (FrI) Fhi Fjk = Fhi Fijk = Fhj

(FrII) Fhi Fjk = Fhji Fjk = Fhki

(j 2 i = ji )

(FrIII) Fhi Fjk = 0 if i ∼ j are unrelated

(i 2 j and j 2 i).

(vi) The distinguished basis of b’s turns the right Frankenstein A-module V (X, A)A into an A-bimodule via Lb ba := bba, and the Frankenstein operators commute with this bimodule action. Thus the Frankenstein operators and left A-multiplications generate a unital associative algebra, the Frankenstein algebra H(X) F(X, A) := LA FH(X) = h,k∈H(X) AFhk ⊆ End(V (X, A)A ), consisting of all Frankenstein transformations, the ﬁnite A-linear combinations h,k ah,k Fhk of Frankenstein operators. The Frankenstein algebra is a free left A-module with the Frankenstein operators as basis, and the Frankenstein module V (X, A)A is naturally a left F(X, A)module. (vii) There is a natural Deep Frankenstein Isomorphism k k h,k ah,k Eh −→ h,k ah,k Fh of the deep matrix algebra E(X, A) with the Frankenstein algebra F(X, A), hence a faithful action of E(X, A) on V (X, A). PROOF (1) These are all special cases of the Frankenstein Action Rules (FAR). Note for (i), (ii) that all bodies have k = ∅ as one of their heads. For (iv), note that the general Frankenstein operator may, without changing the result, pause in mid-operation: chopping oﬀ head k, pausing (temporarily sewing on an empty head), then resuming (removing the empty head) and sewing on the correct head h. (2) First note that the Frankenstein operators act only on the bodies b, and hence commute with left and right multiplications by A, which act only on the coeﬃcients a. This allows us to forget about the coeﬃcient a and prove the relations (FrI-III) only on bodies b. In (FrI), Fhi Fijk (b) vanishes unless b = kb begins with k, in which case it produces

ji k k Fhi (ij b ) = hj b , which coincides with the action of Fhj . In (FrII), Fh Fj (b) vanishes

again unless b = kb begins with k, in which case it produces Fhji (jb ), which vanishes unless ji 2 jb , i.e., i 2 b = i b , so the whole operator vanishes unless b = ki b in

Deep Matrices and their Frankenstein actions

271

which case it produces Fhji (ji b ) = hb , which is precisely the action of Fhki . In (FrIII), as usual Fhi Fjk (b) vanishes unless b = kb , in which case it produces Fhi (jb ), which vanishes by Heads Relatedness 20.1 (i) since we cannot have i beginning jb if i,j are not related, so the operator kills all basic bodies b and is the zero transformation. (3) Since the Frankenstein operators, together with zero, form a semigroup by (2) commuting with left multiplications by A, their ﬁnite A-linear combinations form an algebra F of linear transformations. To see that F is free as a left A-module, suppose some ﬁnite A-linear combination of distinct Frankenstein operators with nonzero coeﬃcients ah,k = 0 is the zero transformation, ah,k Fhk = 0. Following our usual procedure, choose a minimal head k0 among the k’s this time (not among the h’s!), so k 2 k0 =⇒ k = k0 . Since X is inﬁnite and only a ﬁnite number of xj ∈ X appear in the ﬁnite number of ﬁnite strings k, there is at least one y ∈ X which does not appear in any k. Consider the body b := k0 y terminating in all y’s (where y := (y, y, y, . . .) denotes the constant sequence) . Then Fhk (b) = 0 2 k0 since k contains unless k 2 k0 y, which implies k no y’s, which in turnimplies k = k0 by minimality of k0 . Then 0 = ah,k Fhk (b) = k=k0 ah,k0 Fhk0 (k0 y) = k=k0 ah,k0 hy; but the basic bodies hy are all distinct since these h are all distinct (the pairs (h,k) all have the same k = k0 yet are distinct, so the h must be distinct), and none of them involve y, so by A-freedom of the b’s this would force the coeﬃcient ah,k0 of hy to be zero, a contradiction. (4) The rule ϕ( h,k ah,k Ehk ) := h,k ah,k Fhk is a well-deﬁned A-linear bijection of free left A-modules. This map is a homomorphism of algebras since ϕ(AB) = ϕ(A)ϕ(B) on the basis matrix units (both deep and Frankenstein matrix units have the same multiplication rules (DMI-III), (FI-III), so it is an isomorphism of E on F.

20.6

Irreducible Actions

We will identify the irreducible submodules of the Frankenstein action, and thereby hangs a tail. THEOREM 20.8 (Tails) (i) We say that two bodies b, b ∈ B(X) have the same tail, or are tail-equivalent b ∼ b , if they become the same once you chop oﬀ a big enough head: τN (b) = τN (b ). Algebraically this means that b ∼ b ⇐⇒ b = hd, b = h d are obtained from the same tail d by sewing on diﬀerent heads h, h . Note that we do not demand N = N , i.e., that the heads be of the same depth. This gives an equivalence relation on sequences, and the equivalence classes are called the tail classes. We can get from any one body in a tail class to any other by means of Frankenstein operators, b ∼ b ⇐⇒ b = Fhh (b) ∈ F(b)

(for some h, h ∈ H(X)),

since by deﬁnition b ∼ b ⇐⇒ b = h d, b = hd for some tail d ⇐⇒ b = Fhh (b) by the Frankenstein Action Rule (FAR). (ii) For each tail-class τ we deﬁne the tail-submodule of the Frankenstein module V (X, A)A by Vτ (X, A) := b∈τ bA.

272

Kevin McCrimmon

Because the Frankenstein operators only aﬀect a ﬁnite number of indices in a body, they do not change tails (Fhk (b) is 0 or some b ∼ b), so the Frankenstein transformations of the Frankenstein algebra F(X, A) don’t either, and the tail-submodules are invariant under the Frankenstein action. Moreover, by (i) each Vτ (X, A) = F(X, A)bτ is a cyclic right A-module generated by any body bτ in the tail-class τ . (iii) We have a direct decomposition ; V (X, A) = τ Vτ (X, A) of the Frankenstein module V into an uncountable number of invariant submodules Vτ (X, A) for the distinct tail-classes τ. More generally, for any two-sided ideal I A we obtain an invariant F-submodule Vτ (X, I) := b∈τ bI = I Vτ (X, A) (equality holding since I is a right ideal ), which is a left A-module since I is a left ideal, and is Frankenstein-invariant since the tail-class is invariant under the Frankenstein operators Fhk . It will be important in analyzing irreducible actions that the Frankenstein operators act selectively. THEOREM 20.9 (Body Separation) The Frankenstein projections separate bodies: for any ﬁnite collection b1 , . . . , bn of distinct bodies, there exists a Frankenstein projection Fkk such that Fkk (b1 ) = b1 , but Fkk (b2 ) = · · · = Fkk (bn ) = 0. PROOF By Head Separation 20.1 (iii) there is a head k such that k 2 b1 , k 2 bi for i = 1, and the result follows from Frankenstein Algebra 20.7(v). We can now describe all the invariant submodules of the Frankenstein representation THEOREM 20.10 (Frankenstein Submodule) The F(X, A)-invariant submodules of the Frankenstein right A-module are precisely the direct sums W = Vτ (X, Iτ ) (Iτ (W ) := {a ∈ A | a(Vτ ) ⊆ W } A). τ

In particular, the irreducible invariant submodules are precisely all Vτ (X, I) for minimal ideals I A. If A is a simple algebra, the irreducible invariant A-submodules of the Frankenstein module are precisely the tail-submodules Vτ (X, A). PROOF For any F-invariant right A-submodule W the Iτ (W ) as deﬁned above are twosided ideals of A : AIτ A ⊆ Iτ since (AIτ A)(Vτ ) = A(Iτ (A(Vτ ))) ⊆ A(Iτ (Vτ )) (since Vτ is Clearly W ⊇ ;F-invariant) ⊆ A(W ) (by deﬁnition of Iτ ) ⊆ W (since W is F-invariant). τ Iτ Vτ by deﬁnition of Iτ ; the trick is the reverse inclusion. For any w = i bi ai ∈ W we can by the Body Separation Theorem 20.9 pick out each individual b-term using a suitable Frankenstein projection: bi ai = Fhh (w) ∈ Fhh (W ) ⊆ W. Then for any other bi = Fhh (bi ) in the tail-class τi of bi ) (using Tails 20.8 (i)), and for any a ∈ A, we have ai (bi a ) = bi ai a = right A-module). This Fhh (bi ai )a ∈ F(W )a ⊆ W a ⊆ W (using the fact that W is a (V ) ⊆ W, so a belongs to I . Thus w = b a = a (b ) ∈ I V shows that a i τ i τ i i i i τ τi ⊆ i i i i i i ; ; Iτ Vτ , giving the reverse inclusion, and W = τ Iτ Vτ = τ V (X, Iτ ) as claimed.

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273

If A is simple the only minimal ideal is I = A. Alternately, its only ideals are itself and 0, Iτ = A or 0 with Vτ (X, I) = Vτ (X, A) or Vτ (X, 0) = 0, so the only invariant submodules are the sums of certain Vτ ’s, the Vτ ’s are the unique minimal submodules, hence are irreducible.

We also have a complete characterization of the F-endomorphisms of any Frankenstein module. THEOREM 20.11 (Endomorphism) The only F(X, A)-endomorphisms of the Frankenstein right A-module V (X, A) over an arbitrary coordinate ring A are the central coordinate multiplications on the individual tail-submodules Vτ := Vτ (X, A), ; EndF (V ) = τ Center(A)1Vτ , particular the distinct tail-classes provide inequivalent representations of deep matrices: there are no nonzero homomorphism of Vτ into a diﬀerent Vσ , HomF (Vτ , Vτ ) = Center(A)1Vτ ,

HomF (Vτ , Vσ ) = 0

(σ = τ ).

PROOF (1) The crux is that an F-endomorphism ϕ must act diagonally: it can only scale up each body, ϕ(b) = ba, from the fact that we can single out b by the actions >∞ η (b) of the Frankenstein projections determined by its many heads, N =0 FηNN(b) (V ) = bA η (b)

since FηNN(b) (b ) = 0 as soon as b has a diﬀerent N th head ηN (b ) = ηN (b) than b. (Alternately: write ϕ(b) = ba + i bi ai as a sum over distinct bodies, and apply Fkk of k theBody Separation Theorem 20.9 ﬁxing b and killing the bi , to get ϕ(b) = ϕ(Fk (b)) = k k Fk ϕ(b) = Fk ba + i bi ai = ba.) The multiplier a must be the same for all equivalent basis bodies because the Frankenstein operators act transitively on them by Tails 20.8 (i): ϕ(b ) = ϕ(F (b)) = F (ϕ(b)) = F (ba) = F (b)a = b a. Thus ϕ |Vτ = aτ 1Vτ is a left multiplication on each tail-submodule. Since these multiplications by a must commute with left multiplications AE∅∅ ⊆ F, the multipliers must lie in the center of A (and clearly all such central multiplications are F-linear). This establishes (1). (2) follows immediately from this and the direct sum decomposition Tails 20.8 (iii) of V into Vτ ’s. This chapter is dedicated to J. Marshall Osborn on the occasion of his 70th birthday.

References [1] L.A. Bokut, L.A. (1963). Some embedding theorems for rings and semigroups I. Sibirsk Mat. Zh. 4:500-518. [2] Cuntz, J. (1977). Simple C ∗ -algebras Generated by Isometries. Comm. in Math. Phys. 57:173-185. [3] Lin, H., Zhang, S. (1991). On Inﬁnite Simple C ∗ -Algebras. J. Func. Anal. 100:221-231. [4] Cuntz, J. (1981). K-theory for certain C ∗ -algebras. Ann. Math. 113:181-197.

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[5] Cuntz,J., Krieger, W. (1980) A class of C ∗ -algebras and topological Markov chains. Inv. Math. 56:251-268. [6] Goodearl, K.R., Handelman, D. (1975). Simple self-injective rings. Comm. in Alg. 3(9):797-834.

Chapter 21 Trace Reduction on Superspaces Kurt Meyberg Zentrum Mathematik der Technischen Universit¨ at, M¨ unchen, Germany

21.1 21.2 21.3 21.4

Superspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The tensorproduct v ⊗ u∗ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The supertrace . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The reduction map S → S . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

275 276 277 277 280

Abstract In this note we generalize the useful trace reduction, as discussed in [1], to superspaces Key words: supertrace, superspace, trace reduction 2000 MSC: 17A01, 15A03

21.1

Superspaces

A superspace is a Z2 -graded vector space V = V0 ⊕ V1 over a ﬁeld F . The elements vα ∈ Vα , α ∈ Z2 , are called homogeneous of degree α = p(vα ). The elements in V0 are called even, those in V1 odd. On any superspace V = V0 ⊕ V1 we have the conjugation map γ : V → V, γ(v0 + v1 ) = v0 − v 1 . A linear map f : V → W of superspaces (over F ) is called homogeneous of degree α, if f (Vβ ) ⊆ Wα+β , β ∈ Z2 . A linear map of degree zero maps V0 into W0 and V1 into W1 ; the degree-1-maps interchange: V 0 → W1 ; V 1 → W0 . Thus Hom(V, W ) has a natural grading Hom(V, W ) = Hom0 (V, W ) ⊕ Hom1 (V, W ), where Homα (V, W ) = {f | f (Vβ ) ⊆ Wβ+α , β ∈ Z2 }. This grading is compatible with (f, v) → f (v) and with the composition of maps (f, g) → f g in the following sense: p(f v) = p(f ) + p(v) p(f g) = p(f ) + p(g)

(21.1)

275

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Kurt Meyberg

for homogeneous components. Whenever we write p(f ) or p(v) it should be understood that f and v are homogeneous. On the ground ﬁeld F one usually takes the trivial grading F = F0 , F1 = 0. The scalars are even. Then we have the natural grading on the dual space, V ∗ = Hom(V, F ) = V0∗ ⊕ V1∗ . Any λ ∈ V ∗ has a unique decomposition λ = λ0 + λ1 , where λ0 : V0 → F , V1 → 0 and λ1 : V0 → 0, V1 → F . Thus (V ∗ )α = Vα∗ , if we extend the linear form λ ∈ Vα∗ to a linear form on V ∗ by letting (canonically) λ(vβ ) = 0 if β = α. Explicitly, λ(v) = λ0 (v0 ) + λ1 (v1 ), λα (vβ ) = 0 if α = β. We also use the convenient pairing notation , : V ∗ × V → F , λ, v = λ(v). Note, that Vα∗ and Vβ are orthogonal, Vα∗ , Vβ = 0, if α = β. For simplicity we assume that V is ﬁnite dimensional. The dual pairing , : V × V ∗ → F is deﬁned by (21.2) v, λ = (−1)vλ λ, v for homogeneous elements, where we adopted the useful shorthand notation (−1)vλ := (−1)p(v)p(λ) . Since we assume ﬁnite dimensionality, we identify the bidual via (21.2) with V ; V ∗∗ = V . We observe (by deﬁnition) for arbitrary v, λ, v, λ = v0 , λ0 + v1 , λ0 + v0 , λ1 + v1 , λ1 = λ0 , v0 − λ1 , v1 , since Vα∗ , Vβ (α = β) are orthogonal. Thus v, λ = λ, γ(v) = γ(λ), v

(21.3)

for all v ∈ V , λ ∈ V ∗ , where γ denotes the conjugation in V resp. V ∗ (γ(v) = v0 − v1 ).

21.2

The Tensorproduct v ⊗ u∗

Let U , V be subspaces over F . For any v ∈ V , u∗ ∈ U ∗ we deﬁne the linear map v ⊗ u∗ : U → V by (v ⊗ u∗ )(w) = u∗ , wv

(21.4)

for all w ∈ U . If v = v0 + v1 , u∗ = u∗0 + u∗1 then it is easily veriﬁed that (v ⊗ u∗ )0 = v0 ⊗ u∗0 + v1 ⊗ u∗1

(v ⊗ u∗ )1 = v1 ⊗ u∗0 + v0 ⊗ u∗1 are the homogeneous components of v ⊗ u∗ , in particular, p(v ⊗ u∗ ) = p(v) + p(u∗ ) (for homogeneous v, u∗ ). These tensorproducts are always very useful when dealing with traces: For any v ∈ V , λ ∈ V ∗ we have v ⊗ λ ∈ V ⊗ V ∗ = Hom(V, V ) and trace v ⊗ λ = λ, v.

Trace reduction on superspaces

21.3

277

The Supertrace

In superspaces the supertrace will replace the “ordinary” trace. Let V be a superspace, γ the conjugation on V . For any f ∈ HomV the supertrace str f of f is deﬁned by str f = trace γf.

(21.5)

It is easily veriﬁed that str is an even linear form on HomV with the following properties: str f1 =0 str fα gβ = 0 , if α = β

(21.6a) (21.6b)

str f g = (−1)f g str gf = str f str f ∗

(21.6c) (21.6d)

where f ∗ denotes the (super-)adjoint of f , deﬁned by f ∗ λ, v = (−1)f λ λ, f v (for homogeneous components). And again we are using the convenient shorthand notation (−1)uv := (−1)p(u)p(v) . Our basic tensorproducts are equally useful when dealing with supertraces: str v ⊗ λ = tr γ(v ⊗ λ) = tr γ(v) ⊗ λ = λ, γ(v) = v, λ

(21.7)

(see (21.3), tr := trace).

21.4

The Reduction Map S → S

Let U , V be superspaces. f : U → U , g : V → V linear. Then the two (graded) tensorproducts g ⊗ f : Hom(U, V ) → Hom(U, V ) and g × f : Hom U → Hom V are deﬁned (for homogeneous components) by (g ⊗ f )(h) = (−1)f h ghf , h ∈ Hom(U, V ) (g × f )(l) = (str f l)g , l ∈ Hom U

(21.8)

It is an elementary fact that all g ⊗ f span Hom(Hom(U, V )) and all g × f span Hom(Hom U, Hom V ). Moreover, these tensorproducts are compatible with the gradings: p(g ⊗ f ) = p(g) + p(f ), p(g × f ) = p(g) + p(f )

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Special case U1 = V1 = 0: In the “nonsuper” case, U1 = V1 = 0, there is the well known useful trace formula tr g ⊗ f = tr gtr f , and, more generally, the so-called trace reduction map S → S from Hom(Hom(U, V )) onto Hom(Hom U, Hom V ) satisfying trace (S · g ⊗ f ) = trace gS (f ) S (w ⊗ u∗ )v = S(v ⊗ u∗ )w

(21.9a) (21.9b)

for all g ∈ Hom V , f ∈ Hom U , w ∈ U , u∗ ∈ U ∗ , v ∈ V (see [1]). We are going to generalize these reduction formulas to superspaces. LEMMA 21.1 Let U , V be superspaces over F , f ∈ Hom U , g ∈ Hom V and g ⊗ f the graded tensorproduct as deﬁned in (21.8), then str g ⊗ f = str g · str f

(21.10)

PROOF We decompose into homogeneous components, f = f0 + f1 , g = g0 + g1 , h = h0 + h1 (h ∈ Hom(U, V )), then (g ⊗ f1 )(h) = gh0 f1 − gh1 f1 = gΓ(h)f1

(see (21.8))

= (g ◦ f1 )Γ(h), where Γ denotes the conjugation on Hom(U, V ), Γ(h) = h0 − h1 , and ◦ the ordinary (“old,” nonsuper) tensorproduct, (g ◦ f )(h) = ghf . Then str g ⊗ f1 = str (g ◦ f1 )Γ = tr Γ(g ◦ f1 )Γ (see (21.5)) = tr g ◦ f1 = tr gtr f1 = 0, since Γ2 = Id and tr f1 = 0; tr := trace Thus str g ⊗ f = str g ⊗ f0 . But g ⊗ f0 is (by deﬁnition) the “old” tensorproduct (without signchange), g ⊗ f0 = g ◦ f0 . Therefore we can apply the “ordinary” trace reduction (21.9a), (21.9b) to obtain str g ⊗ f0 = tr Γ g ◦ f0 = tr gΓ (f0 ).

(21.11)

We use (21.9b) to compute Γ (f0 ): Γ (w ⊗ u∗ )v = Γ(v ⊗ u∗ )w = [v0 ⊗ u∗0 + v1 ⊗ u∗1 − v0 ⊗ u∗1 − v1 ⊗ u∗0 ](w0 + w1 ) = u∗0 , w0 v0 + u∗1 , w1 v1 − u∗1 , w1 v0 − u∗0 , w0 v1 = u∗0 , w0 γ(v) − u∗1 , w1 γ(v) = u∗ , γ(w)γ(v) = (str w ⊗ u∗ )γ(v)

(see (21.7))

(here γ is the conjugation in V , resp. U ; and use again orthogonality of Uα∗ , Uβ if α = β).

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279

Thus Γ (f0 ) = (str f0 )γ and (21.11) becomes str g ⊗ f0 = str g str f0 .

The following trace reduction theorem in superspaces is a wide-reaching generalization of (21.10). It is expected that it will be of equal value for superalgebras as the ordinary trace reduction has been proven to be for ordinary algebras (see [1]). THEOREM 21.1 Notations as above. For any S ∈ Hom(Hom(U, V )) there is a unique S ∈ Hom(Hom U, Hom V ) such that a) str S · g ⊗ f = (−1)Sg str gS (f ) , S, g homogeneous ∗ ∗ ∗ b) S (w ⊗ u∗ )v = (−1)u w+u v+v w S(v ⊗ u∗ )w for all homogeneous u∗ ∈ U, w ∈ U, v ∈ V c) (g ⊗ f ) = g × f for all g ∈ Hom V , f ∈ Hom U PROOF Since we are dealing with tensorproducts there is a unique linear mapping S → S such that (g ⊗ f ) = g × f (on the generators). That this mapping satisﬁes a) and b) we only have to show these properties for the generators. For homogeneous a ∈ Hom V , b ∈ Hom U we get str (a ⊗ b)(g ⊗ f ) = (−1)bg+bf str ag ⊗ f b (composition rule for this tensorproduct) = (−1)bg+bf str ag str f b (by (21.10) = (−1)(b+a)g str ga str bf (by (21.6c) = (−1)(a+b)g str g [(str bf )a] = (−1)Sg str g S (f ) with S = a ⊗ b, S = a × b. This proves a) and c). For b) we may again restrict to the generators: [(a ⊗ b) (w ⊗ u∗ )](v) = [(a × b)(w ⊗ u∗ )](v) (by deﬁnition of prime) = [str bw ⊗ u∗ ]a(v) (deﬁnition of ×) = bw, u∗ a(v) (see (21.7)) ∗ ∗ = (−1)u b+u w u∗ , bwa(v) (see (21.2)) ∗ ∗ = (−1)u b+u w [a(v ⊗ u∗ )b](w) ∗

= (−1)u

w+vb

[(a ⊗ b)(v ⊗ u∗ )](w)

(see (21.8)).

Since we may assume p(b) = p(u∗ ) + p(w) (otherwise both sides are zero, due to orthogonality) this chain of equations continues: ∗

= (−1)u

w+u∗ v+vw

[(a ⊗ b)(v ⊗ u∗ )](w).

This completes the proof. If U = V we immediately get S = S from part b) of the theorem, and then (a×b) = a⊗b. We state this special case as follows. COROLLARY 21.1 If V is a superspace, f, g ∈ EndV , u, w ∈ V , v ∗ ∈ V ∗ and S : EndV → EndV linear, then

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Kurt Meyberg

a) str S · f ⊗ g = (−1)Sf str f S (g), b) (f ⊗ g) = f × g , (f × g) = f ⊗ g, ∗ ∗ c) S (u ⊗ v ∗ )w = (−1)uv +wv +uw S(w ⊗ v ∗ )u.

References [1] Meyberg, K.: Traceformulas in various algebras and L-projections. Nova J. of Alg. and Geom. 2, 107–135, 1993.

Chapter 22 Monomorphisms between Cayley-Dickson Algebras Guillermo Moreno Departamento de Matem´aticas, CINVESTAV del IPN, M´exico, D.F.

22.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.2 Pure and doubly pure elements in An+1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.3 Monomorphism from Am to An . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

281 282 284 289

Abstract In this chapter we study the algebra monomorphisms from Am = R2 into 2n An = R for 1 ≤ m ≤ n, where An are the Cayley-Dickson algebras. For n ≥ 4, we show that there are many types of monomorphisms and we describe them in terms of the zero divisors in An . m

Key words: Cayley-Dickson, quaternions, octonions, non-associative, alternative, ﬂexible, and G2 2000 MSC: 17A99

22.1

Introduction

The Cayley-Dickson algebras An over the real numbers is an algebra structure on R2 = An for n ≥ 0. By deﬁnition the Cayley-Dickson algebras (C-D algebras) are given by the doubling process of Dickson [1]. For (a, b) and (x, y) in An × An , deﬁne the product in An+1 = An × An as follows: n

(a, b) · (x, y) = (ax − yb, ya + bx). So if A0 = R and x = x for all x in R then A1 = C the complex numbers A2 = H the quaternion numbers and A3 = O the octonion numbers. As is well known, An is commutative for n ≤ 1; associative for n ≤ 2 and alternative for n ≤ 3, also An is normed for n ≤ 3. For n ≥ 4, An is ﬂexible and has zero divisors [4]. Let Aut(An ) be the automorphism group of the algebra An . As is well known, Aut(A1 ) = Z/2 = {Identity, Conjugation}. Aut(A2 ) = SO(3)

the rotation group in R3 .

281

282

Guillermo Moreno Aut(A3 ) = G2

the exceptional Lie group.

(See [3] and [6]). For n ≥ 4, Eakin-Sathaye showed that Aut(An ) = Aut(An−1 ) ×

.

3

Where 3 is the symmetric group of order 6. (See [2] and [5]). In this chapter we will extend the above results in the following sense: Suppose that 1 ≤ m ≤ n. By deﬁnition an algebra monomorphism ϕ : Am → An is a linear monomorphism such that (i) ϕ(e0 ) = e0 and (ii) ϕ(xy) = ϕ(x)ϕ(y) for all x and y in Am where e0 = (1, 0, . . . , 0) is the unit element in Am and An , respectively. We will describe the set M(Am , An ) = {ϕ : Am → An |ϕ

algebra monomorphism}.

We will see that this set is more complicated to describe for n ≥ 4. For n ≤ 3 we will recover known results about the relationship between Aut(An ) and the Stiefel manifolds V2n −1,2 . For n ≥ 3, recall that {e0 , e1 , . . . , e2n −1 } denotes the canonical basis in An and that the doubling process is given by An+1 = An ⊕ An e0 where e˜0 := e2n (half of the way basic in An+1 ). For ϕ ∈ M(Am ; An+1 ) for n ≥ 3, ϕ is of type I if e2n = e˜0 ∈ (Imϕ) ⊂ An+1 and ϕ is of type II if also e2n −1 := ε ∈ (Imϕ) ⊂ An+1 . The main result of this chapter is Theorem 22.1: the set of type II monomorphisms from A3 to An+1 can be described by the set of zero divisors in An+1 for n ≥ 4.

22.2

Pure and Doubly Pure Elements in An+1

Throughout this chapter we will establish the following notational conventions: Elements in An will be denoted by Latin characters a, b, c, . . . , x, y, z. Elements in An+1 will be denoted by Greek characters α, β, γ, . . . For example, α = (a, b) ∈ An × An . When we need to represent elements in An as elements in An−1 × An−1 we use subscripts, for instance, a = (a1 , a2 ), b = (b1 , b2 ), and so on, with a1 , a2 , b1 , b2 in An−1 . Now {e0 , e1 , . . . , e2n −1 } denotes the canonical basis in An . Then by the doubling process, {(e0 , 0), (e1 , 0), . . . , (e2n −1 , 0), (0, e0 ), . . . , (0, e2n −1 )} is the canonical basis in An+1 = An × An . By standard abuse of notation, we denote, also e0 = (e0 , 0), e1 = (e1 , 0), . . . , e2n −1 = (e2n −1 , 0),

Monomorphisms between Cayley-Dickson algebras

283

e2n = (0, e0 ), . . . , e2n+1 −1 = (0, e2n −1 ) in An+1 . For α = (a, b) ∈ An × An = An+1 we denote α = (−b, a) (the complexiﬁcation of α) so e0 = (0, e0 ) and α e0 = (a, b)(0, e0 ) = (−b, a) = α . Notice that α = −α. The trace on An+1 is the linear map tn+1 : An+1 → R given by tn+1 (α) = α + α = 2(real part of α) so tn+1 (α) = tn (a) when α = (a, b) ∈ An × An . DEFINITION 22.1

α = (a, b) in An+1 i s pure if tn+1 (α) = tn (a) = 0.

α = (a, b) in An+1 i s doubly pure if it is pure and also tn (b) = 0; i.e., α is pure in An+1 . n Also 2a, b = tn (ab) for , the inner product in R2 (see [4]). Notice that for a and b pure elements a ⊥ b if and only if ab = −ba. Notation: Im(An ) = {eo }⊥ ⊂ An is the vector subspace consisting of pure elements in n An ; i.e., Im(An ) = Ker(tn ) = R2 −1 . n+1 = Im(An ) × Im(An ) = {e0 , e0 }⊥ = R2n+1 −2 is the vector subspace consisting of A doubly pure elements in An+1 . LEMMA 22.1

n we have that For a and b in A

a and e0 a = − a. 1) a e0 = 2) a a = −||a||2 e0 and aa = ||a||2 e0 so a ⊥ a. with a a pure element. 3) ab = −ab 4) a ⊥ b if and only if ab + ba = 0. 5) a ⊥ b if and only if ab = b a. 6) ab = ab if and only if a ⊥ b and a ⊥ b. PROOF Notice that a is pure if a = −a and if a = (a1 , a2 ) is doubly pure, then a1 = −a1 and a2 = −a2 . a. 1) e0 a = (0, e0 )(a1 , a2 ) = (−a2 , a1 ) = (a2 , −a1 ) = −(−a2 , a1 ) = − 2) a a = (a1 , a2 )(−a2 , a1 ) = (−a1 a2 + a1 a2 , a21 + a22 ) = (0, −||a||2 e0 ) = −||a||2 e0 . Similarly aa = (−a2 , a1 )(a1 , a2 ) = (−a2 a1 + a2 a1 , −a22 − a21 ) = ||a||2 e0 . Now, since −2 a, a = a a+ aa = 0 we have a ⊥ a. 3) ab = (−a2 , a1 )(b1 , b2 ) = (−a2 b1 + b2 a1 , −b2 a2 − a1 b1 ). So ab = (a1 b1 + b2 a2 , b2 a1 − a2 b1 ) = (a1 , a2 )(b1 , b2 ) = ab then − ab = ab. Notice that in this proof we only use that a1 = −a1 ; i.e., a is pure and b doubly pure.

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Guillermo Moreno

= −ba. 4) a ⊥ b ⇔ ab + ba = 0 ⇔ ab = −ba ⇔ ab ⇔ − ab = ba ⇔ ab + ba = 0 by 3). 5) a⊥b⇔ ab + b a = 0 (by 4)) ⇔ −ab + b a = 0. = ba = −ba = ab. 6) If a ⊥ b and a ⊥ b, then by 3) and 4) ab = −ab Conversely, put a = (a1 , a2 ) and b = (b1 , b2 ) in An−1 × An−1 and deﬁne c := (a1 b1 + b2 a2 ) and d := (b2 a1 − a2 b1 ) in An−1 . Then ab = (a1 , a2 )(−b2 , b1 ) = (−a1 b2 + b1 a2 , b1 a1 + a2 b2 ) so ab = (−d, c). = (−d, c) and then Now ab = (a1 , a2 )(b1 , b2 ) = (a1 b1 + b2 a2 , b2 a1 − a2 b1 ) = (c, d), so ab ab = (d, −c). Thus, if ab = ab then c = −c and d = −d. Then tn (ab) = tn−1 (c) = c + c = 0 and a ⊥ b tn ( ab) = tn−1 (d) = d + d = 0 and a ⊥ b.

n . The four dimensional vector subspace COROLLARY 22.1 For each a = 0 in A a, a, e0 } is a copy of A2 = H. (We denote it by Ha ). generated by {e0 , PROOF We suppose that ||a|| = 1, otherwise we take multiplication table.

e0 a a e0

e0 e0 a a a

a a −e0 − e0 e0

a a + e0 −e0 − a

a ||a|| .

Construct the following

e0 e0 −a a −e0

By Lemma 22.1 a e0 = a; e0 a = − a; ae0 = a = −a; e0 a = − a = a; a a = − e0 and aa = e0 . a ↔ ˆi, a ↔ ˆj and But this is the multiplication table of A2 = H identifying e0 ↔ 1, ˆ e0 ↔ k.

22.3

Monomorphism from Am to An

Throughout this chapter 1 ≤ m ≤ n. DEFINITION 22.2 An algebra monomorphism from Am to An is a linear monomorphism ϕ : Am → An such that i) ϕ(e0 ) = e0 (the ﬁrst e0 is in Am and the second e0 in An ) ii) ϕ(xy) = ϕ(x)ϕ(y) for all x and y in Am .

Monomorphisms between Cayley-Dickson algebras

285

By deﬁnition we have that ϕ(re0 ) = rϕ(e0 ) for all r in R so ϕ(Im(Am )) ⊂ ϕ(Im(An )) and ϕ(x) = ϕ(x). Therefore ||ϕ(x)||2 = ϕ(x)ϕ(x) = ϕ(x)ϕ(x) = ϕ(xx) = ϕ(||x||2 ) = ||x||2 for all x ∈ Am m n and ||ϕ(x)|| = ||x|| and ϕ is an orthogonal linear transformation from R2 −1 to R2 −1 . The trivial monomorphism is the one given by ϕ(x) = (x, 0, 0, . . . , 0) for x ∈ Am and 0 in Am (2n−m−1 -times). M(Am , An ) denotes the set of algebra monomorphisms from Am to An . For m = n, M(Am ; An ) = Aut(An ) the group of algebra automorphisms of An PROPOSITION 22.1

M(A1 ; An ) = S(Im(An )) = S 2

n

−2

.

PROOF A1 = C = Span{e0 , e1 }. If x ∈ A1 then x = re0 + se1 and for w ∈ Im(An ) with ||w|| = 1 we have that ϕw (x) = re0 + sw deﬁne an algebra monomorphism from A1 to An . This can be seen by direct calculations, recalling that center (An ) = R for all n and that every associator with one real entry vanish. ϕw (x)ϕw (y) = (re0 + sw)(pe0 + qw) = (rp + sqw2 )e0 + (rq + sp)w = (rp − sq)e0 + (rq + sp)w = ϕw (x)ϕw (y), when y = pe0 + qe1 and p and q in R. Clearly ϕw (e0 ) = e0 . Conversely, for ϕ ∈ M(A1 ; An ), set w = ϕ(e1 ) in An so ||w|| = 1 and ϕw = ϕ. REMARK 22.1

In particular, we have that

Aut(A1 ) = S 0 = Z/2 = {Identity, conjugation} = {ϕe1 , ϕ−e1 }. To calculate M(A2 ; An ) for n ≥ 2 we need to recall (see [5]). DEFINITION 22.3 For a and b in An . We said that a alternate with b, we denote it by a b, if (a, a, b) = 0. We said that a alternate strongly with b, we denote it by a b, if (a, a, b) = 0 and (a, b, b) = 0. Clearly a alternate strongly with e0 for all a in An and if a and b are linearly dependent then a b (by ﬂexibility). Also,by deﬁnition, a is an alternative element if and only if a x for all x in An . By Lemma 22.1 1) and 2) we have that for any doubly pure element a in An (a, a, e˜0 ) = 0 and (by the above remarks) e˜0 alternate strongly with any a in An . For a and b pure elements in An , we deﬁne the vector subspace of An V (a; b) = Span{e0 , a, b, ab}. Also we identify the Stiefel manifold V2n −1,2 as {(a, b) ∈ Im(An ) × Im(An )|a ⊥ b, ||a|| = ||b|| = 1}. LEMMA 22.2

If (a, b) ∈ V2n −1,2 and a b then V (a; b) = A2 = H the quaternions.

286 PROOF

Guillermo Moreno Suppose that (a, b) ∈ V2n −1,2 and that (a, a, b) = 0 then we have ab, a = b, aa = b, ||a||2 e0 = ||a||2 b, e0 = 0 ab, a = a, bb = a, ||b||2 e0 = ||b||2 a, e0 = 0 ||ab||2 = ab, ab = a(ab), b = −a(ab), b = −a2 b, b = −a2 b, b = ||a||2 ||b||2 = 1

so {e0 , a, b, ab} is an orthonormal set of vectors in An . Finally using also that (a, b, b) = 0 and ab = −ba we may check by direct calculations that the multiplication table of {e0 , a, b, ab} coincides with the one of the quaternions and by the identiﬁcation e0 → e0 , a → e1 , b → e2 and ab → e3 we have an algebra isomorphism between A2 = H and V (a; b). PROPOSITION 22.2 In particular

M(A2 ; An ) = {(a, b) ∈ V2n −1,2 |a b} for n ≥ 2. Aut(A2 ) = M(A2 ; A2 ) = V3,2 = SO(3)

and M(A2 , A3 ) = V7,2 . PROOF The inclusion “⊃” follows from Lemma 22.2. Conversely suppose that ϕ ∈ M(A2 , An ) then ϕ(e0 ) = e0 , (ϕ(e1 ), ϕ(e2 )) ∈ V2n −1,2 and V (ϕ(e1 ), ϕ(e2 )) = Imϕ = H ⊂ An . Since A2 is an associative algebra and A3 is an alternative algebra we have that a b for any two elements in An for n = 2 or n = 3. ˜ n = {e0 , e˜0 }⊥ = R2n −2 denotes the vector subspace of REMARK 22.2 Recall that A ˜ n , we have that, if a ∈ S(A ˜ n) doubly pure elements. Since a e˜0 for any element in A i.e., ||a|| = 1 then (a, e˜0 ) ∈ V2n −1,2 and the assignment a → (a, e˜0 ) deﬁne an inclusion from ˜ n ) = S 2n −3 → M(A2 ; An ) ⊂ V2n −1,2 which resembles “the bottom cell” inclusion in S(A V2n −1,2 . To deal with cases 3 = m ≤ n we have to use the notion of a special triple (see [6] and [4]). DEFINITION 22.4 A set {a, b, c} in Im(An ) is a special triple if (i) {a, b, c} is an orthonormal set (ii) a b, a c and b c, i.e., its elements alternate strongly, pairwise. (iii) c ∈ V (a; b)⊥ ⊂ An . Now is easy to see that if {a, b, c} is a special triple then V (a; b); V (a, c); V (b, c) are isomorphic to A2 . For a special triple {a, b, c} consider the following vector subspace of An O(a; b; c) := Span{e0 , a, b, ab, c(ab), cb, ac, c}. PROPOSITION 22.3 For a special triple {a, b, c} in An and n ≥ 3; O(a, b, c) is an eight-dimensional vector subspace isomorphic, as algebra, to A3 = O the octonions and M(A3 , An ) = {(a, b, c) ∈ (An )3 |{a, b, c} special triple}.

Monomorphisms between Cayley-Dickson algebras PROOF also

287

We know that all elements in {e0 , a, b, ab, c(ab), cb, ac, c} are of norm one and c(ab), a = −ab, ac = a(ab), c = a2 b, c = a2 b, c = 0.

Similarly (c(ab)) ⊥ b and (c(ab)) ⊥ c. Thus {e0 , a, b, ab, c(ab), cb, ac, c} is an orthonormal set of vectors and O(a; b; c) is eight-dimensional. To see that O(a; b; c) ∼ = A3 we have to construct the corresponding multiplication table, which is a routine calculation. (See [5]) Conversely, if ϕ ∈ M(A3 , An ) then a = ϕ(e1 ), b = ϕ(e2 ) and c = ϕ(e7 ) form a special triple, when {e0 , e1 , e2 , e3 , e4 , e5 , e6 , e7 } is the canonical basis in A3 , and we recall that e1 e2 = e3 , e7 e3 = e4 , e7 e2 = e5 and e1 e7 = e6 in A3 . REMARK 22.3 For n = 3, A3 is an alternative algebra so a special triple in A3 is every triple such that (i) {a, b, c} is orthonormal (ii) c ⊥ (ab). So Proposition 22.3 gives the construction of G2 = Aut(A3 ) as in [6] and the assignment π

G2 = Aut(A3 ) → M(A2 , A3 ) = V7,2 (a, b, c) → (a, b) π

is the known ﬁbration G2 → V7,2 with ﬁber S 3 . REMARK 22.4 Suppose that n ≥ 4 and that {a, b, c} is a special triple in An so O(a; b; c) is the image of some algebra monomorphism from A3 to An and any orthonormal triple {x, y, z} of pure elements in O(a; b; c) with z ⊥ (xy) is also a special triple in An and O(x; y; z) = O(a; b; c). DEFINITION 22.5 e0 ∈ (Image of ϕ) ⊂ An .

For 1 ≤ m ≤ n. ϕ ∈ M(Am , An ) i s a type I monomorphism if

Since e˜0 = e2n−1 in An then the trivial monomorphism is not a type I monomorphism unless n = m, because by deﬁnition its image is generated by {e0 , e1 , . . . , e2m−1 −1 }. e0 ∈ (Imϕ)} the subset of all type I Denote by M1 (Am ; An ) = {ϕ ∈ M(Am ; An )|˜ monomorphisms, clearly M1 (An , An ) = M(An ; An ) = Aut(An ). Using Proposition 22.1 we may verify that for n ≥ 2. M1 (A1 , An ) = {re0 + s˜ e0 |r2 + s2 = 1} = S 1 . ˜ n we have that V (a; e˜0 ) = Ha and Also, by Lemma 22.2, for a nonzero a ∈ A ˜ n) = S2 M1 (A2 ; An ) = S(A In particular

n

−3

.

S 5 = M1 (A2 ; A3 ) ⊂ M(A2 ; A3 ) = V7,2

is “the bottom cell” of V7,2 . ˜ n , that Also we can check, using the fact that a e˜0 for all a ∈ A M1 (A3 ; An ) = M(A2 ; An ) ∩ V2n −2,2 ˜ n }. where V2n −2,2 = {(a, b) ∈ V2n −1,2 |a and b are in A

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Guillermo Moreno

DEFINITION 22.6

A type I monomorphism ϕ ∈ M1 (Am ; An+1 ) is of type II if e2n−1 := ε ∈ (Im

ϕ) ⊂ An+1 .

By deﬁnition if ϕ ∈ M1 (Am ; An+1 ) then e˜0 ∈ (Im ϕ) ⊂ An+1 and if we also assume that ε ∈ (Imϕ) ⊂ An+1 then ε˜ e0 = ε˜ and Hε := Span{e0 , ε˜, ε, e˜0 } lies in (Im ϕ) ⊂ An+1 ; therefore ϕ ∈ M(Am , An+1 ) is of type II if and only if Hε ⊂ (Im ϕ). Denote M2 (Am , An+1 ) = {ϕ ∈ M(Am ; An+1 )|ϕ is type II}. THEOREM 22.1

For n ≥ 3. M2 (A3 ; An+1 ) = CP 2

n−1

−1

∪ Xn

where CP m is the complex projective space in Cm and X n = {(x, y) ∈ An × An |xy = 0, x = 0 and y = 0}. In particular for n = 3

X 3 = Φ (empty set) and M2 (A3 ; A4 ) = CP 3 .

PROOF Suppose that ϕ : A3 → An+1 is an algebra monomorphism for n ≥ 3 with Hε ⊂ (Im ϕ). So Im ϕ is isomorphic to O = A3 , as algebras, then there is a nonzero ˜ α ∈ H⊥ ε ⊂ An+1 . such that Im ϕ = Oα := Span{e0 , ε˜, ε, e˜0 , α, ˜ αε, ε˜α, α} ⊂ An+1 ˜n × A ˜ n = H⊥ ⊂ An+1 and that a = 0 (similarly we may Suppose that α = (a, b) ∈ A ε assume b = 0). Now Im ϕ = Oα = A3 if and only if (α, α, ε) := α2 ε − α(αε) = 0 i.e.

− ||α||2 ε = α(αε).

Using Lemma 22.1 we have that α(αε) = (a, b)[(a, b)(˜ e0 , 0)] = (a, b)[(˜ a, −˜b)] = ˜ ˜ = (a˜ a − bb, −ba − b˜ a) = (−||a||2 e˜0 − ||b||2 e˜0 , 0) − (0, (b˜ e0 )a − b(˜ e0 a)) = −||α||2 ε + (0, (a, e˜0 , b)). Therefore (a, e˜0 , b) = 0 in An if and only if (α, α, ε) = 0 in An+1 . ⊥ Since An = Ha ⊕ H⊥ a we have that b = c + d where c ∈ Ha and d ∈ Ha with c doubly pure i.e., c ∈ Span{a, a ˜} ⊂ Ha . Since Ha is associative we have that

(a, e˜0 , b) = (a, e˜0 , c + d) = (a, e˜0 , c) + (a, e˜0 , d) = 0 + (a, e˜0 , d) = (a, e˜0 , d) = a ˜d + ad˜ = 2˜ ad by Lemma 22.1 1), 2), and 6). Therefore (α, α, ε) = 0 in An+1 if and only if ad = a ˜d = 0 in An . So, we have two cases: namely, d = 0 and d = 0 in An .

Monomorphisms between Cayley-Dickson algebras

289

˜} so (a, b) determine a complex line Suppose d = 0. Thus b = c ∈ Ha and b ∈ Span{a, a ˜ n = R2n −2 = C2n−1 −1 and α ∈ CP 2n−1 −1 . in A Suppose d = 0 (so b = 0 and a = 0). Thus ad = 0 and (a, d) ∈ X n .

References [1] L.E. Dickson. On quaternions and their generalization and the history of the 8 square theorem, Annals of Math. 20, 155-171, 1919. [2] Eakin-Sathaye. On automorphisms and derivations of Cayley-Dickson algebras, Journal of Pure and Applied Algebra 129, 263-280, 1990. [3] S.H. Khalil and P. Yiu. The Cayley-Dickson algebras: A theorem of Hurwitz and quaternions, Bolet´ın de la Sociedad de L´ odz Vol. XLVIII, 117-169. 1997. [4] G. Moreno. The zero divisors of the Cayley-Dickson algebras over the real numbers, Bol. Soc. Mat. Mex. 4, 13-27. 1998 [5] G. Moreno. Alternative elements hopf.math.purdue.edu/pub/Moreno, 2001

in

the

Cayley-Dickson

algebras,

[6] G. Whitehead. Elements of Homotopy theory, Graduate text in Math. 61, SpringerVerlag.

Chapter 23 The Notion of Lower Central Series for Loops Jacob Mostovoy Instituto de Matem´aticas (Unidad Cuernavaca) Universidad Nacional Aut´ onoma de M´exico, M´exico

23.1 23.2 23.3 23.4 23.5 23.6

The lower central series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Commutators, associators, and associator deviations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The commutator-associator ﬁltration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The lower central series and the commutator-associator subloops . . . . . . . . . . . . . . . . . . . . . Further comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

291 292 292 295 297 297 297

Abstract The commutator calculus is one of the basic tools in group theory. However, its extension to the nonassociative context, based on the usual deﬁnition of the lower central series of a loop, is not entirely satisfactory. Namely, the graded abelian group associated to the lower central series of a loop is not known to carry any interesting algebraic structure. In this note we construct a new generalization of the lower central series to arbitrary loops that is tailored to produce a set of multilinear operations on the associated graded group. Key words: loop, commutator-associator ﬁltration, lower central series 2000 MSC: Primary 20N05; Secondary 20F14

23.1

The Lower Central Series

Let N be a normal subloop of a loop L. There exists a unique smallest normal subloop M of L such that N/M is contained in the center of the loop L/M , that is, all elements of N/M commute and associate with all elements of L/M . We shall denote1 this subloop M by {N, L}. The lower central series of L is deﬁned by setting γ1 L = L and γi+1 L = {γi L, L} for i ≥ 1. This deﬁnition can be found in [1]. For L associative it coincides with the usual deﬁnition of the lower central series. The terms of the lower central series of L are fully invariant normal subloops of L and the successive quotients γi L/γi+1 L are abelian groups. If L is a group, the commutator operation on L induces a bilinear operation (Lie bracket) on the associated graded group ⊕γi L/γi+1 L; this Lie bracket is compatible with the grading. (See Chapter 5 of [4] for a classical account of the commutator calculus in groups.) In general, however, there is no obvious algebraic structure on ⊕γi L/γi+1 L. 1 Usually,

square brackets are used in this situation. We prefer to reserve square brackets for commutators.

291

292

23.2

Jacob Mostovoy

Commutators, Associators, and Associator Deviations

Here we introduce some terminology. The deﬁnitions of this paragraph are valid for L a left quasigroup, that is, a halfquasigroup with left division. Deﬁne the commutator of two elements a, b in L as [a, b] := (ba)\(ab), and the associator of three elements a, b, c in L as (a, b, c) := (a(bc))\((ab)c). The failure of the associator to be distributive in each variable is measured by three operations that we call associator deviations or simply deviations. These are deﬁned as follows: (a, b, c, d)1 := ((a, c, d)(b, c, d))\(ab, c, d), (a, b, c, d)2 := ((a, b, d)(a, c, d))\(a, bc, d), (a, b, c, d)3 := ((a, b, c)(a, b, d))\(a, b, cd). The deviations themselves are not necessarily distributive and their failure to be distributive is measured by the deviations of the second level. The general deﬁnition of a deviation of nth level is as follows. Given a positive integer n and an ordered set α1 , . . . , αn of not necessarily distinct integers satisfying 1 ≤ αk ≤ k + 2, the deviation (a1 , . . . , an+3 )α1 ,...,αn is a function Ln+3 → L deﬁned inductively by (a1 , . . . , an+3 )α1 ,...,αn := (A(aαn )A(aαn +1 ))\A(aαn aαn +1 ), where A(x) stands for (a1 , . . . , aαn −1 , x, aαn +2 , . . . , an+3 )α1 ,...,αn−1 . The integer n is called the level of the deviation. There are (n + 2)!/2 deviations of level n. The associators are the deviations of level zero and the associator deviations are the deviations of level one.

23.3

The Commutator-Associator Filtration

Let us explain informally our approach to generalizing the lower central series to loops. We want to construct a ﬁltration by normal subloops Li (i ≥ 1) on an arbitrary loop L with the following properties: (a) the subloops Li are fully invariant, that is, preserved by all automorphisms of L; (b) the quotients Li /Li+1 are abelian for all i; (c) the commutator and the associator in L induce well-deﬁned operations on the associated graded group ⊕Li /Li+1 ; these operations should be linear in each argument and should respect the grading. Clearly, we also want the ﬁltration Li to coincide with the lower central series for groups. A na¨ıve method of constructing such a ﬁltration would be setting L1 = L and taking Li to be the subloop normally generated by

The notion of lower central series for loops

293

(a) commutators of the form [a, b] with a ∈ Lp and b ∈ Lq with p + q ≥ i, (b) associators of the form (a, b, c) with a ∈ Lp , b ∈ Lq , and c ∈ Lr with p + q + r ≥ i. The subloops Li constructed in this way are fully invariant in L and the quotients Li /Li+1 are abelian groups. It can be seen that the commutator on L induces a bilinear operation on the associated graded group. However, the associator does not descend to a trilinear operation on ⊕Li /Li+1 . The situation can be mended by adding to the generators of Li , for all i, every element of the form (a, b, c, d)α with a ∈ Lp , b ∈ Lq , c ∈ Lr and d ∈ Ls , where p + q + r + s ≥ i and 1 ≤ α ≤ 3. This forces the associator to be trilinear on ⊕Li /Li+1 , but now we are faced with a new problem. Having introduced the deviations into the game, we would like them to behave in some sense like commutators and associators, namely, to induce multilinear operations that respect the grading on ⊕Li /Li+1 . This requires adding deviations of the second level to the generators of the Li , etc. The above reasoning is summarized in the following. DEFINITION 23.1 For a positive integer i, the ith commutator-associator subloop Li of a loop L is L itself if i = 1, and is the subloop normally generated by (a) [Lp , Lq ] with p + q i, (b) (Lp , Lq , Lr ) with p + q + r i, (c) (Lp1 , . . . , Lpn+3 )α1 ,...,αn with p1 + . . . + pn+3 i for all possible choices of α1 , . . . , αn . We refer to the ﬁltration by commutator-associator subloops as the commutator-associator ﬁltration.2 In our terminology the usual commutator-associator subloop becomes the “second commutator-associator subloop.” By virtue of its construction, the commutator-associator ﬁltration has the desired properties: the Li are fully invariant and normal, the quotients Li /Li+1 are abelian, and the associator and the deviations induce multilinear operations on the associated graded group. The bilinearity of the commutator is also readily seen. Indeed, take a, b in Lp and c in Lq . Modulo Lp+q+1 , the commutators [a, c] and [b, c] commute and associate with all elements of L. Also, any associator that involves one element of Lp , one element of Lq and any element of L, is trivial modulo Lp+q+1 . Hence, a(cb) · [b, c] ≡ a(bc) ≡ (ab)c and c(ab) · [a, c] ≡ (ca)b · [a, c] ≡ (ca · [a, c])b ≡ (ac)b ≡ a(cb) modulo Lp+q+1 . we have (c(ab) · [a, c])[b, c] ≡ (ab)c

mod Lp+q+1

and it follows that [a, c][b, c] ≡ [ab, c]

mod Lp+q+1 .

The linearity of the commutator in the second argument is proved in the same manner. One should expect that the algebraic structure on ⊕Li /Li+1 induced by the commutator, the associator and the deviations, generalizes Lie rings in the same way as Sabinin algebras3 generalize Lie algebras. No axiomatic deﬁnition of such a structure is yet known. 2 For

the lack of a better name. known as hyperalgebras, see [5] and [6].

3 Formerly

294

Jacob Mostovoy

While a complete description of the algebraic structure on ⊕Li /Li+1 is a task beyond the scope of the present note, the following result is a step in this direction. THEOREM 23.1

The Akivis identity

[[a, b], c] + [[b, c], a] + [[c, a], b] = (a, b, c) + (b, c, a) + (c, a, b) − (a, c, b) − (c, b, a) − (b, a, c) is satisﬁed in ⊕Li /Li+1 for any L. The proof of the Akivis identity is just a bulky calculation; it occupies the rest of this section. In the course of this proof we shall often write ab instead of b\a. It should not lead to confusion since only left division will be used. Deﬁne w(a, b, c) by (b, a, c) . w(a, b, c) = [[a, b], c] · (a, b, c) We need to prove that for a ∈ Lp , b ∈ Lq and c ∈ Lr w = w(a, b, c) · w(b, c, a)w(c, a, b) ∈ Lp+q+r+1 . We shall work modulo Lp+q+r+1 . First let us prove that w(a, b, c) ≡

a(cb) · [b, c] . b(ac) · [a, b]

(23.1)

Indeed, (ba)([a, b]c) · (ba, [a, b], c) = (ab)c, and, therefore, [a, b]c ≡

(ab)c ba

mod Lp+q+r+1 .

Using similar arguments, together with the fact that the associators involving a, b, and c, commute and associate with everything modulo Lp+q+r+1 , we get (ab)c (ba)\((ab)c) ≡ c[a, b] (ba)c · [a, b] (a, b, c) a(cb) · [b, c] a(bc) · (a, b, c) ≡ · ≡ (b(ac) · [a, b]) · (b, a, c) (b, a, c) b(ac) · [a, b]

[[a, b], c] ≡

and (23.1) follows. Now, without loss of generality we can assume that p ≤ q ≤ r. It is easy to see that, under this assumption, (23.1) implies the following:

a(cb) · [b, c] w(a, b, c) ≡ [a, b]\ b(ac)

b(ac) w(b, c, a) ≡ [b, c]\ · [c, a] c(ba)

c(ba) · [a, b] w(c, a, b) ≡ [c, a]\ . a(cb)

(23.2) (23.3) (23.4)

The notion of lower central series for loops

295

Observe that (b(ac))\(a(cb)) ∈ Lp+q , (c(ba))\(b(ac)) and (a(cb))\(c(ba · [a, b])) are in Lp+r . Multiplying the three expressions (23.2)–(23.4) we obtain [a, b]w =

a(cb) b(ac) c(ba) · [a, b] · . b(ac) c(ba) a(cb)

Further,

a(cb) b(ac) b(ac)

c(ba)

≡

a(cb) c(ba)

and

a(cb) c(ba) · [a, b] c(ba)

a(cb)

≡ [a, b]

and it follows that w ≡ 1.

23.4

The Lower Central Series and the Commutator-Associator Subloops

Let us compare the commutator-associator ﬁltration with the usual lower central series. It is clear that for any L, γ2 L = L2 . More generally, an easy inductive argument establishes that γi L is contained in Li for all i. There is, however, no converse to this statement. THEOREM 23.2 Let F be a free loop. For any positive integer k the term Fk of the commutator-associator ﬁltration is not contained in γ3 F . PROOF Let X be a set freely generating the loop F and assume y is in X. Let y m stand for the product (. . . ((yy)y) . . .)y of m copies of y. We shall prove that for any i ≥ 0 and any positive m, the deviation of ith level (y m , y, . . . , y)1,1,...,1 does not belong to γ3 F . The machinery suitable for this purpose was developed by Higman in [3]. Let L be a loop and assume there is a surjective homomorphism p : F → L with kernel N . Deﬁne A to be the additively written free abelian group, with free generators f (l1 , l2 ) for all l1 = 1 and l2 = 1 in L, and g(x) for all x in X. The product set L × A can be given the structure of a loop by setting (l1 , b1 )(l2 , b2 ) = (l1 l2 , b1 + b2 + f (l1 , l2 )), (l1 , b1 )/(l2 , b2 ) = (l1 /l2 , b1 − b2 − f (l1 /l2 , l2 )), (l2 , b2 )\(l1 , b1 ) = (l2 \l1 , b1 − b2 − f (l2 , l2 \l1 )), where f (l, 1) = f (1, l) = 0 for all l ∈ L. The pair (1, 0) is the identity. Higman denotes this loop by (L, A). There is a homomorphism δ : F → (L, A) deﬁned on the generators by δx := (px, g(x)). Higman proved (Lemma 3 of [3]) that the kernel of δ is precisely {N, F }. Without loss of generality we can assume that F is the free loop on one generator y; take N = γ2 F . The quotient loop L = F/N can be identiﬁed with the group of integers Z. We

296

Jacob Mostovoy

shall see that for any positive m the element δ(y m , y, . . . , y)1,1,...,1 is not zero in (L, A) and, hence, that (y m , y, . . . , y)1,1,...,1 does not belong to {N, F } = γ3 F . For any m ≥ 1 and any n ≥ 0,

LEMMA 23.1

δ(y m , y, . . . , y)1,1,...,1 = (0, f (n + m + 1, 1) + ap,q f (p, q)), 6 78 9 p
where ap,q are integers. The lemma is proved by induction on n. A straightforward calculation gives δy m = (m, mg(y) + f (1, 1) + f (2, 1) + . . . + f (m − 1, 1)). It follows that

δ(y m−2 y 2 ) = (m, mg(y) +

cp,q f (p, q)),

p,q

where p ≤ m − 2, q ≤ 2 and the cp,q are some coeﬃcients whose value is of no importance. Thus δ(y m−2 , y, y) is of the form ap,q f (p, q)) (0, f (m − 1, 1) + p<m−1,q

and the lemma is satisﬁed for n = 0. Now, for arbitrary positive n we have (y m , y, . . . , y)1,...,1 6789 n

⎞

⎛

⎟ ⎜ = ⎝(y m , y, . . . , y)1,...,1 (y, y, . . . , y)1,...,1 ⎠ 6789 6789 n−1

? (y m+1 , y, . . . , y)1,...,1 6789

n−1

n−1

and it only remains to apply δ to both sides and use the induction assumption. REMARK 23.1 Higman’s construction can also be applied to show that the commutator does not always respect the lower central series. A simple calculation shows that in the free loop on four generators a, b, c, d, the element [[a, b], [c, d]] belongs to γ3 but not γ4 . Higman [3] has proved that the terms of the lower central series of a free loop intersect in the identity. We have no proof of a similar statement for the commutator-associator ﬁltration; in view of Theorem 23.2, Higman’s result is not suﬃcient to establish it. Theorem 23.2 reﬂects a fundamental diﬀerence between the lower central series and the commutator-associator ﬁltration. In the proof of Theorem 23.2 we have, in fact, established that the quotient γ2 F/γ3 F is not ﬁnitely generated. On the other hand, we have the following THEOREM 23.3 For any ﬁnitely generated loop L the abelian groups Li /Li+1 are ﬁnitely generated for all i > 0. PROOF For i = 1 the theorem is clearly true: L1 /L2 is generated by the classes of the generators of L. Assume that for all k < n the group Lk /Lk+1 has a ﬁnite set of generators

The notion of lower central series for loops

297

ak,α with α in some ﬁnite index set. Consider a commutator [x, y] with x in Lp and y in Ln−p . The class of [x, y] in Ln /Ln+1 only depends on the classes of x and y in Lp /Lp+1 and Lp /Ln−p+1 , respectively. As the commutator is bilinear on ⊕Li /Li+1 , the class of [x, y] is a linear combination of elements of the form [ap,α , an−p,β ]. Similarly, all classes of associators and deviations can be expressed as linear combinations of associators and deviations of a ﬁnite number of elements. Hence, Ln /Ln+1 is ﬁnitely generated.

23.5

Further Comments

In general, the lower central series of loops is diﬃcult to treat. There is one notable exception, namely, the theory of associators in commutative Moufang loops, which is analogous to a large extent to the theory of commutators in groups, see [1]. There is also another ﬁltration, called distributor series, on commutative Moufang loops. In general, neither of these two ﬁltrations coincides with the commutator-associator ﬁltration. We note that the deviations ﬁrst appeared in the nilpotency theory of commutative Moufang loops as the operations that produce the distributor series. There exists a ﬁltration on any group that is very closely related to the lower central series, namely, the ﬁltration by dimension subgroups. In particular, the graded abelian groups associated to both ﬁltrations become isomorphic after tensoring with the ﬁeld of rational numbers. A detailed comparison of the lower central series and the dimension subgroups can be found in [2]. The dimension ﬁltration can be generalized to loops; it would be interesting to compare it with the commutator-associator ﬁltration. The constructions of this note can be carried out in greater generality. The deﬁnitions of the commutator, the associator and the deviations of all levels only involve left division and, therefore, make sense in any left half-quasigroup. In particular, the commutator-associator ﬁltration can be deﬁned for left loops.

23.6

Acknowledgments

I would like to thank Ted Stanford, Liudmila Sabinina, and Jos´e Mar´ıa P´erez Izquierdo for useful discussions. The author was partially supported by CONACyT grant CO2-44100.

References [1] R. H. Bruck, Survey of Binary Systems, Springer-Verlag, Berlin-G¨ ottingen-Heidelberg 1958. [2] B. Hartley, Topics in the theory of nilpotent groups, In: Group Theory: Essays for Philip Hall. Academic Press, London, 1984.

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[3] G. Higman, The lower central series of a free loop, Quart. J. Math. Oxford (2), 14 (1963), 131–140. [4] W. Magnus, A. Karrass and D. Solitar, Combinatorial Group Theory, Second revised edition. Dover, New York, 1976. [5] P. Miheev and L. Sabinin, Quasigroups and diﬀerential geometry, Quasigroups and loops: theory and applications, 357–430, Heldermann, Berlin, 1990. [6] I. Shestakov and U. Umirbaev, Free Akivis algebras, primitive elements, and hyperalgebras, J. Algebra 250 (2002), no. 2, 533–548.

Chapter 24 Gravity within the Framework of Nonassociative Geometry Alexander I. Nesterov Department of Physics, Guadalajara University, Mexico

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Introduction to nonassociative geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Nonassociative discrete geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Discrete de Sitter spacetime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Toy model of discrete (2+1)-spacetime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24.5.1 Nonassociative discrete geometry of S 2 and (2+1)-spacetime evolution . . . . . . . . 24.6 Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24.1 24.2 24.3 24.4 24.5

299 300 304 306 307 308 310 310

Abstract Recently we proposed a new mathematical theory, nonassociative geometry, providing a uniﬁed algebraic description of continuous and discrete spacetime. It follows from our approach that nonassociativity is an algebraic equivalent of curvature and at distances comparable with Planck length the standard concept of spacetime might be replaced by the nonassociative discrete structure, which at large spacetime scales “looks like” a diﬀerentiable manifold. We give an up-to-date perspective with a general overview of the nonassociative geometry, a description of some discrete models of spacetime, their continuum limit, and the emerged causal structure. Key words: quasigroups, smooth loops, discrete spacetime 2000 MSC: 20N05, 55R65, 53B15, 81T13

24.1

Introduction

The numerous attempts to construct the quantum theory of gravitation and to understand the structure of spacetime has not been successful so far and the problem is still open. Recently the interest to this problem has increased sharply, essentially due to the progress in the string theory and loop quantum gravity (For recent general reviews on subject, see [1, 2, 3, 4, 5]). One should distinguish two general strategies beyond the common treatment [1]: (1) The ﬁrst strategy is to quantize a classical structure and then restore it as some kind of classical limit of the quantum theory (2) The second strategy is to regard the classical structure as one emerging from the other theory.

299

300

Alexander I. Nesterov

According to [1] there are four ways to approach the quantum gravity: 1. Quantizing general relativity. 2. Quantizing a diﬀerent classical theory, having general relativity emerged as a lowenergy (large-distance) limit. 3. Having general relativity emerging as a low-energy limit of a quantum theory that is not a quantization of a classical limit. 4. Having both general relativity and quantum theory emerging from a theory very diﬀerent from both. Loop quantum gravity and superstring theory at the fundamental level use the standard spacetime concept and represent some type of the models (1) and (2). The third approach in the classiﬁcation scheme above deals with quantum theory that is quantum in some “intrinsic” sense [1]. In this approach the classical treatment of spacetime as a pair (M, g), where g is a Lorentz metric on M , with the natural link: set of spacetime points → topology → diﬀerential structure → (M, g), is changed, regarding the regions (areas) as more important objects than points. Ideas of this type imply that quantum physics of the bounded regions will involve only ﬁnite-dimensional Hilbert space [1, 6, 7]. The fourth approach may require a revision of the quantum theory itself in a way that omits the a priori use of continuum numbers in its mathematical foundation. This implies that in quantum theory of gravitation the standard concept of spacetime must be replaced at the Planck scales by some kind of discrete structure, which at large spacetime scales “looks like” a diﬀerentiable manifold [1]. Here we survey, in brief, foundations of nonassociative geometry (for details see [13, 14, 15, 16, 17, 18] ). The key point is to give an algebraic (non local) description of spacetime, which may be used in continuum and discrete cases. Nonassociative geometry provides the uniﬁed algebraic description of continuum and discrete spacetime and admits all basic attributes of spacetime geometry, including generalized Einstein’s equations. By its nature, our approach is closer to the fourth approach to quantum gravity, but may be applied to loop quantum gravity.

24.2

Introduction to Nonassociative Geometry

The recent development of geometry has shown the importance of nonassociative algebraic structures, such as quasigroups, loops, and odules. For instance, it is possible to say that the nonassociativity is the algebraic equivalent of the diﬀerential geometric concept of curvature. The corresponding construction may be described as follows. The main algebraic structures arising in this approach are related to nonassociative algebra and theory of quasigroups and loops. DEFINITION 24.1 Let Q, · be a groupoid with a binary operation (a, b) → a·b and Q be a smooth manifold. A groupoid Q, · is called a quasigroup if equations a·x = b, y·a = b have a unique solution x = a\b, y = b/a. A loop is a quasigroup with a two-sided identity a·e = e·a = a, ∀a ∈ Q. A loop Q, ·, e with a smooth functions φ(a, b) := a·b is called a smooth loop. Let Q, ·, e be a smooth local loop with a neutral element e. We deﬁne La b = Rb a = a·b,

l(a,b) = L−1 a·b ◦ La ◦ Lb ,

(24.1)

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where La is a left translation, Rb is a right translation, l(a,b) is an associator. DEFINITION 24.2 Let M, ·, e be a partial magma with a binary operation (x, y) → x · y and the neutral element e, x · e = e · x = x; M be a smooth manifold (at least C 1 -smooth) and the operation of multiplication (at least C 1 -smooth) be deﬁned in some neighborhood Ue , then M, ·, e is called a partial loop on M . The operation of multiplication is locally left and right invertible. This means that if −1 in some neighborhood of the neutral x · y = Lx y = Ry x then there exist L−1 x and Rx element e: Ra (Ra−1 x) = x. La (L−1 a x) = x, The vector ﬁelds Aj deﬁned on Ue by i ∂ ∂ Aj (x) = (Lx )∗,e j i ≡ Lij (x) i ∂x ∂x

(24.2)

are called the left basic fundamental ﬁelds. Similarly, the right basic fundamental ﬁelds aj are deﬁned by i ∂ ∂ aj (x) = (Rx )∗,e j i ≡ Rji (x) i . (24.3) ∂x ∂x The solution of the equation df i (t) = Lij (f (t))X j , dt

f (0) = e,

(24.4)

is of the form f (t) = ExptX deﬁning the exponential map Exp : X ∈ Te (M ) −→ ExpX ∈ M. The unary operation

tx = Exp(tExp−1 x),

(24.5)

based on the exponential map, is called the left canonical unary operation for M, ·, e. A smooth loop M, ·, e equipped with its canonical left unary operations is called the left canonical preodule M, ·, (t)t∈R , e. If one more operation is introduced, x + y = Exp(Exp−1 x + Exp−1 y),

(24.6)

then we obtain the canonical left prediodule of a loop, M, ·, +, (t)t∈R , e. A canonical left preodule (prediodule) is called the left odule (diodule), if the monoassociativity property tx · ux = (t + u)x

(24.7)

is satisﬁed. In the smooth case for an odule the left and the right canonical operations as well as the exponential maps coincide. DEFINITION 24.3

Let M be a smooth manifold and let L : (x, y, z) ∈ M → L(x, y, z) ∈ M

be a smooth partial ternary operation, such that xa˙ y = L(x, a, z) deﬁnes, in some neighborhood of the point a, the loop with the neutral a; then the pair M, L is called a loopuscular structure (manifold).

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A smooth manifold M with a smooth partial ternary operation L and smooth binary operations ωt : (a, b) ∈ M × M → ωt (a, b) = ta b ∈ M, (t ∈ R), such that xa˙ y = L(x, a, y) and ta z = ωt (a, z) determine in some neighborhood of an arbitrary point a the odule with the neutral element a, is called a left odular structure (manifold) M, L, (ωt )t∈R . Let M, L, (ωt )t∈R and M, N, (ωt )t∈R be odular structures then M, L, N, (ωt )t∈R is called a diodular structure (manifold). If x+ y = N (x, a, y) and ta x = ωt (a, x) deﬁne a vector space, a then such a diodular structure is called a linear diodular structure. A diodular structure is said to be geodiodular if Lutaaxx ◦ Lata x = Laua x

(the ﬁrst geoodular identity) (the second geoodular identity), (the third geoodular identity)

(Lax y = L(x, a, y)), a a Lx ◦ ta = tx ◦ Lx Lax N (y, a, z) = N (Lax y, x, Lax z)

(24.8) (24.9) (24.10)

are true. DEFINITION 24.4 Let M be a C k -smooth (k ≥ 3) aﬃnely connected manifold and the following operations are given on M : Lax y = x a· y = Expx τxa Exp−1 a y, ωt (a, z) = ta z =

N (x, a, y) = x + y = a

(24.11)

Expa tExp−1 a z, Expa (Exp−1 a x

(24.12) +

Exp−1 a y),

(24.13)

Expx being the exponential map at the point x and τxa the parallel translation along the geodesic going from a to x. The construction above equips M with the linear geodiodular structure, which is called a natural linear geodiodular structure of an aﬃnely connected manifold (M, ∇). Any C k -smooth (k ≥ 3) aﬃnely connected manifold can be considered as a geodiodular structure. DEFINITION 24.5 Then the formula

Let M, L be a loopuscular structure of a smooth manifold M .

∇Xa Y =

d a [(Lg(t) )∗,a ]−1 Yg(t) dt

g(0) = a,

5 , t=0

g(0) ˙ = Xa ,

Y being a vector ﬁeld in the neighborhood of a point a, deﬁnes the tangent aﬃne connection. In coordinates the components of aﬃne connection are @ 2 A ∂ (xa˙ y)i i . Γjk (a) = − ∂xj ∂y k x=y=a The equivalence of the categories of geoodular (geodiodular) structures and of aﬃne connections has been shown in [11, 12]. DEFINITION 24.6

Let M, L be a loopuscular structure, then ha(b,c) = (Lac )−1 ◦ Lbc ◦ Lab

(24.14)

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is called an elementary holonomy. An elementary holonomy is, in fact, the parallel translation along a geodesic triangle path. Consequently, it is some integral curvature. Indeed, in the smooth case, diﬀerentiating (ha (x, y))i by xj , y k at a ∈ M we get the curvature tensor at a ∈ M precisely up to numerical factor. For a diodular structure one can consider an elementary holonomy h(a,b) = he(a,b) together with the diodule M, e· , + , (te )t∈R , e, so-called holonomial diodule, and restore this e diodular structure in a unique way: L(x, a, y) = Lex h(a,x) (Lea )−1 y,

(24.15)

N (x, a, y) = Lea ((Lea )−1 x + (Lea )−1 y), e e e −1 ωt (a, y) = ta y = La te (La ) y.

(24.16) (24.17)

In this case the holonomial identities: h(a,b) te x = te h(a,b) x,

h(a,b) (x + y) = h(a,b) x + h(a,b) y e

e

(linearity),

(24.18)

h(a,a·ue b) tb = l(a,ue b) tb (joint identity), h(c·te a,c·ue a) h(c,c·te a) x = h(c,c·ue a) x (h-identity),

(24.19) (24.20)

h(e,q) x = x (e-identity)

(24.21)

are true. Let us consider parallel translation of the geodesic arc v (further it will be called osculating vector) starting at the point a along all edges of a geodesic tetrahedron with one vertex at the point a and let x, y, z be the geodesic arcs that start at a and form three edges of the tetrahedron. The parallel translation of the geodesic v along all edges in a such way, that it goes with conservation of the orientation (clockwise or counterclockwise) around of each face once does not change v, because each edge is passed even-numbered times and in opposed directions. Expressing this idea in terms of the left translations, we ﬁnd that elementary holonomy satisﬁes odular Bianchi identities: ha(z,x) ◦ ha(y,z) ◦ ha(x,y) = (Lax )−1 ◦ hx(y,z) ◦ Lax .

(24.22)

In the linear approximation (24.22) reproduce usual Bianchi identities. Nonassociative geometry is based on the constructions described above. In the table below we compare the basic concepts of the classical diﬀerential geometry and nonassociative geometry. Diﬀerential Geometry vs Nonassociative Geometry Diﬀerential Geometry Tangent space Tangent vector Tangent bundle structure Cotangent space Covector Aﬃne connection ∇X Y Curvature R(X, Y, )Z Bianchi identities

Nonassociative Geometry Osculating space Osculating vector Diodular structure Co-osculating space Osculating covector Nonassociative left translations Lax y Elementary holonomy ha(x,y) z Odular Bianchi identities

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24.3

Nonassociative Discrete Geometry

The program presented below is based on unpublished notes of L.V. Sabinin thought over from the point of view of physics. There are no physical results in the frames of this new approach yet. Thus this is a program of research in a new direction rather than readymade physical theory. But all principal elements of a new formalism are in presence here. And what really we need as the ﬁrst step are some nontrivial examples that can be easily obtained. We are convinced that any discrete geometry useful in physics applications has to bear the main features of the well-established continuous geometry. Since in this case we need to omit the smooth structure, we can appeal to algebraic structures only. Thus it is quite natural to use geoodular (geodiodular) spaces being an adequate algebraic models for aﬃnely connected spaces in the smooth case. 1. We are now going to consider ﬁnite aﬃnely connected spaces. It dictates to use a ﬁnite ﬁeld F instead of reals R. There are some variants when F is of a more general nature, say, a ring or double loop. But in this chapter, for the sake of simplicity, we consider the case when F = F, +, −( ), ·, 0, 1 is a ﬁnite ﬁeld. The structure of ﬁnite ﬁelds is well known. Such ﬁelds are Galois ﬁelds FG(pm ), p being prime number, m = 1, 2, . . . ; in this case pm is a number of elements of the ﬁeld. Note that m in such a ﬁeld xp = x. 2. Thus we intend to consider a geodiodular ﬁnite space M = M, L, N, (ωt )t∈F equipped with ternary operations L(x, y, z) = xy· z, L(x, y, z) = xy· z and a system of binary operations ωt (x, y) = tx y (t ∈ F ). It means that: 1. M, a· , (ta )t∈F is an odule with a neutral a ∈ M , for any a ∈ M , 2. M, + , (ta )t∈F is a n-dimensional vector space (with zero element a ∈ M ), a 3. The following geoodular identities, (the ﬁrst geoodular identity) Ltuaaxx ◦ Lata x = Laua x (the second geoodular identity) Lax ◦ ta = tx ◦ Lax , (the third geoodular identity)

Lax (y

+

a

z) = Lax y

+

x

(Lax y = L(x, a, y)),

Lax z

(24.23) (24.24) (24.25)

are valid. We may consider the operations above as partially deﬁned (if needed). , a, (ta )t∈F is playing the role of tangent space at a ∈ M 3. In our approach M+ a = M, + a because in the classical smooth case Ta M (the tangent space to a ∈ M ) may be identiﬁed, at least locally, with M by means of the exponential map. Any point x ∈ M may be, then, regarded as a vector in M+ a (∀a ∈ M ). Any line (ta b)t∈F may be regarded as a geodesic through a, b ∈ M . As to parallelism, + y → ba· y means that the vector ba· y (of M+ b ) is parallel to a vector y (of Ma ) along the geodesic (ta b)t∈F . The presence of curvature in such a space results in nontrivial (in general) elementary holonomy, Lya ◦ Lxy ◦ Lax = ha (x, y) = id. Thus ha (x, y) = id, for any a, x, y ∈ M means absolute parallelism of our space and if, additionally, x+ y = xa· y (or, L = N , which is the same); it means the absence of torsion, a that is, we come to a classical aﬃne space, see ([16], Ch. 3).

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We do not know, in general, how the absence of torsion can be expressed in purely algebraic terms. However, there are some suﬃcient conditions [16]. 4. One can introduce the notion of a curve in discrete space as an ordered set of points (perhaps, not all diﬀerent), γ = (a1 , a2 , · · · as ) and, further, deﬁne the vector y (of M+ as ) ) along this curve as parallel to a vector x (of M+ a1 def

· · · Laa23 Laa12 x (Lab x = ba· x) y = Laas−1 s One can introduce the holonomy group Ha M at a point a ∈ M considering the parallel translations along all curves with coinciding initial and ﬁnal point. It is easy to see that any element of Ha M is a composition of elementary holonomies at a ∈ M . 5. Having given a ﬁnite diodular space (ﬁnite aﬃnely connected space in other words) we can enrich it by a metric structure. Namely, we can deﬁne additionally nondegenerate symmetric twice covariant tensor ga (x, y) for any M+ a . The condition of covariant permanence for this tensor has the form gb (Lab x, Lab y) = ga (x, y). In this way we obtain a metric diodular structure [16]. In order to apply the above construction to gravity we need to solve some classiﬁcation problem. Since any ﬁnite ﬁeld consists of pm elements (p is prime, m ∈ N, we need to classify spaces with small p (= 2, 3, 5, 7, 11) and m (= 1, 2, 3, 4) such that dim M+ a ≤ 4. Such a classiﬁcation is signiﬁcant for concrete discrete physical models of spacetime. 6. For the description of evolution of spacetime the hypothesis of an increasing number of points is quite natural. It means that we should regard p and m as growing in the time. Probably we need some additional physical reasons to assume the “true” law of dependence of p and m on the time. Finally, we should invent the limit procedure (in rigorous terms) in order to obtain after the time growing process the habitual smooth schemes of spacetime. 7. Now we consider the discrete calculus in discrete spaces. In our opinion, numerous attempts to use for the discrete case the methods of ﬁnite diﬀerences is not very eﬀective. Instead, we may use another approach based on the evident observation that any function given on a direct ﬁnite product of ﬁnite ﬁelds f : F × · · · × F → F, F = F, +, −( ), 0, ·, 1, 78 9 6 l times

is a polynomial. For certainty we should use the minimal degree polynomial representation (for example, m m for FG(pm ), xp = x, that is, the minimal representation for xp is x). Since any polynomial P(x) = a0 + a1 x + · · · + an xn (an = 0) is algebraically diﬀerentiable, P (x) = a1 + · · · + (an n)xn−1 , (similarly for polynomials with several unknowns), we use this derivative in full analogy with the classical smooth case. One should be careful with the properties of such a calculus. For example, for FG(pm ), p (x ) = p(xp−1 ) = 0 (p being caracteristic). The formula (f g) = f g + f g is not valid in general. Also, there is no the existence and uniqueness theorem for a solution of diﬀerential equation with habitual initial conditions and so on. The previously mentioned means that in our case we cannot work with inﬁnitesimal objects only; the algebraic structure of a diodular space should be always taken into account. Thus such a calculus needs to be accurately developed.

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8. Further, we may proceed in the convential way. Thus the notion of a tangent space, vector ﬁelds and so on can be introduced, as well as the aﬃne connection ∇:

5 d a −1 [(L ) Yγ(t) ] , ∇Xa Y = dt γ(t) ∗,a t=0 γ(0) = a,

γ(0) ˙ = Xa ,

Y being a vector ﬁeld. Further, in the canonical way, one may introduce the tensors of torsion and curvature: T (X, Y ) = ∇X Y − ∇X Y − [X, Y ], R(X, Y ) = ∇X ∇Y Z − ∇Y ∇X Z − ∇[X,Y ] Z, where [ , ] is the Lie bracket. In particular, a space of zero torsion may be introduced. 9. One can write Einstein equations [18], including the case with torsion, and try to solve it. Due to the above-described features of diﬀerential equations in this case (the absence of the existence and uniqueness theorem in usual sense) the solution (if any) has some properties that are not valid in the classical smooth case.

24.4

Discrete de Sitter Spacetime

Let us consider a ﬁnite set M = Z4n = {p = (pμ ) | pμ ∈ Zn , μ = 0, . . . , 3, n ∈ N}, where Zn = {p = −n, . . . , n} is the set of integers. We deﬁne a partial loop QHZ4n as a set of quaternions: HZ4n = {ζp = (p0 + i(p1 i + p2 j + p3 k)) : 2

i = −1,

i ∈ C;

p∈

= const,

Z4n },

(24.26)

with the indeﬁnite norm ζp 2 = 2 pμ pμ and the binary operation HZ4n ×HZ4n → HZ4n deﬁned by B

K 1 + ζp+ ζq , ζp , ζq ∈ HZ4n . (24.27) ζp ∗ ζq = ζpq = ζp + ζq 4 We introduce a partial geodiodular ﬁnite space Mn at the neutral element e (zero) as follows: • QHZ4n being the odule with the unary operation multiplication induced by the quaternionic algebra over Z • M+ = HZ4n being the osculating space with the structure of vector space induced by the quaternionic algebra over Z. Employing the geoodular identities one can obtain the geoodular covering of the discrete spacetime. For the diodular metric and elementary holonomy we have ζ −ζ 2

g ζp (ζq , ζq ) = 1− pK ζ +qζ 2 , p q 4 B

K + h(ζp ,ζq ) ζm = 1 − 4 ζq ζp ζm 1 −

K + 4 ζq ζp

(24.28) .

(24.29)

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The smooth spacetime could be regarded as the result of “limit process of triangulating” when n = const, while −→ 0, n −→ ∞. Let us consider q = p + δ, |δ| 2 n. Then we have ζq = ζp + Δζq where Δζq = (δ 0 + i(δ 1 i + δ 2 j + δ 3 k)), and the diodular metric (24.28) takes the form Δζq 2 g ζp (ζq , ζq ) = + O(K2 ), (24.30) + 2 1 − K ζ ζ p p 4 and g ζp (ζq , ζq ) −→ ds2 =

(dζ 0 )2 − (dζ 1 )2 − (dζ 2 )2 − (dζ 3 )2 , 0 2 1 2 2 2 3 2 2 (1 − K 4 ((ζ ) − (ζ ) − (ζ ) − (ζ ) )

while → 0, n → 0. Comparing with the smooth case, we see that to some extent the information concerning the geometry of de Sitter spacetime is hidden in the structure of ﬁnite loop. The similar consideration of the elementary holonomy gives h(ζp ,ζq ) ζm = ζm + where

K Δ(ζp , ζq , ζm ) + O K2 , 4

(24.31)

Δ(ζp , ζq , ζm ) = ζm ζq+ ζp − ζq ζp+ ζm .

In the coordinates (24.31) can be written as μ ν K μ ν λ σ h(ζp ,ζq ) ν ζm = ζm + εμνκδ εκδ λσ ζm ζp ζq + O K2 4 Approaching the limit, while n −→ ∞, −→ 0, one restores the curvature tensor of de Sitter spacetime in the normal coordinates: Rμνλσ = −

K εμνκδ εκδ λσ . 2

The above examples show how the continuum and discrete structure of spacetime might be described in the framework of the nonassociative geometry. We deal with points and essentially nonassociative operation. This leads us to the concept of the nonassociative (discrete) spacetime, when at distances comparables with Planck length the standard concept of spacetime might be replaced by the diodular discrete structure, which at large spacetime scales, “looks like” a diﬀerentiable manifold.

24.5

Toy Model of Discrete (2+1)-Spacetime

In this section we consider the toy model of discrete (2+1)-spacetime with the discrete sphere S 2 as a two-dimensional space. We start with a brief description of the nonassociative geometry of smooth sphere [20, 21]. Let C be a complex plane and ζ, η ∈ C. The nonassociative multiplication is deﬁned by Lζη = ζ η =

ζ +η , 1 − ζη

ζ, η ∈ C

(24.32)

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where a bar denotes complex conjugation and neutral element e coincides with the origin of coordinates. The obtained loop is isomorphic to the local two-parametric loop associated with two-sphere S 2 . The isomorphism between points of the sphere and the complex plane C is established by the stereographic projection from the south pole of the unit sphere, ζ = tan(θ/2)eiϕ . The entire sphere may be covered by two local loops, one of them with the neutral element at the north pole (see above) and another with the neutral element at the south pole. The normal complex coordinates X = X 1 + iX 2 at the north pole are determined as follows: X = τ eiϕ = 2 arctan |ζ|eiϕ , where τ is the length of the geodesic arc from the north pole. The exponential mapping: Te (S 2 ) → S 2 in the complex coordinates {ζ} is given by ζ=

tan |X| X, |X|

and in the spherical coordinates (θ, ϕ) we have θ = |X|, ϕ = arg X. The associator is given by 1 − ζη l(ζ,η) ξ = ξ (24.33) 1 − ηζ Since the sphere is a symmetric space, the elementary holonomy is determined by associator: h(ζ,η) = l(ζ,L− 1ζ η) [13, 16]. The computation yields h(ζ,η) ξ =

1 + ζη ξ. 1 + ηζ

(24.34)

The basis of left fundamental vectors and the dual basis of one-forms are found to be Γ1 = (1 + |η|2 )∂η , Γ2 = (1 + |η|2 )∂η¯ dη d¯ η , θ2 = . θ1 = 1 + |η|2 1 + |η|2

(24.35) (24.36)

Finally, the metric being associated with the left fundamental basis forms is deﬁned as follows: 2dηd¯ η ds2 = 2θ1 θ2 = (1 + |η|2 )2 and we can see that is the metric of the two-sphere S 2 (up to the factor 2).

24.5.1

Nonassociative discrete geometry of S 2 and (2+1)-spacetime evolution

We consider the natural geodesic triangulation of the sphere, which is speciﬁed as follows. The simplex triangulating S 2 is geodesic triangle, and the geodesic lattice will be assumed to consist of central vertex at the north pole and the geodesic triangles attached to this vertex. For each surface vertex (j) we assign the polar coordinates (θj , ϕj ), assuming θj = πj/n, ϕ = 2πj/n (j = 1, 2, . . . n). With this choice there are n2 initial points allocated on the surface of the sphere at the points with coordinates: πp 2πip (24.37) ζp = tan e n . 2n

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Obviously these points deﬁne some triangulation of the sphere. The next step is to “multiply” points, applying (24.32): ζp + ζq ζpq = . (24.38) 1 − ζ p ζq Writing ζpq as

θpq iϕpq , ζpq = tan e 2

(24.39)

one obtains from (24.38) the following formulae deﬁning left translations in the “spherical coordinates”: |ζp |2 + |ζq | + ζ p ζq + ζ q ζp θpq = 2 arctan (24.40) 1 + |ζp |2 |ζq |2 − (ζ p ζq + ζ q ζp ) i ϕpq = arg(ζp + ζq ) − ln l(ζp ,ζq ) , (24.41) 2 πp 2πip πq 2πiq ζq = tan (24.42) ζp = tan e n , e n , n n where the associator l(ζp ,ζq ) η =

1 − ζp ζ q 1 − ζ p ζq

η

is introduced. Starting with the initial set of points, one needs to multiply all of them, involving new obtained points in this process. One may describe this as the following sequence of triangulations: (24.43) {ζp } −→ {ζp , ζpq } −→ {ζp , ζpq , ζp(ql) , ζ(pq)l } −→ . . . The spacetime evolution in our model will be understood as a transition from a triangulation of S 2 at a given time instant to a later one. There exists a natural causal structure produced by the ordering sequence of triangulations (24.43). As result we obtain a discrete spacetime, which includes many of the attributes of the continuum spacetime: causal structure, spacelike slices and “many-ﬁngered time.” In some sense the information about the geometry of the sphere is hidden in the structure of the ﬁnite loop. The smooth sphere could be regarded as the limiting result of the dynamical triangulating, because in the process of evolution the number of points increases and the triangulations become ﬁner. In our toy model of two-dimensional spacetime the “universe” with topology S 2 is created from a few initial points. This leads to a new undestanding of “big-bang”: in our approach the initial singularity does not exist. A discrete model MF n of Friedmann-Robertson-Walker (FRW) spacetime is discussed in [22]. The important feature of FRW discrete model is the appearance of the natural arrow of time as follows. Let us assume MF n as being a triangulated topological space. For the given triangulation, we identify the elements xp ∈ MF n as the vertices of the simplices, n being a number vertices. Then the causal ordering in FRW space is related to the reﬁnement of the triangulation of 3-dimensional space, and the evolution of FRW universe is interpreted in terms of a sequence F F · · · → MF n → Mn+1 → · · · → Mn . . . F F where · · · ≺ MF n ≺ Mn+1 ≺ · · · ≺ Mn . . . Certainly the above used loopuscular structure is given ad hoc and the open question is how to introduce the elementary holonomy in the realistic model? This is the subject of a future study.

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24.6

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Acknowledgments

This work was partly supported by SEP-PROMEP (Grant No. 103.5/04/1911)

References [1] Butterﬁeld, J. and Isham, C.J.: Spacetime and the Philosophical Challenge of Quantum Gravity, gr-qc/9903072. [2] Isham, C.J.: Structural Issues in Quantum Gravity, gr-qc/9510063. [3] Rovelli, C.: Quantum spacetime: what do we know?, gr-qc/9903045. [4] Rovelli, C.: Loop Quantum Gravity. Living Reviews in Relativity (refereed electronic journal), http://www.livingreviews.org/Articles/Volume1/1998-1rovelli, grqc/9709008. [5] Rovelli, C.: String, loops and others: a critical survey of the presence approaches to quantum gravity, gr-qc/9803024. [6] Isham, C.J.: An introduction to general topology and quantum topology. In: Lee, H.C.: editor, Physics, Geometry and Topology, pp. 129-190, Plenum Press, New York, 1990. [7] Sorkin, R.D.: Finitary substitute for continuous topology. Int. Journ. Theor. Phys. 30, 923-947 (1991). [8] Kikkawa, M.: On local loops in aﬃne manifolds. J. Sci. Hiroshima Univ. Ser A-I Math. 28, 199 (1961). [9] Sabinin, L. V.: On the equivalence of categories of loops and homogeneous spaces, Soviet Math. Dokl. 13, 970 (1972a). [10] Sabinin, L. V.: The geometry of loops, Mathematical Notes, 12, 799 (1972b). [11] Sabinin, L. V.: Odules as a new approach to a geometry with a connection, Soviet Math. Dokl. 18, 515 (1977). [12] Sabinin, L. V.: Methods of Nonassociative Algebra in Diﬀerential Geometry, in Supplement to Russian translation of S.K. Kobayashi and K. Nomizy “Foundations of Diﬀerential Geometry,” Vol. 1. Moscow: Nauka, 1981 [13] Sabinin, L. V.: Diﬀerential equations of smooth loops, In: Proc. of Sem. on Vector and Tensor Analysis, 23 133. Moscow: Moscow Univ., 1988. [14] Sabinin, L. V.: Diﬀerential Geometry and Quasigroups, Proc. Inst. Math. Siberian Branch of Ac. Sci. USSR, 14, 208 (1989). [15] Sabinin, L. V.: On diﬀerential equations of smooth loops, Russian Mathematical Survey, 49 172 (1994).

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[16] Sabinin, L. V.: Smooth Quasigroups and Loops. Dordrecht: Kluwer Academic Publishers, 1999. [17] A.I. Nesterov, and L.V. Sabinin, Nonassociative geometry: Towards discrete structure of spacetime, Phys. Rev.D 62, 081501(R), (2000), hep-th/0010159. [18] A.I. Nesterov and L.V. Sabinin, Non-aasociative geometry and discrete structure of spacetime, Comment. Math. Univ. Carolinae 41, 2, 347 (2000), hep-th/0003238. [19] Eguchi, T., Gilkey, P.B. and Hanson A.J.: Gravitation, Gauge Theory and Diﬀerential Geometry, 66, 213-393 (1980). [20] Nesterov, A.I. and Sabinin, L.V.: Smooth Loops and Thomas Precession. Hadr. Journ. 20, 219–237 (1997). [21] Nesterov, A.I.: Quasigroups, Asymptotic Symmetries and Conservation Laws in General Relativity. Phys. Rev. D 56, R7498–R7502 (1997). [22] Nesterov, A.I. and Sabinin, L.V.: Nonassociative geometry: Friedmann-RobertsonWalker spacetime, hep-th/0406229

Chapter 25 Algebras Satisfying Symmetric Triality Relations Susumu Okubo Department of Physics and Astronomy, University of Rochester, New York

25.1 Symmetric triality algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25.2 Examples of normal STA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25.3 Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

313 316 321 321

Abstract Algebras satisfying symmetric triality relations are shown to lead to a construction of Lie algebras that will give Freudenthal’s magic square. Some examples of these algebras are given and studied. Keywords: symmetric triality algebra, nonassociative algebras, structurable algebras, Freundenthal’s magic square 2000 MSC: 17B25; 17A75

25.1

Symmetric Triality Algebra

Recently, Burton and Sudbery [2] reformulated the standard construction of exceptional Lie algebras by Tit [11] as a construction using two composition algebras. Subsequently, a more symmetrical approach has been found by Elduque [3] in terms of symmetric triality relations based upon two symmetric composition algebras. Let A be an algebra over a ﬁeld F with its product denoted by juxtaposition xy for x, y ∈ A. Let a triple (d0 , d1 , d2 ) be dj : A ⊗ A → End A

(j = 0, 1, 2).

(25.1)

Suppose that dj (u, v) ∈ End A for u, v ∈ A satisfy a symmetric triality relation dj (u, v)(xy) = (dj+1 (u, v)x) y + x (dj+2 (u, v)y) ,

(25.2)

where we imposed the cyclicity condition dj±3 (u, v) = dj (u, v) = −dj (v, u)

(25.3)

for convenience. We call any algebra A satisfying Eq. (25.2) to be a symmetric triality algebra (abbreviated hereafter as STA) with respect to the triple (d0 , d1 , d2 ). If we have d0 (u, v) = d1 (u, v) = d2 (u, v) := d(u, v), then Eq. (25.2) simply implies that d(u, v) is a derivation of A. We may call such a case a trivial STA. It is easy to show that any STA over a ﬁeld F of characteristic not 2 is trivial, if it is unital.

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For reasons to be explained shortly, a more interesting case is that dj (u, v) will satisfy the following additional conditions: (1) d1 (x, y) = R(y)L(x) − R(x)L(y), (2) d2 (x, y) = L(y)R(x) − L(x)R(y), (3) d0 (x, y)z + d0 (y, z)x + d0 (z, x)y = 0, (4) (5)

d0 (x, yz) + d1 (z, xy) + d2 (y, zx) = 0, [d (u, v), dm (x, y)] = dm (d −m (u, v)x, y) + dm (x, d −m (u, v)y) ,

(25.4) (25.5) (25.6) (25.7) (25.8)

where L(x) and R(x) in Eqs. (25.4) and (25.5) are left and right multiplication operators, respectively, given by L(x)y = R(y)x := xy,

(25.9)

and where and m in Eq. (25.8) stand for any integers. We then call any STA satisfying Eqs. (25.4)–(25.9) to be a normal STA [9]. As we shall see shortly, we can always construct a normal STA from any structurable algebra [1]. Before going into detail, we note the following. If we set D(x, y) := d0 (x, y) + d1 (x, y) + d2 (x, y),

(25.10)

then Eq. (25.2) shows that it is a derivation of A, i.e., we have D(u, v)(xy) = (D(u, v)x)y + x(D(u, v)y).

(25.11)

Moreover, Eq. (25.7) also gives D(x, yz) + D(z, xy) + D(y, zx) = 0.

(25.12)

Kamiya [6] has called any algebra with derivation D(u, v) satisfying Eq. (25.12) to be a generalized structurable algebra, which includes Jordan, Malcev, and structurable algebras. From any normal STA, we can always construct a large Lie algebra as follows (see [9] for detail): First, we introduce three copies of A, depicted as ρj (x)(x ∈ A, j = 0, 1, 2) and consider a vector space L given by L = ρ0 (A) ⊕ ρ1 (A) ⊕ ρ2 (A) ⊕ T,

(25.13)

where T is a vector space spanned by some Tj (x, y) satisfying (1)

Tj (x, y) = −Tj (y, x) = Tj±3 (x, y), (j = integer)

(25.14a)

(2)

T0 (x, yz) + T1 (z, xy) + T2 (y, zx) = 0.

(25.14b)

Let (i, j, k) to refer to any cyclic permutation of indices (0, 1, 2). We now assume the following commutation relations in L:

(1) (2)

[ρi (x), ρi (y)] = γj γk−1 T3−i (x, y), [ρi (x), ρj (y)] = − [ρj (y), ρi (x)] = −γj γi−1 ρk (xy),

(3) (4)

[T (u, v), ρj (x)] = − [ρj (x), T (u, v)] = ρj (d +j (u, v)x) , (25.15c) [T (u, v), Tm (x, y)] = Tm (d −m (u, v)x, y) + Tm (x, d −m (u, v)y) , (25.15d)

(25.15a) (25.15b)

Algebras satisfying symmetric triality relations

315

where γj (j = 0, 1, 2) are some nonzero constants. If we identify Tj (x, y) either with dj (x, y) itself or with the triple T (dj (x, y), dk (x, y), di (x, y)) as in [1] and [3], then Eqs. (25.14b) and (25.15d) are readily satisﬁed by Eqs. (25.7) and (25.8). Conversely, a consistency between Eqs. (25.14b) and (25.15c) requires an implicit validity of Eq. (25.7). We now state the following theorem (see [9]). THEOREM 25.1 We assume ρj (x) to be F -linear in x. If A is a normal STA, then L is a Lie algebra. Conversely, if ρj (x) = 0 for some x ∈ A implies x = 0, then A must be a normal STA, if L is a Lie. In general, it is rather diﬃcult to test whether a given algebra is a normal STA or not. However, for some cases, we can utilize the following propositions (see [9] for details): PROPOSITION 25.1 Let A be an STA (which does not need to be normal) such that d1 (x, y) and d2 (x, y) are given by Eqs. (25.4) and (25.5). Moreover, suppose that A satisﬁes at least one of the following conditions: (a) AA = A or (b) if xA = 0 for some x ∈ A, then x = 0, or (c) if Ay = 0 for some y ∈ A, then y = 0. In that case, d (u, v) satisfy the Lie equation, Eq. (25.8), as well as Eq. Eq1.6. PROPOSITION 25.2 form < .|. >, satisfying

Let A possess a symmetric bilinear associative nondegenerate

< dj (u, v)x|y > = − < x|dj (u, v)y > = < d3−j (x, y)u|v >

(25.16)

for some dj (u, v) ∈ End A. Then any one of the following 3 statements implies the validity of all others. (1) We have Eq. (25.2), i.e., dj (u, v)(xy) = (dj+1 (u, v)x) y + x (dj+2 (u, v)y)

(25.17)

for all j = 0, 1, 2. (2) Eq. (25.17) holds valid only for one value of j, say, j = 0. (3) The validity of Eq. (25.7), i.e., d0 (x, yz) + d1 (z, xy) + d2 (y, zx) = 0. REMARK 25.1 Assuming < .|. > to satisfy the conditions of Proposition 25.2, Eq. (25.16) will hold for j = 1, 2, if dj (x, y) for j = 1, and 2 are given by Eqs. (25.4) and (25.5), respectively. Moreover, if we have AA = A, then d0 (x, y) is determined, in principle, in terms of d1 (x, y) and d2 (x, y) by Eq. (25.7). Then, we can also prove the validity of Eq. (25.16) for j = 0, if Eq. (25.2) holds valid for j = 0. PROPOSITION 25.3

If d0 (x, y) has a special form of d0 (x, y)z = Λ(x, z)y − Λ(y, z)x,

(25.18a)

where Λ(x, y) ∈ End A is symmetric in x, y, i.e., Λ(x, y) = Λ(y, x), then the condition, Eq. (25.6) is automatically satisﬁed.

(25.18b)

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Susumu Okubo

Using all these propositions, we can considerably simplify the test of normal STA. However, another way of constructing it is to relate it to a structurable algebra [1] as follows. Suppose that A is an involutive STA with involution mapping x → x so that x = x,

xy = y x.

(25.19)

∗

We then introduce a new algebra A with the 2nd product x ∗ y in the same vector space of A by x ∗ y := xy = y x. ∗

(25.20) ∗

Then, A is also involutive with x ∗ y = y ∗ x = xy. We call A to be the conjugate algebra of A, where the STA relation, Eq. (25.2) becomes the Lie-related triple relation (see [1]), dj (u, v)(x ∗ y) = (dj+1 (u, v)x) ∗ y + x ∗ (dj+2 (u, v)y) ,

(25.21)

where Q for Q ∈ End A is deﬁned as usual by Qx = Q x. We can now prove the following Theorem [9]. THEOREM 25.2 A necessary and suﬃcient condition for A∗ to be a structurable algebra is that its conjugate algebra A is a involutive normal STA with the para-unit e. Here, by the para-unit e, we imply the validity of ex = xe = x in A or equivalently e∗x = x∗e = x in A∗ . The Lie algebra L constructed in Theorem 25.1 possesses an automorphism of order 3, if we choose the constants γj in Eqs. (25.15a)–(25.15d) to be γj = 1 for all j = 0, 1, 2. Let θ : L → L be given by ρj (x) → ρj+1 (x),

(25.22a)

Tj (u, v) → Tj−1 (u, v)

(25.22b)

for j = 0, 1, 2. Then, Eqs. (25.15a)–(25.15d) remain invariant with θ3 = 1. Note that dj (u, v) remains unchanged for this transformation.

25.2

Examples of Normal STA

By Theorem 25.2, we can always construct normal STAs from many known examples of structurable algebras (see [1]). However, we will here give other types of examples of some interests. Example 25.1 Let < .|. > be a bilinear symmetric associative nondegenerate form for an algebra A. If we have (xy)x = x(yx) = < x|x > y

(25.23)

for x, y ∈ A, then A is called a symmetrical composition algebra. It is a normal STA with d0 (x, y) given by d0 (x, y)z = 4{< x|z > y − < y|z > x}.

(25.24)

Algebras satisfying symmetric triality relations

317

We note that any symmetrical composition algebra is either 8-dimensional pseudo-octonion algebra or any para-Hurwitz algebra, which is the conjugate of a Hurwitz algebra (see [4], [7], [10]). We simply remark here that the Lie algebras L constructed in Theorem 25.1 are A1 , A2 , C3 , and F4 , depending upon Dim A = 1, 2, 4, and 8, respectively. We also note that for the case of either the pseudo-octonion or para-octonion case with Dim A = 8, Eq. 25.8 gives the same so(8) Lie algebra relation of [d (u, v), d (x, y)] = d (d0 (u, v)x, y) + d (x, d0 (u, v)y)

(25.25)

for all = 0, 1, 2, realizing the classical triality relationship based upon the Dynkin diagram of the Lie algebra D4 . Example 25.2 Let V1 and V2 be two independent symmetric composition algebra. Then, the tensor product algebra A = V1 ⊗ V2 is a normal STA with respect to a triple given by Dj (x1 ⊗ x2 , y1 ⊗ y2 ) 1 = {dj (x1 , y1 ) ⊗ < x2 |y2 > 12 ⊕ < x1 |y1 > 11 ⊗ dj (x2 , y2 )} 2

(25.26)

where 1j (j = 1, 2) stands for the identity map in End (Vj ). Varying choices of V1 and V2 , Theorem 25.1 will give Freudenthal’s magic square as in Table 25.1 below. Table 25.1. Freundenthal’s magic square

Dim V1

\ Dim V 1 2 4 8

2

1

2

4

8

A1 A2 C3 F4

A2 A2 ⊗ A2 A5 E6

C3 A5 D6 E7

F4 E6 E7 E8

As has been noted by Elduque [3], we can choose V1 and/or V2 to be either a para-octonion or pseudo-octonion algebra for the case of Dim V1 = 8 and/or Dim V2 = 8, to obtain new realizations of F4 , E6 , E7 and E8 . Example 25.3 We note that the exceptional Lie algebra G2 does not appear in the magic square of Table 25.1. However, it can be derived from the following 4-dimensional normal STA. Let {e, f, g, g} be a basis of an algebra A with the multiplication table of

(1)

e2 = e, f 2 = e, ef = f e = −f,

(25.27a)

(2) (3)

eg = ge = g, eg = ge = g, f g = −gf = −g, f g = −gf = g,

(25.27b) (25.27c)

(4)

gg = −2g, g g = −2g, 3 3 gg = (e + f ), gg = (e − f ), 2 2

(25.27d)

(5)

(25.27e)

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Susumu Okubo

where we assume the underlying ﬁeld F to be of characteristic neither 2 nor 3. This 4dimensional algebra A is a normal STA. To check it, we ﬁrst note that it has a bilinear symmetric associative nondegenerate form < .|. >, if we set < e|e > = − < f |f > = 1, 3 < g|g > = < g|g > = , 2

(25.28a) (25.28b)

with all other entries being zero. Moreover, we note AA = A. Furthermore, the subalgebra A0 spanned by two elements e and f constitutes a 2-dimensional symmetric composition algebra, which is a normal STA. Then, the resulting Lie algebra by Theorem 25.1 for A0 is A2 = su(3), which can be speciﬁed as ρ0 (e) = B12 − B21 , ρ1 (e) = B20 − B02 , ρ2 (e) = B01 − B10 (25.29a) 2 1 0 2 ρ0 (f ) = B1 + B2 , ρ1 (f ) = B2 + B0 , ρ2 (f ) = B01 + B10 (25.29b) T0 (e, f ) = 2 B11 − B22 , T1 (e, f ) = 2 B00 − B11 , T2 (e, f ) = 2 B22 − B00 (25.29c) where Bνμ (μ, ν = 0, 1, 2) will satisfy the su(3) Lie algebra relation of μ α Bν , Bβ = δβμ Bνα − δνα Bβμ , 2

(25.30a)

Bλλ = 0

(25.30b)

λ=1

for μ, ν, α, β = 0, 1, 2. We now come to the 4-dimensional algebra A. By the reason to be explained shortly, d0 (x, y) must be given by d0 (x, y) = [R(x), L(y)] − [R(y), L(x)] = −d1 (x, y) − d2 (x, y).

(25.31)

Then, we can verify A to be a normal STA. For this, we ﬁrst note the validity of dj (e, g) = dj (e, g) = dj (f, g) = dj (f, g) = 0

(25.32a)

Tj (e, g) = Tj (e, g) = Tj (f, g) = Tj (f, g) = 0

(25.32b)

as well as for j = 0, 1, 2 from studies of Eqs. (25.8) and (25.14b). Then, we can realize the solution of Eqs. (25.14a) and (25.14b) and (25.15a–25.15d) to be given by ρj (g) = Aj , T0 (g, g) =

ρj (g) = −Aj ,

−3B00 ,

T1 (g, g) =

(j = 0, 1, 2) −3B22 ,

T2 (g, g) =

(25.33a) −3B11

,

(25.33b)

where they satisfy commutation relations of 1 [Bνμ , Aλ ] = δλμ Aν − δνμ Aλ , 3 μ λ 1 λ μ Bν , A = −δν A + δνμ Aλ , 3

(25.34a) (25.34b)

Algebras satisfying symmetric triality relations [Aμ , Aν ] = −2

2

μνλ Aλ ,

319 (25.34c)

λ=0

[Aμ , Aν ] = 2 [Aμ , Aν ] =

2

μνλ Aλ ,

(25.34d)

λ=0 3Bμν ,

(25.34e)

for μ, ν, λ = 0, 1, 2, where μνλ = μνλ is the Levi-Civita symbol. Then, 14-dimensional algebra consisting of Bνμ , Aα , and Aβ is the Lie algebra G2 as has been noted in [8]. The present 4-dimensional normal STA is rather a peculiar algebra. We can easily verify that it does not possess any nontrivial derivation, although it admits the symmetric triality relation. This fact also explains the validity of Eq. (25.31), since the derivation D(x, y) given by Eq. (25.10) must then vanish. Also, this case gives [d (u, v), dm (x, y)] = 0 identically, and the Lie algebras speciﬁed by Eq. (25.8) and Eq. (25.15d) are simply u(1) ⊕ u(1). This algebra possesses two involutions x → x ˜; (a)

f˜ = −f

(b)

g˜ = g

but x ˜ = x for x = e, g

and g

and and g˜ = g

but e˜ = e and f˜ = f.

For the case of (a), e is the para-unit of A and hence its conjugate algebra must be a structurable algebra with respect to this involution. Example 25.4 We have also found a 3-dimensional normal STA of some interest. Let {e0 , e1 , e2 } be its basis with the multiplication table of (1) (2)

e0 e0 = e0 , e1 e1 = e1 , e2 e2 = e2 , e0 e1 = −e2 , e1 e2 = −e0 , e2 e0 = −e1 ,

(25.35a) (25.35b)

(3)

e1 e0 = e2 e1 = e0 e2 = 0.

(25.35c)

Introducing a bilinear form < .|. > in A by < ej |ek > = δjk ,

(j, k = 0, 1, 2),

(25.36)

it deﬁnes a bilinear symmetric associative nondegenerate form of A. Moreover, if we set d0 (x, y)z := < x|z > y − < y|z > x,

(25.37)

then we can verify that A is a normal STA. The Lie algebra given in Theorem 25.1 is however A1 ⊕ A1 ⊕ A1 ⊕ A1 . This algebra has some interesting properties. For a generic element x ∈ A, we set x = λ0 e0 + λ1 e1 + λ2 e2

(25.38)

for λj ∈ F and introduce a linear form t : A → F and a quadratic one Q : A ⊗ A → F by

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Susumu Okubo

t(x) = λ0 + λ1 + λ2 , Q(x) = λ0 λ1 + λ1 λ2 + λ2 λ0 .

(25.39a) (25.39b)

3Q(x) = [t(x)]2 − t(x2 ),

(25.40a)

We then have

2

< x|x > = (t(x)) − 2Q(x).

(25.40b)

First, it is quadratic, i.e., we have x2 = t(x)x − Q(x)f,

(25.41a)

f := e0 + e1 + e2 ,

(25.41b)

f f = 0.

(25.41c)

xx2 + x2 x = t(x)x2 + t x2 x,

(25.42)

x2 x2 = < x|x2 > x.

(25.43)

N (x) := < x|x2 > = t(x)t x2 = λ30 + λ31 + λ32 − 3λ0 λ1 λ2 .

(25.44)

where f is given by which satisﬁes It satisﬁes also a cubic relation of

as well as a quartic relation,

Note also that we have

Especially, if we introduce a commutative product x · y by x·y =

1 (xy + yx), 2

(25.45)

then Eq. (25.43) implies that the new commutative algebra A+ is a cubic-admissible algebra of [5]. Then, the relation N x2 = (N (x))2 gives a rather interesting identity of 3 2 3 3 3 ˜0 + λ ˜1 + λ ˜ 2 − 3λ ˜0λ ˜1λ ˜2 λ0 + λ31 + λ32 − 3λ0 λ1 λ2 = λ

(25.46a)

with ˜ 0 = λ2 − λ1 λ2 , λ 0

˜ 1 = λ2 − λ2 λ0 , λ 1

˜ 2 = λ2 − λ0 λ1 . λ 2

(25.46b)

Also, it presents a 6-dimensional realization of Duﬃn-Kemmer-Petiau algebra (see [9]).

Algebras satisfying symmetric triality relations

25.3

321

Acknowledgments

This paper is supported in part by the U.S. Department of Energy Grant. No. DE-FG0291ER40685.

References [1] B.N. Allison and J.R. Faulkner, Nonassociative Coeﬃcient Algebras for Steinberg Unitary Lie Algebras, J. Algebra 161 (1993), 1-19. [2] C.H. Barton and A. Sudbury, Magic Squares of Lie Algebras, preprint arXiv: Math. RA/0001083 and RA/0203010. [3] A. Elduque, The Magic Square and Symmetric Composition, Rev. Mat. Iberoamericana, 20 (2004), 477-493. [4] A. Elduque, S ymmetric Composition Algebras, J. Algebra 196 (1997), 282-300. [5] A. Elduque and S. Okubo, On Algebras Satisfying x2 x2 = N (x)x, Math. Zeit. 235 (2000), 275-314. [6] N. Kamiya, On Generalized Structurable Algebras and Lie Related Triples, Advances in Applied Cliﬀord Alg. 5 (1995), 127-140. [7] M.A. Knus, A.S. Merkurjev, M. Rost and J.P. Tignal, The Book of Involutions, American Mathematical Society, Colloq. Pub. 44, Providence (1998). [8] S. Okubo, Introduction to Octonion and Other Non-associative Algebras in Physics, Cambridge University Press, Cambridge (1995). [9] S. Okubo, Symmetric Triality Relations and Structurable Algebras, Linear Algebra and its Applications 396 (2005), 189-222. [10] S. Okubo and J.M. Osborn, Algebras with Non-degenerate Associative Symmetric Bilinear Forms Permitting Compositions (I) and (II), Comm. Algebra 9 (1981), 12331261 and 2015-2073. [11] J. Tits, Alg´ebres Alternatives, Alg´ebres de Jordan et Alg´ebres de Lie Exceptionnelles, I Construction, Nederl. Akad. Wetensch. Proc. Ser. A69 = Indag. Math. 28, (1966), 223-237.

Chapter 26 Operads and Nonassociative Deformations Eugen Paal Department of Mathematics, Tallinn University of Technology, Estonia

26.1 26.2 26.3 26.4 26.5 26.6 26.7 26.8

Introduction and outline of the chapter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Operad (composition system) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Gerstenhaber brackets and associator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Coboundary operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Generalized Maurer-Cartan equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Operadic Sabinin principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bianchi identity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

323 323 324 325 326 327 327 328 328

Abstract Deformation equation and its integrability condition (the Bianchi identity) of a nonassociative deformation in operad algebra are considered. Sabinin’s principle is reformulated in operadic terms. Key words: Operad, cohomology, deformation, Sabinin’s principle 2000 MSC: 17A01, 18D50

26.1

Introduction and Outline of the Chapter

The Sabinin principle states that nonassociativity is an algebraic equivalent of the differential geometric concept of curvature (e.g., [1, 2]). To see the equivalence, one must represent an associator in a suitable category. In this chapter, the equivalence is clariﬁed from the operad theoretical point of view. By using the Gerstenhaber brackets and a coboundary operator in an operad, the (formal) associator can be represented as a curvature form in diﬀerential geometry. This equation is called a deformation equation. Its integrability condition is the Bianchi identity. So the Sabinin principle can be seen in operadic terms as well: an associator is an operadic equivalent of the curvature. It is shown that the Bianchi identity does not produce any restrictions on the deformation.

26.2

Operad (Composition System)

Let K be a unital associative commutative ring, char K = 2, 3, and let C n (n ∈ N) be unital K-modules. For homogeneous f ∈ C n , we refer to n as the degree of f and write . . (when it does not cause confusion) f instead of deg f . For example, (−1)f =(−1)n , C f =C n

323

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Eugen Paal

. . and ◦f =◦n . Also, it is convenient to use the reduced degree |f |=n − 1. Throughout the . chapter we assume that ⊗=⊗K . DEFINITION 26.1 A linear operad (composition system) with coeﬃcients in K is a . sequence C ={C n }n∈N of unital K-modules (an N-graded K-module), such that the following conditions hold. (1) For 0 ≤ i ≤ m − 1 there exist partial compositions, ◦i ∈ Hom(C m ⊗C n , C m+n−1 ),

|◦i | = 0

(2) For all h⊗f ⊗g ∈ C h ⊗C f ⊗C g , the composition relations hold, ⎧ |f ||g| ⎪ (h◦j g)◦i+|g| f if 0 ≤ j ≤ i − 1, ⎨(−1) (h◦i f )◦j g = h◦i (f ◦j−i g) if i ≤ j ≤ i + |f |, ⎪ ⎩ |f ||g| (h◦j−|f | g)◦i f if i + f ≤ j ≤ |h| + |f |. (−1) (3) There exists a unit I ∈ C 1 such that I ◦0 f = f = f ◦i I,

0 ≤ i ≤ |f |

In the 2nd item, the ﬁrst and third parts of the deﬁning relations turn out to be equivalent. Example 26.1 (endomorphism operad [3]) Let L be a unital K-module and . . ELn =EndnL = Hom(L⊗n , L). Deﬁne the partial compositions for f ⊗g ∈ ELf ⊗ELg as . ⊗(|f |−i) f ◦i g =(−1)i|g| f ◦(id⊗i ), L ⊗g⊗ idL

0 ≤ i ≤ |f |

. Then EL ={ELn }n∈N is an operad (with the unit idL ∈ EL1 ) called the endomorphism operad of L. Thus algebraic operations can be seen as generating elements of an endomorphism operad. It is convenient to call homogeneous elements of an abstract operad the operations as well.

26.3

Gerstenhaber Brackets and Associator

The total composition • C f ⊗C g → C f +|g| is deﬁned by |f |

. f ◦i g f •g =

∈ C f +|g| ,

|•| = 0

i=0

. The pair Com C ={C, •} is called a composition algebra of C. The composition algebra multiplication • is nonassociative and satisﬁes the Gerstenhaber identity: . (h, f, g)=(h•f )•g − h•(f •g) = (−1)|f ||g| (h, g, f )

Operads and nonassociative deformations

325

The Gerstenhaber brackets [·, ·] are deﬁned in Com C by . [f, g]=f •g − (−1)|f ||g| g•f = −(−1)|f ||g| [g, f ],

|[·, ·]| = 0

. The commutator algebra of Com C is denoted as Com− C ={C, [·, ·]}. By using the Gersten− haber identity, one can prove that Com C is a graded Lie algebra. The (graded) Jacobi identity reads (−1)|f ||h| [[f, g], h] + (−1)|g||f | [[g, h], f ] + (−1)|h||g| [[h, f ], g] = 0 Let {L, μ} be a nonassociative algebra with a multiplication μ : L⊗L → L. The multiplication μ can be seen as an element of the component EL2 of an endomorphism operad EL . One can easily check that the associator of μ reads 1 . . A=μ◦(μ⊗ idL − idL ⊗μ) = μ•μ = [μ, μ]=μ2 , 2

μ ∈ EL2

So the total composition and Gerstenhaber brackets can be used for representing the associator in operadic terms. This was ﬁrst noticed by Gerstenhaber [3]. PROPOSITION 26.1 If K is a ﬁeld of characteristic 0, then the binary operation μ ∈ C 2 generates a power-associative subalgebra in Com C. PROOF Use the Albert criterion [4] that a power-associative algebra over a ﬁeld K of characteristic 0 can be given by the identities μ2 •μ = μ•μ2 ,

(μ2 •μ)•μ = μ2 •μ2

Both identities easily follow from the corresponding Gerstenhaber identities (μ, μ, μ) = 0,

26.4

(μ2 , μ, μ) = 0

Coboundary Operator

Let h ∈ C be an operation from an operad C. By using the Gerstenhaber brackets, deﬁne an adjoint representation h → ∂h of Com− C by . . ∂h f = adright f =[f, h], h

|∂h | = |h|

It follows from the Jacobi identity in Com− C that ∂h is a (right) derivation of Com− C, ∂h [f, g] = [f, ∂h g] + (−1)|g||h| [∂h f, g] and the following commutation relation holds: . [∂f , ∂g ]=∂f ∂g − (−1)|f ||g| ∂g ∂f = ∂[g,f ]

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Eugen Paal

. Let h=μ ∈ C 2 be a binary operation. Then, since |μ| = 1 is odd, we have ∂μ2 =

1 1 [∂μ , ∂μ ] = ∂[μ,μ] = ∂ 21 [μ,μ] = ∂μ•μ = ∂μ2 = ∂A . 2 2

So associativity μ2 = 0 implies ∂μ2 = 0. Then ∂μ is called a coboundary operator. In particular, for C = EL we obtain the Hochschild coboundary operator [5]: −∂μ f = μ◦(idL ⊗f ) −

|f |

⊗(|f |−i)

(−1)i f ◦(id⊗i L ⊗μ⊗ idL

)

i=0

+ (−1)|f | μ◦(f ⊗ idL ).

26.5

Generalized Maurer-Cartan Equation

. For an operad C, let μ, μ0 ∈ C 2 be two binary operations. The diﬀerence ω =μ − μ0 is . called a deformation, and μ is said to be a deformation of μ0 . Let ∂ =∂μ0 and denote the (formal) associators of μ and μ0 as follows: 1 . A=μ•μ = [μ, μ], 2

1 . A0 =μ0 •μ0 = [μ0 , μ0 ]. 2

The deformation is called associative if A = 0 = A0 . To ﬁnd the deformation equation, calculate 1 [μ, μ] 2 1 = [μ0 + ω, μ0 + ω] 2 1 1 1 1 = [μ0 , μ0 ] + [μ0 , ω] + [ω, μ0 ] + [ω, ω] 2 2 2 2 1 1 1 = A0 − (−1)|μ0 ||ω| [ω, μ0 ] + [ω, μ0 ] + [ω, ω] 2 2 2 1 = A0 + [ω, μ0 ] + [ω, ω]. 2

A=

So we obtain the deformation equation 1 A − A0 = ∂ω + [ω, ω] 6 78 9 2 6 78 9 deformation

operadic curvature

The deformation equation can be seen as a diﬀerential equation for ω with given associators A0 , A. Note that if the associator is ﬁxed, i.e., A = A0 , we obtain the Maurer-Cartan equation, well known from the theory of associative deformations: A = A0

⇐⇒

1 ∂ω + [ω, ω] = 0 2

Thus the deformation equation may be called the generalized Maurer-Cartan equation as well.

Operads and nonassociative deformations

26.6

327

Operadic Sabinin Principle

We know that ∂ 2 = ∂A0 . Hence, if (formal) associativity constraint A0 = 0 holds, then ∂2 = 0 The deformation equation for such a nonassociative deformation of an associative multiplication μ0 reads 1 A = ∂ω + [ω, ω] 2 One can see that the associator of the deformed algebra is a formal (operadic) curvature while the deformation is working as a connection. By reformulating the Sabinin principle, one can say that associator is an operadic equivalent of the curvature.

26.7

Bianchi Identity

Now diﬀerentiate the deformation equation, 1 ∂(A − A0 ) = ∂ 2 ω + ∂[ω, ω] 2 1 1 = ∂ 2 ω + (−1)|∂||ω| [∂ω, ω] + [ω, ∂ω] 2 2 1 1 2 = ∂ ω − [∂ω, ω] + [ω, ∂ω] 2 2 1 1 2 = ∂ ω − [∂ω, ω] − (−1)|∂ω||ω| [∂ω, ω] 2 2 = ∂ 2 ω − [∂ω, ω]. Again using the deformation equation, we obtain ∂(A − A0 ) = ∂ 2 ω − [∂ω, ω] 1 = ∂ 2 ω − [A − A0 − [ω, ω], ω] 2 1 = ∂ 2 ω − [A − A0 , ω] + [[ω, ω], ω]. 2 It follows from the Jacobi identity that ∂A0 = [A0 , μ0 ] =

1 [[μ0 , μ0 ], μ0 ] = 0, 2

[[ω, ω], ω] = 0.

By using these relations we obtain ∂A = ∂ 2 ω − [A − A0 , ω]. Recall that ∂ 2 = ∂Ao and calculate ∂A + [A, ω] = ∂A0 ω + [A0 , ω] = [ω, A0 ] + [A0 , ω]

328

Eugen Paal = −(−1)|ω||A0 | [A0 , ω] + [A0 , ω] = 0.

Thus the associator of the deformed algebra satisﬁes the Bianchi identity ∂A + [A, ω] = 0 The Bianchi identity is an integrability condition of the deformation equation. To clarify the algebraic meaning of the Bianchi identity, let us note that ∂A + [A, ω] = [A, μ0 ] + [A, μ − μ0 ] = [A, μ] =

1 [[μ, μ], μ] = 0, 2

. where the latter equality is evident from the Jacobi identity as well. But A=μ•μ and so the Bianchi identity strikingly reads (μ•μ)•μ = μ•(μ•μ), which can (also) be easily seen from the Gerstenhaber identity. Hence the Bianchi identity does not produce any restrictions on the deformation.

26.8

Acknowledgments

Research was in part supported by Grant 5634 of the Estonian Science Foundation.

References [1] Nesterov, A. I.; Sabinin, L. V. Non-associative geometry and discrete structure of spacetime. Comment. Math. Univ. Carolinae, 2000, 41, 347-357. [2] Nesterov, A. I.; Sabinin, L. V. Nonassociative geometry: Towards discrete structure of spacetime. Phys. Rev. 2000, D62, 081501. [3] M. Gerstenhaber. The cohomology structure of an associative ring. Ann. of Math. 1963, 78, 267-288. [4] A. A. Albert. Power-associative rings. Trans. Amer. Math. Soc. 1948, 64, 552-593. [5] G. Hochschild. Cohomology groups of an associative algebra. Ann. Math. 1945, 46, 58-67.

Chapter 27 Units of Alternative Loop Rings: A Survey C´ esar Polcino Milies Instituto de Matem´atica e Estat´ıstica, Universidade de S˜ ao Paulo, Brazil

27.1 27.2 27.3 27.4 27.5 27.6

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Deﬁnitions and basic facts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Finiteness conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Normal complements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Unit loops torsion over the center . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

329 330 334 339 341 344 344

Abstract In this chapter, we give an account of known results regarding the structure of the loop of units of an alternative loop ring. Key words: loop, alternative loop ring, loop of units, normal complements 2000 MSC: Primary 20N05; Secondary 17D05, 16S34, 16U60

27.1

Introduction

Let R be a commutative (and associative) ring with unity and let L be a loop (see Deﬁnition 27.1 below). The loop algebra of L over R was introduced in 1944 by R.H. Bruck [Bru44] as a means of obtaining a family of examples of nonassociative algebras and it is deﬁned in a way similar to that of a group algebra; i.e., as the free R-module with basis L, with a multiplication induced distributively from the operation in L. In 1955, L. Paige proved that in a characteristic diﬀerent from 2, in a commutative loop algebra, the very weak identity, x2 x2 = x3 x, implies full associativity [Pai55]. In other words, there are no “interesting” nonassociative commutative loop algebras that are not already group algebras. Twenty years ago, in 1983, E.G. Goodaire [Goo83] showed that there do exist alternative loop algebras that are not group algebras and deﬁned RA loops as those loops whose loop algebra over a ring with no 2-torsion is alternative but not associative (it follows, as a consequence of his characterization, that if the loop algebra over a ring with no 2-torsion is alternative, then the loop algebra of the given loop, over all such rings, is also alternative) and O. Chein and E.G. Goodaire gave a full description of those loops in [CG86]. In a subsequent paper [CG90] they also deﬁned RA2 loops as those loops whose loop algebra over a ring of characteristic 2 is alternative. In particular, they showed that RA loops are also RA2 loops. In the last two decades, alternative loop rings have been extensively studied. Results obtained up to 1996 were organized into a book [GJM96] and later results appeared in the survey [Pol98]. In this chapter we shall give a quick summary of some of the earlier results

329

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C´esar Polcino Milies

that are needed in the sequel and then discuss subsequent progress on the properties of the loop of units of these rings, which were all obtained in joint work with E.G. Goodaire.

27.2

Deﬁnitions and Basic Facts

We ﬁrst give the formal deﬁnition of a loop. DEFINITION 27.1 A loop is a set L together with a (closed) binary operation (a, b) → ab for which there is a two-sided identity element 1 and such that the right and left translation maps Rx : a → ax and Lx : a → xa are bijections for all x ∈ L. This requirement implies that, for any a, b ∈ L, the equations ax = b and ya = b have unique solutions. In a loop L we can deﬁne the commutator of two elements in a way that is similar to the same concept for groups. We also deﬁne the associator of three elements. DEFINITION 27.2 Given elements a, b, c in a loop L, the commutator (a, b) and associator (a, b, c) are the elements (uniquely) deﬁned by the following equations: ab = ba(a, b)

(loop) commutator

(ab)c = [a(bc)](a, b, c)

(loop) associator

The commutator subloop is the subloop generated by the set of all commutators and the associator subloop is the subloop generated by all associators. The nucleus and center of L are the subloops N(L) and Z(L), respectively, deﬁned by N(L) = {x ∈ L | (a, b, x) = (a, x, b) = (x, a, b) = 1, for all a, b ∈ L}, Z(L) = {x ∈ N(L) | (a, x) = 1 for all a ∈ L}. DEFINITION 27.3 A loop L is called a Moufang loop if it satisﬁes any of the following three identities (which are equivalent). ((xy)x)z = x(y(xz))

left Moufang identity

((xy)z)y = x(y(zy))

right Moufang identity

(xy)(zx) = (x(yz))x

middle Moufang identity

Now we turn to rings. DEFINITION 27.4 A (not necessarily associative) ring is a set R with two operations, denoted + and ·, such that (R, +) is an abelian group, (a, b) → a · b is a binary operation on R, and both distributive laws hold: a(b + c) = ab + ac, (a + b)c = ac + bc, for all a, b, c ∈ R. If, in addition, (R, +) is a module over a commutative, associative ring Φ such that α(ab) = (αa)b = a(αb) for all α ∈ Φ and all a, b ∈ R, then (R, +, ·) is said to be an

Units of alternative loop rings

331

algebra over Φ. A ring R is alternative if x(xy) = (xx)y and (xy)y = x(yy) for all x, y ∈ R. It can be shown that the Moufang identities hold in an alternative ring R. Consequently, it follows that the set U(R) of all invertible elements is a Moufang loop. One of the most useful properties of alternative rings is the fact that if three elements associate, then the subring that they generate is associative. Thus alternative rings are diassociative in the sense that the subring generated by any two elements is always associative. Similarly, Moufang loops are diassociative: the subloop generated by any pair of elements is always associative, and thus, a group. DEFINITION 27.5 Let L be a loop and let R be a commutative and associative ring with 1. The loop ring of L over R is the free R-module RL with basis L and multiplication given by extending, via the distributive laws, the multiplication in L. In other words, the elements of RL are formal sums, g∈L αg g, where the αg ∈ R are almost all 0 and unique in the sense that αg g = βg g implies αg = βg for all g ∈ L. Addition and multiplication are given by αg g + βg g = (αg + βg )g

αg g

( αh βk )g. βg g = hk=g

The support of an element α =

∈L

α , α ∈ R, in a loop ring RL is the set

supp(α) = { ∈ L | α = 0}. By an alternative loop ring, we mean a loop ring that happens to be alternative. As a subloop of the loop of units of an alternative loop ring RL, the loop L that deﬁnes RL must of course be Moufang, as noted earlier. That such (nonassociative) loops actually exist was ﬁrst shown by E.G. Goodaire [Goo83]. We quote the following. THEOREM 27.1 Let L be a loop. Then L is a loop with an alternative but not associative loop ring if and only if it has the following properties: (i) If three elements associate in some order then they associate in all orders and (ii) If g, h, k ∈ L do not associate, then gh.k = g.kh = h.gk. It follows from this characterization that if the loop ring of a loop L over a ring R whose characteristic is not 2 is alternative then the loop ring RL is alternative for any ring R with char(R) = 2. DEFINITION 27.6 An RA (ring alternative) loop is a loop L whose loop ring RL over some ring R of characteristic not 2 is alternative, but not associative. THEOREM 27.2

Let L be an RA loop. Then:

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C´esar Polcino Milies

(i) g 2 ∈ N(L), ∀g ∈ L. (ii) N(L) = Z(L). (iii) For any pair of elements g, h ∈ L we have that (g, h) = 1 if and only if (g, h, L) = 1. (iv) Given g, h, k ∈ L, if (g, h, k) = 1 then (g, h, k) = (g, h) = (g, k) = (h, k) is a central element of order 2. (v) The commutator and associator subloops coincide and are a central subgroup of order 2. Using the theorem above, it is not hard to show the following. COROLLARY 27.1 The direct product L × K of loops is an RA loop if and only if precisely one of L and K is an RA loop while the other is an abelian group. The next theorem, which is due to O. Chein and E.G. Goodaire [CG90, Section 3] (see also [GJM96, Theorem IV.3.1]) gives a construction for RA loops that is fundamental for the development of the theory. THEOREM 27.3 A loop L is RA if and only if it is not commutative and, for any two elements a and b of L which do not commute, the subloop of L generated by its center together with a and b is a group G such that (i) for any u ∈ / G, L = G ∪ Gu is the disjoint union of G and the coset Gu; (ii) G has a unique nonidentity commutator s, which is necessarily central and of order 2; (iii) the map g → g ∗ =

g if g is central sg otherwise

is an involution of G (i.e., an antiautomorphism of order 2); (iv) multiplication in L is deﬁned by g(hu) = (hg)u (gu)h = gh∗ u (gu)(hu) = g0 h∗ g where g, h ∈ G and g0 = u2 is a central element of G such that g0∗ = g0 . The loop described by this theorem shall be denoted by M (G, ∗, g0 ). COROLLARY 27.2 Let L be an RA loop. Then L/Z(L) ∼ = C2 × C2 × C2 , where C2 denotes the cyclic group of order 2. For any elements a, b ∈ L that do not commute and any u ∈ L that does not associate with a and b, we have that the subloop G = a, b, Z(L) is a group and L = M (G, ∗, u), where L is the involution deﬁned in the theorem above; in particular G/Z(G) ∼ = C2 × C2 . As an interesting consequence of the corollary above, it can be shown that for any two elements x, y in an RA loop L we have that xy = yx if and only if either x or y or xy is

Units of alternative loop rings

333

a central element. Because of this fact, we say that L has the LC (lack of commutativity) property. After Corollary 27.1 we see that to classify RA loops we need only to consider the indecomposable ones and it is very easy to show that these are 2-loops (see [GJM96, Corollary V.1.3]). We shall ﬁrst study groups G such that G/Z(G) ∼ = C2 × C2 and then see how to construct indecomposable RA loops from them. Groups G such that G/Z(G) ∼ = Cp × Cp , for an integral prime p, were studied by G. Leal and C. Polcino Milies. We quote the main result, specializing for p = 2. THEOREM 27.4 [LePol93, Lemma 1.1] A group G is such that G/Z(G) ∼ = C2 × C2 if and only if G can be written in the form G = D × A, where A is abelian and D is an indecomposable 2-group generated by its center and two elements x and y that satisfy i) Z(D) = C2m1 × C2m2 × C2m3 , where C2mi is cyclic of order 2mi for i = 1, 2, 3, m1 ≥ 1 and m2 , m3 ≥ 0; ii) (x, y) ∈ C2m1 ; iii) x2 ∈ C2m1 × C2m2 and y 2 ∈ C2m1 × C2m2 × C2m3 . Using this description, E. Jespers, G. Leal and C. Polcino Milies [JLM95] classiﬁed all ﬁnite groups of this type. THEOREM 27.5 Let G be a ﬁnite group. Then G/Z(G) ∼ = C2 × C2 if and only if G can be written in the form G = D × A, where A is abelian and D = Z(D), x, y is of one of the following ﬁve types of indecomposable 2-groups.

Type

Z(D)

D1

t1

D2

t1

D3

t1 × t2

D4

t1 × t2

D m1 m1 x, y, t1 | (x, y) = t12 −1 , x2 = y 2 = t21 m1 m1 x, y, t1 | (x, y) = t12 −1 , x2 = y 2 = t1 , t2 = 1 m1 m1 m2 x, y, t1 , t2 | (x, y) = t12 −1 , x2 = t21 = t22 = 1, y 2 = t2 m1 m1 m2 x, y, t1 , t2 | (x, y) = t12 −1 , x2 = t1 , y 2 = t2 , t21 = t22 =

1

D5 t1 × t2 × t3 x, y, t1 , t2 , t3 | m1 m1 m2 m3 (x, y) = t12 −1 , x2 = t2 , y 2 = t3 , t21 = t22 = t23 = 1

Then, it is possible to describe all ﬁnite indecomposable RA loops. THEOREM 27.6 [JLM95] [GJM96, Theorem V.3.1] Let L = M (G, ∗, g0 ) be a ﬁnite indecomposable RA loop. Then G is either one of the ﬁve groups speciﬁed in Theorem 27.5 or the direct product D5 × w of D5 and a cyclic group w and L is one of the following seven types of loops.

334

C´esar Polcino Milies Type G L1 D1 L2 D2 L3 D3 L4 D4 L5 D5 L6 D5 L7 D5 × w

x2 1 t1 1 t1 t2 t2 t2

y2 1 t1 t2 t2 t3 t3 t3

g0 1 t1 1 t1 1 t1 w

Some of the results in the following sections make use of the concept of augmentation ideal, one reference for which is [GJM96, §VI.1]. Let R be any commutative and associative ring with 1. Let N be a normal subloop of an RA loop (or a group) L and let N : RL → R[L/N ] denote the linear extension to RL of the natural homomorphism L → L/N . This map is a ring homomorphism whose kernel is the ideal Δ(L, N ) = n∈N RL(n − 1). In the special case N = L, we write = L , calling this the augmentation map on RL and set Δ(L) = Δ(L, L). We call Δ(L) the augmentation ideal of RL. The identity 1 (2 − 1) = (1 2 − 1) − (1 − 1) shows that Δ(L) = { α ( − 1) | α ∈ R}.

∈L

For any normal subloop N of L, note that Δ(L, N ) = (RL)Δ(N ).

27.3

Finiteness Conditions

In this section, we shall study necessary and suﬃcient conditions for the loop of units of an alternative loop ring to satisfy some fundamental ﬁniteness conditions: namely, FC, nilpotence, or solvablility. In general, in an arbitrary loop, it is not true that the relation x ≡ y if y = a−1 xa for some a is an equivalence relation, because this relation is not transitive. However, as in group theory, we call a Moufang loop L an FC loop if, for all ∈ L, the set {x−1 x | x ∈ L} is ﬁnite. For any prime p, by a p-element, we mean an element whose order is a power of p. A p -element is an element of ﬁnite order relatively prime to p. We shall denote by P and A the sets of p- and p -elements, respectively. It is easy to see that the following hold (see [GP96a, Lemma 1.2]; in the case p = 2, see also [GJM96, Proposition V.1.1]). LEMMA 27.1 The set T of torsion elements of an RA loop L is a normal locally ﬁnite subloop of L. If T is commutative, it is a group. LEMMA 27.2 Let L be any subloop of a torsion RA loop. Then L = L0 × B is the direct product of a 2-loop L0 and a (central) abelian group B all of whose elements have an odd order. Thus, for any prime p, the sets P of p-elements and A of p -elements of L are subloops of L. Furthermore, L = P × A.

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We shall ﬁrst consider loop rings over a ﬁeld F . Assume that an RA loop L = M (G, ∗, g0 ) is torsion. Since G is a nonabelian group and as U(F L) FC implies U(F G) FC, it follows from [CoPo91, Theorem A ] that G must also be ﬁnite. So we obtain THEOREM 27.7 Let L be a torsion RA loop and let F be any ﬁeld. Then U(F L) is an FC loop if and only if both F and L are ﬁnite. We turn our attention to RA loops that contain elements of inﬁnite order. THEOREM 27.8 Let F be a ﬁeld of characteristic p > 0 and let L be a nontorsion RA loop that contains an element of order p. Let T denote the torsion subloop of L. Then U(F L) is an FC loop if and only if p = 2 and T = s × A, where L = {1, s} and A is a ﬁnite abelian group of an odd order (and, thus, T is central). We now turn to the case where char F = p > 0 and L contains no p-elements. Notice that since o(s) = 2, we must then assume that p is odd. THEOREM 27.9 Let F be a ﬁeld of characteristic p > 0 and let L be an RA loop with torsion subloop T = L and with no p-elements. Then U(F L) is an FC loop if and only if T is an abelian group and one of the following conditions holds: r

(i) F T is ﬁnite and for all t ∈ T and all x ∈ L, we have tx = tp for some nonnegative integer r, which is a multiple of [F : P], where P denotes the prime ﬁeld of F . (ii) T is ﬁnite and central. (iii) T = Z(2∞ ) × B with B ﬁnite, and there exists an integer k such that F does not contain roots of unity of order 2k . ufer group; that is, the additive group Z(p∞ ) = Q(p) /Z Here, Z(p∞ ) denotes the the p-Pr¨ (p) n where Q = {a/p | a, n ∈ Z}. The case of characteristic 0 is very similar. THEOREM 27.10 Let F be a ﬁeld of characteristic 0 and L a nontorsion RA loop. Then U(F L) is FC if and only if T is central and, if T is inﬁnite, then T = Z(2∞ ) × B where B is ﬁnite and there exists an integer k such that F does not contain roots of unity of order 2k .

We recall that a Moufang loop L is said to be nilpotent (centrally nilpotent in the sense of [Bru58, Chapter VI]) if there exists a ﬁnite series: L = L0 ⊃ L1 ⊃ L2 ⊃ · · · ⊃ Ln = {1} of subloops Li all normal in L, such that Li−1 /Li is contained in the center of L/Li for 1 ≤ i ≤ n.

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The following result on group rings with nilpotent unit groups will be of use. THEOREM 27.11 [Seh78, Theorem VI.3.1] Let G be a group having an element of order p in its center and let F be a ﬁeld of characteristic p > 0. Then U(F G) is nilpotent if and only if G is a ﬁnite p-group. If L = M (G, ∗, g0 ) is an RA loop and U(F L) is nilpotent, then U(F G) is also nilpotent. Since L = G = {1, s}, Theorem 27.11 readily shows that if p is odd and L contains a p-element, then U(F G) cannot be nilpotent. On the other hand, if p = 2, it is easy to see that U(F L) is always nilpotent. For this, let u, v, w ∈ U(F L) and compute (u, v) − 1 = u−1 v −1 (uv − vu) ∈ Δ(L, L ) = F L(1 − s) (u, v, w) − 1 = (vw)−1 u−1 (uv = cw˙ − u · vw) ∈ Δ(L, L ) = F L(1 − s) so that γ1 (L), the subloop of U(F L) generated by the commutators and associators is contained in 1 + F L(1 − s). Next, let u, v ∈ U(F L) and 1 + δ ∈ γ1 (L) for some δ = α(1 − s) ∈ Δ(L, L ), α ∈ F L. Then (1 + δ, u) − 1 = (1 + δ)−1 u−1 ((1 + δ)u − u(1 + δ) = (1 + δ)−1 u−1 (δu − uδ) = (1 + δ)−1 u−1 (1 − s)(αu − uα) = 0 because it is easy to see that αu−uα ∈ Δ(L, L ) and in characteristic 2, we have (1−s)2 = 0. Similarly, (1 + δ, u, v) − 1 = (uv)−1 (1 + δ)−1 ((1 + δ)u · v − (1 + δ) · uv) = (uv)−1 (1 + δ)−1 (δu · v − δ · uv) = (uv)−1 (1 + δ)−1 (1 − s)(αu · v − α · uv) = 0 because αu · v − α · uv ∈ Δ(L, L ). Thus γ2 (L), the subloop of U(F L) generated by the set {(a, u), (a, u, v) | a ∈ γ1 , u, v ∈ U(F L)} is {1}. So we have the following. THEOREM 27.12 Let F be a ﬁeld of characteristic p > 0 and let L be an RA loop that contains an element of order p. Then U(F L) is nilpotent if and only if p = 2. Now let us assume that char F = p ≥ 0, but that L contains no element of order p, in the case p > 0. In this situation, the following theorem will be of use. THEOREM 27.13 [Seh78, Theorem VI.3.6] Let F be a ﬁeld of characteristic p ≥ 0. Let G be a group that contains no element of order p, in the case p > 0. Then U(F L) is nilpotent if and only if G is nilpotent and one of the following conditions holds: (i) T (G) is a central subgroup. (ii) |F | = p is a Mersenne prime (that is, p = 2β − 1 for some positive integer β), T (G) is an abelian group of exponent 2(p − 1) and for all x ∈ G and all t ∈ T (G), we have that x−1 tx = t or tp .

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Suppose L is an RA loop. Recall that if T = T (L) is not central, then we can write L = M (G, ∗, g0 ) for some group G such that T = T (G) ∪ T (G)z, where z ∈ T . If U(F L) is nilpotent, then so is U(F G), so Theorem 27.13 shows that T (G) is abelian. Then, since T (G), z ⊆ L, it follows again that T (L) = T (G), z is actually a group. But then U(F T ) is also nilpotent and we obtain that T is also commutative, and hence an abelian group. THEOREM 27.14 Let F be a ﬁeld of characteristic p > 0 and let L be an RA loop that contains no elements of order p. Let T denote the torsion subloop of L. Then U(F L) is nilpotent if and only if one of the following conditions holds: (i) T is central; (ii) |F | = p is a Mersenne prime, T is an abelian group of exponent 2(p − 1) and for all x ∈ L and t ∈ T , we have that x−1 tx = t or tp . THEOREM 27.15 Let F be a ﬁeld of characteristic 0 and let L be an RA loop. Then U(F L) is nilpotent if and only if the torsion subloop of L is a central group.

Solvability for Moufang loops is deﬁned just as it is for groups [Gla68]. A Moufang loop L is solvable if there is a ﬁnite subnormal series of subloops extending from 1 to L with each quotient loop a commutative group. Subloops and homomorphic images of solvable loops are solvable. Moreover, if M is a normal subloop of a Moufang loop L, then L is solvable if and only if M and L/M are solvable. If L is RA, the LC property implies that 2 ∈ Z(L) for any ∈ L. Thus L is an extension of its center by an elementary abelian 2-group (in fact, an extension of its center by C2 × C2 × C2 ), and hence solvable. (Here, C2 denotes the cyclic group of order 2.) It turns out that many conditions for the solvability of a unit loop U(RL) involve the set of torsion elements of L. In contrast to the general situation, the torsion elements of an RA loop always form a subloop [GJM96, Lemma VIII.4.1], [GP95, Lemma 2.1]. We give necessary and suﬃcient conditions for the unit loop of an alternative loop ring to be solvable. A Moufang loop is Hamiltonian if it is not commutative and every subloop is normal. A group is Hamiltonian if and only if it is the direct product Q8 × E × A, where Q8 is the quaternion group of order 8, E is elementary abelian of exponent 2 and A is a group all of whose elements are of an odd order [Hal59, Theorem 12.5.4]. A Moufang loop that is not associative is Hamiltonian if and only if it is the direct product C × E × A where C = M16 (Q8 ) is the Cayley loop [GJM96, §4.1] and E and A are as before [GJM96, Theorem II.4.8], [Nor52]. As above, we ﬁrst consider loop rings over ﬁelds. Since all torsion indecomposable RA loops are 2-loops [GJM96, Corollary V.1.3], [CG86, Theorem 6], it is not surprising that characteristic 2 generally represents a very special case in theorems concerning RA loops. The results below were given in [GP01a] THEOREM 27.16 Let L be an RA loop and F a ﬁeld. If char F = 2, then [U(F L)] is an abelian group, so U(F L) is solvable. COROLLARY 27.3 Let L be a torsion RA loop and F a ﬁeld. Then U(F L) is solvable if and only if char F = 2.

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Since a Hamiltonian Moufang loop is a torsion loop, the following corollary is immediate. COROLLARY 27.4 Let F be a ﬁeld and let L be a Hamiltonian Moufang loop that is not associative. Then U(F L) is solvable if and only if char F = 2. We now wish to study solvability of the unit loop of a loop algebra F L over an arbitrary ﬁeld without the assumption that L is torsion. The corresponding problem for group rings remains unsettled in general, although A. Bovdi has made progress for nilpotent nontorsion groups [Bov92]. In positive characteristic p, whether or not L contains p-elements turns out to be critical, so our main results are described by two theorems, Theorems 27.17 and 27.18 below. We begin with several lemmas, the ﬁrst two giving conditions under which an alternative loop algebra F L contains F, Zorn’s vector matrix algebra over F , which is the set of 2 × 2 matrices of the form @ A ax , y b where a, b ∈ F We add two @ a1 y1

and x, y ∈ F 3 , the set of ordered triples of elements of F . such matrices entrytwise and multiply them according to the following rule: A@ A @ A x1 a1 x2 + b2 x1 − y1 × y2 a2 x2 a1 a2 + x1 .y2 = , b1 y2 b 2 a2 y1 + b1 x2 + x1 × x2 b 1 b 2 + y 1 x2

with . and × denoting the usual dot and cross products, respectively, in F 3 . This is an alternative algebra also known as the split Cayley-Dickson algebra over F . LEMMA 27.3 Let F be a ﬁeld of characteristic diﬀerent from 2. Let L be an RA loop with torsion subloop T . Suppose there exists t ∈ T such that the subloop t generated by t is not normal in L. Then F L contains Zorn’s vector matrix algebra F; hence U(F L) is not solvable. LEMMA 27.4 Let L be an RA loop that contains the quaternion group Q8 and let F be a ﬁeld of characteristic 3. Then U(F L) contains F and hence is not solvable. The above lemmas lead to the following results. LEMMA 27.5 Let L be an RA loop that contains the quaternion group Q8 and let F be a ﬁeld of characteristic 3. Then U(F L) contains F and hence is not solvable. THEOREM 27.17 Let F be a ﬁeld of characteristic p ≥ 0 and suppose L is an RA loop that contains no elements of order p (in the case p > 0). Then U(F L) is solvable if and only if (i) p = 2 or (ii) T is an abelian group and every idempotent of F T is central in F L. THEOREM 27.18 Let F be a ﬁeld of characteristic p > 0. Let L be an RA loop that is not torsion and that contains elements of order p. The U(F L) is solvable if and only if (i) p = 2 or (ii) the set P of p-elements in L is a ﬁnite central group and U(F [L/P ]) is solvable.

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We now turn to the case of integral alternative loop rings. The following theorem is the analog for alternative loop rings that are not associative of a theorem of Sehgal for group rings [Seh78, Theorem VI.4.8]. THEOREM 27.19 solvable if and only if

Let L be an RA loop with torsion subloop T . Then U(ZL) is

T is either an abelian group or a Hamiltonian Moufang 2-loop and every subloop of T is normal in L. (27.1) As in immediate consequence, in the case of a ﬁnite RA loop, we see that solvability is equivalent to many other ﬁnite conditions (see [GJM96, Corollary 2.14]). THEOREM 27.20 equivalent:

Let L be a ﬁnite RA loop. Then the following conditions are

(i) U(ZL) is an RA loop. (ii) U(ZL) is nilpotent. (iii) U(ZL) is nilpotent of class 2. (iv) U(ZL) is a group of order 2. (v) U(ZL) is n-Engel, for some n ≥ 2. (vi) U(ZL) is 2-Engel. (vii) U(ZL) is FC. (viii) U(ZL) is a torsion loop. (ix) U(ZL) is solvable. (x) L is a Hamiltonian Moufang 2-loop.

27.4

Normal Complements

The loop of units in ZL contains L and it is of interest to see how L sits inside U(ZL). If there exists a normal subloop N of U(ZL) such that L ∩ N = {1} and U(ZL) = ±LN , then N is called a normal complement of L. The search for a normal complement that is torsion-free is of great interest since, as we show below, a positive answer to this question leads to a diﬀerent proof of the isomorphism conjecture. It so happens that when L is a ﬁnite RA loop, normal complements always exist. In the case of group rings, the existence of a normal complement is not equivalent to the isomorphism problem since it is well known that a ﬁnite metabelian group is determined by its integral loop rings (see [PolShg, 9.3.13]) while it has been shown by K.W. Roggewnkamp and L.L Scott [RS83] that normal complements do not always exist in this case.

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Our ﬁrst lemma is of a technical nature, but is needed for the proof of the main theorem in this section. LEMMA 27.6 Let A be a normal subloop of an RA loop L and let δ ∈ Δ(L, A). Then there exists a ∈ A such that δ ≡ a − 1 (mod Δ(L)Δ(A)). With this in mind, one can prove the following. THEOREM 27.21

Let L be a ﬁnite RA loop with center Z. Then N = 1 + Δ(L)Δ(Z) ∩ U1 (ZL)

is a torsion-free normal complement for L in the loop U1 of normalized units in ZL. We now show, using the theorem above, that ﬁnite RA loops are determined by their integral alternative loop rings. The original proof of the following “isomorphism theorem” appeared in [GP89]. See also [GJM96, Theorem IX.1.1]. THEOREM 27.22 L1 ∼ =L.

Let L and L1 be ﬁnite RA loops and suppose that ZL1 ∼ =ZL. Then

PROOF Note ﬁrst that L and L1 have the same order, since each is the rank of the same free Z-module. Suppose ϕ : ZL1 → ZL is the given isomorphism and let N be a torsion-free normal complement for L1 in U1 (ZL1 ). Then ϕ(N ) is torsion-free in U1 (ZL) and so L ∩ ϕ(N ) = {1}. Since [U1 (ZL) : ϕ(N )] = |L1 | = |L| = [Lϕ(N ) : ϕ(N )], we have U1 (ZL) = Lϕ(N ). Thus L U1 (ZL1 ) ∼ U1 (ZL) ∼ Lϕ(N ) ∼ = L. L1 ∼ = = = = N ϕ(N ) ϕ(N ) L ∩ ϕ(N )

In view of Theorem 27.21, it is natural to ask if L can ever be a direct factor of U1 . As we shall see, with L ﬁnite, this happens only when U1 = L. THEOREM 27.23 Let L be a ﬁnite RA loop. Then L is normal in U(ZL) if and only if U(ZL) is itself an RA loop and this occurs if and only if U(ZL) = ±L. REMARK 27.1 The condition that U(ZL) be RA is equivalent to many other conditions on this unit loop, including nilpotency, sovability, and the FC properties as noted in the previous section. Example 27.1 There do indeed exist RA loops L with U(ZL) = ±L also RA. By Corollary XII.2.14 of [GJM96] (see also [GP95, Theorem 3.3]), it is suﬃcient to construct an RA loop with a torsion subloop T , which is an abelian group with the property that if x ∈ L does not centralize T , then x−1 tx = t−1 for all t ∈ T . To construct such an L, let A = z × b, z 8 = 1, be the direct product of a cyclic group of order 8 and an inﬁnite cyclic group b. Let G be the group generated by A and elements x, y subject to the relations ax = xa, ay = ya for a ∈ A, x2 = z 4 , y 2 = b, (x, y) = z 4 .

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Then G = {1, z 4 }, Z = A and G/Z = x × y∼ =C2 × C2 . The loop M (G, ∗, b) = G ∪ Gu, where u2 = b, is RA with torsion subloop T = z, x and y −1 x−1 y = (y, x)x−1 = z 4 x−1 = x. In contrast to Theorem 27.23, L is never normal in the unit loop of a loop algebra over a ﬁeld F and U(F L) is never RA. THEOREM 27.24 normal in U(F L).

Let L be a ﬁnite RA loop and let F be a ﬁeld. Then L is not

REMARK 27.2 Let R be any commutative associative ring with 1 and of characteristic p > 0. Since R contains Fp , it can be shown that L is never normal in U(RL). Such is not the case with group rings over ﬁnite rings; for example, the symmetric group S3 is normal in U(F2 S3 ) [Seh78, §6.2, p. 215]. THEOREM 27.25 normal in U(F L).

Let L be a ﬁnite RA loop and let F be a ﬁeld. Then L is not

The results we quote in this section come from [GP01b] but it should be noted that Theorems 27.21 and 27.22 also appear in a paper of S. O. Juriaans and L. G. X. de Barros [dBJ97].

27.5

Unit Loops Torsion over the Center

In this section, we consider the possibility that the unit loop U(RL) might be torsion over its center ; that is, for any μ ∈ U(RL), μn is central for some n = n(μ). We answer the question when R = Z and when R is a ﬁeld. In each case, we consider also whether there is a uniform bound on the exponents n. For group rings, these questions have been investigated by Sehgal [Seh78, Section II.2], Cliﬀ and Sehgal [CS80], Coelho [Coe82], and Bist [Bis94]. The results below were obtained in [GP02]. We ﬁrst remark the following. PROPOSITION 27.1 2-loop.

U(ZL) is torsion if and only if L is a Hamiltonian Moufang

In the case of integral alternative loop rings, we can state the following. THEOREM 27.26 Let L be an RA loop with torsion subloop T . Then the following conditions are equivalent: (i) U(ZL) is torsion over its center, Z(U(ZL)). (ii) T is an abelian group or a Hamiltonian Moufang 2-loop and, if x ∈ L does not centralize T , then x−1 tx = t−1 for all t ∈ T . (iii) U(ZL) is torsion of bounded exponent 2 over Z(U(ZL)). (iv) U(ZL) is nilpotent.

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Now we turn to loop rings over ﬁelds. PROPOSITION 27.2 Let L be an RA loop and K a ﬁeld. Then U(KL) is torsion if and only if L and K ∗ = K \ {0} are both torsion. COROLLARY 27.5 inﬁnite order.

If U(KL) is not torsion, there exist central units in K or in L of

LEMMA 27.7 Let K be a ﬁeld and let L be an RA loop with torsion subloop T . Suppose U(KL) is torsion over its center, but not torsion. Then every idempotent of KL is central. We have noted and, indeed, made frequent use of the fact that the set of torsion elements of an RA loop L is a subloop. Actually, as noted in Lemma 27.2, for any prime p, the sets Tp of p-elements (order a power of p) and Tp of p -elements (order relatively prime to p) are also normal subloops of L and T = Tp × Tp COROLLARY 27.6 Let K be a ﬁeld of characteristic p > 0 and let L be an RA loop with torsion subloop T . Suppose U(KL) is torsion over its center but not torsion. Let Tp denote the subloop of p -elements in T , that is, those of order relatively prime to p. Then (i) every subloop of Tp is normal in L; (ii) Tp is an abelian group. To obtain these results we make use of the fact that if G is an associative subloop of an RA loop, then either G is abelian or else the quotient G/Z(G) of G by its center is C2 × C2 , the Klein 4-group [GJM96, Proposition III.3.6 and Corollary IV.2.2], [CG86, Theorem 5]. We also require the following lemma, the proof of which can be deduced from Lemma XII.1.1 of [GJM96] and its proof. See also [GP95, Lemma 2.3]. LEMMA ; 27.8 Let K be a ﬁeld and let L be an RA loop with torsion subloop T . Suppose KT = Di is the direct sum of division rings and that every idempotent of KT is central in KL. Then the following are true. (i) Any sum S of a subset of the division rings Di is normal in KT in the sense that Sα = αS, (Sα)β = S(αβ), (αS)β = α(Sβ) and α(βS) = (αβ)S for any α, β ∈ KL. (ii) Each unit μ ∈ KL can be written in the form μ = μq q, where q ∈ L, the μq are in KT and, if q = q , the set of division rings required to write μq1 as a sum in 1 2 ; Di has empty intersection with the set required to write μq2 as a sum; in particular, μq1 μq2 = 0 if q1 = q2 . Then, one can prove the following. THEOREM 27.27 Let L be an RA loop with torsion subloop T and let K be a ﬁeld of characteristic 0. Suppose U(KL) is not torsion. Then the following conditions are equivalent. (i) U(KL) is torsion over its center. (ii) T is central.

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(iii) U(KL) is torsion of exponent 2 over its center; that is, [U(KL)]2 ⊆ Z(U(KL)). (iv) U(KL) is torsion of bounded exponent over its center. THEOREM 27.28 Let K be a ﬁeld of positive characteristic p = 2 and let L be an RA loop with torsion subloop T . Suppose U(KL) is not torsion. Then U(KL) is torsion over its center if and only if T is an abelian group, every idempotent of KT is central in KL and, if T is not central, then K is algebraic over its prime ﬁeld. Notice that one can describe, in terms of L and K, precisely when all the idempotents of KT are central in KL [GJM96, §XIII.1], [GP96a]. COROLLARY 27.7 Let K be a ﬁeld of characteristic p = 2 and let L be an RA loop that has central torsion and that contains no elements of order p. Then U(KL) is torsion over its center if and only if it is torsion of exponent 2 over its center. We wish now to explore the case of bounded exponent. For the proof of the next theorem, the following lemma of S. Coelho [Coe82] is needed. LEMMA 27.9 Let U(KL) denote the unit loop of an alternative loop algebra over a ﬁeld p > 0 and suppose [U(ZL)]n ⊆ Z(U(ZL)). Write n = pa n with % K of characteristic pa % p n . Then x is central for any nilpotent x ∈ KL. We recall that, for a normal subloop N of a loop L, the ideal Δ(L, N ) = n∈N RL(n−1) is the kernel of the map N : RL → R[L/N ] induced by the natural homomorphism L → L/N . THEOREM 27.29 Let K be a ﬁeld of positive characteristic p = 2 and let L be an RA loop with torsion subloop T . Let P and A = Tp denote, respectively, the sets of pand p -elements in T . Suppose U(KL) is not torsion. Then the following statements are equivalent. (i) U(KL) is torsion of bounded exponent over its center. a

(ii) T is abelian, there exists a such that xp is central for all x ∈ Δ(L, P ) and either • T is central and U(KL) has exponent 2pa over its center, or • K is ﬁnite, Am = {1} for some m and U(KL) has exponent 2rpa over its center, where r = |K(ζ)| − 1, ζ a primitive m-th root of unity. Theorems 27.26 and 27.29 have assumed that the characteristic of K is diﬀerent from 2. The remaining case is answered by the following very easy result. THEOREM 27.30 Let L be an RA loop and let K be a ﬁeld of characteristic 2. Then U(KL) is torsion of bounded exponent 2 over its center.

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Acknowledgments

This research was supported by FAPESP and CNPq of Brasil, Proc. 00/07291-0b and Proc. 300243/79-O(RN), respectively.

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[Goo83] E.G. Goodaire, Alternative loop rings, Publ. Math. Debrecen, 30 (1983), 31–38. [GP89] Edgar G. Goodaire and C´esar Polcino Milies, Isomorphisms of integral alternative loop rings, Rend. Circ. Mat. Palermo XXXVII (1988), 126-135. [GP95] Edgar G. Goodaire and C´esar Polcino Milies, On the loop of units of an alternative loop ring, Nova J. Algebra Geom. 3 (1995), no. 3, 199–208. [GP95a] E.G. Goodaire and C. Polcino Milies, Ring Alternative loops and their loop rings, Resenhas Inst. Mat. Est. Univ. S˜ ao Paulo, 2, 1(1995), 47-82 [GP96a] Edgar G. Goodaire and C´esar Polcino Milies, Central idempotents in alternative loop algebras, Nova J. Math. Game Theory Algebra 5 (1996), no. 3, 207–214. [GP96b] Edgar G. Goodaire and C´esar Polcino Milies, Finite conjugacy in alternative loop algebras, Comm. Algebra 24 (1996), no. 3, 881–889. [GP96c] Edgar G. Goodaire and C´esar Polcino Milies, Finite subloops of units in an alternative loop ring, Proc. Amer. Math. Soc. 124 (1996), no. 4, 995–1002. [GP96d] Edgar G. Goodaire and C´esar Polcino Milies, The torsion product property in alternative algebras, J. Algebra 184 (1996), 58–70. [GP97] Edgar G. Goodaire and C´esar Polcino Milies, Nilpotent Moufang unit loops, J. Algebra 190 (1997), 88–99. [GP01a] Edgar G. Goodaire and C´esar Polcino Milies, Alternative loop rings with solvable unit loops, J. of Algebra, 240 (2001), 25-39. [GP01b] Edgar G. Goodaire and C´esar Polcino Milies, Normal subloops in the integral loop ring of an RA Loop, Can. Math. Bull., 44, 1 (2001), 27-35 [GP02] Edgar G. Goodaire and C´esar Polcino Milies, Moufang unit loops torsion over their centres, Quaestiones Mathematicae, 25, 1 (2002), 1-12. [GPa86] Edgar G. Goodaire and M. M. Parmenter, Units in alternative loop rings, Israel J. Math. 53 (1986), no. 2, 209–216. [Gon86] Jairo Zacarias Gon¸calves, Group rings with solvable unit groups, Comm. Algebra 14 (1986), no. 1, 1–20. [Hal59] M. Hall, Jr., The Theory of Groups, MacMillan, New York, 1959. [HP80] B. Hartley and P. F. Pickel, Free subgroups in the unit groups of integral group rings, Canad. J. Math. 32 (1980), 1342–1352. [Jen41] S. A. Jennings, The structure of the group ring of a p-group over a modular ﬁeld, Trans. Amer. Math. Soc. 50 (1941), 175–185. [JLM95] E. Jespers, G. Leal, and C. Polcino Milies, Loop algebras of indecomposable RA loops, Comm. Algebra 22 (1994), no. 4, 1369–1379. [LePol93] G. Leal and C. Polcino Milies, Isomorphic group (and loop) algebras, J. of Algebra, 155 (1993), 195-210. [MN72] Kaoru Motose and Yasushi Ninomiya, On the solvability of unit groups of group

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[MT71] Kaoru Motose and Hisao Tominaga, Group rings with solvable unit groups, Math. J. Okayama Univ. 15 (1971), 37–40. [Nor52] D. A. Norton, Hamiltonian loops, Proc. Amer. Math. Soc. 3 (1952), 56–65. [Pai55] L.J. Paige, A theorem on commutative power associative loop algebras, Proc. Amer. Math. Soc., 6 (1955), 279-280. [Pai56] L.J. Paige, A class of simple Moufang loops, Proc. Amer. Math. Soc. 7 (1956), 471–482. [Pas77] D. S. Passman, Observations on group rings, Comm. Algebra 5 (1977), 1119–1162. [Pol82] C. Polcino Milies, Units of group rings—a short survey, Groups–St. Andrew’s 1981, London Math. Soc. Lecture Notes, vol. 71, Cambridge University Press, Cambridge–New York, 1982, pp. 281–297. [Pol98] C. Polcino Milies, Alternative loop rings and related topics, In Algebra, Some Recent Advances, ed. by I.B.S. Passi, Indian National Acad. Sci., Hindustan Book Agency, New Delhi, 1998. [PolShg] C. Polcino Milies and S. K. Sehgal, An Introduction to Group Rings, Kluwer Acad. Publ., Dordrecht, 2002. [RS83] K. W. Roggenkamp and L. L. Scott, Units in metabelian group rings: non-splitting examples for normalized units, J. Pure Appl. Algebra 27 (1983), 299–314. [Seh75] S. K. Sehgal, Nilpotent elements in group rings, Manuscripta Math. 15 (1975), 65–80. [Seh78] S. K. Sehgal, Topics in Group Rings, Marcel Dekker, New York, 1978. [ZSSS82] K. A. Zhevlakov, A. M. Slin’ko, I. P. Shestakov, and A. I. Shirshov, Rings That Are Nearly Associative, Academic Press, New York, 1982 (translated by Harry F. Smith.

Chapter 28 Vinberg Algebras Associated to Some Nilpotent Lie Algebras Elisabeth Remm Laboratoire de Math´ematiques et Applications, Universit´e de Haute, Alsace, France

28.1 Vinberg algebras associated with Lie algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28.1.1 Generalities: Aﬃne connections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28.1.2 Aﬃne structures on Lie algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28.1.3 Vinberg algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28.1.4 Classical examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28.2 Vinberg algebras associated with graded ﬁliform complex Lie algebras . . . . . . . . . . . . . . . . 28.2.1 Filiform Lie algebras, characteristically nilpotent ﬁliform Lie algebras . . . . . . . . . 28.2.2 Aﬃne structures on ﬁliform complex Lie algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28.2.3 Noncomplete aﬃne structures on ﬁliform Lie algebras . . . . . . . . . . . . . . . . . . . . . . . . . . 28.3 Nilpotent Lie algebras with a contact form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28.3.1 Nilpotent contact Lie algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28.3.2 Aﬃne structures on nilpotent contact Lie algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28.4 Application: Aﬃne structures on 7-dimensional characteristically nilpotent contact Lie algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28.4.1 Algebra η74 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28.4.2 Algebra η712 (λ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28.4.3 Algebra η714 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28.4.4 Algebra η721 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28.4.5 Algebra η728 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28.5 Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

348 348 348 349 349 349 350 351 352 355 355 356 360 360 360 361 362 362 363 363

Abstract We are going to study the existence of left invariant ﬂat aﬃne connections on Lie groups. The corresponding covariant operator gives on the Lie algebra g of the Lie group an aﬃne structure characterized by the vanishing of torsion and curvature. This aﬃne structure induces on the underlying vector space of g a Vinberg product whose commutator is the Lie bracket on g. For nilpotent algebras, the above existence problem was considered by J. Milnor. In this chapter, we will construct such Vinberg algebras by a central extension process. Key words: Vinberg algebras, aﬃne structure, left invariant aﬃne connection, ﬁliform Lie algebras, contact Lie algebras 2000 MSC: 17Bxx, 53Cxx

347

348

28.1 28.1.1

Elisabeth Remm

Vinberg Algebras Associated with Lie Algebras Generalities: Aﬃne connections

DEFINITION 28.1 We will say that a connection ∇ on a manifold M is aﬃne if for any vector ﬁeld X, the corresponding endomorphism ∇X of the space D1 (M ) of vector ﬁelds on M satisﬁes the following conditions

∞

∇f X+gY = f ∇X + g∇Y ;

(28.1)

∇X (f Y ) = f ∇X (Y ) + (Xf )Y

(28.2)

1

for f, g ∈ C (M ), X, Y ∈ D (M ). Suppose that M is a Lie group G and consider an aﬃne connection ∇ on G. For every left translation Φ, we can deﬁne another connection ∇Φ by −1

Φ Φ , ∇Φ X (Y ) = (∇X Φ (Y ))

X, Y ∈ D1 (G).

The connection ∇ is called left invariant if for any Φ ∇Φ = ∇. An aﬃne connection on G is left invariant if and only if the left translations are aﬃne maps. Recall that the torsion T and the curvature C of an aﬃne connection ∇ are the tensors deﬁned by T (X, Y ) = ∇X (Y ) − ∇Y (X) − [X, Y ] , C(X, Y ) = ∇X ∇Y − ∇Y ∇X − ∇[X,Y ] . In this chapter we consider left invariant connections with the vanishing torsion and curvature.

28.1.2

Aﬃne structures on Lie algebras

Let (G, ∇) be an aﬃne Lie group and g the Lie algebra of G. Since the operator ∇ is left invariant, it induces a bilinear map on g, denoted again by ∇. We will write ∇(X, Y ) = ∇X Y to keep up with the classical notation of connection. In this case the bilinear map ∇ on g is characterized by the properties given in the following deﬁnition. We say that the Lie algebra g with such a map ∇ is aﬃne. DEFINITION 28.2

A bilinear map ∇ on the Lie algebra g satisfying ∇X Y − ∇Y X = [X, Y ]

(28.3)

∇X ∇Y − ∇Y ∇X = ∇[X,Y ]

(28.4)

is called an aﬃne structure on g. DEFINITION 28.3 An aﬃne structure on g is complete if the endomorphisms ψX : g → g deﬁned by ψX (Y ) = Y + ∇Y X are bijective for all X ∈ g. An equivalent deﬁnition is that the right translation RX : g → g deﬁned by RX (Y ) = ∇Y X is nilpotent for any X ∈ g or that the trace of RX is zero for all X ∈ g (see [11]).

Vinberg algebras associated to some nilpotent Lie algebras

28.1.3

349

Vinberg algebra

Let (A, μ) be an algebra with a multiplication μ. We denote by aμ : A⊗3 → A the associator of μ that is aμ (X, Y, Z) := μ((μ (X, Y )), Z) − μ(X, μ(Y, Z)).

(28.5)

DEFINITION 28.4 The algebra (A, μ) is a Vinberg algebra (also called left-symmetric algebra) if for all X, Y, Z ∈ A, aμ (X, Y, Z) = aμ (Y, X, Z). PROPOSITION 28.1 Vinberg algebra.

(28.6)

If ∇ is an aﬃne structure on a Lie algebra g then (g, ∇) is a

The statement of the proposition is obvious because (28.4) directly translates into axiom (28.6) with μ(X, Y ) := ∇X (Y ) and replacing the bracket using axiom (28.3). Equation (28.3) expresses the fact that the Lie product on g is the commutator of the Vinberg product.

28.1.4

Classical examples

Let us ﬁrst recall some classical examples of nilpotent Lie algebras with an aﬃne structure. • As proved in [3], nilpotent Lie algebras of dimension ≤ 7 admit an aﬃne structure. • Every 2-step, 3-step, and some classes of 4-step nilpotent Lie algebras can be endowed with an aﬃne structure [16], [4]. • Symplectic Lie algebras: Let θ be a symplectic form on g, that is, a 2-cocycle θ ∈ Z 2 (g) of maximal rank. Then the operator ∇X deﬁned by θ(adX(Y ), Z) = −θ(Y, ∇X Z)

(28.7)

gives an aﬃne structure. • Lie algebras with a regular derivation: A Lie algebra g having a regular derivation is necessarily nilpotent, and in this case there is a regular diagonalizable derivation. Let f be such a derivation. For all X ∈ g, the operator ∇X = f −1 ◦ adX ◦ f

(28.8)

deﬁnes an aﬃne structure on g [16].

28.2

Vinberg Algebras Associated with Graded Filiform Complex Lie Algebras

In this section we show that any graded ﬁliform complex Lie algebra has an aﬃne structure.

350

28.2.1

Elisabeth Remm

Filiform Lie algebras, characteristically nilpotent ﬁliform Lie algebras

i i+1 Let us recall that the ideals C k g are deﬁned by C 0 g = g and g for i ≥ 1. C g =k C g, The nilindex of a nilpotent Lie algebra is the integer k = min i ∈ N : C g = 0 . An n-dimensional nilpotent Lie algebra g is called ﬁliform if its

DEFINITION 28.5 nilindex equals n − 1.

In this case the descending central sequence has the following form: g ⊃ C 1 g ⊃ · · · ⊃ C n−2 g ⊃ C n−1 g = {0} with

Example 28.1 by

dim C 1 g = n − 2, dim C i g = n − i − 1, for i = 1, ..., n − 1.

The n-dimensional nilpotent Lie algebra Ln = Span(X1 , ...Xn ), deﬁned [X1 , Xi ] = Xi+1 for i ∈ {2, ..., n − 1} ,

(28.9)

with all remaining brackets trivial, is ﬁliform. As stated in [7], any ﬁliform Lie algebra is a linear deformation of Ln . The nilpotent Lie algebra Qn = Span(X1 , ..., Xn ) with n = 2p and with bracket

[X1 , Xj ] = X1+j , j = 2, ..., n − 1 [Xi , Xn−i+1 ] = (−1)i+1 Xn , i = 2, ..., p

(28.10)

is also ﬁliform. DEFINITION 28.6 A nilpotent Lie algebra is characteristically nilpotent if all its derivations are nilpotent. REMARK 28.1 A noncharacteristically nilpotent complex Lie algebra has a nonnilpotent derivation whose semisimple part is also a derivation. Thus a noncharacteristically nilpotent complex Lie algebra is a nilpotent complex Lie algebra that has a nontrivial diagonal derivation. Therefore the noncharacteristic nilpotent Lie algebras are nontrivially graded and conversely any nontrivially graded nilpotent complex Lie algebra is noncharacteristically nilpotent. DEFINITION 28.7 An abelian subalgebra of the algebra of derivations Der(g) consisting of semisimple derivations is called an exterior torus of derivations of g. All maximal tori of derivations of g are conjugated in Der(g). The dimension of a maximal torus of derivations of g is the rank of g. For a ﬁliform Lie algebra g, the rank of g is less than or equal to 2. If a ﬁliform Lie algebra is characteristically nilpotent, its rank is zero.

Vinberg algebras associated to some nilpotent Lie algebras

28.2.2

351

Aﬃne structures on ﬁliform complex Lie algebras

In this paragraph, we prove that any complex noncharacteristically nilpotent and ﬁliform Lie algebra can be equipped with an aﬃne structure. For this, we will use the following theorem. THEOREM 28.1 Let g be a nilpotent Lie algebra with a derivation f whose restriction to the derived subalgebra C 1 g = [g, g] is nonsingular (that is invertible as a linear map). Then g admits an aﬃne structure. PROOF Let f be the restriction of f to C 1 g. Let g be the inverse of f. We can extend g to an endomorphism g of g such that C 1 g is an invariant subspace of g. Let us consider ∇ deﬁned by ∇X Y = g ◦ adX ◦ f (Y ). It satisﬁes ∇X (Y ) − ∇Y (X) = g ◦ adX ◦ f (Y ) − g ◦ adY ◦ f (X) = g(f [X, Y ]) because f is a derivation. As g and f are inverse to each other on the derived algebra, we see that ∇X (Y ) − ∇Y (X) = [X, Y ]. Likewise ∇X ∇Y (Z) − ∇Y ∇X (Z) = g[X, [Y, f (Z)]] − g[Y, [X, f (Z)]] = −g[f (Z), [X, Y ]] = ∇[X,Y ] (Z).

Example 28.2 Let g be a n-dimensional nilpotent Lie algebra. If its rank is nonzero and if 0 is a root then the associated root space has a trivial intersection with the derived subalgebra. Therefore such algebras admit aﬃne structures. Let us consider now the case of ﬁliform Lie algebras: PROPOSITION 28.2 ([7]) Every n-dimensional complex ﬁliform Lie algebra of rank 2 is isomorphic to Ln or Qn deﬁned in Example 28.1. Every n-dimensional complex ﬁliform Lie algebra of rank 1 is isomorphic to one of the following Lie algebras : , 2 ≤ k ≤ n − 3 and the bracket i) Akn (λ1 , ..., λt−1 ) := Span {X1 , ..., Xn } with t = n−k+1 2 deﬁned by: ⎧ ⎨ [X1 , Xi ] = Xi+1 , i = 2, ..., n − 1 [Xi , Xi+1 ] = λi−1 X2i+k−1 , 2 ≤ i ≤ t ⎩ [Xi , Xj ] = aij Xi+j+k−2 , 2 ≤ i ≤ j et i + j + k − 2 ≤ n k ii) Bn (λ1 , ..., λt−1 ) := Span {X1 , ..., Xn } with n = 2m, t = n−k , 2 ≤ k ≤ n − 3 and 2 ⎧ [X1 , Xi ] = Xi+1 i = 2, ..., n − 2 ⎪ ⎪ ⎨ i+1 Xn , i = 2, ..., n − 1 [Xi , Xn−i+1 ] = (−1) ⎪ , X ] = λ X , i = 2, ..., t [X i i+1 i−1 2i+k−1 ⎪ ⎩ [Xi , Xj ] = aij Xi+j−k−2 , 2 ≤ i, j ≤ n − 2 et i + j + k − 2 ≤ n − 2 , j = i + 1

352

Elisabeth Remm

iii) Cn (λ1 , ..., λt ) := Span {X1 , ..., Xn } , n = 2m + 2, t = m − 1 and ⎧ ⎨ [X1 , Xi ] = Xi+1 i = 2, ..., n − 2 i−1 [Xi , Xn−i+1 ] = (−1) Xn , i = 2, ...m + 1 ⎩ i+1 λk Xn , i = 2, ..., n − 2 − 2k and k = 1, ..., m − 1 [Xi , Xn−i−2k+1 ] = (−1) with remaining brackets trivial, [−] denotes the integer part function, (λ1 , .., λt ) are parameters (none all simultaneously zero) satisfying polynomial relations implied by the Jacobi identity, and the constants aij satisfy the equations aij = ai,j+1 + ai+1,j , ai,i+1 = λi . The Lie algebras Ln , Qn , Akn (λ1 , ..., λt−1 ) and Bnk (λ1 , ..., λt−1 ) admit nonsingular derivations. Thus they are equipped with aﬃne structures (constructed in (28.8)). THEOREM 28.2

Every noncharacteristically nilpotent ﬁliform Lie algebra is aﬃne.

PROOF It remains to be proved that Lie algebras of type Cn (λ1 , ..., λt ) are aﬃne. For an algebra of this type, every diagonalizable derivation is a scalar multiple of the derivation f given by f (Y1 ) = 0, f (Yi ) = Yi , i = 2, ..., n − 1, f (Yn ) = 2Yn . Such a derivation is singular, but its restriction to the derived subalgebra is nonsingular. From Theorem 28.1, there exists an aﬃne structure on Cn (λ1 , ..., λt ) . REMARK 28.2 There exist characteristically nilpotent Lie algebras (ﬁliform or not) with aﬃne structures. Let us consider for example the 7-dimensional Lie algebra spanned by {X1 , ..., X7 } with bracket deﬁned by [X1 , Xi ] = Xi+1 , 2≤i≤5 [X2 , X5 ] = −X6 [X2 , X7 ] = −X5 − X6 [X3 , X4 ] = X6 [X3 , X7 ] = −X6 . Then Der(g) is of dimension 10 and it is isomorphic to the Lie algebra spanned by {Z1 , ..., Z10 } with the product given by [Z1 , Z2 ] = Z3 [Z1 , Z3 ] = Z4 [Z1 , Z4 ] = Z5 [Z1 , Z7 ] = −Z4 [Z1 , Z8 ] = −Z6

[Z2 , Z6 ] = −Z5 [Z2 , Z8 ] = −Z6 [Z2 , Z9 ] = −Z4 − 2Z6 [Z2 , Z10 ] = −Z5

[Z3 , Z8 ] = −Z5 [Z3 , Z9 ] = −Z5 [Z7 , Z8 ] = 2Z5 − 2Z6 + 2Z10 [Z7 , Z9 ] = Z5 − 2Z6 + 2Z10 [Z8 , Z9 ] = 2Z6 − 2Z10 .

The algebra Der(g) is nilpotent, so g is characteristically nilpotent. It is easy to see that g has an aﬃne structure. Notice that Der(g) itself is not characteristically nilpotent.

28.2.3 28.2.3.1

Noncomplete aﬃne structures on ﬁliform Lie algebras Faithful representations associated with an aﬃne connection

Let ∇ be an aﬃne structure on an n-dimensional K-Lie algebra g with K = R or C. Let us consider the (n + 1)-dimensional linear representation ρ : g → End(g ⊕ K) given by ρ(X)(Y, t) = (∇(X, Y ) + tX, 0) .

(28.11)

Vinberg algebras associated to some nilpotent Lie algebras

353

It is easy to verify that ρ is faithful. DEFINITION 28.8 A representation ρ is nilpotent if the endomorphism ρ(X) is nilpotent for every X in g. PROPOSITION 28.3 Suppose that g is a nonabelian complex nilpotent indecomposable Lie algebra and let ρ be a faithful representation of g on a vector space M . Then there exists a faithful nilpotent representation of the same dimension. PROOF Let us consider the g-module M associated to ρ. As g is nilpotent, M can be decomposed as M=

k

Mλi

(28.12)

i=1

where Mλi is a g-submodule, and λi ’s are linear forms on g. For all X ∈ g, the restriction ρ |Mi (X) has the following form: ⎞ ⎛ λi (X) ∗ · · · ∗ ⎜ .. ⎟ ⎜ 0 ... ... . ⎟ ⎟. ⎜ ρ |Mi (X) = ⎜ . ⎟ . . .. .. ∗ ⎠ ⎝ .. 0 · · · 0 λi (X) Let Cλi = Ca be the one dimensional g-module deﬁned by μ: g → End(Cλi ) X → ρ(X) · a = λi (X)a. Let us equip the tensor product Mλi ⊗ C−λi with the g-module structure given by the formula X · (Y ⊗ a) = ρ |Mi (X) (Y ) ⊗ a − Y ⊗ (λi (X)a). ; C= Then M (Mλi ⊗ C−λi ) is a nilpotent g-module. We denote by ρ the representation C associated to M . It is faithful if and only if ρ(X) = 0 for every 0 = X ∈ Z(g). Let X = 0 be in Z(g) such that ρ(X) = 0. This implies that ρ(X) is a diagonal endomorphism. By hypothesis Z(g) ⊂ C 1 (g). Then ∃ (Yi , Zi ) ∈ (g, g) such that [Yi , Zi ] = X. i

Then ρ(X) = i [ρ(Yi ), ρ(Zi )] is nilpotent and the eigenvalues of ρ(X) are all 0. Therefore ρ(X) = 0 and ρ is not faithful. Then ρ(X) = 0 and ρ is a faithful representation. 28.2.3.2

Nilpotent aﬃne structures on ﬁliform Lie algebras

Let g be a complex ﬁliform aﬃne Lie algebra of dimension n ≥ 3. Let ρ be the (n + 1)dimensional faithful representation of g constructed in (28.11). Let M = g ⊕ C be the corresponding complex g-module. As g is ﬁliform, decomposition (28.12) has one of the following forms [12]: 1) M = M0 and M is indecomposable, 2) M = M0 ⊕ Mλ , λ = 0.

354 THEOREM 28.3

Elisabeth Remm For n ≥ 3, there exist nonnilpotent faithful Ln -modules.

Proof. i) Let us suppose n ≥ 4. Let {X1 , ..., Xn } be the canonical basis of Ln given in Example 28.1 and {e1 , ..., en , en+1 } the basis of the (n + 1)-dimensional module M given by ei = (Xi , 0) and en+1 = (0, 1). Let us consider the representation ρ, depending on some scalars a, α and β, deﬁned by ⎛

a a 0 ··· ···

⎜ ⎜a ⎜ ⎜0 ⎜ ⎜ .. ⎜. ⎜ ⎜. ρ(X1 ) = ⎜ .. ⎜ ⎜ .. ⎜. ⎜ ⎜ ⎜0 ⎜ ⎝α

a 0 0 0 .. 1 .. . 2 . .. .. .. . . . . . i−3 . . . i−2 . .. .. . . 0 β 0 ··· ··· 0 0 0 0 0 0 0

⎛

a a 0 ··· ···

⎜ ⎜ a ⎜ ⎜ −1 ⎜ ⎜ ⎜ 0 ⎜ ⎜ . ρ(X2 ) = ⎜ .. ⎜ ⎜ .. ⎜ . ⎜ ⎜ ⎜ 0 ⎜ ⎝ β

a 0 1 0 1 2

..

. .. .. .. . . . .. 1 . i−2 .. . 0 α 0 ··· ··· 0 0 0 0 0 0

.. ..

.

. ··· 0

while for j ≥ 3, ⎧ 1 ρ(Xj )(e1 ) = − j−1 ej+1 if j + 1 ≤ n; ⎪ ⎪ ⎪ 1 ⎪ )(e ) = e if j + 1 ≤ n; ρ(X ⎪ j 2 j+1 ⎪ j−1 ⎪ ⎪ 1 ⎪ )(e ) = e ρ(X j 3 ⎨ j(j−1) j+2 if j + 2 ≤ n; ··· ⎪ ⎪ ⎪ ρ(Xj )(ei−j+1 ) = (j−2)!(i−j−1)! ei , ⎪ (i−2)! ⎪ ⎪ ⎪ ⎪ ρ(Xj )(ei−j+1 ) = 0, i = n + 1, .., n + j − 1. ⎪ ⎩ ρ(Xj )(en+1 ) = ej .

⎞ 01 .. ⎟ . 0⎟ ⎟ 0 0⎟ ⎟ .. ⎟ . 0⎟ ⎟ .. ⎟ . 0⎟ ⎟ .. ⎟ . 0⎟ ⎟ . . .. ⎟ . . 0⎟ ⎟ n−3 ⎠ n−2 0 0 0 00 ⎞ ··· 0 0 .. ⎟ . 1⎟ ⎟ 0 0⎟ ⎟ .. ⎟ . 0⎟ ⎟ .. ⎟ . 0⎟ ⎟ .. ⎟ . 0⎟ ⎟ . . .. ⎟ . . 0⎟ ⎟ 1 ⎠ n−2 0 0 0 00

ρ(Xj )(e1 ) = 0 otherwise. ρ(Xj )(e2 ) = 0 otherwise. ρ(Xj )(e3 ) = 0 otherwise. i = j + 2, ..., n.

We easily verify that this describes a faithful representation which is nonnilpotent if a = 0. ii) If n = 3, then L3 is the 3-dimensional Heisenberg algebra. We consider it in detail in Remark 28.3. 3. Associated noncomplete aﬃne structure The representation of Theorem 28.2 is associated to an aﬃne structure on the ﬁliform Lie algebra Ln given by ∇Xi = ρ(Xi ) |Ln ,

Vinberg algebras associated to some nilpotent Lie algebras

355

where Ln is interpreted in the obvious way as a subset of M = Ln ⊕ C. This structure is complete if and only if the endomorphisms RX deﬁned by RX : g → g Y → ∇Y (X) are nilpotent for all X ∈ g [11]. ⎛ a ⎜ ⎜a ⎜ ⎜ ⎜0 ⎜ ⎜0 ⎜ ⎜. ⎜ .. ⎜ ⎜ ⎜ ⎜0 ⎜ ⎜ ⎝0 α

But the matrix of RX1 equals ⎞ a 0 ··· 0 ··· 0 0 ⎟ .. .. a . . 0⎟ ⎟ ⎟ .. .. −1 . . 0⎟ ⎟ 0 − 21 · · · 0 · · · 0 1 ⎟ ⎟ ⎟ .. .. .. . . 0 ··· . 0⎟ ⎟ ⎟ .. .. . . ⎟ 1 . − j−1 . 0⎟ 0 . ⎟ . ⎟ .. .. . . 0 0⎠ 0 .. · · · 1 β 0 0 0 0 − n−2 0

Its trace is 2a, therefore for a = 0, it is not nilpotent. We have proved the following. PROPOSITION 28.4 There exist noncomplete aﬃne structure on the ﬁliform Lie algebra Ln , n ≥ 3, which are noncomplete. REMARK 28.3 The algebra L3 coincides with the Heisenberg algebra. The nonnilpotent faithful representation associated to the noncomplete aﬃne structure is given by ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ a a0 a a 0 000 a 0 ⎠ , ∇X3 = ⎝ 0 0 0 ⎠ . ∇X1 = ⎝ a a 0 ⎠ , ∇X2 = ⎝ a αβ0 β−1 α+1 0 000 The corresponding aﬃne representation is ⎛ a(x1 + x2 ) ⎜ a(x1 + x2 ) ⎜ ⎝ αx1 + (β − 1)x2 0

28.3

a(x1 + x2 ) a(x1 + x2 ) βx1 + (α + 1)x2 0

0 0 0 0

⎞ x1 x2 ⎟ ⎟. x3 ⎠ 0

Nilpotent Lie Algebras with a Contact Form

In this section we consider K-Lie algebras with K = R or C.

28.3.1

Nilpotent contact Lie algebras

DEFINITION 28.9 Let g be a (2p + 1)-dimensional algebra. A contact form on g is a linear form ω on g∗ such that ω ∧ (dw)p = 0. In this case, (g, ω) or simply g, is called a contact Lie algebra.

356

Elisabeth Remm

PROPOSITION 28.5 dimensional.

([6]) The center of a contact nilpotent Lie algebra is one-

PROOF Let (g, ω) be a (2p + 1)-dimensional contact nilpotent Lie algebra. As g is a nilpotent Lie algebra, the center Z(g) is not trivial. For every X ∈ Z(g) we have dω(X, Y ) = −ω [X, Y ] = 0, ∀Y ∈ g. Let us suppose that dim Z(g) > 1. As codim kerω = 1, we have Z(g) ∩ kerω = 0. Let X = 0 be in Z(g) ∩ kerω. This vector satisﬁes ω(X) = 0 and X dω = 0, where denotes the inner product. The vector X belongs to the characteristic subspace and ω ∧ dω p = 0. Thus Z(g) ∩ kerω = 0, which proves that dim Z(g) ≤ 1. COROLLARY 28.1 plectic Lie algebra.

Let g be a contact nilpotent Lie algebra. Then g/Z(g) is a sym-

Corollary 28.1 implies that any contact Lie algebra is a one-dimensional central extension of a symplectic Lie algebra: 0 → V → (g2p+1 , ω) → (g2p , θ) → 0, where V is a one-dimensional K-vector space. Since any symplectic nilpotent Lie algebra admits an aﬃne structure, we have the following corollary. COROLLARY 28.2 Any nilpotent contact Lie algebra is a one-dimensional central extension of an aﬃne Lie algebra.

28.3.2

Aﬃne structures on nilpotent contact Lie algebras

Let g be a (2p+1)-dimensional nilpotent Lie algebra equipped with a contact form ω. Let g be the symplectic algebra g = g/Z( g) and consider a symplectic form θω on g satisfying π ∗ (θω ) = dω. Let Φ be the set of all 2-cocycles of g of maximal rank. We have that θω ∈ Φ. Under the identiﬁcation g = g⊕V, the bracket of g is given by

[(X, α) , (Y, λ)]g = [X, Y ]g , θω (X, Y ) . Let {e1 , ..., en , en+1 } be a basis of g adapted to the previous decomposition, that is {e1 , ..., en } g are parameterized by is a basis of g and en+1 a basis element of V . Therefore elements of pairs (X, α) := X + αen+1 with X ∈ g and α ∈ K. Let θ ∈ Φ be an arbitrary 2-cocycle of maximal rank on g, and ∇θ the aﬃne structure induced on g by this symplectic form as deﬁned in (28.7). The aim is to use ∇θ to construct an aﬃne structure on g when possible. For this, let us consider an extension of ∇θ : g⊗g → g C θ g⊗ g→ g satisfying into an operator ∇ : " Cθ ((X, 0) , (Y, 0)) = ∇θ (X, Y ) , ϕ (X, Y ) ∇ (28.13) Cθ ((0, λ) , (X, 0)) , Cθ ((X, 0) , (0, λ)) = ∇ ∇ where ϕ is a bilinear map on g such that ϕ (X, Y ) − ϕ (Y, X) = θω (X, Y ) .

(28.14)

Vinberg algebras associated to some nilpotent Lie algebras LEMMA 28.1

357

Cθ satisﬁes the following identity: The operator ∇

Cθ ((X, α) , (Y, λ)) − ∇ Cθ ((Y, λ) , (X, α)) = [(X, α) , (Y, λ)] ∇ g for all X, Y ∈ g and α, λ ∈ K. PROOF

We have for all X, Y ∈ g and α, λ ∈ K:

Cθ ((X, α) , (Y, λ)) − ∇ Cθ ((Y, λ) , (X, α)) ∇ Cθ (X, en+1 ) + α∇ Cθ (en+1 , Y ) = ∇θ (X, Y ) , ϕ (X, Y ) + λ∇ Cθ (en+1 , en+1 ) − ∇θ (Y, X) , ϕ (Y, X) +αλ∇ Cθ (en+1 , X) − λα∇ Cθ (en+1 , en+1 ) Cθ (Y, en+1 ) − λ∇ −α∇

= [X, Y ]g , θω (X, Y ) .

[(X, α) , (Y, λ)]g = [X, Y ]g , θω (X, Y ) ,

But which implies that

Cθ ((X, α) , (Y, λ)) − ∇ Cθ ((Y, λ) , (X, α)) = [(X, α) , (Y, λ)] . (∇ g

Cθ is associated to a torsion-free linear connection Lemma 28.1 means that the operator ∇ on g. C is another bilinear map on C , with π the We notice that if ∇ g such that π ∗ ∇θ = ∇ projection of g on g, the vanishing of the torsion of the linear connection associated with C implies that ∇ C also satisﬁes conditions (28.13). This justiﬁes their choice. We are going ∇ Cθ vanishes. to study which conditions guarantee that the curvature of ∇ ⊗3 g as follows: To this end, we introduce the map C : g →

Cθ ((Y, λ) , (Z, ρ)) Cθ (X, α) , ∇ C((X, α) , (Y, λ) , (Z, ρ)) =∇

Cθ ((X, α) , (Z, ρ)) − ∇ Cθ [(X, α) , (Y, λ)] , (Z, ρ) . Cθ (Y, λ) , ∇ −∇ g It follows from the deﬁnitions that C((X, α) , (Y, λ) , (Z, ρ)) = Cθ ((X, α) , ∇θ (Y, Z) , ϕ (Y, Z) + ρ∇ Cθ (Y, en+1 ) + λ∇ Cθ (en+1 , Z) + λρ∇ Cθ (en+1 , en+1 )) ∇ Cθ (X, en+1 ) + α∇ Cθ (en+1 , Z) + αρ∇ Cθ (en+1 , en+1 )) Cθ ((Y, λ) , ∇θ (X, Z) , ϕ (X, Z) + ρ∇ −∇ Cθ (([X, Y ] , θω (X, Y )), (Z, ρ)), −∇ g

for all X, Y, Z ∈ g and α, λ, ρ ∈ K. The next lemma follows immediately from this equation. Cθ satisﬁes for every X, Y, Z ∈ g: The operator ∇ : C(X, Y, Z) = ϕ(X, ∇θ (Y, Z)) − ϕ(Y, ∇θ (X, Z)) − ϕ([X, Y ]g , Z) en+1

LEMMA 28.2

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Elisabeth Remm

Cθ (Y, en+1 ) − θω (X, Y )∇ Cθ (Z, en+1 ), Cθ (X, en+1 ) − ϕ(X, Z)∇ +ϕ(Y, Z)∇

Cθ (Y, en+1 ) − ∇ Cθ X, ∇ Cθ ∇θ (X, Y ) , en+1 C(X, en+1 , Y ) = ∇ Cθ (en+1 , en+1 ) , −ϕ (X, Y ) ∇

Cθ en+1 , ∇ Cθ (Y, en+1 ) − ∇ Cθ Y, ∇ Cθ (en+1 , en+1 ) . C(en+1 , Y, en+1 ) = ∇ (Recall that X ∈ g is identiﬁed with (X, 0) ∈ g)

LEMMA 28.3 C(X, Y, en+1 ) = 0. PROOF

Let X and Y be in g. If C(X, en+1 , Y ) = C(Y, en+1 , X) = 0 then

The proof is based on a direct veriﬁcation using the formulas of Lemma 28.2.

Cθ (X, ∇ Cθ (Y, en+1 )) − ∇ Cθ (Y, ∇ Cθ (X, en+1 )) C(X, Y, en+1 ) = ∇ Cθ ([X, Y ] + θω (X, Y )en+1 , en+1 ) −∇ Cθ (∇θ (X, Y ), en+1 ) + ϕ(X, Y )∇ Cθ (en+1 , en+1 ) − ∇ Cθ (∇θ (Y, X), en+1 ) =∇ Cθ (en+1 , en+1 ) − ∇ Cθ ([X, Y ] + θω (X, Y )en+1 , en+1 ) −ϕ(Y, X)∇ Cθ ([X, Y ] , en+1 ) + θω (X, Y )∇ Cθ (en+1 , en+1 ) − ∇ Cθ ([X, Y ] , en+1 ) =∇ Cθ (en+1 , en+1 ) −θω (X, Y )∇ = 0.

Let us give some necessary conditions for the map C to vanish. Let π be the canonical projection of g on g, that is, π(X, α) = X. We introduce the vector VX of g deﬁned by Cθ (X, en+1 )). VX := π(∇ LEMMA 28.4 PROOF

Suppose that dimg ≥ 4. If C = 0 then VX = 0 for all X ∈ g.

If C vanishes, the ﬁrst equation of Lemma (28.2) implies ϕ (Y, Z) VX − ϕ (X, Z) VY − θω (X, Y ) VZ = 0

(28.15)

for all X, Y, Z ∈ g. Suppose that dimg ≥ 4. For any Z ∈ g, the codimension of the kernel of the linear form ϕ(., Z) is at most 1, so it is not isotropic for θω . Thus there are X, Y ∈ g such that ϕ(X, Z) = 0 = ϕ(Y, Z) but θω (X, Y ) = 0. Hence Equation (28.15) gives VZ = 0. PROPOSITION 28.6 Let g be a contact nilpotent Lie algebra of dimension greater g/Z( g) than or equal to 5. If the aﬃne structure ∇θ deﬁned by a symplectic cocycle θ on g = Cθ on extends to an aﬃne structure ∇ g, then Cθ (X, T )) = 0 π(∇

Vinberg algebras associated to some nilpotent Lie algebras

359

for all X∈ g and T ∈ Z( g). Cθ (X, en+1 ) = aX en+1 . The equality PROOF For all vectors X in g, VX = 0 and ∇ C(X, Y, Z) = 0 implies that

ϕ X, ∇θ (Y, Z) − ϕ Y, ∇θ (X, Z) − ϕ [X, Y ]g , Z = −aX ϕ (Y, Z) + aY ϕ (X, Z) + aZ θω (X, Y ) . Similarly C(X, en+1 , Y )) = 0 implies that Cθ Cθ (X, aY en+1 ) − a θ ∇ ∇ (X,Y ) en+1 − ϕ (X, Y ) ∇ (en+1 , en+1 ) = 0. This gives the following equation: Cθ (en+1 , en+1 ) = (aY aX − a θ ϕ (X, Y ) ∇ ∇ (X,Y ) )en+1 and the equation, Cθ (en+1 , en+1 ) = (aY aX − a θ ϕ (Y, X) ∇ ∇ (Y,X) )en+1 , obtained by permuting the vectors X and Y . We combine these two equations to Cθ (en+1 , en+1 ) = (a θ θω (X, Y )∇ ∇ (Y,X) − a∇θ (X,Y ) )en+1 = a[X,Y ] en+1 . Cθ (en+1 , en+1 ) = ρen+1 , with ρ ∈ K, and This shows in particular that ∇ ρθω (X, Y ) = a[X,Y ] .

(28.16)

Finally, C(en+1 , Y, en+1 ) = 0 implies

Cθ (en+1 , en+1 ) Cθ Y, ∇ Cθ (en+1 , en+1 ) = ∇ aY ∇ thus

Cθ (en+1 , en+1 ) = ρ aY en+1 . aY ∇

This last equation is always satisﬁed (see (28.16)). Let us suppose ρ = 0. In this case θω (X, Y ) = 0 implies a[X,Y ] = 0. Let us consider X in Z(g). As θω is of maximal rank, there exists Y such that θω (X, Y ) = 0. Since [X, Y ] = 0 implies a[X,Y ] = 0, this leads to a contradiction. Therefore ρ = 0. Cθ (en+1 , en+1 ) = 0. Then a Conclusion. As ρ = 0, ∇ [X,Y ] = 0 and the map α : g →K deﬁned by α(X) = aX is a one-dimensional linear representation of g. We obtain the following. THEOREM 28.4 Let ( g, ω) be a nilpotent contact Lie algebra with dim g ≥ 5 and ∇θ the aﬃne structure on the symplectic algebra g/Z( g) associated to a symplectic form Cθ be the operator induced by ∇θ on θ (θ = θω in general). Let ∇ g and α : g →K the corresponding one-dimensional linear representation of g. Cθ is an aﬃne structure if and only if If α is trivial, ∇

360

Elisabeth Remm

Cθ (U, en+1 ) = 0 for all U ∈ g, and 1) ∇

θ 2) ϕ satisﬁes ϕ X, ∇ (Y, Z) − ϕ Y, ∇θ (X, Z) − ϕ [X, Y ]g , Z = 0, i.e., if ϕ is a 2-cocycle of the cochain complex deﬁning the cohomology of the Vinberg algebra associated with ∇θ with values in a trivial module. Cθ is an aﬃne structure if and only if If α is nontrivial, ∇ Cθ (en+1 , en+1 ) = 0 and ∇ Cθ (X, en+1 ) = 0 for all X ∈ Kerα, and 1) ∇

θ 2) ϕ X, ∇ (Y, Z) − ϕ Y, ∇θ (X, Z) − ϕ [X, Y ]g , Z = α(Z)θω (X, Y ) for all X, Y ∈ Ker(α).

28.4

Application: Aﬃne Structures on 7-Dimensional Characteristically Nilpotent Contact Lie Algebras

We are going to use the notations of the list of 7-dimensional nilpotent Lie algebras given in [8]. As proved in Section 28.2, every nontrivially graded 7-dimensional nilpotent Lie algebra admits an aﬃne structure. We apply the construction of an aﬃne structure on a contact Lie algebra given in Section 28.3 to describe such a structure for any 7dimensional characteristically nilpotent contact Lie algebra. From the classiﬁcation [8], these Lie algebras are isomorphic to η74 , η712 (λ) (where λ is a scalar parameter), η714 , η721 , and η728 (Classiﬁcation [8] also contains algebra η719 , but this one turns out to be isomorphic to η712 (0).) For each of these algebras we describe the symplectic form on the quotient used in the construction of the aﬃne structure and we compute the corresponding aﬃne structure.

28.4.1

Algebra η74

This algebra is equipped with the aﬃne structure constructed from the symplectic 6dimensional algebra g/K{X7 } with the symplectic chosen form: θ = 9(ω1 ∧ ω6 + ω2 ∧ ω4 ) + 8(ω2 ∧ ω5 − ω3 ∧ ω4 ). The aﬃne structure is deﬁned by ⎛ ⎛ ⎞ 0 0 0 0000 0 00 ⎜ −9 0 0 0 0 0 0⎟ ⎜ 0 00 8 ⎜ 81 ⎜ ⎟ ⎜− ⎜ 0 00 ⎟ ⎜ 64 19 0 0 0 0 0 ⎟ ⎜ ⎟ , ∇X2 = ⎜ − 9 0 0 0 − 1 0 0 0 0 ∇X1 = ⎜ 8 ⎜ ⎜ 818 ⎟ ⎜ 0 81 − 9 1 0 0 0 ⎟ ⎜ 64 8 ⎜ ⎜ 64 0 0 ⎟ ⎝ 0 0 0 0 0 0 0⎠ ⎝ 0 00 a2 0 0 a1 a2 0 0 0 18 0

0 0 0 0 0 − 89 3 8

0 0 0 0 0 0 1 3

⎞ 00 0 0⎟ ⎟ 0 0⎟ ⎟ 0 0⎟ ⎟. 0 0⎟ ⎟ 0 0⎠ 00

The values of ∇Xi for i = 2, .., 7, can be obtained from the relations ∇Xi = [∇X1 , ∇Xi−1 ].

28.4.2

Algebra η712 (λ)

The center of this algebra is spanned by X6 . We need to distinguish two cases. For λ = 0, we use the symplectic form: θ = λ(ω1 ∧ ω7 + ω2 ∧ ω4 ) + 4(ω3 ∧ ω4 − ω2 ∧ ω5 ).

Vinberg algebras associated to some nilpotent Lie algebras This algebra ⎛ 0 ⎜ 0 ⎜ ⎜ 0 ⎜ 0 ∇X1 = ⎜ ⎜ ⎜ 0 ⎜ ⎝ 0 λ2 8

361

is equipped with the following aﬃne structure: ⎛ ⎞ ⎞ 0 0 0 000 0 λ4 0 0 0 0 0 ⎜ 0 0 0 0 0 0 0⎟ 0 0 0 0 0 0⎟ ⎜ ⎟ ⎟ ⎜ 0 0 0 0 0 0 0⎟ 1 0 0 0 0 0⎟ ⎜ ⎟ ⎟ λ λ ⎟ ⎟ ∇X2 = ⎜ 42 1 0 0 0 0 ⎟ , 42 −1 0 0 0 0 0 ⎟ , ⎜ λ λ ⎜ λ −λ 0 0 0 0 0⎟ ⎟ 1 0 0 0 ⎜ 16 4 ⎟ ⎟ 16 4 ⎝ 0 0 0 4 0 0 0⎠ 0 0 0 0 0 0⎠ a2

λ2 λ 8 4

λ

a2 b2 0 0 0

100

λ 4

0

∇Xi = [∇X1 , ∇Xi−1 ], i = 3, 4, 5, 6, and ∇X7 = [∇X2 , ∇X3 ] − ∇X5 . For λ = 0, we consider the symplectic form: √ θ = 4ω1 ∧ ω4 + 2 5(ω1 ∧ ω5 + ω2 ∧ ω4 + ω3 ∧ ω7 ) + ω3 ∧ ω4 − ω2 ∧ ω5 . It induces the aﬃne structure ⎛ √ 1 − 125 24 0 0 0 ⎜ 5 √5 0 0 ⎜ − 6 12 0 ⎜ 0 0 0√ 0 ⎜ 0 ⎜ 0 0 − 125 0 ∇X1 = ⎜ ⎜ 0 1 ⎜ 0 0 0 0 ⎜ 6 ⎜ √2 1 1 ⎝ 5 1 2√5 24 0 − 192 35 0

⎛

∇X2

1 24 √ 5 12

− 481√5

⎜ 1 − 24 ⎜ ⎜ −1 √ 1 ⎜ 2 5 ⎜ 0 =⎜ ⎜ 0 ⎜ 0 0 ⎜ ⎜ 1 − √ 1 ⎝ 2 5 0

√

0 − 21055

0

5 21

given by ⎞ 0 0 ⎟ 0 0⎟ ⎟ 0 0⎟ ⎟ 5 − 0⎟ √6 ⎟, 5 0⎟ ⎟ √3 ⎟ 5 0 ⎠ 12 0 0

0

0

0

0

0 0

0 0

0 0

0 0

1 0 24 √ 5 0 12 − 481√5 0 √ 34 8√ − 241755 105 21 5

0 0 1

√

5 12 5 6 1 − 24 √ − 4355

0

⎞

0⎟ ⎟ 0⎟ ⎟ ⎟ 0⎟ ⎟, 0⎟ ⎟ 0⎟ ⎠ 0

∇Xi = [∇X1 , ∇Xi−1 ], i = 3, 4, 5, 6, and ∇X7 = [∇X2 , ∇X3 ] − ∇X5 .

28.4.3

Algebra η714

The center of this algebra is K{X6 }. The symplectic form used in the construction of the aﬃne structure on η714 is 1 1 θ = ω1 ∧ ω 5 − ω 1 ∧ ω 7 + ω 2 ∧ ω 4 + ω 3 ∧ ω 4 + ω 3 ∧ ω 7 . 6 6 Then we compute ⎛ 0 0 0 0 ⎜ −6 0 0 0 ⎜ ⎜ 0 0 0 0 ⎜ ∇X1 = ⎜ ⎜ 0 01 01 −6 ⎜ 0 36 6 1 ⎜ ⎝ 0 1 1 6 6 a1 a2 0 23

0 0 0 0 0 0 0 −6 0 1 0 6 −3 12

⎛ ⎞ 0 0 ⎜ 0 0⎟ ⎜ ⎟ ⎜ −1 0⎟ ⎜ ⎟ ⎟ 0 ⎟ , ∇X2 = ⎜ ⎜ 01 ⎜ 0⎟ ⎜ 36 ⎟ ⎝ 1 ⎠ 0 6 a2 0

0 0 0 0

1 36 1 6

0

0 0 0 0 0 0 2 9

0 0 0 0 1 0 − 21

0 0 0 0 0 0 0 0 0 1 0 0 0 − 43

⎞ 0 0⎟ ⎟ 0⎟ ⎟ 0⎟ ⎟, 0⎟ ⎟ 0⎠ 0

362

Elisabeth Remm

∇Xi = [∇X1 , ∇Xi−1 ], i = 3, 4, 5, 6, and ∇X7 = [∇X2 , ∇X3 ] − ∇X4 .

Algebra η721

28.4.4

The center is K{X6 } and the symplectic form that we consider is θ = ω1 ∧ ω5 −

6 5 5 6 ω1 ∧ ω7 + ω2 ∧ ω4 + ω2 ∧ ω5 + ω3 ∧ ω4 . 11 11 11 11

The aﬃne structure is given by ⎛ 0 0 0 0 0 ⎜ 0 0 0 0 0 ⎜ 11 5 ⎜− − 0 0 0 ⎜ 6 56 1 0 0 0 ∇X1 = ⎜ 6 ⎜ ⎜ 0 − 5 −1 − 6 0 6 5 ⎜ ⎝ 0 − 55 − 11 − 11 0 36 6 5 37 19 a1 a2 − 197 72 − 15 30

0 0 0 0 0 0 − 45

⎛ ⎞ 0 0 ⎜ 0 0⎟ ⎜ 11 ⎟ ⎜− 0⎟ ⎜ 56 ⎟ ⎟ 0 ⎟ , ∇X2 = ⎜ ⎜ 65 ⎜− 0⎟ 6 ⎜ 55 ⎟ ⎝− ⎠ 0 36 a2 0

0 0 − 65

5 3 − 35 − 55 18 b2

0 0 0 1 −1 − 11 6 − 251 72

0 0 0 0 0 0 0 0 0 0 −1 0 0 − 16

0 0 0 0 0 0 0

⎞ 0 0⎟ ⎟ 0⎟ ⎟ 0⎟ ⎟. 0⎟ ⎟ 0⎠ 0

The remaining ∇Xi for i = 3, 4, 5, 6 can be obtained from the relation ∇Xi = [∇X1 , ∇Xi−1 ] and ∇X7 = [∇X2 , ∇X3 ] − ∇X4 − ∇X5 .

Algebra η728

28.4.5

This algebra is an extension of the 6-dimensional symplectic algebra g/K{X6 } whose symplectic form is θ = 4(ω1 ∧ ω5 + ω2 ∧ ω4 + ω3 ∧ ω4 + ω3 ∧ ω7 ) − 3ω2 ∧ ω7 . The induced aﬃne structure is ⎛ ⎛ ⎞ 0 0 0 0 0 0 0 0 0 ⎜−4 0 0 0 0 0 0⎟ ⎜ 0 0 7 ⎜ 3 ⎜ ⎟ ⎜− 0 0 0 0 0 0⎟ ⎜ −1 0 ⎜ 7 3 3 4 ⎜ 3 3 ⎟ 4 ⎜ ⎟ ∇X1 = ⎜ 0 7 7 − 7 0 − 7 0 ⎟ , ∇X2 = ⎜ ⎜ 7 7 ⎜ 0 0 0 0 0 0 0⎟ ⎜ 0 0 ⎜ ⎜ 4 4 ⎟ ⎝ 0 4 4 4 0 4 0⎠ ⎝ 7 7 7 7 7 7 2 1 a2 − 71 a1 a2 0 0 7 4 0

0 00 0 0 00 0 0 00 0 3 7 0 0 0 0 10 1 4 7 0 0 0 5 − 71 73 0 28

⎞ 0 0⎟ ⎟ 0⎟ ⎟ 0⎟ ⎟, 0⎟ ⎟ 0⎠ 0

∇Xi = [∇X1 , ∇Xi−1 ], i = 3, 4, 5, 6 and ⎛

∇X7

0 ⎜ 0 ⎜ ⎜ 0 ⎜ 4 =⎜ ⎜−7 ⎜ 0 ⎜ 4 ⎝ 7 1 4

0 0 0 0 0 0

5 28

0 0 0 0 0 0 − 74

⎞ 0000 0 0 0 0⎟ ⎟ 0 0 0 0⎟ ⎟ 0 0 0 0⎟ ⎟. 0 0 0 0⎟ ⎟ 0 0 0 0⎠ 0000

REMARK 28.4 There exists a 7-dimensional nilpotent Lie algebra without semisimple derivation and which is not a contact algebra. Such an algebra is also a central extension of a symplectic 6-dimensional Lie algebra. But the cocycle θ deﬁning the extension is

Vinberg algebras associated to some nilpotent Lie algebras

363

degenerated and the corresponding 7-dimensional Lie algebra has no contact form. We may use the previous construction in this case as well, but the vector VX for X ∈ g need not be trivial.

28.5

Acknowledgments

The author expresses her thanks to Martin Markl for proofreading this chapter and to the scientiﬁc reviewer for many interesting and useful comments.

References [1] Auslander L., The structure of complete locally aﬃne manifolds. Topology 3 (1964) suppl. 1, 131-139. [2] Benoist Y., Une nilvari´et´e non aﬃne. J. Diﬀ. Geom., 41 (1995), 21-52. [3] Burde D., Aﬃne structures on nilmanifolds. Int. J. of Math, 7 (1996), 599-616. [4] Dekimpe K., Hartl M., Aﬃne structures on 4-step nilpotent Lie algebras. J. Pure Appl. Algebra 120 (1997), no. 1, 19-37. [5] Fried D., Goldman W., Three dimensional aﬃne crystallographic groups. Adv. Math., 47 (1983), 1-49. [6] Goze M., Sur la classe des formes et syst`emes invariants a ` gauche sur un groupe de Lie. CRAS Paris A-B 283 (1976), no. 7, A iii, A499-A502. [7] Goze M., Khakimdjanov Y., Nilpotent Lie algebras. Kluwer, 1995. [8] Goze M., Remm E., Classiﬁcation of 7-dimensional nilpotent Lie algebras. Available at http://www.math.uha.fr, (2002). [9] Goze M., Remm E., Non complete aﬃne structures on ﬁliform Lie algebras. Inter. Jour. of Math and Math Sci. (http://ijmms.hindawi.com) Vol 29 (2), 2002, 71-78. [10] Goze M., Remm E., Aﬃne structures on abelian Lie algebras. Linear Algebra and its Applications, 360 (2003), 215-230. [11] Helmstetter J., Radical d’une alg`ebre sym´etrique a ` gauche. Ann. Inst. Fourier, 29 (1979), 17-35. [12] Khakimdjanov Y., Medina A., Groupes de Lie nilpotent a ` structure aﬃne invariante a gauche. Transform. Groups, 6, No. 2, 165-174 (2001). ` [13] Kuiper N., Sur les surfaces localement aﬃnes. Colloque G´eom´etrie diﬀ´erentielle Strasbourg (1953), 79-87.

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[14] Malcev A., Commutative subalgebras of semi-simple Lie algebras. Bull. Acad. Sci. URSS. S´er. Math. [Izvestia Akad. Nauk SSSR] 9 (1945), 291-300. [15] Remm E., Structures aﬃnes sur les alg`ebres de Lie et op´erades Lie-admissibles. Thesis, Universit´e de Haute Alsace, Mulhouse, December 2001. [16] Scheuneman J., Aﬃne structures on three-step nilpotent Lie algebras. Proc. Amer. Math. Soc. 46 (1974), 451-454.

Chapter 29 Algebraic and Diﬀerential Structures in Renormalized Perturbation Quantum Field Theory M. Rosenbaum Instituto de Ciencias Nucleares, Universidad Nacional Aut´ onoma de M´exico, D.F.

29.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29.2 Rooted trees and Feynman graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29.2.1 Toy model of renormalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29.3 The Hopf algebra of rooted trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29.3.1 Toy model from the Hopf algebra point of view . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29.4 Hopf algebra of renormalization in ﬁeld theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29.5 The diﬀerential calculus of renormalization, the Birkhoﬀ algebraic decomposition, and the forest formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29.5.1 The forest formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29.5.2 The Birkhoﬀ algebraic decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29.6 Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

365 367 368 370 373 373 377 379 379 380 380

Abstract Some of the algebraic and diﬀerential geometric structures underlying the process of renormalization in perturbation quantum ﬁeld theory are discussed. It is shown, in particular, that the combinatorics resulting from the perturbative expansion of the transition amplitude and the relation of this expansion to the Hausdorﬀ series leads naturally to consider the Hopf algebra of decorated rooted trees, its relation to the BPHZ Forest formula of renormalization and the geometrical interpretation of the latter in terms of left invariant forms. Key words: Hopf algebras; rooted trees; renormalization in PQFT 2000 MSC: 05C05; 16W30; 57T05; 81T15; 81T75

29.1

Introduction

To date very few calculations can be done in quantum ﬁeld theory without having to resort to perturbations about a coupling parameter of the theory. Perturbative quantum ﬁeld theory leads naturally to Green functions, the terms of which are diagrammatically represented by Feynman graphs. Analytically, Feynman graphs correspond to generally ill-deﬁned integrals that result in a plethora of inﬁnities in the theory. To cure this disease, the inﬁnities need to be subtracted from the theory and the procedures for doing so are

365

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known as renormalization. Not all the theories devised by theoretical physicists and mathematicians are amenable to renormalization. Only those, such as quantum electrodynamics and quantum chromodynamics, for which the coupling parameters are dimensionless are renormalizable. On the other hand general relativity, where the coupling is the dimension-full Newton gravitational constant, has resisted so far all attempts toward renormalization. This means in turn that one of the basic cornerstones of physics: the theory of gravitation, with all its implicit geometrical beauty, cannot be extended to the quantum realm for the investigation of phenomena at distances of the order of the Planck length. If one adheres to the reductionist ideal of a uniﬁed theory of fundamental interactions in physics, then an acceptable way to incorporate gravitation at the quantum level description of physics must be found. There are several ongoing eﬀorts in this direction, such as string theory, nonperturbative QFT, and noncommutative geometry (which may turn out to be more closely related than it appears). An additional right step toward this end can be provided by the understanding of the mathematics underlying the above-mentioned process of renormalization, which is regarded by many physicists as a rather ad hoc procedure that exhibits our present lack of understanding of quantum ﬁeld theory proper. In 1998 Kreimer [1] wrote a very interesting paper where he shows that the mathematical structure behind the process of renormalization is a Hopf algebra, for which the Hopf algebra of rooted trees is the universal object. Shortly after, Connes and Kreimer [2] studied further this Hopf algebra of renormalization HR and established its remarkable relation to the Hopf algebra HCM associated to diﬀeomorphisms of a manifold [3] (for an introductory review of this relation see [4]). Since then there have been a number of papers (see e.g., [5], [6], [7], [8] and references within) that exhibit the mathematical richness of HR and show that renormalization is not such an ad hoc procedure. Rooted trees have been known in mathematics for quite a long time. They appear, for example, quite naturally, in diﬀerential equations, as was established by Cayley [9] in 1857 when considering solutions to the ﬂow of a vector ﬁeld given by d i x (s) = f i (x(s)). ds

(29.1)

Taking higher-order derivatives of (29.1) one gets ∂f i j d2 xi (s) ∂f i dxj = = f ds2 ∂xj ds ∂xj d3 xi (s) ∂ 2 f i j k ∂f i ∂f j k = f f + f . ds3 ∂xj ∂xk ∂xj ∂xk

(29.2) (29.3)

Moreover, using the notation fji1 ...jk := we can write (29.3) as

∂kf i , ∂xj1 . . . ∂xjk

(29.4)

d3 xi (s) i = f k fjk f j + fji fkj f k . (29.5) ds3 The right side of the above equation can be identiﬁed with a sum of rooted trees with decorated vertices as shown in Fig. 29.1. Consequently we have the one-to-one relationship between diﬀerentials and rooted trees: dn xi (s) ⇔ sum of all rooted trees with n vertices, where each summand is weighted by the dsn

Algebraic and diﬀerential structures in renormalized perturbation QFT fji

i fjk

d3 xi (s) = ds3

fj

367

fkj

+ fk

fk

FIGURE 29.1. Diﬀerential equations and rooted trees.

N (1) = N 2 (1) = N 3 (1) =

+

N 4 (1) =

+ 3

+

+

(29.6)

FIGURE 29.2. Natural forest growth operator. Connes-Moscovici weight factors [6]. These factors are generated by the action of the natural forest growth operator N , which produces the sum of trees by attaching a leaf to any one of the vertices of a given tree while preserving the original root, as shown in Fig. 29.2. Rooted trees also appear when considering the Runge-Kutta numerical algorithms for solving ﬂow equations [10], [11], as well as when considering the solution of diﬀerential equations, of the form dY ds = A(s)Y (s), by means of Picard’s method of successive approximations. The method leads to iterated integrals of the form,

b

ω1 . . . ωn = a

b

b

f1 (s) a

(ω2 . . . ωn )ds,

(ωi = fi (s)ds).

(29.7)

a

This multiple integral corresponds to a ladder tree (a tree where all its vertices have fertility one) and each subintegral decorates a vertex of the rooted tree.

29.2

Rooted Trees and Feynman Graphs

So far we have used a fairly intuitive conception of trees, a mathematical more precise deﬁnition of rooted trees is the following:

DEFINITION 29.1 A rooted tree is a graph with a designated vertex called a root, such that there is a unique path from the root to any other vertex in the tree. In Fig. 29.3 we show examples of rooted trees. The important thing to note is that in this mathematical object the root is at the top of the diagram (contrary to what happens with trees in nature!).

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M. Rosenbaum root

6 78 9 leaves

branch

FIGURE 29.3. Rooted tree nomenclature. In general if n is the number of vertices in a given tree then, modulo isomorphisms, there are an exponentially growing ﬁnite number of rooted trees with n-vertices; we show below the elements in the ﬁrst few sets: n

ti

0

1

1 2 3 4 5

, , , , 9 trees with ﬁve vertices each .

(29.8)

As we mentioned in the introduction rooted trees appear naturally when considering diagrammatic expressions of iterated integrals, but iterated integrals appear naturally, in turn, in the Green functions of perturbative quantum ﬁeld theory (pQFT). It is not surprising then to ﬁnd that the algebra of rooted trees should underlay the process of renormalization of these theories. We shall see later on that the algebra of rooted trees is actually a commutative but not a co-commutative Hopf algebra. However, before embarking on these matters let us ﬁrst review brieﬂy what happens with the ultraviolet (high energy) divergences, which permeate pQFT, from the point of view of a very simpliﬁed toy model.

29.2.1

Toy Model of Renormalization

In pQFT the occuring UV divergences can be of 3 types: disjoint, nested, and overlapping. It is well known by practitioners of QFT that the overlapping divergences can be resolved into the disjoint or nested types, which in turn correspond analytically to iterated integrals. Although highly simpliﬁed, the toy model we are about to discuss allows us nonetheless to study the most basic properties of the renormalization procedure. Let us therefore consider integrals of the form ∞ dy , x1 (c) = y+c 0

(29.9)

where c > 0 is a parameter corresponding to the external parameters in a bona ﬁde QFT (such as momentum, charge, mass). Clearly integrals of the form (29.9) are ill-deﬁned because they diverge at the upper limit of integration. We confront this type of divergences in QFT (UV divergences in loop inte-

Algebraic and diﬀerential structures in renormalized perturbation QFT

369

grations). These type of integrals correspond to a rooted tree with only one vertex. Now, to a tree t with m vertices enumerated downward such that the root has number one, corresponds an analytical expression of the form xt (c) =

∞

0

m − y1− dy1 = ym . . . y2− dym . . . dy2 , y1 + c i=2 yi + yj (i)

(29.10)

where j(i) is the number of the vertex to which the ith vertex is connected. Thus, for example, the integral ∞ Dr − j=1 xtj (y)y dy xt (c) = y+c 0 corresponds to the rooted tree where the root of each of the tj , j = 1, . . . , r, trees is attached to a new vertex, which becomes the root of fertility r of the resulting new tree. Let us consider in detail the ill-deﬁned iterated integral, x2 (c) =

0

∞

∞

0

y1− y2− dy1 dy2 . (y1 + c)(y2 + y1 )

(29.11)

Following the guidance from ﬁeld theory, this expression for x2 (c) (which corresponds to a bare Green function) can be “renormalized” by the following steps: 1) Calculate ﬁrst x ¯2 (c) := x2 (c) − x1 (c)x1 (1), i.e., subtract a “counterterm” given by multiplying the x1 (c), obtained from x2 (c) by setting the subdiverge x1 (y1 ) in x2 (c) equal to one, times x1 (c = 1). Here x1 (c) is given by (29.9). 2) Calculate xR 2 (c) by taking the limit, x2 (c) − x ¯2 (1)]. xR 2 (c) = lim [¯ →0

One thus ﬁnds −2 xR − B(, 1 − )c− ) 2 (c) = lim [B(, 1 − )(B(2, 1 − 2)c →0

−B(, 1 − )(B(2, 1 − 2) − B(, 1 − ))] =

1 2 ln (c), 2

(29.12)

where the B’s above are the Beta functions given by B(n, 1 − n) = π/ sin nπ. It is easy to verify that (29.12) is indeed ﬁnite. Note that when performing the integrations in (29.11) the poles appear as a Laurent series. This mimics very nicely the regularization procedure used for some of the schemes of renormalization in QFT. Within some of these schemes (minimal subtraction and on-shell renormalization) one also sets the external parameter equal to one (for the toy model this corresponds to setting c = 1). Having exhibited the nature of the UV singularities that typically appear in QFT, by means of the above-described toy model, let us now return to the description of the Hopf algebra associated with rooted trees.

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29.3

M. Rosenbaum

The Hopf Algebra of Rooted Trees

Let Σn be the set of trees of degree ≤ n. Let Hn be the polynomial commutative algebra generated by the symbols ti ∈ Σn . Here i is in general a composite index that uniquely characterizes a given tree. A Poincar´e-Birkhoﬀ-Witt basis for Hn is given by {f i } = {1, ti , ti tj , . . . }, where multiplication is merely given by the juxtaposition of the symbols ti and the unit 1 is obtained by the inclusion map i : C → Hn according to i(1) = 1. We can give to this algebra the structure of a coalgebra by deﬁning a coproduct on Hn as P C (ti ) ⊗ RC (ti ), (29.13) Δ(ti ) = ti ⊗ 1 + 1 ⊗ ti + adm cuts C

where the admissible cuts C are those cuts that involve at most one edge on a given branch of the tree, RC (t) is the remanent tree left after the cuts that contains the root and P C (t) is the monomial of trees representing the branches left after the cuts and that do not contain the original root. An example of the action of the coproduct is

=1⊗ Δ + ⊗1+2 ⊗ + ⊗ . (29.14) The counit of the coalgebra is given by (1) = 1 (ti ) = 0, ∀ti = 1. One can check that the coproduct (29.13) satisﬁes the coassociativity axiom (id ⊗ Δ) ◦ Δ(ti ) = (Δ ⊗ id)Δ(ti ),

(29.15)

as well as the counit and connection axioms (id ⊗ ) ◦ Δ = ( ⊗ id) ◦ Δ = id, Δ(ti tj ) = Δ(ti ) · Δ(tj ), respectively. Now note that the above deﬁnitions for Δ and on generators extend uniquely to the whole algebra as algebra homomorphisms Δ, : Hn → Hn ⊗ Hn . Further let HR ≡ ∪Hn . With the above structure the universal enveloping algebra of rooted trees is a bialgebra. To arrive at a Hopf algebra we need to introduce an additional bijective map S : HR → HR , the antipode S, which is also an algebra antihomomorphism, S(a · b) = S(b) · S(a),

∀a, b ∈ HR .

On generators the antipode is deﬁned by S(1) = 1,

(29.16)

Algebraic and diﬀerential structures in renormalized perturbation QFT S(P C (ti ))RC (ti ). S(ti ) = −ti −

371 (29.17)

admC(ti )

It can be veriﬁed that (29.17) satisﬁes the antipode axiom m ◦ (S ⊗ id)Δ(ti ) = (ti ) = m ◦ (id ⊗ S)Δ(ti ).

(29.18)

Note that the above deﬁnition for the antipode is recursive. We can give an alternate nonrecursive deﬁnition, which will also serve as the launching point for introducing the concept of twisting of the antipode and for formulating the process of renormalization in terms of this Hopf algebra. The alternate deﬁnition is C (−1)n P C (ti )RC (ti ), (29.19) S(ti ) = all full cuts

where nC denotes the number of full cuts. The simplest way to clarify the notation in (29.19) and to verify its equivalence with (29.17) is by means of some examples where we use doted boxes to encase the branches below a cut edge, together with a dotted box to denote a full cut. We thus have, for example,

S

S

S

S

−

−

−

+

+

−

+

−

+

+

=− +

=− +

−

=−

+

−

+2 −

Let us now verify the antipode axiom on the above examples. We get m ◦ (S ⊗ id) ◦ Δ( ) = m(1 ⊗ − ⊗ 1) = 0 m ◦ (S ⊗ id) ◦ Δ( ) = m(1 ⊗ − ⊗ 1 + ⊗ 1 − ⊗ ) = 0 m ◦ (S ⊗ id) ◦ Δ(

) = m(1 ⊗ − −2 ⊗ +

⊗1+2 ⊗1− ⊗ ) = 0

m ◦ (S ⊗ id) ◦ Δ( ) = m(1 ⊗ − ⊗ 1 + 2 − ⊗ − ⊗ +

⊗1−

⊗1 ⊗1

⊗ ) = 0.

(29.20)

From these examples we can make the general inference that terms cancel in pairs. It is also important to observe that the use of full cuts in formula (29.19) for the antipode induces a natural bracket structure on trees. Consequently, instead of using the dotted boxes as a merely mnemonic procedure for calculating the antipode, let us formally include this bracket structure in the expressions for the antipode of the above examples, as well as in the expressions resulting from application of the antipode axiom.

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We thus have S

−→ −[ ] S −→ − + [ ] + , + , + , S −→ − + [ ] + [ ] − [ ] S −→ − [ ] + 2 [ ] − [ ][ ] , and m ◦ (SR ⊗ id)Δ( ) = −[ ] + m ◦ (SR ⊗ id)Δ( ) = − + + [ ] − [ ]

+ , = − + [ ] −[ ] + [ ] −[ ] m ◦ (SR ⊗ id)Δ + , + [ ] − [ ] + , + 2 [ ] − 2 [ ] + [ ][ ] − [ ][ ] . m ◦ (SR ⊗ id)Δ( ) = − Note that here, since full cuts involve an extra bracket relative to normal cuts, pairings in the above expressions for the antipode axiom are of the form X − [X], and no longer cancel. When we relate rooted trees to analytic expressions (iterated integrals for Feynman diagrams) we begin to see how the logic of renormalization begins to emerge. To this end, observe ﬁrst that by means of the bracket structure representing the twisting of the antipode we are able to take into account the degree of freedom allowed in renormalization due to the fact that arbitrary scales are introduced as a result of the dimensional regularization procedure. In fact, as long as the evaluation of the brackets leaves the divergent part of the analytical expression unchanged, it is considered a valid renormalization scheme. More speciﬁcally, a renormalization scheme is valid as long as the poles in the Feynman integrals remain untouched by the process of bracket evaluation, and the following multiplicative constraints are satisﬁed: [XY ] − [[X]Y ] − [X[Y ]] + [X][Y ] = 0.

(29.21)

With the twisted antipode SR , and an algebra homomorphism φ : ti → φti , which maps rooted trees to Feynman diagrams, Eq. (29.19) is modiﬁed to read m ◦ (SR ⊗ id)(φ ⊗ φ)Δ(ti ) ∼ (φti ),

(29.22)

which means that the right-hand side is zero modulo ﬁnite terms (for ti = 1). These ﬁnite terms are the physically meaningfully values of the renormalized quantities. We shall discuss more fully the meaning of the twisted antipode when considering the ﬁrst-order diﬀerential calculus of renormalization in Section 29.5. However, with the tools we already have on hand, let us go back to the toy model discussed in the previous section and see how the antipode axiom for the twisted Hopf algebra reproduces the results we had obtained there.

Algebraic and diﬀerential structures in renormalized perturbation QFT

29.3.1

373

Toy model from the Hopf algebra point of view

Recall the iterated integral in (29.11) and make the identiﬁcation x2 (c) = φt2 . Using the results above we have that m ◦ (SR ⊗ id)Δ(φt2 ) = m ◦ (SR ⊗ id)(φt2 ⊗ 1 + 1 ⊗ φt2 + φt1 ⊗ φt1 ) = SR (φt2 ) + φt2 + SR (φt1 )φt1 = −[x2 (c)] + [[x1 (c)]x1 (c)] + x2 (c) −[x1 (c)]x1 (c) =

(29.23)

xR 2 (c).

Now, when evaluating x2 (c) in terms of Beta functions we get a Laurent series with poles of leading order two and no ﬁnite terms. Thus the on-shell and MS scheme, which both satisfy the multiplicative constraints (29.21), give the same result: [x2 (c)] = x2 (1), [x1 (c)] = x1 (1). Evaluating (29.9) and (29.11) at c = 1 and substituting into (29.23) yields 0

∞

0

∞

m ◦ (SR ⊗ id)Δ(φt2 ) = 1 1 1 1 1 1 ( − )− ( − ))dydy1 . ( y1 + c y1 + y 1 + y y1 + 1 y1 + y 1 + y

(29.24)

It is easy to verify that when performing the integrations in (29.24) and taking afterward the lim → 0 one obtains the same result as in (29.12). Consequently the twisted antipode axiom indeed provides the renormalized function, which in the on-shell or MS scheme, is, for this particular example, xR 2 (c = 1) = 0.

29.4

Hopf Algebra of Renormalization in Field Theory

In pQFT the expansion of the eﬀective action in powers of the coupling constants of the theory leads to Feynman diagrams, which are connected and which are one-particle irreducible (1PI), i.e., diagrams that cannot be separated into two independent ones by cutting on one of their edges. As we have mentioned before, Feynman diagrams are in a one-to-one relation to rooted trees (or sums of rooted trees if the diagrams are overlapping) where each of the vertices in the tree is decorated by a subdiagram that is primitive, that is, a diagram corresponding to an analytic expression with a nonvanishing superﬁcial degree of divergence, but which in turn is free of subdivergences. Clearly, from the point of view of the Hopf algebra of rooted trees, primitives correspond to a one-vertex tree and their coproduct is simply Δ(ti ) = 1 ⊗ ti + ti ⊗ 1. We now give some examples of Feynman diagrams, taken from speciﬁc cases of ﬁeld theory, and their correspondence to decorated rooted trees. Consider ﬁrst the diagram in Fig. 29.4 where the primitive subdiagram γ1 is a one-loop vertex graph representing a boson which at the ﬁrst vertex (counting from left to right) interacts to split into two bosons. These propagate and at the second vertex one of the bosons emits a third boson, which in turn is absorbed at the third vertex by the other boson. The external and internal edges (associated with propagators) are oriented clockwise and are labeled by momenta in such a way that momentum is conserved at each vertex (with the convention that momenta on ingoing edges are added while the ones on outgoing edges are subtracted). According

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M. Rosenbaum

to Feynmam rules, the analytic expression corresponding to this diagram for a massive φ3 theory in Euclidean space of six dimensions is ∞ d6 q 3 γ1 ∼ λ . (29.25) (q 2 + m2 )((q − p1 )2 + m2 )((q − k)2 + m2 ) 0 This subdivergence is nested in another divergence of the same type represented by a similar graph and the analytic expression for the total diagram is ∞ d6 q 5 Γ1 ∼ λ (q 2 + m2 )((q − p1 )2 + m2 ) 0 (29.26) ∞ 6 d k × . 2 + m2 )(k 2 + m2 )((p − k)2 + m2 )((k − p )2 + m2 ) ((q − k) 1 2 0 In the corresponding rooted tree the primitive decorations of each of the two vertices are then γ1 , and in the language of parenthesized words of [1] we have that Γ1 = ((γ1 )Γ1 /γ1 ), where the quotient Γ1 /γ1 denotes the diagram resulting from shrinking γ1 to a point in Γ1 .

p2

Γ1 /γ1 = γ1

k

Γ1 =

p1

q q−k

k − p2

⇔

q − p1

γ1

k − p1 p1 − p2

k q

γ1 =

p1

q−k

⇔

γ1

q − p1 p1 − k

FIGURE 29.4. A primitive 1PI vertex diagram from φ3 boson ﬁeld theory. The renormalization of this rooted tree by means of the twisted antipode axiom (29.22) is given by (29.24) which, for the particular decorations considered, reads (Γ1 )R = Γ1 − [Γ1 ] + [[γ1 ]γ1 ] − [γ1 ]γ1 . In Figs. 29.5 and 29.6 we show other Feynman graphs where each decoration of the corresponding rooted trees is diﬀerent. Fig. 29.5 represents an iterated integral where each

Algebraic and diﬀerential structures in renormalized perturbation QFT

375

primitive subdivergence is nested inside the other one, while in Fig. 29.6 we exemplify the case of having both nested and disjoint subdivergences. Γ2 /((γ0 )γ2 ) ⇔

Γ2 =

((γ0 )γ2 )/γ0 γ0

γ0 =

γ2 =

γ1 =

FIGURE 29.5. A rooted tree decorated by primitive 1PI subdiagrams corresponding to a graph from φ3 boson ﬁeld theory.

Γ3 /γ3 ∪ ((γ1 )(γ2 )γ0 ) = γ0 γ2

Γ3 = γ1

⇔

γ3

γ3 γ1

γ1 =

γ2 =

γ3 =

γ0 =

γ2

FIGURE 29.6. A rooted tree decorated by primitive 1PI subdiagrams corresponding to nested and disjoint subdivergences in a graph from φ3 boson ﬁeld theory.

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The renormalization of the diagrams in Figs. 29.5 and 29.6 proceeds along similar lines as in the case discussed before, although the algebra becomes increasingly more complicated. Thus the forest of counterterms generated by the action of the twisted antipode on Fig. 29.6, and which we display in Fig. 29.7, involves one diagram for a full cut, 4 diagrams with 2 full cuts, 6 diagrams for 3 full cuts, 4 diagrams for 4 full cuts, and one diagram for 5 full cuts, for a total of 16 terms. This is indeed beginning to look like a forest!.

S −

+

+

+

+

−

−

−

−

−

−

+

+

+

+

−

FIGURE 29.7. Forest of counterterms generated by the antipode.

For the sake of completeness, we give only the ﬁnal result for these two examples. The renormalization of the graph in Fig. 29.5 is (Γ2 )R = (Γ2 − [Γ2 ]) − ([(γ0 )γ2 ]Γ2 /γ0 ∪ γ2 [[(γ0 )γ2 ]Γ2 /γ0 ∪ γ2 ]) − ([γ0 ]Γ2 /γ0 − [[γ0 ]Γ2 /γ0 ]) + ([[γ0 ]γ2 ]Γ2 /γ0 ∪ γ2 − [[[γ0 ]γ2 ]Γ2 /γ0 ∪ γ2 ]). The renormalization of the graph in Fig. 29.6 is (Γ3 )R = (Γ3 − [Γ3 ]) − ([γ3 ]Γ3 /γ3 − [[γ3 ]Γ3 /γ3 ])

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377

− ([γ1 ]Γ3 /γ1 − [[γ1 ]Γ3 /γ1 ]) − ([γ2 ]Γ3 /γ2 − [[γ2 ]Γ3 /γ2 ]) − ([(γ1 )(γ2 )γ0 )](γ3 )γ0 − [[(γ1 )(γ2 )γ0 )](γ3 )γ0 ]) + ([γ1 ][γ3 ]((γ2 )γ0 )γ0 − [[γ1 ][γ3 ]((γ2 )γ0 )γ0 ]) + ([γ2 ][γ3 ]((γ1 )γ0 )γ0 − [[γ2 ][γ3 ]((γ1 )γ0 )γ0 ]) + ([((γ1 )(γ2 )γ0 )[γ3 ]γ0 − [[((γ1 )(γ2 )γ0 )[γ3 ]γ0 ]) + ([γ1 ][γ2 ]((γ0 )(γ3 )γ0 ) − [[γ1 ][γ2 ]((γ0 )(γ3 )γ0 )]) + ([γ1 ][(γ2 )γ0 ]((γ3 )γ0 ) − [[γ1 ][(γ2 )γ0 ]((γ3 )γ0 )]) + ([γ2 ][(γ1 )γ0 ]((γ3 )γ0 ) − [[γ2 ][(γ1 )γ0 ]((γ3 )γ0 )]) − ([γ1 ][γ2 ][γ0 ]((γ3 )γ0 ) − [[γ1 ][γ2 ][γ0 ]((γ3 )]) − ([γ1 ][((γ2 )γ0 )][γ3 ]γ0 − [[γ1 ][((γ2 )γ0 )][γ3 ]γ0 ]) − ([γ2 ][((γ1 )γ0 )][γ3 ]γ0 − [[γ2 ][((γ1 )γ0 )][γ3 ]γ0 ]) − ([γ1 ][γ2 ][γ3 ](γ0 )(γ0 ) − [[γ1 ][γ2 ][γ3 ](γ0 )(γ0 )]) + ([γ1 ][γ2 ][γ0 ][γ3 ]γ0 − [[γ1 ][γ2 ][γ0 ][γ3 ]γ0 ]). Note how rapidly complicated the evaluation of the renormalized functions becomes with increasing n. So this example also serves to emphasize the merits of the underlying Hopf algebra of the renormalization process, since eﬃcient numerical codes can be readily devised for the calculation of the coproduct and antipode axiom.

29.5

The Diﬀerential Calculus of Renormalization, The Birkhoﬀ Algebraic Decomposition, and The Forest Formula

As we have seen from the general discussion and examples given above, the renormalization of Feynman graphs procedure in pQFT involves the judicious introduction of counterterms in order to cancel the undesired singularities arising from the ill-deﬁned iterated integrals occurring in the Green functions of the theory. We have also seen that from the point of view of Hopf algebra, these counterterms are generated by a twisted antipode that preserves the Hopf structure of the algebra and at the same time allows for the freedom implied in the various possible normalization schemes. Let us now express all this in the language of a ﬁrst-order diﬀerential calculus of the Hopf algebra of renormalization. For this purpose, consider the inﬁnite-dimensional vector space V spanned by the elements φt , and consider the Karoubi diﬀerential, δφt = 1 ⊗ φt − φt ⊗ 1 ∈ V 2 ⊂ V ⊗ V,

(29.27)

V 2 = {q ∈ V ⊗ V |mq = 0}.

(29.28)

where The set V 2 = {δφt } is a subbimodule of the bimodule Γ over HR . We can construct a basis (θi )i∈I in the vector space of all left-invariant elements of V 2 . These are necessarily of the form (29.29) θi = P (δφti ) = (S ⊗ id)Δ(φti ), where P is the unique projector to the subspace of left invariant elements of Γ such that P (φti δφtj ) = (φti )P (δφtj ).

(29.30)

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M. Rosenbaum

Now recalling the expression (29.13) for the coproduct and making use of (29.29) we can write θi =

φS(P

C

(t))

⊗ δφR

C

(t)

,

(29.31)

adm cuts C

where the sum in the right side of the above expression now includes the empty and full cuts of the tree. In order to see more clearly what the twisting of the antipode really means, it is pertinent at this point to make a parenthetical remark concerning the homomorphism map φ : t → φt which we introduced in Sec. 29.3. Although rather technical and probably beyond the essential purposes of this chapter, it will also serve to make contact with the subject of operads. Indeed, looking at Fig. 29.8 one notes that we could shift the place of insertion of the self-energy subgraph:

and that this information is not stored in the representation by decorated rooted trees. In fact, the hierarchy that determines the recursive mechanism of renormalization is independent of this additional information. Nonetheless one could formally restore it into the mathematical structure of the Hopf algebra of Feynman graphs by making use of the operad formalism or by formulating the Hopf algebra directly on graphs. We refer the interested reader to [7] for an ample and clear discussion of this subject.

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭

⇔

FIGURE 29.8. A decorated rooted tree corresponding to diﬀerent insertions of a self-energy subgraph.

Algebraic and diﬀerential structures in renormalized perturbation QFT

29.5.1

379

The forest formula

After this brief diversion, let us return to the mainstream of our discussion. Thus note ﬁrst that we can rewrite (29.29) as θi = (S ⊗ id)(φ ⊗ φ)Δ(ti ),

(29.32)

If we were to let the composition of φ and S to be given by φ ◦ S = S ◦ φ, then it would follow immediately from (29.32) that m(θi ) = φ ◦ m(S ⊗ id)Δ(ti ) = φ ◦ (ti ) = 0, which is a valid, albeit trivial renormalization. To arrive at a more general and physically useful procedure let us introduce the idempotent τR such that φR := τR ◦ φ satisﬁes SR ◦ φ = φR ◦ S,

(29.33)

i.e., τR evaluates the resulting analytical expressions under some renormalization scheme that leaves their divergent part unchanged and satisﬁes the multiplicative constraints discussed in Sec. 29.3. (This choice of scheme, known as the mass independent or minimal substraction renormalization, will guarantee that the antihomomorphism axiom will be preserved for the twisted antipode). The operator SR is then the twisted antipode discussed before and we have from (29.32) that (29.34) (τR ⊗ id)θi = (SR ⊗ id)(φ ⊗ φ)Δ(ti ). The term on the right of the above expression is equivalent to the “forest formula” of pQFT, which yields the renormalized Green functions for a given nonoverlapping Feynman diagram.

29.5.2

The Birkhoﬀ algebraic decomposition

Let us now consider in more detail the algebra homomorphism φ described above, and recall that regularization in pQFT consists essentially in mapping the ill-deﬁned Green functions of the theory onto a Riemann sphere of radius ε. Thus φ ∈ HomK-alg. (K(HT ), A) is an homomorphism map from the K-algebra of decorated rooted trees K(HT ) to the K-algebra A = {f ∈ Holom(C − 0)} with 0 a pole of ﬁnite order, i.e., the unital K-algebra of analytic functions on the Riemann sphere. Moreover, φ is considered as a K-algebra under the convolution product φ φ4(ti ) = mK (φ ⊗ φ4)Δti .

(29.35)

Let A = A− ⊕ A+ denote the Birkhoﬀ sum of the K-linear multiplicative subspaces: A− = { polynomials in z −1 without constant term}, A+ = { restriction to (C − 0) of functions in Holom(C)}, and let R be a Rotta-Baxter projection operator, which maps elements in A to elements in A− and satisﬁes R(ab) + R(a)R(b) = R[(R(a)b + aR(b)]. (29.36) Furthermore, deﬁne φ− (ti ) ∈ A− by means of the iterative equation, ˜ i ), φ− (ti ) = −R[φ(ti ) + mK (φ− ⊗ φ)Δ(t

380

M. Rosenbaum φ− (1) = 1K ,

(29.37)

˜ denotes the coproduct with the primitives part removed. Now let where Δ φ ◦ S(ti ) = SR ◦ φ(ti ) = φ− (ti ),

(29.38)

and apply the homomorphism φ to the antipode axiom with ti ∈ ker . We then have (1)

φ(m ◦ (S ⊗ id)Δti ) = mK ◦ [(φ ⊗ φ) ◦ (S ⊗ id)(ti (1)

(2)

⊗ ti )] (2)

= mK ◦ [(φ(ti ) ⊗ φ(ti )] = (φ− φ)(ti ) = φ+ (ti ) ∈ A+ ,

(29.39)

where in deriving the last equality we have made use of the Birkhoﬀ algebraic decomposition, φ+ = φ− φ.

(29.40)

It is interesting to observe that the Rotta-Baxter operator R considered here, and which satisﬁes (29.36), corresponds to the idempotent τR introduced in the previous subsection, provided we choose as a renormalization scheme the mass independent procedure so that Eq. (29.36) is made to correspond to the multiplicative constraints (29.21). We then see that Eqs. (29.37) correspond to (29.17) with the additional bracket structure discussed in Sec. 29.3, and (29.38) are the counterterms generated by the forest formula in the mass independent renormalization procedure of Weinberg and ’t Hooft. We thus now have a deeper algebraic understanding of the forest formula and the renormalization of the Feynman diagrams in pQFT: Regularization followed by renormalization in the mass independent scheme amounts to an algebraic Birkhoﬀ decomposition of the Green functions with the Rotta-Baxter projector, and (29.40) is equivalent to the forest formula.

29.6

Acknowledgments

The author is grateful to Dr. J.C. Lopez-Vieyra for invaluable help with the drawings. This work was partially supported by CONACyT Project No. G25427-E.

References [1] D. Kreimer, On the Hopf algebra structure of perturbative quantum ﬁeld theories, Adv. Theor. Math. Phys. 2, 303-334, 1998. [2] A. Connes and D. Kreimer, Hopf algebras, renormalization and noncommutative geometry, Commun. Math. Phys. 199, 203-242, 1998. [3] A. Connes and H. Moscovici, Hopf algebras, cyclic cohomology and the transverse index theorem, Commun. Math. Phys. 198, 198-246, 1998. [4] M.Rosenbaum and J.D. Vergara, The Dirac operator, the Hopf algebra of renormalization and the structure of space-time, In Cliﬀord Algebras and their Applications in Mathematical Physics, Volume 1 Algebra and Physics, pp. 283-301, Eds. R. Ablamowicz and B. Fausler, Birkhaser, Boston, 2000.

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[5] D. Kreimer, On overlapping divergences, Commun. Math. Phys. 204, 669-689, 1999. [6] D. Kreimer, Chen’s Iterated Integral represents the Operator Product Expansion, Adv. Theor. Math. Phys. 3.3 (1999). [7] D. Kreimer, Combinatorics of (perturbative) quantum ﬁeld theory. [8] A. Connes and D. Kreimer, Renormalization in quantum ﬁeld theory and the RiemannHilbert problem I: the Hopf algebra structure of graphs and the main theorem, Commun. Math. Phys. 210 , 249-273 (2000); A. Connes and D. Kreimer, Renormalization in quantum ﬁeld theory and the Riemann-Hilbert problem II: the β-function, diﬀeomorphisms and the renormalization group, Commun. Math. Phys. 216, 215-241 (2001). [9] A. Cayley, On the theory of the analytical forms called trees, Phil. Mag., 13, 172-176, 1857. [10] J.C. Butcher, The Numerical Analysis of Ordinary Diﬀerential Equations. Wiley, Chichester, 1987. [11] Ch. Brouder, Runge-Kutta methods and renormaslization. Eur. Phys. J. C12, 521-534, 2000.

Chapter 30 Survey on Smooth Quasigroups Development Lev V. Sabinin and Larissa V. Sbitneva Universidad Aut´ onoma del Estado de Morelos, Cuernavaca, M´exico

30.1 The development of investigations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30.2 Smooth quasigroups and loops and their applications in geometry . . . . . . . . . . . . . . . . . . . . 30.2.1 Smooth quasigroups and loops: General theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30.2.2 Smooth P L-loops and transsymmetric spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30.3 Non-associative geometry and discrete structure of space-time . . . . . . . . . . . . . . . . . . . . . . . . 30.4 Smooth loops action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30.5 Some applications of the smooth quasigroup and loop theory to generalized geometric algebra, mechanics, and physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30.6 Mirror symmetries of Lie algebras, Lie groups, and homogeneous spaces . . . . . . . . . . . . . . 30.7 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30.8 Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

383 384 384 385 388 389 390 391 391 392 392

Abstract We survey the recent research in algebra (quasigroups and loops, algebraic and smooth), geometry of Lie groups, Lie algebras and homogeneous spaces with applications, and mathematical physics (loopuscular approach to discrete space-time). Key words: smooth quasigroups and loops, odules, Bol loops, M -loops, nonlinear geometric algebra, F-quasigroups, transsymmetric spaces, hyperalgebra, hyporeductivity, pseudoreductivity 2000 MSC: 20N05, 53A60, 53C22

30.1

The Development of Investigations

We will survey mainly the results published after the IV International Conference on the Nonassociatve Algebra and its Applications (S˜ ao Paulo, Brazil, 1998). The remarkable development of smooth quasigroups and loops theory since the pioneering works of Mal’cev in 1955 was presented by Lev V. Sabinin in the lecture “Smooth Quasigroups and Loops: Forty-ﬁve Years of Incredible Growth” at the International Conference Loops-99 (Prague, Czech Rep., 1999) (see [8], where the large bibliography on the subject is given). Now we give a brief account of the further development of smooth quasigroups and loops theory and its applications. The fundamental monograph “Smooth Quasigroups and Loops” [1], based essentially on the original results of L. Sabinin and his scientiﬁc school, was published by Kluwer Academic Publishers in 1999. This monograph, discussed in Section 30.2, is now the standard book

383

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Lev V. Sabinin and Larissa V. Sbitneva

of reference in the ﬁeld. In the fundamental work “From Symmetric to Transsymmetric Spaces” L. Sabinin, L. Sabinina, L. Sbitneva [13] use the fruitful combination of diﬀerential-geometric methods with smooth quasigroups methods in order to construct a comprehensive theory of transsymmetric spaces. The studies of smooth M -loops and P L-loops [7] generalizing smooth Bol loops and having important applications to diﬀerential geometry of transsymmetric spaces have been prolonged. The foundations of their inﬁnitesimal theory have been established. Much attention has been paid to nonassociative structures (hyporeductive loops) related to hyporeductive spaces. The hyporeductive spaces and their structures, inﬁnitesimal (in the language of Lie algebras) local and global (in the language of Lie groups), and loopuscular (in terms of hyporeductive smooth loops) have been investigated. In Section 30.3, the problems of discrete aﬃnely connected spaces (nonassociative geometry) are intensively studied in algebraic-geometric context as well as in the context of discrete space-time in gravity theory. The purely algebraic formulation of (generalized) Einstein equations was discovered. The discrete de Sitter space was investigated, and the process of its evolution to the smooth de Sitter space was considered [10][14]. The general theory of ﬁnite aﬃnely connected spaces over ﬁelds of Galois was developed in [12]. Some kind of speciﬁc mathematical analysis in such spaces was proposed. The possible applications of this construction to the problem of Discrete Space-Time were detected. Section 30.4 studies smooth loops actions proposed in [7] that were initiated in the form suitable for applications to mathematical physics [18], [22]. In Section 30.5, the investigations of the generalized geometric algebra [7] are developed as well as the applications of the algebraic theory of geodesic maps for aﬃnely connected spaces [8] (i.e., for smooth geoodular spaces) to foundations of geometry and to mechanics. Section 30.6 renews the studies in the mirror geometry of Lie groups, Lie algebras, and homogeneous spaces. The complete account of this ﬁeld is presented in the monograph by Lev V. Sabinin, “Mirror Geometry of Lie Groups, Lie Algebras, and Homogeneous Spaces” [15].

30.2 30.2.1

Smooth Quasigroups and Loops and Their Applications in Geometry Smooth quasigroups and loops: General theory

For the ﬁrst time, the comprehensive up-to-date treatment of the smooth quasigroups and loops theory with applications to diﬀerential geometry is given in the monograph of Lev Sabinin [1]. Based on a generalization of the Lie-group theory, the book establishes new backgrounds for diﬀerential geometry in the form of non-linear geometric algebra and ”loopuscular” geometry (nonassociative geometry). This monograph contains the complete theory of smooth quasigroups and loops as well as its geometric and algebraic applications. The general inﬁnitesimal theory, as well as smooth Bol loops and algebras, smooth Moufang loops, and Malcev algebras are treated. The concept of aﬃne connection is reformulated in the purely algebraic form of loopuscular (nonassociative) geometry. The theories of reductive, symmetric, and generalized symmetric spaces (s-spaces) are presented in the above new algebraic setting.

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Appendices contain some topics in the process of current research. See, for example, the concept of antiproduct of a loop giving a new treatment of the three-web theory, Chern connection. The studies of smooth quasigroups are motivated, in particular, by applications to physics (theory of anomalies, requiring a change of a group by loop). This is useful in applications in such diverse ﬁelds as mathematical physics, relativity, Poisson and sympletic mechanics, quantum gravity, and dislocation theory.

30.2.2

Smooth P L-loops and transsymmetric spaces

There are various objects in geometry as well as various types of smooth loops related to binary-ternary linear algebras of diﬀerent types, for example, generalized symmetric spaces, transsymmetric spaces, hyporeductive and pseudoreductive loops, M -loops, half Bol loops [1]. Many signiﬁcant diﬀerential-geometric structures have such algebras as a proper inﬁnitesimal objects. Typical examples are as follows: 1. It is well known that a triple Lie system is a proper inﬁnitesimal counterpart for a symmetric space (T = 0, ∇R = 0, T , R being torsion and curvature, respectively). 2. Any triple Lie algebra is a binary-ternary linear algebra and is a proper inﬁnitesimal object for a reductive space (∇T = 0, ∇R = 0, T , R being torsion and curvature, respectively). Such an algebra reduces to a Lie algebra if its ternary operation is trivial, and to a Lie triple system, if its binary operation is trivial. 3. A right Bol algebra is a proper inﬁnitesimal counterpart to a C 3 -smooth (local) right Bol loop. There is the structural inﬁnitesimal theory of this case analogous to Lie group theory [1]. Let us note that the geometric equivalent of such a theory, the right Bol loops–Bol algebras theory, is reduced to the theory of aﬃnely connected manifolds without curvature (R = 0, σ τ + Tμν Tτσλ ), (T being a torsion). satisfying the property: ∇α (∇λTμν 4. The analog of a Lie algebra for the nonlinear Lie (or Poisson) bracket (having the local (or global) counterpart as a smooth loop) is a ν-hyperalgebra [1], [2]. Due to the obvious geometric nature of given examples one may develop the structural theories taking into consideration the homogeneous space structure of the reductive space and reducing all to the Lie algebras theory. One new remarkable class of transsymmetric spaces, the perfect transsymmetric spaces, was introduced and explored in terms of Lie groups–Lie algebras [6]. Being reductive spaces of a special kind transsymmetric spaces may be considered within the frames of homogeneous spaces theory. By the Baer-Sabinin construction, they may be treated within the frames of smooth quasigroups and loops theory [17]. In the language of smooth quasigroups theory the geometry of transsymmetric spaces is related to M -loops [7]. There are many open problems in this area [13]. In [7] L. Sabinin introduced nonsingular smooth left M -loops and P L-loops generalizing smooth Bol loops and developed the inﬁnitesimal theory of M -loops. The diﬀerential equations of smooth M -loops have been obtained. The studies of smooth P L-loops generalizing the class of M -loops have been prolonged and the foundations of their inﬁnitesimal theory have been established. Here we outline the theory of smooth local P L-loops. DEFINITION 30.1

Let Q, ·, ε be a (partial or global) loop, x · y = Lx y, x · ε =

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Lev V. Sabinin and Larissa V. Sbitneva

ε · x = x, and Q, ∗, ε be another loop x ∗ y = Px y, x ∗ ε = ε ∗ x = x such that −1 Pa ◦ Lx ◦ Pqa = La∗(x·Jqa) ,

q : Q → Q,

qε = ε,

x ∗ Jx = ε,

(30.1)

In this case we call Q, ·, ε a P L-loop with a parameter q, or qP L-loop. REMARK 30.1 If Pb = Lb we have a so-called Belousov loop (qBe-loop), or in old terminology, an M -loop, with the deﬁning identity: La ◦ Lx ◦ L−1 qa = La·(x·Jqa) .

(30.2)

Furthermore, if L−1 y = Ly −1 we obtain La ◦ Lx ◦ LJqa = La·(x·Jqa) ,

(30.3)

being the left Jq-half Bol identity [1]. Moreover, if Jq = id we arrive at the left Bol identity: La ◦ Lx ◦ La = La·(x·a) . PROPOSITION 30.1 qβλ α ¯ λμ (y)

(30.4)

For any C 1 -smooth local qP L-loop the diﬀerential equation,

∂(x · y)ν ∂(x · y)ν + (¯ αβμ (x) − qβλ Aμλ (x)) =α ¯ βν (x · y), μ ∂(y) ∂(x)μ

(30.5)

where α ¯ λμ (y) = [

∂(a ∗ y)μ ]a=ε , ∂(a)λ

Aμλ (x) = [

∂(x · a)μ ]a=ε , ∂(a)λ

qβλ (y) = [

∂(q)λ (a) ]a=ε , ∂(a)β

is valid. DEFINITION 30.2 We say that a smooth qP L-loop Q, ·, ε is regular if x → ρ(x) = x ∗ Jqx is invertible near ε (equivalently, if Id − q∗ , ε is invertible), and almost regular if ¯ λμ (y), αλμ (y) = α that is, [

∂(a · y)μ ∂(a ∗ y)μ ] = [ ]a=ε . a=ε ∂(a)λ ∂(a)λ

Any regular qP L-loop is almost regular.

COROLLARY 30.1

PROOF Indeed, substituting in (30.5) x = ε and using the invertibility of Id − q∗ , ε , we obtain the result. PROPOSITION 30.2 PROOF

Any regular qP L-loop is a q˜Be-loop (˜ q M -loop).

The deﬁning identity 30.1, −1 Pa ◦ Lx ◦ Pqa = La∗(x·Jqa) ,

x ∗ Jx = ε,

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387

−1 = La∗Jqa = Lρ(a) at x = ε. implies that Pa ◦ Pqa −1 −1 −1 And after calculation we obtain Lρa ◦ Lx ◦ L−1 = ρqa = (Pa ◦ Pqa ) ◦ Lx ◦ (Pqa ◦ Pq 2 a x) −1 −1 −1 Pa ◦ (Pqa ) ◦ Lx ◦ Pq2 a ) ◦ Pqa == Pa ◦ Ly ◦ Pqa = Lz Thus q˜b = ρqρ−1 b. Lb ◦ Lx ◦ L−1 q b) , q˜b = Lb·(x·˜

PROPOSITION 30.3 If for a C 1 -smooth local loop Q, ·, ε, x·y = Lx y, the diﬀerential equation (30.5) is valid, then this loop is a smooth qP L-loop. The above propositions give us, actually, the analog of the ﬁrst (direct and converse) Lie theorem. Further examination of the integrability conditions allows us to introduce proper inﬁnitesimal objects. This consideration leads to a Lie algebra g and a subalgebra h ⊂ g ˙ with the decomposition g = h+m, which is not reductive ([m, h] m) in this case. But this decomposition is hyporeductive. Thus it is possible to introduce a proper inﬁnitesimal object (unary-binary-ternary algebra on Tε Q) in the same way as for hyporeductive and pseudoreductive loops. As far as hyporeductive homogeneous spaces and corresponding hyporeductive algebras is concerned see [1]. Furthermore, we note that general (not almost regular) smooth qP L-loops are of special interest, but this case needs some new approaches diﬀerent from the generalization of the Bol loop case because here we cannot expect to reduce the case to the investigation of certain Lie algebras. REMARK 30.2 A local loop Q, ·, ε with the property Lx ◦Ly ◦L−1 x = Lx·(y·x−1 ) ∀ x, y) gives us an example of “singular” (q∗ − Id = 0) P L-loop. A local Bol loop (30.4) gives us an example of “non-singular” P L-loop. Nonsingular smooth left M -loops and P L-loops have a direct concern to the diﬀerential geometry of transsymmetric spaces. Moreover, any transsymmetric space may be considered as a nonsingular smooth left φM -loop with Iφ-Bruck identity [7]. In the quasigroups language transsymmetric spaces can be described as a smooth correct left F-quasigroup, i.e., a smooth correct quasigroup Q, ·, \ with F -identity, x · (y · z) = (x · y) · (F x · z), [17], which generalizes the left-distributive identity related to the generalized symmetric spaces [1], [13]. In general, a geodesic loop of the canonical aﬃne connection does not coincide with the loop of canonical loopuscular structure, in contrast to the case of symmetric spaces [17]. However, the special case when the canonical loopuscular structure coincides with its tangent geoodular structure leads to the concept of perfect transsymmetric space [13], [17]. For perfect transsymmetric spaces any geodesic loop is a left Bol loop (30.4) (as in the case of symmetric spaces) [17]. Finally, we note that in the papers [25] and [26] R. Carrillo Catalan and L. Sabinina describe the tangent algebras for Moufang loops with Al -property. The relations between curvature and torsion for the so-called Kikkawa spaces (i.e., manifolds with aﬃne connection, such that all geodesic loops of some neighborhood (at some point) are right-monoalternative) have been established.

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Lev V. Sabinin and Larissa V. Sbitneva

Nonassociative Geometry and Discrete Structure of Space-time

The problem of constructing a proper geometric discrete model of space-time is very important in order to understand an adequate quantum nature of gravity. The most prospective here is a direct approach: to reformulate the concept of aﬃnely connected (and Riemannian) space in purely algebraic terms, as certain algebraic structure, in such a way that, if this structure is smooth we obtain the conventional aﬃnaly connected (or Riemannian) space. Then we may use this structure in the discrete, ﬁnite, or topological setting as an appropriate model of space-time. Such a reformulation, the nonassociative (or loopuscular) geometry, has been achieved by L. Sabinin. (See, for example, [1]). The new approach to discrete space-time relativity based on the nonassociative geometry is given in the article of L. Sabinin, “Loop-Theoretic Foundations of Diﬀerential Geometry and Relativity” [14]. The methods of nonassociative geometry are applied to the model of discrete (in particular, ﬁnite) space-time. With the usage of operators of elemetary holonomy the remarkable generalization of Einstein equations is suggested, which gives a ready-made formalism for a discrete space-time model, and some consequences are discussed. This gives a pure algebraic construction of relativity, prospective for future exploration. The methods of further exploration are outlined. It is necessary to involve some restrictive relations in order to obtain an analog of a weak gravitational ﬁeld in purely algebraic terms. In the smooth case such assumptions give us projectively ﬂat spaces of aﬃne connection; in particular, they contain spaces of constant curvature. Thus in algebraic context one may carry out a further investigation of this situation and, furthermore, in the smooth case, to compare Einstein gravity with generalized Einstein gravity. L. Sabinin and A. Nesterov have applied the nonassociative geometry to de Sitter model of space-time. In the article “Nonassociative Geometry: Towards Discrete Structure of Space-Time” [10] they compare the classical de Sitter space-time with its “nonassociative” ﬁnite counterpart. It is shown that as approaching limit, while the number of points is growing, the classical smooth de Sitter space-time is obtained. This scheme is applicable to space-time in general. The growing of the number of points may be interpreted as depending on time in the process of universe evolution. This leads us to the concept of the nonassociative (discrete) space-time when, at distances comparable with Planck length, the standard concept of space-time might be replaced by the diodular discrete structure which at a large space-time scale “looks like” a diﬀerentiable manifold. Further, the nonassociative geometry is applied to the Friedmann-Robertson-Walker universe in [16]. In the article of L. Sabinin, “Nonassociative Geometry and Discrete Space-Time” [12] the methods of nonassociative (loopuscular) geometry are used in order to introduce ﬁnite (discrete) space-time. This problem has a direct relation to the quantum description of gravity. After a brief introduction into nonassociative geometry allowing the purely algebraic treatment of the concept of aﬃne connections and Riemannian spaces, the authors consider the so-called ﬁnite diodular structure (ﬁnite aﬃnely connected space) as a hypothetic model of discrete space-time. This model involves Galois ﬁelds GF (pm ) depending on two parameters, p (prime number) and a positive integer m. The growing of p and (or) m may be interpreted as an evolution of the universe in time. An appropriate, correct limit process (p → ∞, m → ∞) may give a smooth model of the standard theory of gravity. It is very

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instructive to classify all 4-dimensional spaces for small p = 2, 3, 5, 7, 11 and m = 1, 2, 3, 4 in order to obtain some models of space-time for early states of the universe. In order to develop some kind of calculus in discrete space-time we observe that any function given on a direct ﬁnite product of ﬁnite ﬁelds is a polynomial (it is unique, provided that we use a polynomial of minimal degree). It allows us to use the algebraic diﬀerentiation of polynomials and develop some calculus. The properties of this calculus are very unusual (in general, the Leibnitz formula is not valid). Now, it is possible to introduce an aﬃne connection ∇, torsion T , and curvature, as well as Einstein equations that do not have a solution in some cases. We may also generalize the above scheme changing ﬁelds by rings or double loops. Apart from the physical interpretations, the above construction has a purely mathematical meaning, introducing us to the geometry of “ﬁnite aﬃnely connected spaces” as a part of nonlinear geometric algebra.

30.4

Smooth Loops Action

In the article, “Smooth Quasigroups and Loops. Recent Achievements and Open Problems” [7], the concept of smooth loop action was introduced. This concept may give an alternative approach to the construction of diﬀerentiable groupoid. The right and left natural actions are considered. The concept of a strict action is introduced and the corresponding diﬀerential equations are deduced and investigated. In this way analogs of the ﬁrst and the second Lie theorem are established. The problem with the third Lie theorem in this case is more sophisticated and needs a properly constructed inﬁnitesimal object. Thus, here is an open problem. Recent interest of physicists to Moufang symmetries has generated the problem of Bol symmetries. The approach to smooth Bol representation via Lie triple families was initiated. In the paper of L. Sbitneva [18] the notion of a left Bol loop and Bol-Bruck actions was introduced in a purely algebraic form. It was shown that in the smooth case a left Bol loop action coincides with the so-called (local) Nono family (local triple family in the terminology of Nono [23]. Further, in the paper “Smooth Bol Loop Actions” [22], L. Sbitneva showed that the problem is reduced to the problem of linear representation for Lie groups, which is in direct connection to the problem of smooth loop actions. The inﬁnitesimal theory of smooth left Bol loop actions is considered and the theory of linear representations (actions) for a smooth left Bol is developed. It implies, in particular, that we are dealing with a linear representation of a triple Lie system. In the paper of L. Sbitneva “Exceptional Smooth Bol loops” [22], a remarkable class of smooth Bol loops, so-called exceptional Bol loops, is introduced. The Bol-Bruck and Moufang loops belong to this class. Basic diﬀerential equations are given in matrix form. The explicit form for the structure functions and basic vector ﬁelds are obtained. The approach to an analog of the Campbell-Hausdorﬀ formula is outlined in this case. The results of the paper “Analytic Decompositions of Smooth Bol-Bruck Loops” by Lev V. Sabinin, L.V. Sbitneva [11] have been used. This research contributes into the solution of the following problem: how one may restore a smooth Bol loop by its algebra by means of matrix series. Another approach to the problem of the existence of Campbell-Hausdorﬀ formula for a local Bol loop can be found in the paper, “The Campbell-Hausdorﬀ Series of Local Analytic Bruck Loops,” by G. Nagy [27].

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Lev V. Sabinin and Larissa V. Sbitneva

Some Applications of the Smooth Quasigroup and Loop Theory to Generalized Geometric Algebra, Mechanics, and Physics

The very interesting problem of describing spaces of constant curvature (and, more generally, projectively ﬂat spaces) in the purely algebraic way has been studied by O. Matveyev and his collaborators. The precise algebraic equivalent of an aﬃnely connected manifold was constructed. The classes of locally ﬂat, locally symmetric, locally reductive aﬃnely connected spaces were described in the terms of smooth local partial loops with additional operations. The tangent algebras of some classes aﬃnely connected manifolds were investigated. The algebraic theory of geodesic maps of aﬃnely connected manifolds has been developed since 1985. The notion of manifold with geodesics has been introduced as one parameter family of smooth local partial quasigroups with certain identities. It was proved that this notion is the precise equivalent of aﬃnely connected manifold with a zero torsion tensor ﬁeld. In the analytic case one geodesic quasigroup with speciﬁc algebraic properties restores the unique connection with zero torsion. On the other hand the quasigroup construction of the horizontal lift of any aﬃne connection to the tangent bundle shows that any aﬃnely connected space with any torsion can be treated as the manifold with geodesics. The locally symmetric and locally abelian (ﬂat) manifolds with geodesics may be characterized by the possibility of reconstructing the geodesic loop and the geodesic odule in a purely algebraic way. This gives us the opportunity to develop the theory of prosymmetric (projective symmetric) and proabelian aﬃnely connected manifolds in the loop and the quasigroup terms. During the last ﬁve years these classes of spaces and their subclasses were intensively investigated (O.A. Matveyev, E.L. Nesterenko [28], [29], [30], [31]. The remarkable spaces, such as almost symmetric and antisymmetric aﬃnely connected manifolds, have common geodesic lines (with a preservation an aﬃne parameter) with symmetric spaces. The proabelian manifolds of dimensions 2 and 3 are under special investigation. This scientiﬁc direction is in close relation to the algebraic construction of aﬃne geometry. In 1992 Professor L.V. Sabinin published purely algebraic proof that a ﬂat geoodular space can be treated as an aﬃne space and vice versa. This result is reformulated in terms of the notion of the manifold with geodesics. All geometry conﬁgurations in an aﬃne space are described by the identities in the abelian space with geodesics. In particular, the theorems of Chevy and Menelay may be reproved in quasigroup terms. The notion of a manifold with geodesics is generalized to the notion of a manifold with trajectories, which is introduced as a family of local quasigroups, reﬂecting the geometric properties of the solutions of a diﬀerential equation on a manifold in the form of the algebraic identities. Aﬃne connections are associated with a manifold with trajectories. Mechanical systems are considered from this viewpoint. The classes of locally symmetric and locally ﬂat manifolds with trajectories are investigated by O.A. Matveyev, A.V. Panshina [33]). The methods of the quasigroup theory in the real projective geometry have been discussed since 2001 The algebraic axiomatic is introduced in dimensions one and two. This result gives algebraic models of the projective line and plane. This is also very interesting from the viewpoint of teaching geometry at the university level. Along with the above research on foundations of generalized geometry the paper of N. Casta˜ neda, “Sabinin Spaces and Moufang Loops” [36], presents some generalization of aﬃne spaces, the so-called quasiaﬃne spaces. Some axioms of geometry are presented as certain

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identities in the loopuscular space (the space with the families of loops with some speciﬁc identities). This relates to the question how one may generalize Desargues axioms in order to obtain an aﬃnely connected space. The axiomatic description of the relativistic law of composition of velocities in special relativity has a representation in terms of smooth loops and odules, as is shown in the paper, “Nonassociative Geometry of Special Relativity” [19]. The left Bol and left Bruck identities hold for the law of composition of relativistic 3-dimensional velocities. The BolBruck odule is introduced. This consideration allows us to give an algebraic interpretation of special relativity in the frames of nonassociative algebra. (See also [20].)

30.6

Mirror Symmetries of Lie Algebras, Lie Groups, and Homogeneous Spaces

In the monograph by Lev V. Sabinin, “Mirror Geometry of Lie Groups, Lie Algebras and Homogeneous Spaces” [15], a new and powerful method in the theory of homogeneous spaces, mirror geometry, based on the original works of the author, is presented with different applications. This new theory of Lie groups and Lie algebras based on mirror geometry and systems of mirrors allows us to study homogeneous spaces with mirrors (further generalization of symmetric spaces of E. Cartan) and to solve some classiﬁcation problems in the theory of homogeneous Riemannian spaces.

30.7

Problems

We summarize the problems to be solved. 1. To construct the representation theory for loops and quasigroups (which is very important in applications to physics and mechanics). 2. To give systematic and rigorous treatment of transsymmetric spaces (generalizing symmetric spaces of E. Cartan). 3. To develop the inﬁnitesimal theory of binary Bol loops, to describe the corresponding inﬁnitesimal object. 4. To elaborate the structural theory of Bol algebras, to classify the simple Bol algebras. 5. To start with the topological theory of some remarkable classes of smooth loops. In particular, to investigate the generalized ﬁfth problem of Hilbert for Bol loops (when does a topological Bol loop become smooth?). 6. To explore new prospective smooth loops like M -loops and P L-loops. 7. To develop the mathematical apparatus for discrete space-time theory. 8. To develop the algebraic theory of relativity. 9. To develop the algebraic theory of prevector spaces. 10. To develop generalized geometric algebra (foundations of geometry): in particular, how to generalize Desargues axioms in order to obtain an aﬃnely connected space.

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30.8

Acknowledgments

This research was developed within the framework of project E-32236, “Nonassociative Algebra and Diﬀerential Geometry,” supported by CONACYT and, in part, within the framework of the project, “Mirror Symmetries of Homogeneous Spaces and Lie Groups,” supported by PROMEP, Mexico.

References [1] Lev V. Sabinin, Smooth Quasigroups and Loops Monograph. Kluwer Academic Publishers, Dordrecht, The Netherlands, 1999. [2] Lev V. Sabinin, Smooth Loops I Algebras, Groups and Geometries 15, no. 2, pages 127–153. Hadronic Press, 1998. [3] Lev V. Sabinin, Quasigroups, Geometry and Nonlinear Geometric Algebra, Acta Applicandae Mathematicae 50, pages 45–66. Kluwer Academic Publishers, 1998. [4] Lev V. Sabinin, L.L. Sabinina, On Bol-Bruck loops Webs and Quasigroups, 19981999, pages 106–108, English. Tver University Press, 1999. [5] Lev V. Sabinin, C. Castillo, Generalized Bol loops, Herald of Friendship of Nations University (Math.) 6, no. 1, pages 182–190, 1999. [6] Lev V. Sabinin, L.L. Sabinina, L.V. Sbitneva, Perfect transsymmetric spaces. Publicaciones Mathematicae 54, no. 3-4, pages 303–311, Debrecen, Hungary, 1999. [7] Lev V. Sabinin, Smooth quasigroups and loops. Recent achievements and open problems 0-8247-0406-1 Nonassociative algebra and its Applications. Eds. R. Costa, A. Grishkov, H. Guzzo, L. Peresi. In: Lecture Notes in Pure and Applied Mathematics. Vol. 211, pages 337–344, Marcel Dekker, New York, 2000. [8] Lev V. Sabinin, Smooth Quasigroups and loops: Forty-ﬁve years of incredible growth, Commentationes Mathematicae Universitatis Carolinae 41, no. 2, pages 377–400, Czech Republic, 2000. [9] Lev V. Sabinin, A.I. Nesterov, Non-associative geometry and discrete structure of space-time, Commentationes Mathematicae Universitatis Carolinae 41, no. 2, pages 347–357, Czech Republic, 2000. [10] Lev V. Sabinin, A.I. Nesterov, Nonassociative geometry: Towards discrete structure of spacetime, Physical Review D. 62, 081501 (R), 081501-1–081501-5. Rapid communications, American Physical Society, 2000. [11] Lev V. Sabinin, L.V. Sbitneva, Analytic decompositions of smooth Bol-Bruck loops, Webs and Quasigroups, pages 94–97, Tver University Press, 2000. [12] Lev V. Sabinin, Nonassociative Geometry and Discrete Space-Time, International Journal of Theoretical Physics. 40, no. 1, pages 351–358, 2001.

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[13] Lev V. Sabinin, Ludmila Sabinina, Larissa Sbitneva, From symmetric spaces to transsymmetric spaces. In: Diﬀerential Geometry and its Applications, Elsevier Science, The Nethertlands, 2002. [14] Lev V. Sabinin, Loop-Theorertic Foundations of Diﬀerential Geometry and Relativity. Webs and Quasigroups, pages 67–72, Tver University Press, 2002. [15] Lev V. Sabinin, Mirror Geometry of Lie Groups, Lie Algebras and Homogeneous Spaces Monograph. Kluwer Academic Publishers, Dordrecht, The Netherlands, 2004. [16] Lev V. Sabinin, A.I. Nesterov, Nonassociative geometry: Friedmann-RobertsonWalker universe, hep-th/0406229. [17] L. Sbitneva, Algebraic structure of transsymmetric spaces, In: Nonassociative Algebra and its Applications. Eds. R. Costa, A. Grishkov, H. Guzzo, L. Peresi. In: Lecture Notes in Pure and Applied Mathematics, Vol. 211, pages 337–344. Marcel Dekker. New York, 2000. [18] L. Sbitneva, Bol loop actions, Commentationes Mathematicae Universitatis Carolinae, Vol. 41, no. 2, 405–408, Czech Republic, 2000. [19] L. Sbitneva, Nonassociative Geometry of Special Relativity, International Journal of Theoretical Physics (IJTP), Plenum Publ., vol. 40, no. 1, 359–362, 2001. [20] L.V. Sabinin, L.L. Sabinina, L.V. Sbitneva, On the notion of Gyrogroup. Aequationes Mathematicae 56, no. 1, 1998, 11–17. [21] L. Sbitneva, Exceptional smooth Bol loops, Int. Journ. of Mathematics and Mathematical Sciences (IJMMS), vol. 30, no. 9, Hindawi Publishers, 2002. [22] L. Sbitneva, Smooth Bol Loop Actions, Webs and Quasigroups, pp. 73–77, Tver University Press, 2002. [23] T. Nono Sur les familles triples locales de transformations locales de Lie, J. Sci. Hiroshima Univ., Ser. A-I Math. 25 1961, 357–366, [24] Sabinina L.L., On smooth left square distributive quasigroups, Cap. 31 en el Libro: Nonassociative Algebra and its Applications. Vol. 211, 345–348, Marcel Dekker Inc., 2000. [25] Sabinina L., On Kikkawa Spaces, Russian Math. Surveys 58, No. 4, 796–797, 2003. [26] Carrillo Catalan R, Sabinina L, On smooth power alternative loops, Communications in Algebra, Vol. 32, No 8, pp. 2969–2976, 2004. [27] G.P. Nagy, The Campbell-Hausdorﬀ series of local analytic Bruck loops, Abh. Math. Sem. Univ. Hamburg 72, 79–87, 2002, (English). [28] O.A. Matveyev, E.L. Nesterenko, On the quasigroup properties of prosymmetric spaces with zero curvature, Werbs and Quasigroups. Tver, pp. 78–84, 2002. [29] O.A. Matveyev, E.L. Nesterenko, To the theory of prosymmetrik reductive spaces, Vestnik of Peoples’ Friendship University of Russia, no. 7 (1), pp. 114–126, 2000, (Russian).

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Lev V. Sabinin and Larissa V. Sbitneva [30] E.L. Nesterenko, Reductive proabelian spaces, The actual problems of the mathematics and the system of it teaching. Penza University, pp.76–78, 2001, (Russian). [31] E.L. Nesterenko, The algebraic properties of aﬃnely connected on the tangent differentiation, Fundamental Problems of Physics and mathematics, Moscow Government Technology University, pp. 31–45, 2004, (Russian). [32] Matveyev O.A., On quasigroup theory of manifolds with trajectories, Webs and quasigroups, Tver, pp. 129–138, 2000, (English). [33] Matveyev O.A., Panshina A.V., On locally symmetric and abelian mechanical systems, Actual problems of mathematics and its teaching, Penza, pp. 62–68, 2001, (Russian). [34] Matveyev O.A. The quasigroup theory methods in the projective geometry, Actual problems of mathematics and its teaching, Penza, pp. 58–62, 2001, (Russian). [35] Matveyev O.A., Soldatenkov R.M., To the theory of quasigroups in the projective geometry, The fundamental physical and mathematical problems and modelling of technical technological systems, no. 8, Moscow, pp. 27–31, 2005, (Russian). [36] N. Casta˜ neda, Sabinin spaces and Moufang Loops, Algebras Groups and Geometries 18, 259–280, 2001.

Chapter 31 The Lie Multiplication in the Lie Duals of the Witt Algebras Earl J. Taft Department of Mathematics, Rutgers University, Piscataway, New Jersey Hao Zhifeng Department of Applied Mathematics, South China University of Technology, Guangzhou, P.R. China 31.1 31.2 31.3 31.4 31.5 31.6

Lie bialgebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Lie bialgebra structures on the Witt algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Continuous Lie duals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The continuous duals of the Witt algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (i) Lie multiplication in (W1 )0 and (W (i) )0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

395 396 396 397 398 398 399

Abstract For each i ≥ −1, the one-sided Witt algebra W1 = Der k[x], k a ﬁeld of characteristic zero (i) (i) has a Lie bialgebra structure W1 . The Lie bialgebra dual (W1 )0 can be identiﬁed as the (i) space of k-linearly recursive sequences. We consider the Lie algebra structure of (W1 )0 . In particular, for any linearly recursive sequences f = (fn )n≥−1 and g = (gn )n≥−1 , we obtain a recursive relation satisﬁed by [f, g] in terms of recursive relations satisﬁed by f and by g. Analogous results hold for the two-sided Witt algebra W = Der k[x, x−1 ] for each integer i Key words: Witt algebras, Lie bialgebras, linearly recursive sequences. 2000 MSC: 22B62

31.1

Lie Bialgebras

Let k be a ﬁeld of characteristic zero. A Lie algebra L over k has a skew-symmetric product [ , ] i.e., a k-linear map from L ∧ L (the skew-symmetric rank 2 tensors) to L, which satisﬁes the Jacobi identity [[x, y], z] + [[y, z], x] + [[z, x]y] = 0. A Lie coalgebra M over k has a k-linear comultiplication δ from M to M ∧ M satisfying the co-Jacobi identity (1 + σ + σ 2 )(1 ⊗ δ)δ = 0, where σ = (123) acts in the usual way on M ⊗ M ⊗ M . If L is both a Lie algebra and a Lie coalgebra over k, we say that L is a Lie bialgebra if δ is in Z 1 (L, L ∧ L), where L acts on L ∧ L by the adjoint action [a ∧ b, x] = [a, x] ∧ b + a ∧ [b, x]. The condition is that δ([x, y]) = [δx, y] − [δy, x]. If δ = δr is in B 1 (L, L ∧ L) for r in L ∧ L, we say L is a coboundary Lie bialgebra. The condition is that δr (x) = [r, x]. A suﬃcient condition for this is that r satisfy the classical Yang-Baxter equation (CYBE).

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(CY BE)

[r12 , r13 ] + [r12 , r23 ] + [r13 , r23 ] = 0,

where, if r = ai ⊗ bi , r12 = ai ⊗ bi ⊗ 1, r13 = ai ⊗ 1 ⊗ bi and r23 = 1 ⊗ ai ⊗ bi in U (L) ⊗ U (L) ⊗ U (L), U (L) the universal, enveloping algebra of L. A coboundary Lie bialgebra L is said to be triangular if δ = δr for r in L ∧ L satisfying (CYBE):

31.2

Lie Bialgebra Structures on the Witt Algebras

The one-sided Witt algebra W1 = Der k[x] has a basis ei = xi+1 d/dx for i ≥ −1. The Lie product is given by [ei , ej ] = (j − i)ei+j . W1 is a well-known simple Lie algebra. For (i) each i ≥ −1, e0 ∧ ei satisﬁes (CYBE). We denote by W1 the corresponding triangular coboundary Lie bialgebra structure on W1 . The Lie comultiplication δi is described by δi (en ) = n(en ∧ ei ) + (n − i)(e0 ∧ en+i ). The two-sided Witt algebra W = Der k[x, x−1 ] has a basis ei = xi+1 d/dx for i in Z, the integers. The Lie product is again [ei , ej ] = (j − i)ei+j . W is also a simple Lie algebra. For each i in Z, e0 ∧ ei satisﬁes (CYBE). We denote by W (i) the corresponding triangular coboundary Lie bialgebra structure on W . The Lie comultiplication δi is again described by δi (en ) = n(en ∧ ei ) + (n − i)(e0 ∧ en+i ). (i)

See [10] for a discussion of the structure of W1 and W (i) . These are also discussed in (i) [6]. It is proved in [7] that if k is algebraically closed, then W1 gives all the Lie bialgebra structures on W1 up to isomorphism. However, W has Lie bialgebra structures that are not isomorphic to any W (i) .

31.3

Continuous Lie Duals

If (M, δ) is a Lie coalgebra, then the dual space M ∗ = Hom (M, k) is a Lie algebra under the convolution product f ∗ g = (f ⊗ g)δ, i.e., if δm = mi ⊗ ni , then (f ∗ g)(m) = f (mi )g(ni ). If (L, [ , ]) is a Lie algebra, consider [, ]∗ : L∗ → (L ⊗ L)∗ . Identifying L∗ ⊗ L∗ in (L ⊗ L)∗ ∗ in the usual [, ]∗ (V ) ⊂ V ⊗ V . This means that if way, we call a subspace V of L good if ∗ fi ⊗ gi for f, fi , gi in V , then f ([xy]) = fi (x)gi (y) for x, y in L. Let L0 be [, ] (f ) = ∗ the sum of all good subspaces of L . Then (L0 , δ) is a Lie coalgebra, where for f in a good subspace V of L∗ , δ(f ) = [, ]∗ (f ). See [5] for a discussion of this dual. We also note that one can also describe L0 as ([, ]∗ )−1 (L∗ ⊗ L∗ ), in analogy with the associative theory (see [1] and [2]). If L is a Lie bialgebra, then L∗ is a Lie algebra and L0 is a Lie coalgebra. One can show that L0 is a Lie subalgebra of L∗ and that L0 is then a Lie bialgebra. See [10, Proposition 3].

The Lie multiplication in the Lie duals of the Witt algebras

31.4

397

The Continuous Duals of the Witt Algebras

We identify W1∗ as a k-linear space with sequences over k by identifying f in W1∗ with the sequence (fn )n≥−1 , where fn = f (en ). A sequence (fn )n≥−1 is called linearly recursive if it satisﬁes a recursive relation fn = h1 fn−1 + h2 fn−2 + · · · + hr fn−r for some r ≥ 1, some h1 , h2 , · · · , hr in k, and all n ≥ r − 1. We say (fn ) satisﬁes the recursive relation xr − h1 xr−1 · · · − hr−1 x − hr . Examples are the unit sequences e∗i , i ≥ −1, where e∗i has 1 in the ith position and 0 elsewhere, the geometric sequences (an )n≥−1 for a = 0 in k, and more generally (an ni )n≥−1 for a = 0 in k, i ≥ 0. In [8] it is shown that W10 is the space of linearly recursive sequences. Let γ denote the in W10 . We give some examples of comultiplication formulas. γ(e∗i ) = comultiplication ∗ ∗ (l − j)ej ⊗ el . For a geometric series (an ), γ((an )) = (an ) ⊗ (an n) − (an n) ⊗ (an ).

j+l=i

More generally,

γ(an ni ) =

i+1 @ i j=0

j

−

i j−1

A (an nj ) ⊗ (an ni+1−j ).

(i)

Recall that the Lie bialgebra W1 is W1 as a Lie algebra, with comultiplication δi the coboundary of e0 ∧ ei , so δi (en ) = n(en ∧ ei ) + (n − i)(e0 ∧ en+1 ). (i)

Then (W1 )0 is a Lie bialgebra, which as a vector space can be identiﬁed with the linearly (i) recursive sequences. The Lie algebra structure on (W1 )0 comes from the Lie coalgebra (i) (i) structure δi on W1 , via the convolution product on (W1 )∗ . We consider the Lie algebra (i) structure on (W1 )0 considered as the space of linearly recursive sequences. (i)

If i = 0, then (W1 )0 is an abelian Lie algebra. Let i = 0. Then one can easily see that = (j − 2i)e∗j−i for j = 0, [e∗j , e∗i ] = je∗j for j = 0, i, and all other products are zero. This enables one to describe a given coordinate of a product [f, g] of two sequences f and (i) (i) (i) g in (W1 )∗ . It follows from this description that (W1 )∗ , and hence, (W1 )0 , are solvable Lie algebras.

[e∗0 , e∗j ]

In [10], it was left as an open problem to describe the Lie product [f, g] of f and g linearly (i) recursive sequences, i.e., in (W1 )0 , as a linearly recursive sequence. This means giving a recursive relation satisﬁed by [f, g] in terms of recursive relations satisﬁed by f and by g. We give an answer to this question in the next section. An analogous discussion is possible for (W (i) )0 as linearly recursive sequences (fn ) for n in Z, for each i in Z. (see [9]). The only diﬀerence is that here the recursive relation fn = h1 fn−1 + h2 fn−2 + · · · + hr fn−r for ﬁxed r ≥ 1, h1 , · · · , hr in k holds for all n in Z, and requires that hr = 0. In particular, the e∗i for i in Z are not in (W (i) )0 , although one can use (i) Lie multiplication formulas analogous to those for (W1 )∗ to describe the coordinates of a Lie product [f, g] for f and g in (W (i) )∗ . In the next section we describe a recursive relation satisﬁed by [f, g], when f and g are in (W (i) )0 , in terms of recursive relations satisﬁed by f and by g.

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Earl J. Taft and Hao Zhifeng

(i)

Lie Multiplication in (W1 )0 and (W (i) )0 (0)

We ﬁrst assume that k is algebraically closed. W1 has a basis {e∗i }i≥−1 ∪ {(an nj )n≥−1 |a in k ∗ , j ≥ 0}. e∗i satisﬁes xi+2 and (an nj ) satisﬁes (x − a)j+1 . (i) Let f = (fn )n≥−1 and g = (gn )n≥−1 be linear recursive sequences in (W1 )0 . Let f satisfy p(x) = xr0 (x−a1 )r1 · · · (x−ak )rk (x−c1 )rk+1 · · · (x−cn )rk+n , and let g satisfy q(x) = xs0 (x− a1 )s1 · · · (x − ak )sk (x − d1 )sk+1 · · · (x − dm )sk+m , where a1 , · · · , ar , c1 , · · · , cn , dr , · · · , dm are distinct nonzero elements of k; r0 , s0 ≥ 0, r1 , · · · , rk+n , s1 , · · · , sk+m > 0. If i = 0, then [f, g] = 0. We ﬁrst consider i ≥ 1, distinguishing two cases. Case 1: r0 = s0 = 0, k = 1; n = m = 0, r1 = 1 or s1 = 1. Then [f, g] satisﬁes (x − a1 )max(r1 ,s1 ) . Case 2: All other cases for i ≥ 1. [f, g] satisﬁes xmax(r0 ,s0 ) (x − a1 )max(r1 ,s1 )+1 · · · (x − ak )max(rk ,sk )+1 (x − c1 )rk+1 +1 · · · (x − cn )rk+n +1 (x − d1 )sk+1 +1 · · · (x − dm )sk+m +1 . For i = −1, the results are similar to the above, but there are more cases. For example, if r0 = 0 or s0 = 0, the relation for [f, g] starts with xmax(r0 ,s0 )+1 . W (0) has a basis {(an nj )n in Z |a in k ∗ , j ≥ 0}. (an nj ) satisﬁes (x − a)j+1 . For f = (f )n in Z and g = (gn )n in Z in (W (i) )0 , the relations obtained for [f, g] are similar to those (i) obtained for (W1 )0 , except that r0 and s0 do not appear. The proofs are technical, ﬁrst using the description of the general coordinate for the convolution product of basis elements in (W (i) )0 . Then the results for (W (i) )0 are used (i) (i) to get the results for (W1 )0 , after discussing the product of basis elements for (W1 )0 . Details appear in [4]. By a ﬁeld extension argument, one can drop the condition that k be algebraically closed, (i) obtaining the following uniﬁed statement for both (W1 )0 and (W (i) )0 : [f, g] satisﬁes the least common multiple [LCM (p(x), q(x))](1) , where the (1) means that all exponents are raised by one. Some special situations where the (1) can be dropped are discussed in [3].

31.6

Examples

Our results are sharp, i.e., if p(x) and q(x) are the minimal recursive relations for f and g respectively, then there are example where our obtained relation for [f, g] is the minimal recursive relation of [f, g]. However, there also exist examples where the minimal recursive relation for [f, g] has a smaller degree than the one we obtain. √ √ (1) 0, 1, 2,√ 3, · · · ) satisﬁes x(x − 1)2 and g = ( 5, 0, 1, Example 31.1 In (W1 )0 , f = ( 3, √ 4, 9, 16, · · · ) satisﬁes x(x − 1)3 . [f, g] = ( 5 − 3, 0, 0, −4, −18, · · · ), whose nth coordinate for n ≥ 0 is n2 − n3 . Our obtained relation for [f, g] is x(x − 1)4 , which is the minimal recursive relation of [f, g]. √ (1) Example 31.2 In (W1 )0 , f = (π, 1,√ 1, 1, · · · ) satisﬁes x(x − 1) and g = ( 3, π, −1, 0, 1, 2, 3, · · · ), satisﬁes x(x − 1)2 .[f, g] = ( 3 − π, −1, 0, 1, 2, · · · ), whose nth coordinate for n ≥ 0 is n − 1. Our obtained relation for [f, g] is x(x − 1)3 , but the minimal recursive relation of [f, g] is x(x − 1)2 .

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References [1] B. Diarra, On the deﬁnition of the dual Lie coalgebra of a Lie algebra, Publicacions Matem`atiques 39 (1995), 349–354. [2] G. Griﬃng, The dual coalgebra of certain infinite-dimensional Lie algebras, Comm. Algebra 30 (2002), 5715–5724. [3] Z. Hao and L. Feng, A note on the Taft’s problem, Internat. J. of Pure and Appl. Math. 4 (2003), 47–55. [4] Z. Hao and E. J. Taft,The recursive relation on Lie multiplications of Lie duals of Witt and Virasoro algebras (to appear). [5] W. Michaelis, Lie coalgebras, Adv. in Math. 57 (1985), 1–54. [6] W. Michaelis, A class of infinite-dimensional Lie bialgebras containing the Virasoro algebra, Adv. in Math. 107 (1994), 365–392. [7] S.-H. Ng and E. J. Taft, Classification of the Lie bialgebra structures on the Witt and Virasoro algebras, J. Pure Appl. Algebra 151 (2000), 67–88. [8] W. Nichols, The structure of the dual Lie coalgebra of the Witt algebra, J. Pure Appl. Algebra 68 (1990), 359–364. [9] W. Nichols, On Lie and associative duals, J. Pure Appl. Algebra 87 (1993), 313– 320. [10] E. J. Taft, Witt and Virasoro algebras as Lie bialgebras, J. Pure Appl. Algebra 87 (1993), 301–312.

Chapter 32 Simple Decompositions of Simple Lie Superalgebras T.V. Tvalavadze Department of Mathematics and Statistics, Memorial University of Newfoundland, Canada

32.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.2 Decomposition of simple Lie superalgebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.3 Decompositions of simple superalgebra of the type sl(m, n) into the sum of basic Lie subalgebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.3.1 General properties of subalgebras in the decomposition . . . . . . . . . . . . . . . . . . . . . . . 32.3.2 Explicit form of the ﬁrst subalgebra in the decomposition . . . . . . . . . . . . . . . . . . . . . 32.3.3 Explicit form of the second subalgebra in the decomposition . . . . . . . . . . . . . . . . . . 32.3.4 Decompositions of sl(m, n) into the sum of osp(p, k) and osp(l, q) . . . . . . . . . . . . 32.3.5 Main theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.4 Decompositions of simple superalgebras of the type sl(m, n) into the sum of basic and strange Lie subalgebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.4.1 General properties of subalgebras in the decomposition . . . . . . . . . . . . . . . . . . . . . . . 32.4.2 Explicit form of the even part of a strange subalgebra . . . . . . . . . . . . . . . . . . . . . . . . . 32.4.3 Main theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.5 Decompositions of simple superalgebras of the type sl(m, n) into the sum of two strange Lie subalgebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.6 Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

401 402 404 404 409 413 415 415 416 416 421 426 427 429 429

Abstract In this chapter we completely describe all isomorphism types of decompositions of the simple Lie superalgebra sl(m, n) as the sum of two basic simple subalgebras. Key words: reductive (super)algebras (roots) 2000 MSC: 17B05, 17B20

32.1

Introduction

In this chapter we consider simple Lie superalgebras decomposable as the sum of two proper simple subalgebras. Any of these superalgebras have the form of the vector space sum L = A + B where A and B are proper simple subalgebras, which need not be ideals of L, and the sum need not be direct. The structure of these sums has attracted considerable attention, mostly in the cases where A, B are (semi)simple or A, B are nilpotent. The ﬁrst ones arise in the work of Onishchik (see [ON]). Using topological methods he has determined

401

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all decompositions L = L + L where L, L and L are real or complex ﬁnite-dimensional semisimple Lie algebras. Based on the recent paper [BK] devoted to describing simple associative algebras, which can be written as the sum of two proper simple subalgebras over an arbitrary algebraically closed ﬁeld we look at similar question for Lie superalgebras. The main result of this chapter is the following. THEOREM 32.1 Let S be a Lie superalgebra of the type sl(m, n) over the algebraically closed ﬁeld of zero characteristic, and S = S1 + S2 where S1 , S2 are proper classical nonexceptional subalgebras of S. If both m and n are odd, then S has no decompositions into the sum of S1 , S2 . If one of the indices, for example n, is even and the other is odd, then the only possible decomposition is the following: S = S1 + S2 where S1 has the type sl(m, n − 1) and S2 has the type osp(m, n). If both m and n are even, then S admits two types of decompositions: 1. S1 has the type sl(m, n − 1) and S2 has the type osp(m, n), 2. S1 has the type sl(m − 1, n) and S2 has the type osp(n, m).

32.2

Decomposition of Simple Lie Superalgebras

In this section our main goal is to study decompositions of simple Lie algebras over an algebraically closed ﬁeld F of zero characteristic into the sum of two semisimple subalgebras. The classiﬁcation of such decompositions over the ﬁeld of complex numbers was obtained by Onishchik [ON]. THEOREM 32.2 (Onishchik) Any nontrivial irreducible factorization G = G G of connected simple compact Lie group G into the product of two connected subgroup G and G is equivalent to one of the following factorizations: SU2n = Spn · SU2n−1 , n ≥ 2 SO7 = G2 · SO6 , SO7 = G2 · SO5 , SO2n = SO2n−1 · SUn , n ≥ 4, SO4n = SO4n−1 · Spn , n ≥ 2, SO16 = SO15 · Spin9 SO8 = SO7 · Spin7

These factorizations induce the following decompositions of Lie algebras into sum of reductive subalgebras over C : A2n−1 = A2n−2 + Cn B3 = G2 + D3

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B 3 = G2 + B 2 Dn = Bn−1 + An−1 D2n = B2n−1 + Cn D8 = B7 + B5 D4 = B3 + B3 DEFINITION 32.1 Two decompositions L = A1 +B1 and L = A2 +B2 are equivalent if and only if A1 ∼ = A2 and B1 ∼ = B2 . In the following proposition we discuss the decomposition of a Lie superalgebra over an arbitrary algebraically closed ﬁeld F of zero characteristic. PROPOSITION 32.1 Let L = L1 + L2 be a decomposition of simple Lie algebra L over an algebraically closed ﬁeld F of zero characteristic into the sum of two proper simple subalgebras L1 and L2 . Then this decomposition is equivalent to one of the above decompositions. PROOF First we assume that F is an algebraically closed subﬁeld of the ﬁeld of complex numbers C. Let L = L1 + L2 is the given decomposition over F . Then this decomposition induces the following decomposition: L ⊗F C = L1 ⊗F C + L2 ⊗F C It is clear that L ⊗F C is a simple complex Lie algebra, and L1 ⊗F C, L2 ⊗F C are simple complex Lie algebras of the same types as L1 and L2 , respectively. Thus, we have the decomposition of the complex simple Lie algebra into the sum of two semisimple subalgebras. By Theorem 32.2, this decomposition is equivalent to one of above decompositions. Now we consider the general case where F is an arbitrary algebraically closed ﬁeld F of zero characteristic. Let {εi }i=1,...,n , {vj }j=1,...,m , {wk }k=1,...,l be the Chevalley bases for L, L1 , L2 , respectively. For every element εi we denote by εi and εi the elements such that εi = εi + εi , εi ∈ L1 , εi ∈ L2 . ¯ (the algebraic closure of Q) by adjoining all coeﬃcients α Next we extend the ﬁeld Q ij ¯ and αij where εi = j αij vj and εi = j αij wj . Let P denote this ﬁnite extension of Q. ¯ in C can be extended to embedding of P in It is well known that embedding of Q C. Let us deﬁne the following algebras over P : LP = spanP {εi }, LP 1 = spanP {vj }, P P = span {w }. It follows from the construction of the ﬁeld P that L = LP LP k P 2 1 + L2 is a decomposition of simple Lie algebras over P . Next we can consider ﬁeld P as subﬁeld of C. However, as mentioned in ﬁrst part of the proof, all such decompositions are equivalent to the decompositions listed in the hypothesis of the proposition. REMARK 32.1 Proposition 32.1 holds true if L1 and L2 are the direct sums of a semisimple algebra and one-dimensional Lie algebra. To prove this we consider Chevalley bases plus a nonzero element from a one-dimensional Lie algebra.

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32.3 32.3.1

T.V. Tvalavadze

Decompositions of Simple Superalgebra of the Type sl(m, n) into the Sum of Basic Lie Subalgebras General properties of subalgebras in the decomposition

LEMMA 32.1 Let a Lie superalgebra S of the type sl(m, n) be decomposed into the sum of two proper simple subalgebras K and L of the type sl(p, k) and sl(l, q), respectively. Then either p = m and q = n or l = m and k = n. PROOF Without any lose of generality we only prove that p = m and q = n. Let S0 be the even part of S. It is well known that the even part of sl(m, n) has the form: sl(m) ⊕ sl(n) ⊕ U if n = m, n, m ≥ 1 or sl(m) ⊕ sl(n) if n = m ≥ 2, where U is a onedimensional Lie algebra. Therefore if n = m, S0 is decomposable into the direct sum of three ideals P , Q, and U , S0 = P ⊕ Q ⊕ U , and if n = m, S0 is decomposable into the direct sum of two ideals P and Q, S0 = P ⊕ Q. Next, we consider two projections π1 and π2 of S0 onto the ideals P and Q as follows π1 : S0 → P and π2 : S0 → Q. Since K is isomorphic to sl(p, k), it follows that K0 is isomorphic to either sl(p) ⊕ sl(k) ⊕ U , p = k or sl(p) ⊕ sl(k), p = k. Similarly, L0 is isomorphic to either sl(l) ⊕ sl(q) ⊕ U , l = q or sl(l) ⊕ sl(q), p = k. In both cases, K0 and L0 are reductive subalgebras. It follows that the projections π1 (K0 ), π1 (L0 ), π2 (K0 ), and π2 (L0 ) are also reductive as homomorphic images of reductive algebras. Since S = K + L, S0 is decomposable into the sum of two subalgebras K0 and L0 , S0 = K0 + L0 . Therefore, π1 (S0 ) = π1 (K0 ) + π1 (L0 ) and π2 (S0 ) = π2 (K0 ) + π2 (L0 ). Moreover, π1 (S0 ) = P and π2 (S0 ) = Q where P and Q are simple Lie algebras of the types sl(m) and sl(n). Now we have the decompositions of simple Lie algebras P and Q into the sum of two reductive subalgebras. By Remark 32.1, sl(n) cannot be decomposed into the sum of two proper reductive subalgebras of these types. Hence, one of the subalgebras coincides with sl(n). Next we consider the following decomposition: P = π1 (K0 ) + π1 (L0 ). Without any loss of generality, assume that π1 (K0 ) coincides with P . Then π1 (K0 ) is isomorphic to sl(m). Conversely π1 (K0 ) is a homomorphic image of K0 , where either K0 ∼ = sl(p) ⊕ sl(k) ⊕ U or K0 ∼ = sl(p) ⊕ sl(k). Hence sl(p), sl(k) and U are the only possible simple homomorphic images of K0 . It follows, by the previous argument, that p = m or k = m. Without any loss of generality, we assume that p = m. Now let Q = π2 (K0 ) + π2 (L0 ). Then, we are going to prove that π2 (K0 ) is a proper subalgebra of Q. Let us assume the contrary, that is either π2 (K0 ) = 0 or π2 (K0 ) = Q ∼ = sl(n). First let π2 (K0 ) = 0. This implies that π1 (K0 ) = P and K0 = P , which is a contradiction. Next let π2 (K0 ) = Q ∼ = sl(n). Since either K0 ∼ = sl(m) ⊕ sl(k) ⊕ U ∼ or K0 = sl(m) ⊕ sl(k), it follows that either m = n or k = n. Suppose that k = n. Then m = n. Therefore, π1 (K0 ) ∼ = F ⊕H ⊕U = sl(m) and π2 (K0 ) ∼ = sl(m). Since K0 ∼ ∼ ∼ ∼ (K0 = F ⊕ H) where F = sl(m), H = sl(k) and k = m, we have that π1 (F ) = P and π2 (F ) = Q. Because of [F, H] = 0, we have that [π1 (F ), π1 (H)] = 0 and [π2 (F ), π2 (H)] = 0. Therefore, π1 (H) = 0 and π2 (H) = 0, which is wrong. Hence k = n. Since K ∼ = sl(p, k), S ∼ = sl(m, n) and p = m, according to the classiﬁcation of simple Lie superalgebras, we obtain that K = S, but K is a proper subalgebra of S, which is a contradiction. Therefore, π2 (L0 ) coincides with S0 ∼ = sl(n). Acting in the same manner as in the case of P , we obtain that either l = n or q = n. Without any loss of generality, we assume that q = n.

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Now we consider the decomposition of a Lie superalgebra S into the sum of simple subalgebras K and L of the types osp(p, k) and sl(l, q). LEMMA 32.2 Let a simple Lie superalgebra S of the type sl(m, n) be decomposed into the sum of two proper simple subalgebras K and L of the types osp(p, k) and sl(l, q), respectively. Then, without any loss of generality, p = m, l = m − 1, and q = n. PROOF Let π1 and π2 be projections of S0 onto P and Q, respectively. Since K is isomorphic to osp(p, k), then, according to [FS], subalgebra K0 is isomorphic to sp(p) ⊕ so(k). As mentioned above, subalgebra L0 is isomorphic to sl(l) ⊕ sl(q) ⊕ U , l = q and sl(l) ⊕ sl(q), l = q. In both cases, K0 and L0 are reductive (as a direct sum of reductive subalgebras). Therefore, projections π1 (K0 ), π1 (L0 ), π2 (K0 ), π2 (L0 ) are also reductive. Also, we know that S can be represented as the sum of K and L, S = K + L Hence, S0 = K0 + L0 . This implies that π1 (S0 ) = π1 (K0 ) + π1 (L0 ),

(32.1)

π2 (S0 ) = π2 (K0 ) + π2 (L0 ).

(32.2)

Moreover, π1 (S0 ) = P and π2 (S0 ) = Q, where P and Q are simple Lie subalgebras of the types sl(m) and sl(n), respectively. Thus, we obtain two decompositions of Lie algebras P and Q into the sum of reductive subalgebras. By Remark 32.1, sl(n) cannot be decomposed as the sum of two proper reductive subalgebras of these types. Hence, one of subalgebras coincides with algebra sl(n). By Remark 32.1, the only possible decomposition of sl(n) into the sum of two proper reductive subalgebras is sl(n) = A + B,

(32.3)

where A ∼ = sp(n), B ∼ = sl(n−1). Notice that both decompositions (32.1) and (32.2) are nontrivial. Indeed, if both decompositions are trivial then π1 (L0 ) = π1 (S0 ), π2 (L0 ) = π2 (S0 ). Hence, acting in the same manner as in the previous Lemma 32.2, we obtain that l = m, q = n. Thus, L ∼ = sl(l, q) is not a proper subalgebra of S ∼ = sl(m, n), which is impossible. Without any loss of generality we assume that the ﬁrst decomposition is nontrivial. Then π1 (K0 ) ∼ = sp(m). According to (32.3), π1 (L0 ) ∼ = sl(m − 1). Next, we consider the decomposition (32.2). Since π1 (K0 ) ∼ = sp(m) and π2 (K0 ) are homomorphic images of K0 ∼ = sp(m) ⊕ so(k), π2 (K0 ) is not isomorphic to sp(n). Hence, in the following decomposition π2 (S0 ) = π2 (K0 )+π2 (L0 ), π2 (L0 ) coincides with the projection π2 (S0 ). Thus K ∼ = osp(m, k) and L ∼ = sl(m − 1, n). LEMMA 32.3 [A1 , A1 ] = A0 .

Let A = A0 ⊕ A1 be a simple Lie superalgebra. Then [A0 , A1 ] = A1 and

The proof of this lemma is a straightforward calculation. Let us consider S in the standard realization, that is, sl(m, n). In our later discussion we will consider that K is isomorphic to either sl(m, k) or osp(m, k). In the ﬁst case K0 ∼ = sl(m) ⊕ sl(k) ⊕ U , m = k and K0 ∼ = sl(m) ⊕ sl(k), m = k. Algebra K0 can be represented as the direct sum of ideals F , H and U , where F ∼ = sl(m), H ∼ = sl(k) and U is isomorphic to either one-dimensional Lie algebra or zero.

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In the second case K0 ∼ = sp(m) ⊕ so(k). Algebra K0 could be represented as the direct sum of ideals F and H where F ∼ = sp(m), H ∼ = so(k). We know that S0 can be represented as the direct sum of ideals P , Q and U . The projector π2 induces the representation of K0 in the vector space W of the dimension n (Q is of the type sl(n)). Since algebra F ⊕ H ⊂ K0 is semisimple, it follows that the representation π2 acts in W completely reducible. Let W = V1 ⊕ V2 ⊕ . . . ⊕ Vs be the decomposition of W into the direct sum of subspaces, which are invariant and irreducible with respect to the action of F ⊕ H. Then two cases are possible: 1. For each subspace Vi , i = 1, . . . , s, there exists a subalgebra (F or H) such that the action of the subalgebra on this subspace is trivial. In this case we will call the embedding of K into S an embedding of the ﬁrst type. 2. There exists a subspace Vi0 , 1 ≤ i0 ≤ s, such that the action of algebras F and H on this subspace is nontrivial. In this case we will call the embedding of K into S an embedding of the second type. LEMMA 32.4

The projection π1 (H) is 0.

PROOF First let K ∼ = sl(m, k). We are given that π1 (K0 ) is isomorphic to sl(m). If m = k, then π1 (F ) = 0 since H ∼ = sl(k). If m = k, then either π1 (F ) = 0 or π1 (H) = 0. It follows from H ∼ = sl(m) and F ∼ = sl(m) that, say π1 (F ) = 0. Since π1 (F ) = 0 and π1 (K0 ) ∼ = sl(m), then π1 (F ) = π1 (K0 ). Next we consider the case when K ∼ = osp(m, k). We know that the projection π1 (K0 ) is isomorphic to sp(m). Besides the ideal F is also isomorphic to sp(m). We obtain that π1 (F ) = π1 (K0 ), since π1 (F ) = 0. The commutant [F, H] equals zero because subalgebras F and H are terms of the direct sum. This implies that [π1 (F ), π1 (H)] = 0. Since π1 (F ) = π1 (K0 ), it follows that the π1 (H) is a proper ideal in π1 (K0 ). Hence π1 (H) = 0. REMARK 32.2 Let X and Y be two irreducible sets of matrices of orders k × k and m × m, respectively. Let C be an arbitrary nonzero matrix of order m × k. Consider all linear combinations of products Y1 Y2 . . . Yi CX1 X2 . . . Xj where Xi ∈ X, Yj ∈ Y . The set of such combinations coincides with M at(m, k). Indeed the set consisting of X is irreducible. This implies that the associative enveloping algebra of the set equals to M at(k, k). In a similar manner, the associative enveloping algebra of set Y equals to M at(m, m). Since C is nonzero we can choose nonzero coeﬃcient ci0 j0 = 0. Clearly, products of the form Eii0 CEj0 j = ci0 j0 Eij generate M at(m, k). Let V = W + W be a Z2 -graded vector space of (n + m)-vector columns, dim W = m, dim W = n. LEMMA 32.5 Let the embedding of K into S be of the ﬁrst type. Then the projection π2 of F into Q is trivial, π2 (F ) = 0. Moreover, there exists a basis of V such that K has the form: ⎞ ⎛ Y C1 . . . Cs 0 ⎜ D1 X1 . . . 0 0 ⎟ ⎟ ⎜ ⎜ .. .. . . .. ⎟ , ⎜ . . . .⎟ ⎟ ⎜ ⎝ Ds 0 Xs 0 ⎠ 0 0 ... 0 0

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where X1 , . . ., Xs is a set of square matrices of order k, C1 , . . ., Cs is a set of matrices of order m × k, D1 , . . ., Ds is the set of matrices of order k × m, Y is a set of matrices of order m. In addition, the last block can be absent. PROOF Let W1 be the direct sum of subspaces Vi , i = 1, . . . , s such that the action of H on these subspaces are nontrivial, and W2 be the direct sum of subspaces Vi such that the actions of H on these subspaces are nontrivial. We have that W = W1 ⊕ W2 . Let us choose the basis for W from the elements of W1 and W2 . Then π2 (H) can be represented in the following form: X0 , (32.4) 0 0 and π2 (F )

0 0 0Y

,

(32.5)

where X is matrix realization of the representation π2 (H) in W1 , Y is matrix realization of the representation π2 (F ) in W2 . Let us show that Y is trivial. Indeed, there exists a basis for W such that H ⊕ F consists of all matrices of the form: ⎞ ⎛ Y˜ 0 0 ⎝ 0 X 0 ⎠, 0 0 Y where Y˜ is a matrix of order m with a zero trace, and X, Y are matrices from X and Y , respectively. In the new basis, the component K1 takes the following form: ⎛ ⎞ 0 C1 C2 ⎝ D1 0 0 ⎠ . (32.6) D2 0 0 We know that the projection π2 (H) has the form (32.4). According to Lemma 32.4, the projection π1 (H) is zero. Hence H has the form: ⎧⎛ ⎞⎫ ⎨ 0 0 0 ⎬ ⎝0 X 0⎠ , (32.7) ⎭ ⎩ 0 0 0 where X ∈ X. Since the projection π2 (F ) has the form (32.5), F is of the form: ⎧⎛ ⎞⎫ ⎨ Y˜ 0 0 ⎬ ⎝0 0 0⎠ , ⎭ ⎩ 0 0Y

(32.8)

where Y ∈ Y , and Y˜ is a matrix of order m with a zero trace. We denote the set of all matrices Y˜ as Y˜ . Since [K0 , K1 ] ⊆ K1 , it follows that the commutator of a matrix of the form (32.7) and a matrix of the form (32.6) is an element of K1 . Clearly, this commutator is equal to ⎛ ⎞ 0 C1 X 0 ⎝ −XD1 0 0 ⎠ . (32.9) 0 0 0

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Next we consider the commutator of the previous matrix and a matrix of the form (32.8): ⎛

⎞ Y˜ C1 X 0 0 ⎝ XD1 Y˜ 0 0⎠. 0 0 0

(32.10)

Since the representation π2 of H in W1 is completely reducible, W1 is decomposable into the direct sum of subspaces, W1 = W1 ⊕ W2 ⊕ . . . ⊕ Ws , and H acts irreducibly on each subspace. Let dim W1 = k1 , dim W2 = k2 , . . ., dim Ws = ks . There exists a basis for W1 such that X takes the form: ⎞ ⎛ X1 0 . . . 0 ⎜ 0 X2 0 0 ⎟ ⎟ ⎜ (32.11) ⎜ .. . ⎟, ⎝ . 0 . . . .. ⎠ 0

0 . . . Xs

where X1 , X2 , . . ., Xs are matrix realizations of the irreducible representations of H. Without any loss of generality, we consider only X1 . First we choose a basis for subspaces W and W2 . Then we ﬁx a basis for V , which is the union of the bases of the subspaces W1 , W2 and W . Notice, if we represent K1 in the new basis, it is still of the form (32.6). By the above argument, we can represent a matrix C1 from (32.6) in the following form: C1 = C11 C12 . . . C1s , where C11 ∈ M at(m, k1 ), . . ., C1s ∈ M at(m, ks ). Hence, K1 consists of the following elements: ⎞ ⎛ 0 C11 . . . C1s C2 ⎟ ⎜ D11 ⎟ ⎜ ⎟ ⎜ .. ⎟, ⎜ . 0 ⎟ ⎜ ⎠ ⎝ Ds 1 D2

(32.12)

Then Y˜ C1 X has the form: Y˜ C1 X = Y˜ C11 X1 Y˜ C12 X2 . . . Y˜ C1s Xs .

(32.13)

Let us prove that for any i, 1 ≤ i ≤ s, there exists an element (32.12) from K1 such that C1i is nonzero. Assume that i = 1. Consider the commutator of two arbitrary elements from K1 : ⎡⎛ ⎞ ⎛ ⎞⎤ ⎛ ⎞ 0 C11 ∗ 0 0 0 C¯11 ∗ 0 ¯ 1 0 0 ⎠⎦ = ⎝ 0 D1 C¯ 1 + D ¯ 1C 1 ∗ ⎠ . ⎣⎝ D11 0 0 ⎠ , ⎝ D 1 1 1 1 1 ∗ 0 0 ∗ 0 0 0 ∗ ∗ On the one hand, H is the subset of [K1 , K1 ], since, by Lemma 32.3, H ⊂ K0 = [K1 , K1 ]. On the other hand, H has the form (32.11). Hence the set X1 in (32.11) belongs to the set ¯ 1 C 1 . Since X1 is not a zero of all linear combinations of elements of the form D11 C¯11 + D 1 1 subspace, there exists an element from K1 such that C1 is not zero. In the same manner we can prove this for D1 . To apply Remark 32.2, we set X = X1 , C = C11 , Y = Y˜ (from (32.8),(32.11)). As a result we obtain that C11 and D11 could be arbitrary matrices of appropriate orders. First we look at the case when K ∼ = osp(m, k).

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Since k1 is the dimension of the irreducible representation of H and k is the dimension of the standard representation of H then k1 ≥ k. We know that C11 and D11 are arbitrary matrices of orders m × k1 and k1 × m, respectively. Hence the dimension of the subspace of the form (32.10) is greater than or equal to mk1 ≥ mk. Since dim K1 = mk, this subspace coincides with K1 . Therefore, K1 has the form (32.10). Further, we ﬁx two arbitrary elements from K1 : ⎞ ⎛ 0 C1 C2 ⎝ D1 0 0 ⎠ D2 0 0 ⎛

⎞ 0 C1 C2 ⎝ D1 0 0 ⎠ . D2 0 0

and

The commutator of these two elements ⎡⎛ ⎞ ⎛ 0 C1 C2 0 C1 ⎣⎝ D1 0 0 ⎠ , ⎝ D1 0 D2 0 0 D2 0

has the following form: ⎞ ⎞⎤ ⎛ C2 0 ∗0 ⎠. 0 ⎠⎦ = ⎝ 0 ∗ 0 0 0 0 D2 C2 + D2 C2

On the one hand, F is subset of [K1 , K1 ] because by Lemma 32.3, F ⊂ K0 = [K1 , K1 ]. On the other hand, algebra F has the form (32.5). Hence the set Y in (32.5) belongs to the set of all linear combinations of the elements of the form D2 C 2 + D 2 C2 . Since C2 = C2 = D2 = D2 = 0, it follows that Y is zero. Hence, in the case K ∼ = osp(m, k), the lemma is proved. Next we consider the case K ∼ = sl(m, k). Since K ∼ = sl(m, k) then, by [FS], K0 -module K1 is a direct sum of two irreducible submodules M and N each of them has the dimension mk. We choose two nonzero elements xm ∈ M and xn ∈ N of the form (32.12), then the submodules M and N , generated by xm and xn , have the form (32.10). Since M and N are irreducible, M = M , N = N . Hence K1 has the form (32.10). Conversely, by Remark 32.2, if the submodule M (or N ) contains an element x of the form (32.12), where C1s = 0, then M (or N ) contains all elements of the form (32.12), where C1s is a matrix of order m × k1 . Therefore, the dimension of M (or N ) is greater than or equal to mk1 . However, the dimension of M (or N ) is mk. Since k1 is the dimension of an irreducible representation of H, and k is the dimension of the standard representation of H, k1 ≥ k. It follows that the component K1 has the form (32.10), and k1 = k. Acting in the same matter as in the case K ∼ = osp(m, k), we obtain that Y is zero. Therefore, the lemma is proved.

32.3.2

Explicit form of the ﬁrst subalgebra in the decomposition

Now we assume that there exists subspace U = Vi0 , 1 ≤ i0 ≤ s, such that F ⊕ H acts irreducibly in U , moreover F and H act nontrivially in U . LEMMA 32.6

The dimension of U is greater than or equal to mk.

PROOF Let π2 |U be a restriction of π2 to U . To simplify our notation we use π2 for the action H ⊕ F in U .

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To begin with we consider the representation π2 of the subalgebra H in U . This representation is completely reducible because H is simple. Let U = U1 ⊕ U2 ⊕ . . . ⊕ Ur be the decomposition of U into the direct sum of invariant subspaces, such that H acts irreducibly in each subspace. Hence, by [BN], all Ui , i = 1, . . . , r are pairwise isomorphic. In particular, dim U1 = dim U2 = . . . = dim Ur = k1 . Hence, there exists a basis for U such that π2 (H) takes the following block-diagonal form: ⎛ ⎞ X 0 ... 0 ⎜ 0 X ... 0 ⎟ ⎜ ⎟ ⎜ .. .. . . .. ⎟ , ⎝ . . . . ⎠ 0 0 ... X where X is the set of matrices of order k1 . Let m1 be the number of the diagonal blocks. Next we take a close look at π2 (F ). Since π2 (H)π2 (F ) = π2 (F )π2 (H), π2 (F ) has the form: ⎛ ⎞ y11 Ik1 y12 Ik1 . . . y1m1 Ik1 ⎜ y21 Ik1 y11 Ik1 . . . y2n1 Ik1 ⎟ ⎜ ⎟ ⎜ ⎟, .. .. .. .. ⎝ ⎠ . . . . ym1 1 Ik1 yn1 2 Ik1 . . . ym1 m1 Ik1 where yij ∈ F . The dimension of an irreducible representation of F is greater than or equal to m because either F ∼ = sl(m) or F ∼ = sp(m). Therefore m1 ≥ m. In a similar manner we can show that k1 ≥ k. We obtain that the dimension U is equal to m1 k1 ≥ mk. COROLLARY 32.1 Let S = K + L, and the embedding of the subalgebra K into S is of the second type. It follows that the embedding of the subalgebra L into S is of the ﬁrst type. PROOF According to the previous lemma, n ≥ mk. If the embedding of L into S is of the second type, then m ≥ ln. Hence, n ≥ mk ≥ lnk. This contradicts the fact that H∼ = sl(k), k > 1. LEMMA 32.7 Let S = K + L where K ∼ = sl(m, k) or osp(m, k), k < n, L ∼ = sl(l, n) and the embedding of K into S is of the ﬁrst type. Then there exists a basis for V (deﬁned above) such that K takes the form: ⎧⎛ ⎞⎫ ⎨ Y C 0 ⎬ ⎝D X 0⎠ , (32.14) ⎩ ⎭ 0 0 0 where X is a set of matrices of order k, C is of order m × k, D is of order k × m and Y is of order m. PROOF According to Lemma 32.5, W1 = W1 ⊕W2 ⊕. . .⊕Ws . We know that dim W1 = dim W2 = . . . = dim Ws = k. Next we are going to prove that s = 1. Let us assume the contrary that is s = 1. We take a close look at C11 and C12 from (32.12). We prove that C12 = λC11 where λ = 0 ∈ F . Indeed, if C11 and C12 are not proportional then denote C11 as A and C12 as B. Then there exist nonzero coeﬃcients ai0 j0 , ak0 l0 of the matrix

Simple decompositions of simple Lie superalgebras

411

A and coeﬃcients bi0 j0 , bk0 l0 of the matrix B such that bi0 j0 = λai0 j0 and bk0 l0 = μak0 l0 where λ = μ. Notice that there exists an appropriate basis for V such that algebra H ⊕ F takes the following forms: ⎧⎛ ⎞⎫ Y 0 . . . 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎨⎜ 0 X 0 0 ⎟⎪ ⎜ ⎟ H ⊕F = ⎜ . ⎟ . ⎪ ⎝ .. 0 X .. ⎠⎪ ⎪ ⎪ ⎪ ⎪ ⎭ ⎩ 0 0 ... ∗ or ⎧⎛ ⎞⎫ ⎪ ⎪ Y 0 ... 0 ⎪ ⎪ ⎪ ⎨⎜ 0 X 0 0 ⎟⎪ ⎬ ⎜ ⎟ H ⊕F = ⎜ . ⎟ , . ⎪⎝ .. 0 −X t .. ⎠⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎭ 0 0 ... ∗ where, in the case K ∼ = sl(m, k), Y ∈ sl(m), X ∈ sl(k), in the case K ∼ = osp(m, k), Y ∈ sp(m), X ∈ so(k). Indeed, up to equivalence, there exist only two irreducible representations of sl(n) in V , dimV = n. The standard realizations of these representations have the following forms ρ1 : sl(n) → gl(n), ρ1 (X) = X and ρ2 : sl(n) → gl(n), ρ1 (X) = −X t . Similarly, so(n) has a standard representation of the following form ρ : so(n) → gl(n), ρ(X) = X. In both cases, by (32.10) and (32.13), K1 contains elements of the form: ⎞ ⎛ 0 Y AX Y BX ∗ ⎟ ⎜∗ ⎟ ⎜ (32.15) ⎠ ⎝∗ 0 ∗ where Y is an arbitrary matrix from sp(m), X is an arbitrary matrix from so(k). Since sp(m) and so(k) are irreducible subsets of M at(m) and M at(k), respectively, their associative enveloping algebras are M at(m) and M at(k), respectively. Then, by Remark 32.2, K1 contains an element of the form (32.15) where Y and X are arbitrary matrices of appropriate orders. Next we choose an element x1 ∈ K1 such that Y = E1i0 and X = Ej0 1 . Then Y AX = E1i0 AEj0 1 = ai0 j0 E11 , Y BX = E1i0 BEj0 1 = bi0 j0 E11 = λai0 j0 E11 . If we set Y = E1k0 and X = El0 1 , then Y AX = E1k0 AEl0 1 = ak0 l0 E11 , Y BX = E1k0 BEl0 1 = bk0 l0 E11 = μak0 l0 E11 . Hence K1 contains an element of the form (32.12) where C11 = C12 =

1 ai0 j0

(ai0 j0 E11 ) −

1 ak0 l0

(ak0 l0 E11 ) = 0,

1 1 (λai0 j0 E11 ) − (μak0 l0 E11 ) = (λ − μ)E11 = 0. ai0 j0 ak0 l0

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Let K0 -submodule M be a submodule in K1 generated by x1 . Then the submodule M contains an element of the form (32.12) where C11 = 0 and C12 = 0. Likewise, M contains an element of the form (32.12) where C11 = 0 and C12 = 0. Therefore, by Remark 32.2 and (32.13), M contains an element of the form (32.12) where C11 and C12 are arbitrary matrices of order m × k. Hence the dimension of M is greater than or equal to 2mk. According [FS], if K ∼ = sl(m, k) then K0 - module K1 is the direct sum of two irreducible submodules of the dimension mk, and if K ∼ = osp(m, k) then K0 - module K1 is an irreducible submodules of a dimension mk. Therefore the submodule M cannot be a submodule of a dimension 2mk because it is irreducible, a contradiction. Hence C12 = λC11 where λ = 0 ∈ F . First we consider the case K ∼ = sl(m, k). As was shown above, K1 contains an element of the form (32.12) where C11 and D11 are arbitrary matrices. Let D11 = E11 = (eij )i=1...m i=1...k where e11 = 1, and the other elements are zero. Then there exists an element x ∈ K1 such that C11 = Z where Z is an arbitrary matrix of order m × k and D11 = E11 . Hence C12 = λC11 = λZ where λ = 0. Let us consider an element [x , x ] ⊂ K0 : ⎞ ⎛ ⎞⎤ ⎛ ⎡⎛ ⎞ 0 Z λZ ∗ ∗ 0 0 Z λZ ∗ ⎟ ⎜ E11 ⎟⎥ ⎜ E11 Z λE11 Z ∗ ⎟ ⎢⎜ E11 ⎟,⎜ 2 ⎟⎥ = ⎜ ⎢⎜ 2 ⎟. ⎝ ⎠ ⎣⎝ D1 0 D1 0 ⎠⎦ ⎝ 0 D12 Z λD12 Z ∗ ⎠ ∗ ∗ ∗ ∗ ∗ Notice that λE11 Z is not identical zero since Z is an arbitrary matrix and λ = 0. However, according to the form (32.11), λE11 Z must be equal to zero. Therefore, we prove the lemma in the case K ∼ = sl(m, k). Next we consider the case K ∼ = osp(m, k). The dimension of K1 is mk and the dimension of L1 is 2ln. Since dim S1 ≤ dim K1 + dim L1 it follows that 2mn ≤ mk + 2ln. Moreover k < n. Therefore we obtain 2ln ≥ 2mn − mk > 2mn − mn = mn. It follows that l > m/2. Now we prove that the embedding of L ∼ = sl(l, n) into S ∼ = sl(m, n) is an embedding of ∼ the ﬁrst type. Since L = sl(l, n), it follows that L0 = H ⊕ F ⊕ U where H ∼ = sl(l), F ∼ = sl(n) and U was deﬁned above. Let us assume that the embedding of L into S is of the second type. Then, by Lemma 32.5, we have that m ≥ ln. We have proved that l > m/2; hence n ≤ 1. Since the embedding of K into S is of the second type, F = 0 and H = 0. Moreover F = 0 ∼ = sl(n). Thus n > 1, which is a contradiction. Therefore, the embedding of L into S is of the ﬁrst type. By using Lemma 32.5 for L and π1 , we obtain that π1 (F ) = 0 and L takes the form: ⎛

0 0 ⎜ 0 Z1 ⎜ ⎜ .. ⎜. ⎜ ⎝0 0 0 B1

⎞ ... 0 0 0 A1 ⎟ ⎟ . . .. .. ⎟ , . . . ⎟ ⎟ ⎠ . . . Zr Ar . . . Br Y

where Z1 , . . ., Zr are matrices of order l, A1 , . . ., Ar are matrices of order l × n, B1 , . . ., Br are matrices of order n × l and Y is a matrix of order n × n. Moreover the ﬁrst zero block can be absent. Since m > l > m/2, it follows that r = 1 where r is the number of blocks. Hence the ﬁrst rows of all matrices from L are zero. We know that S1 = K1 + L1 . This implies that the ﬁrst rows of all matrices from K1 are arbitrary. On the other hand, the component K1 has the form (32.12) where C12 = λC11 . This implies that the ﬁrst row of matrices from K1 cannot be arbitrary, which is a contradiction.

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413

REMARK 32.3 Let K ∼ = sl(m, k) be a subalgebra of S ∼ = sl(m, n), and the embedding of K into S is of the ﬁrst type. Then K has the form (32.14). Indeed, in the proof of Lemma 32.6 for the case K ∼ = sl(m, k) we do not use the condition S = K + L.

32.3.3

Explicit form of the second subalgebra in the decomposition

LEMMA 32.8 Let S = K + L, and the embedding of L into S be of the ﬁrst type. Then the embedding of K into S is of the ﬁrst type. PROOF Case 1. K ∼ = osp(m, k). Let us assume the contrary, that is the embedding of K into S is of the second type. It is well known that the dimension of the odd component of S ∼ = sl(m, n) is 2mn. The dimensions of the odd components of K ∼ = osp(m, k) and L ∼ = sl(l, n) are mk and 2ln, respectively. Hence the dimension of the odd component of L is less than or equal to 2(m − 1)n. Clearly dim S1 ≤ dim K1 + dim L1 . Thus mk = dim K1 ≥ dim S1 − dim L1 ≥ 2n. Conversely, by Lemma 32.5, n ≥ dim U ≥ mk, which is a contradiction. Case 2. K ∼ = sl(m, k). As above we assume the contrary, that is the embedding of K into S is of the second type. The dimension of the odd component of S ∼ = sl(m, n) is 2mn. The dimensions of the odd components of K ∼ = sl(m, k) and L ∼ = sl(l, n) are 2mk and 2ln, respectively. Hence the dimension of the odd component of L is less than or equal to 2(m − 1)n. We know that dim S1 ≤ dim K1 + dim L1 . Hence 2mk = dim K1 ≥ dim S1 − dim L1 ≥ 2n. By Lemma 32.5, n ≥ dim U ≥ mk. Finally, n = mk. Using the notation from Lemma 32.5 we obtain that m1 = m, k1 = k, and, moreover, there exists a basis for W such that the projection π2 (H) takes the form: ⎛

X ⎜0 ⎜ ⎜ .. ⎝ .

0 ... X ... .. . . . . 0 0 ...

0 0 .. .

⎞ ⎟ ⎟ ⎟, ⎠

(32.16)

X

where X ∈ X = sl(k) On the other hand, π2 (F ) has the form: ⎛

y11 Ik y12 Ik ⎜ y21 Ik y11 Ik ⎜ ⎜ .. .. ⎝ . . ym1 Ik yn1 2 Ik

⎞ . . . y1m Ik . . . y2n Ik ⎟ ⎟ .. ⎟ , .. . . ⎠ . . . ymm Ik

where (yij )i,j=1...m ∈ Y = sl(m). Let us use tensor notation Im ⊗ X and Y ⊗ Ik for π2 (H) and π2 (F ), respectively. Next we consider components K0 and K1 in the matrix form: A 0 , (32.17) 0 B and

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0 C D 0

,

(32.18)

where A, B, C and D are matrices of appropriate orders. We want to prove that K1 contains an element of the form (32.18) where D has the following form: ⎛ ⎞ → → w . . . λ1m − w λ11 − ⎜ .. ⎟ .. .. (32.19) ⎝ . ⎠, . . → → w ... λ − w λ − m1

mm

→ w ∈ F k . Equivalently, D has the where Λ = (λij )i,j=1...m is a matrix of the order m and − → − following tensor form Λ ⊗ w . Indeed the embedding of L into S is of the ﬁrst type. Hence, by Lemma 32.8, there exists a basis for V such that the ﬁrst row and column of matrices from L are zero. On the other hand, S = K + L and S ∼ = sl(m, n). This implies that the → ﬁrst row and column of K can be arbitrary. Next we choose any nonzero element − w from k F . Then, as shown above, K contains an element x1 of the form (32.18) such that the block D has the following form: ⎛− −−−→ ⎞ → w − v→ 11 . . . v1m−1 ⎜ .. .. .. .. ⎟ , ⎝ . . . . ⎠ − − − → − − − → w v . . . v −→ m1

mm−1

j=1..m−1 ∈ F k and the ﬁrst column of C is zero. where (− v→ ij )i=1..m → − Now we will show that for any i, j vectors − v→ ij and w are linearly dependent. → − Let us assume the contrary, that is, there exist i, j such that − v→ ij and w are linearly − → → w = 0. We independent. Then we ﬁx a matrix X0 ∈ sl(m) such that X0 vij = 0 and X0 − know that A @ A 0 0 C 0 AC − CB , = . 0 B D 0 BD − DA 0

Let x0 be an element from H such that in matrix realization (32.17) B has the form (32.16) with X equals X0 . Then A from (32.16) is zero because, by Lemma 32.4, π1 (H) = 0. Set x = [x0 , x1 ]. Next we point out the explicit form of the matrix BD − DA. Since A = 0, we have BD − DA = BD and BD has the form: ⎛ − −−−→ ⎞ w X0 − v→ X0 → 11 . . . X0 v1m−1 ⎟ ⎜ .. .. .. .. ⎠. ⎝ . . . . → − − − → − − − − → X w X v ... X v 0

0 m1

0 mm−1

− w = 0, the ﬁrst row of the matrix BD −DA is zero. However, BD −DA is nonzero Since X0 → v→ because X0 − ij = 0. ˜ Let K be a subset of matrices from K1 such that the ﬁrst rows or columns of these ˜ = 2mk, and x is not an element of K. ˜ matrices are nonzero. It is easy to verify that dimK Thus we have 2n + 1 = 2mk + 1 linearly independent elements from K1 . This contradicts → w . Let us the fact that the dimension of K1 equals to 2mk. Finally, D has the form Λ ⊗ − consider an element from H of the form (32.17) such that A = 0, B = Im ⊗ X. Therefore → → → w ) = Im Λ ⊗ X − w = Λ ⊗ X− w = Λ ⊗ z where z = Xw. Since X BD − DA = (Im ⊗ X)(Λ ⊗ − is an arbitrary element from sl(k), z can be an arbitrary vector from F k . Next we consider an element from F of the form (32.17) such A = Y , B = Y ⊗ Ik . Then → → → → → w − ΛY ⊗ − w = (Y Λ − ΛY ) ⊗ − w. w ) − (Λ ⊗ − w )(Y ⊗ I) = Y Λ ⊗ Ik − BD − DA = (Y ⊗ Ik )(Λ ⊗ −

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415

Let Z denote the commutator [Y, Λ] = Y Λ − ΛY . We can repeat be previous calculation → → but this time use D = Z ⊗ − w instead of D = Λ ⊗ − w . Therefore K1 contains an element → − of the form (32.18) where D equals to [Y, Z] ⊗ w . Next we consider the ideal generated by the matrix Λ in sl(m). Since Λ is not a scalar matrix, this ideal coincides with sl(m). Clearly K1 contains elements of the form (32.18) where D is an arbitrary matrix of the form Z ⊗ z, Z ∈ sl(m) and z ∈ F n . Hence the dimension of K1 is greater than or equal to (m2 − 1)n, which is a contradiction.

32.3.4

Decompositions of sl(m, n) into the sum of osp(p, k) and osp(l, q)

Now we consider the decomposition of a simple Lie superalgebra S ∼ = sl(m, n) into the ∼ sum of simple subalgebras K ∼ osp(p, k) and L osp(l, q) . Notice that = = K0 ∼ = sp(p) ⊕ so(k), L0 ∼ = sp(l) ⊕ so(q).

(32.20)

Since S = K + L, it follows that S = K0 + L0 . Let π1 and π2 denote two projectors of S0 into ideals P and Q deﬁned above. Hence π1 (S0 ) = π1 (K0 ) + π1 (L0 ) and π2 (S0 ) = π2 (K0 )+π2 (L0 ). According to (32.20) π1 (K0 ) and π1 (L0 ) are either simple algebras or direct sums of two simple algebras or zero. However, according to Remark 32.1, π1 (S0 ) ∼ = sl(n) cannot be decomposed into the sum of two simple subalgebras of these forms. Therefore S∼ = sl(m, n) cannot be decomposed into the sum of simple subalgebras K ∼ = osp(p, k) and L∼ osp(l, q). =

32.3.5

Main theorem

THEOREM 32.3 A simple Lie superalgebra S ∼ = sl(m, n) cannot be decomposed into the sum of the following subalgebras: 1. K ∼ = sl(p, k) and L ∼ = sl(l, q). 2. K ∼ = osp(p, k) and L ∼ = sl(l, q) where k < n. 3. K ∼ = osp(p, k) and L ∼ = osp(l, q). PROOF According to Lemma 32.8, the embedding of K and L into S is of the ﬁrst type. Therefore, by Lemma 32.6, K and L have nonzero annihilators. Then the associative algebras generated by K and L also have nonzero annihilators in M at(m, n). Moreover, the sum of these associative algebras equals to M at(m, n) because K + L = sl(m, n). This contradicts the fact that a simple associative algebra cannot be decomposed into the sum of proper simple subalgebras [BK]. ∼ sl(m, n), where n is even, in the Example 32.1 We consider a Lie superalgebra S = standard matrix realization. Let the ﬁrst subalgebra S1 be the following: ⎧⎛ ⎞⎫ 0 ⎪ ⎪ ⎪ ⎪ ⎨⎜ ⎟⎬ ⎜ X 0⎟ ⎝ 0 ⎠⎪ ⎪ ⎪ ⎪ ⎩ ⎭ 0 0 00 where X is a matrix of order (m + n − 1) × (m + n − 1). The second subalgebra S2 consists

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of all matrices of the form:

⎧⎛ ⎞⎫ A C D ⎬ ⎨ ⎝ −Dt E F ⎠ ⎭ ⎩ C t H −E t

where A is a skew-symmetric matrix of order m, H and F are symmetric matrices of order n n n n n 2 × 2 , E is a matrix of order 2 × 2 , and C, D are matrices of order m × 2 . Then S1 ∼ = sl(m, n − 1), S2 ∼ = osp(m, n) and S = S1 + S2 is a decomposition of simple Lie superalgebra into the sum of simple subalgebras.

32.4 32.4.1

Decompositions of Simple Superalgebras of the Type sl(m, n) into the Sum of Basic and Strange Lie Subalgebras General properties of subalgebras in the decomposition

In this section we study decompositions of the Lie superalgebra S ∼ = sl(m, n) into the sum of two simple subalgebras K ∼ = P (k) (or Q(k)) and L ∼ = sl(l, q). LEMMA 32.9 Let S ∼ = sl(m, n) be decomposable as the sum of two proper simple subalgebras K ∼ = P (k) (or Q(k)) and L ∼ = sl(l, q). Then, without any loss of generality, k = m − 1 q = n. PROOF Let π1 and π2 be the projectors of S0 into P and Q, respectively. Since K is isomorphic to either P (k) or Q(k), K0 is isomorphic to sl(k + 1). We know that if l = q, then L0 ∼ = sl(l) ⊕ sl(q) ⊕ U , and if l = q, then sl(l) ⊕ sl(q). Hence K0 and L0 are reductive subalgebras. Thus π1 (K0 ), π1 (L0 ), π2 (K0 ) and π2 (L0 ) are also reductive as homomorphic images of reductive algebras. Since S0 = K0 + L0 , π1 (S0 ) = π1 (K0 ) + π1 (L0 ) and π2 (S0 ) = π2 (K0 ) + π2 (L0 ). Moreover π1 (S0 ) = P and π2 (S0 ) = Q where P and Q are simple Lie algebras of the types sl(m) and sl(n), respectively. Thus we obtain the decompositions of simple Lie algebras P and Q into the sum of two reductive subalgebras. On the other hand, by Remark 32.1, sl(n) cannot be decomposed into the sum of two subalgebras of these forms. Therefore one of these subalgebras coincides with sl(n). First we consider the decomposition of P . Acting in the same manner as in Lemma 32.1, we obtain that both equations π1 (L0 ) = P and π2 (L0 ) = Q cannot hold true. Let π1 (L0 ) = P . Hence π1 (K0 ) = P ∼ = sl(m). However π1 (K0 ) ∼ = sl(k + 1). Thus m = k + 1. Next we prove that either q = n or l = n. Let us assume the contrary, that is, q < n and l < n. Then π2 (L0 ) is a proper subalgebra of Q. Hence π2 (K0 ) = Q. Since π2 (K0 ) ∼ = sl(k + 1), it follows that n = k + 1 = m and π1 (K0 ) ∼ = π2 (K0 ) ∼ = sl(m). It is well known that any automorphism of sl(m) is either ϕ C1 (X) = CXC −1 or ϕ C2 (X) = t −1 −CX C where C is a nondegenerate matrix of order m. Therefore, acting by appropriate automorphisms ϕ1 or ϕ2 , K0 can be reduced to one of the following form:

5 X 0 ϕ1 (K0 ) = 0 −X t

Simple decompositions of simple Lie superalgebras

or ϕ2 (K0 ) =

X 0 0 X

417

5 .

Let us consider the ﬁrst case. We have sl(m) = π1 (ϕ1 (K0 )) + π1 (ϕ1 (L0 )), sl(m) = π2 (ϕ1 (K0 )) + π2 (ϕ1 (L0 )). Therefore, for any Y , Z ∈ sl(m) we obtain

Y = X + Y ,

(32.21)

Z = −X t + Z ,

(32.22)

where Y ∈ π1 (ϕ1 (L0 )), Z ∈ π2 (ϕ1 (L0 )). Hence −(Z)t = −(−X t + Z )t = X − Z . t

(32.23)

By subtracting Eq. (32.23) from Eq. (32.21), we have that Y − (−(Z)t ) = Y + Z t = X + Y − X + Z = Y − Z . t

t

Clearly Y + Z t is an arbitrary element from sl(m), and Y , −Z t are arbitrary elements from π1 (ϕ1 (L0 )), π2 (ϕ1 (L0 )), respectively. Thus we obtain the proper decomposition of sl(m) into the sum of two simple subalgebras sl(l) and sl(q), which is a contradiction. Similarly the second case is not possible. Therefore q = n. LEMMA 32.10 Let a subalgebra K of a Lie superalgebra S be isomorphic to either P (m − 1) or Q(m − 1), and π1 (K0 ) = P . Then π2 (K0 ) = 0. PROOF Let us assume the contrary, that is π2 (K0 ) = 0. Then there exists a basis such that K0 consists of matrices of the form: X0 , (32.24) 0 0 where X is an arbitrary matrix of the order m with zero trace. Let us consider the commutator of an element from K0 and an element from K1 : @ A X0 0 C 0 XC , = . 0 0 D 0 −DX 0 Hence,

@

0 XC −DX 0

,

0 XC −DX 0

A

=2

−XCDX 0 0 −DXXC

.

(32.25)

Since K is isomorphic to P (m − 1), according to the classiﬁcation of simple Lie superalgebras, m ≥ 3. Hence we can choose three matrix unit Eik , Ekj , Eik where i, j, k are pairwise diﬀerent. Therefore (Eik + Ekj )(Eik + Ekj ) = Eij . Let X be Eik + Ekj . Then XX equals to Eij where i = j. Conversely, if X is equal to Eii − Ejj , then XX = (Eii − Ejj )(Eii − Ejj ) = Eii − Ejj . Now we will show that K1 contains an element of the form: 0 C , (32.26) D 0 where C = 0.

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Again let us assume the contrary. Then any element from K0 has the form (32.26) where C = 0. We consider the commutator of two elements of the form (32.26): @ A 0 0 0 0 00 , = . D 0 D 0 00 However, by Lemma 32.3, [K1 , K1 ] = K0 = 0, which is a contradiction. Similarly, K0 contains an element of the form (32.26) where D = 0. Therefore K0 contains an element of the form (32.26) where C = 0 and D = 0. Let di0 j0 be a nonzero entry of D. Now let us ﬁx the matrix C. The following two cases are possible. 1. C contains a nonzero coeﬃcient ck0 l0 where k0 = j0 . 2. The coeﬃcients ck0 l , k0 = j0 are zero and there exists a nonzero coeﬃcient cj0 l0 = 0. First we look at the case 1. Let, in the kk0 . Then XX = formula (32.25),X be Ej0 k + E Ej0 k0 , k0 = j0 . Hence −DXXC = −( ij dij Eij )Ej0 k0 ( pr cpr Epr ) = ir dij0 ck0 r Eir = 0 since di0 j0 ck0 l0 Ei0 l0 = 0. However, according to (32.24), −DXXC = 0, which is a contradiction. Next we consider the second case. Let, in the formula (32.25), X be Ei0 i0 − Ej0 j0 then XX = Ei0 i0 − Ej0 j0 . Hence −DXXC = −D(Ei0 i0 − Ej0 j0 )C = −DEi0 i0 C + DEj0 j0 C. Since ck0 l = 0 for k0 = j0 it follows that Ei0 i0 C = 0. Hence −DXXC = DEj0 j0 C = − ir dii0 ci0 r Eir + ir dij0 cj0 r Eir = 0. This contradicts the fact that −DXXC = 0. LEMMA 32.11 Let S = K + L, and K be isomorphic to either P (m − 1) or Q(m − 1), L be isomorphic to sl(l, n). Then there exists a basis such that L1 takes the form

5 0 C , D 0 where C is a matrix of order m × n, D a matrix of order n × m. Moreover, the ﬁrst row of C and the ﬁrst column of D are zero. PROOF At ﬁrst, we prove that the embedding L into S is of the ﬁrst type. Let us assume the contrary, that is the embedding L into S is of the second type. Then, by Lemma 32.5, m ≥ ln. Since dim S1 ≤ dim K1 + dim L1 , 2mn ≤ m2 + 2ln. According to Lemma 32.10, m ≤ n. By using the previous equations, we obtain that m ≤ 2l. However, according to the classiﬁcation of simple Lie superalgebras, m ≥ 3. Thus l ≥ 2. This contradicts the fact that m ≥ ln. Thus the embedding L into S is of the ﬁrst type. Therefore, by Remark 32.3, there exists a basis for V such that the ﬁrst row and the ﬁrst column of L0 are zero. LEMMA 32.12

Let K be a simple Lie superalgebra, and K0 has the form: X 0 , 0 −X t

(32.27)

where X = sl(m). ˜ the K0 -submodule of K1 generated by x1 . Choose an element x1 ∈ K1 and denote by K (a) If 0 C , (32.28) x1 = D 0

Simple decompositions of simple Lie superalgebras ˜ where C is a nonzero symmetric matrix of order m, then Kcontains 0 C , D 0

419

(32.29)

where C is an arbitrary symmetric matrix of order m. (b) If x1 is of the form (32.28), where C is a nonzero skew-symmetric matrix of order m, ˜ contains the matrices of the form (32.29) where C is an arbitrary skew-symmetric then K matrix of order m. (c) If x1 is of the form (32.28), where C is not a symmetric or skew-symmetric matrix, ˜ contains the matrices of the form (32.29) where C is an arbitrary matrix of order then K m. PROOF

The commutator of any two elements from K0 and K1 has the form: @ A X 0 0 C 0 XC + CX t . , = −(X t D + DX) 0 0 −X t D 0

Case (a). Let K1 contain an element x1 (32.29) where C = C is a nonzero symmetric matrix. We will prove that for any symmetric matrix C there exists an element from K1 of the form (32.29) where C = C . Indeed, we choose an element x0 is in (32.27) from K0 such that X = X1 where X1 is a symmetric matrix with a zero trace. Then [x0 , x1 ] has the form (32.29) and X1 C + CX1 t = X1 C + CX1 = 2X1 . C. Notice that X1 . C is a product in the Jordan algebra of symmetric matrices. Next we choose an element y0 of the form (32.27) from K0 such that X = X2 where X2 is a symmetric matrix. Clearly [y0 , [x0 , x1 ]] has the form (32.29) where C = 4X2 . (X1 . C ). Thus the set of all matrices C forms the ideal of a Jordan algebra of all symmetric matrices. Because of the simplicity of this Jordan algebra, the ideal generated by C coincides with the entire algebra. Therefore, for any symmetric matrix C there exists an element from K1 of the form (32.29) such that C = C . Case (b). The proof of this case is similar to the proof of the case (a), but instead of the Jordan product X1 C + CX1 t = X1 C + CX1 = 2X1 . C we take the Lie product X1 C + CX1 t = X1 C − CX1 = [X1 , C]. Case (c). Let us represent the matrix C in the form C = Csymm +Cskew where Csymm is a nonzero symmetric matrix, Cskew is a nonzero skew-symmetric matrix. Notice that XCsymm + Csymm X t is a symmetric matrix, and XCskew + Cskew X t is a skew-symmetric matrix. + Cskew . As Hence XC + C X t = (XCsymm + Csymm X t ) + (XCskew + Cskew X t ) = Csymm was shown in previous cases, K1 contains an element of the form (32.28) such that Csymm is an arbitrary symmetric matrix. Similarly, K1 contains an element of the form (32.28) is an arbitrary skew-symmetric matrix. such that Cskew The dimension of the algebra of all symmetric matrices of order m is (m2 + m)/2, and the dimension of an algebra of all skew-symmetric matrices is (m2 − m)/2. This implies + Cskew where that K1 contains an element of the form (32.29) such that C = Csymm 2 2 Cskew = 0 since (m + m)/2 > (m − m)/2. Hence K1 contains elements of the form (32.29) where C is an arbitrary symmetric matrix. In particular C = Csymm . Since C − Csymm = (Csymm + Cskew ) − Csymm = Cskew = 0, it follows that K1 contains elements of the form (32.29) where C is an arbitrary skew-symmetric matrix. Finally, an arbitrary matrix can be obtained as a sum of appropriate symmetric and skew-symmetric matrices.

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REMARK 32.4 Let S = K + L. Then K1 contains an element of the form (32.29), where C is an arbitrary symmetric matrix of order m. Indeed, according to Lemma 32.11, the ﬁrst columns and rows of matrices from L are zero. Hence, the ﬁrst columns and rows of matrices from K are also arbitrary. In particular, j=1...m K1 contains an element of the form (32.29) where C = (cij )i=1...m , c11 = 1, c1i = 0, i > 1. Clearly C is not a skew-symmetric matrix. Therefore, by Lemma 32.12(a) and (c), K1 contains elements of the form (32.29) where C is an arbitrary symmetric matrix of order m. Similarly, K1 contains an element of the form (32.29) where D is an arbitrary symmetric matrix of order m. REMARK 32.5 Let S = K + L, and K1 contain a nonzero element x1 of the form (32.29) where D is a nonzero matrix of order m. Then C is a nonzero skew-symmetric matrix. Let us assume the contrary, that is, the matrix C is not skew-symmetric. Then, according Lemmas 32.12(a) and 32.12(c), K1 contains a subspace of elements of the form (32.29) where C is an arbitrary symmetric matrix and D is zero. The dimension of this subspace is m(m + 1)/2. By the previous remark, K1 contains a subspace of elements of the form (32.29) where D is an arbitrary symmetric matrix. Thus there exist two subspaces of K1 such that their intersection is zero. This contradicts the fact that dim K1 ≤ m2 < m(m + 1)/2 + m(m + 1)/2. Similarly, K1 contains a nonzero element x1 of the form (32.29) where C is a nonzero matrix of order m. Then D is a nonzero skew-symmetric matrix. LEMMA 32.13 matrices:

Let K be a simple Lie superalgebra and K0 consist of the following

X 0 0 X

,

where X ∈ sl(m). Let K1 contain an element x1 of the form:

0 C D 0

,

where C is not a scalar matrix. ˜ has ˜ be a K0 -submodule of K0 -module K1 generated by the element x1 . Then K Let K the form:

5 0 C (32.30) D 0 for any where C ∈ sl(m). PROOF The proof of the lemma follows from the fact that tr(XC − CX) = 0 and the commutator of two elements from K0 and K1 is @ A X 0 0 C 0 XC − CX , = . 0 X D 0 XD − DX 0

Simple decompositions of simple Lie superalgebras

32.4.2

421

Explicit form of the even part of a strange subalgebra

Let S = K + L where S is isomorphic to sl(m, n), K is isomorphic to either P (m − 1) or Q(m − 1) and L is isomorphic to sl(l, n). Let us consider the projection π2 deﬁned above. We know that π2 induces a representation of K0 in the vector space W of dimension n. This representation is completely reducible since K0 ∼ = sl(m). Let W = W1 ⊕ . . . ⊕ Ws be the decomposition of W into the direct sum of irreducible subspaces. Let ρi be a restriction of this representation to a subspace Wi , 1 ≤ i ≤ s. Next we consider the superalgebra S in the standard matrix realization, and V is a Z2 -graded vector space of the dimension (n + m) such that V1 = W . LEMMA 32.14 The highest weight of the representation (Wi , ρi ), i = 1, . . . , s is neither (1,0, . . . ,0) nor (0, . . . ,0,1). Thus (Wi , ρi ) is not a standard representation. PROOF Let us assume the contrary, that is, there exists i = 1 . . . s such that the highest weight of a representation (Wi , ρi ) is either (1, 0, . . . , 0) or (0, . . . , 0, 1). Let i = 1. It is well known that the representation ρ : sl(n) → gl(n) with the highest weight (0, . . . , 0,1) is equivalent to ρ(X) = −X t and the representation ρ : sl(n) → gl(n) with the highest weight (1,0, . . . , 0) is equivalent to ρ(X) = X. Hence there exists a basis for V such that algebra K0 takes the form: ⎛ ⎞ X 0 0 ⎝ 0 −X t 0 ⎠ , (32.31) 0 0 ∗ or

⎛

⎞ X 0 0 ⎝ 0 X 0⎠, 0 0 ∗

(32.32)

where X is an arbitrary matrix of order m with a zero trace. First, let K0 has the form (32.31). To simplify our notation we consider only the ﬁrst blocks of matrices from K. In other words, instead of a matrix, ⎛ ⎞ AC∗ ⎝D B ∗⎠, ∗ ∗ ∗ we write

AC DB

.

According to Remark 32.4, K1 contains an element of the form (32.29) where C = E11 . On the other hand, by Remark 32.5, we have that D is nonzero. Notice that D cannot be a skew-symmetric matrix. Indeed, if D is skew-symmetric, then by Lemma 32.12, K1 contains a subspace (32.29) where C is an arbitrary symmetric matrix and D is an arbitrary skew-symmetric matrix. Hence the dimension of this subspace is m2 . On the other hand, by Remark 32.4, K1 contains a subspace (32.29) where D an arbitrary symmetric matrix. So, K1 contains two disjoint subspaces of dimensions m2 and m(m + 1)/2, respectively. This contradicts the fact that dim K1 ≤ m2 . Therefore D is not skew-symmetric. j=1...m Let D = (dij )i=1...m . Two cases are possible. 1. There exists i0 such that di0 i0 = 0.

422

T.V. Tvalavadze

Deﬁne X = Ei0 i where i = 1 i = i0 . Then XC + CX t = Ei0 i E11 + E11 Eii0 = 0 since i = 1. Conversely X t D + DX = Eii0 D + DEi0 i . The coeﬃcient (i, i0 ) of the matrix Eii0 D is equal to di0 i0 = 0 and the coeﬃcient (i, i0 ) of the matrix DEi0 i is equal to zero since i = i0 . 2. There exist i0 and j0 such that i0 = j0 and di0 j0 = −dj0 i0 . Let j0 = 1. The commutator of two elements is given by A @ 0 C 0 XC + CX t X 0 . , = −(X t D + DX) 0 D 0 0 −X t Let X = Ei0 j0 . Then XC + CX t = Ei0 j0 E11 + E11 Ej0 i0 = 0. However X t D + DX = Ej0 i0 D+DEi0 j0 . The coeﬃcient (j0 , j0 ) of the matrix Ej0 i0 D is equal to di0 j0 and coeﬃcient (j0 , j0 ) of the matrix DEi0 j0 is equal to dj0 i0 . Since di0 j0 = −dj0 i0 , it follows that X t D+DX is a nonzero matrix. Thus K1 contains a matrix of the form (32.29) where C is zero, D is nonzero. Hence, by Remark 32.5, D is a skew-symmetric matrix. Therefore, according to Lemma 32.12, K1 contains a subspace (32.29) where C is zero and D is an arbitrary skew-symmetric matrix. Similarly, K1 contains a subspace (32.29) where C is an arbitrary skew-symmetric matrix and D is zero. It is clear that the intersection of these subspaces is zero. On the other hand, according to Lemma 32.9, K1 contains a subspace (32.29) where C is an arbitrary symmetric matrix. Thus we ﬁnd three subspaces in K1 such that pairwise intersections of these subspaces are zero. This implies that dim K1 ≥ m(m − 1)/2 + m(m − 1)/2 + m(m + 1)/2 > m2 , which is a contradiction. Next let K0 have the form (32.32). Let us use the formula: @ A X 0 0 C 0 XC − CX , = . (32.33) 0 X D 0 XD − DX 0 We prove that K1 contains a nonzero element of the form (32.30) such that either C or D is zero. According to Lemma 32.13, K0 contains an element x (32.30) where C = E12 . Notice that if D = 0 then x is a required element. Let D = 0. Then three cases are possible. Case (a). D = C = E12 . According (32.33), K0 -submodule N ⊂ K1 generated by x has the form: 0 Y , (32.34) Y 0 Y is an arbitrary matrix from sl(m). The dimension of K0 -submodule N is m2 − 1. It is well known [FS] that if K ∼ = P (m − 1), then the K0 -module K1 is the direct sum of two irreducible submodules of dimensions m(m − 1)/2 and m(m + 1)/2, respectively. If K ∼ = Q(m − 1), then K0 -module K1 is an irreducible submodule of dimension m2 − 1. Hence the submodule N coincides with K1 . So any element from K1 has the form (32.34). This contradicts the fact that by Lemma 32.11 the ﬁrst rows and columns of all matrices from K are arbitrary. Case (b). There exist i0 and j0 such that i0 = j0 and di0 j0 = 0. Clearly then we can choose either i0 = 1 or j0 = 2 because i0 = j0 . At ﬁrst, we consider the case i0 = 1. Let X, in the formula (32.33), be Eii0 where i = 2 and i = i0 . Since, by the classiﬁcation of strange superalgebras, m ≥ 3, X is well-deﬁned. Then XC − CX = Eii0 E12 − E12 Eii0 = 0. On the other hand, XD − DX = Eii0 D − DEii0 . The coeﬃcient (i, j0 ) of Eii0 D is equal to di0 j0 , and the coeﬃcient (i, j0 ) of DEii0 is equal to zero since i0 = j0 . Hence (XD − DX) is a zero matrix.

Simple decompositions of simple Lie superalgebras

423

Now we consider the case when j0 = 2. Let X = Ej0 i where i = j0 and i = 1. Then XC − CX = Ej0 i E12 − E12 Ej0 i = 0. On the other hand, XD − DX = Ej0 i D − DEj0 i . The coeﬃcient (i0 , i) of Ej0 i D is equal to zero since i0 = j0 , and the coeﬃcient (i0 , i) of DEj0 i is equal to di0 ,j0 . Hence (XD − DX) is a zero matrix. Case (c). There exists i0 such that di0 i0 = 0. Since the trace of D is zero and di0 i0 = 0 , we can choose j0 such that di0 i0 − dj0 j0 = 0. Notice that either i0 = 1, j0 = 2 or i0 = 2, j0 = 1. Let i0 = 1, j0 = 2 and X = Ej0 i0 . Then XC − CX = Ej0 i0 E12 − E12 Ej0 i0 = 0. On the other hand, XD − DX = Ej0 i0 D − DEj0 i0 . The coeﬃcient (j0 , j0 ) of Ej0 i0 D is equal to di0 i0 , and the coeﬃcient (j0 , j0 ) of DEj0 i0 is equal to dj0 j0 . Since the coeﬃcient (j0 , j0 ) is equal to di0 i0 − dj0 j0 = 0, XD − DX is a nonzero matrix Thus K1 contains a matrix of the form (32.30) such that C is a zero matrix, and D is a nonzero matrix. According to Lemma 32.13, K1 contains a subspace (32.30) where C is a zero matrix and D is an arbitrary matrix with trace zero. On the other hand, by Lemma 32.13, K1 contains a subspace (32.30) such that C is an arbitrary matrix with a zero trace. Clearly the intersection of these subspaces is a zero vector. This implies that dim K1 > m2 , which is a contradiction. LEMMA 32.15 The highest weight of the representation (Wi , ρi ), i = 1 . . . s is equal to either (0, 1, 0, . . . , 0) or (0, . . . , 0, 1, 0). PROOF We know that dim S1 ≥ dim K1 + dim L1 where dim S1 = 2mn, dim L1 = 2ln. Hence dim K1 ≥ dim S1 − dim L1 = 2mn − 2ln ≥ 2n. Moreover the dimension of K1 is greater than or equal to m2 since either K1 ∼ = P (m − 1) or K1 ∼ = Q(m − 1). Therefore, n ≤ m2 /2. We prove that if the highest weight (α1 . . . αm−1 ) of the irreducible representation (Wi , ρi ) is diﬀerent from (1, 0, . . . , 0), (0, 1, 0, . . . , 0), ((0, . . . , 0, 1) and (0, . . . , 0, 1, 0)) then the dimension of this representation is greater than m2 /2. Indeed, by Lemma 32.14, the highest weight of the representation (Wi , ρi ) is neither (1, 0, . . . , 0) nor (0, . . . , 0, 1). Next we consider several cases then the highest weight takes special forms 1. Let the highest weight (α1 . . . αm−1 ) of (Wi , ρi ) be (2, 0, . . . , 0) ((0, . . . , 0, 2)). Then, by [FS], the dimension of Wi is m(m + 1)/2 > m2 /2. 2. Let the highest weight (α1 . . . αm−1 ) of (Wi , ρi ) be (10 . . . 01). Then, by [FS], the dimension of Wi is m2 − 1 > m2 /2 where m > 1. 3. Let us consider the fundamental representation of sl(m − 1) with the highest weight λk = (0, . . . , 0, 1, 0, . . . , 0) where k > 2 and k < m − 2. Hence m ≥ 6. It is well known that m , without any loss the dimension of this representation is equal to Ckm . Since Ckm = Cm−k of generality, k ≤ (m + 1)/2. Ckm =

(m − k + 1) . . . m (m − 2)(m − 1)m m! = ≤ , k!(m − k)! 1 · 2... · k 1·2·3

because k ≤ m − k + 1. On the other hand, since (m − 2)(m − 1)/3 > m where m ≥ 6, it follows that m2 (m − 2)(m − 1)m Ckm ≥ > . 1·2·3 2 Now we are ready to consider the general case. Let (V, ρ) and (V , ρ ) be two irreducible representation of a simple Lie algebra g with the highest weights ω and ω , respectively. Then, according to [FS], the dimension of an

424

T.V. Tvalavadze

irreducible representation (U, ρ ) with the highest weight ω + ω is greater than or equal to each of dimensions V and V . Therefore, it is easy to verify that an irreducible representation (Wi , ρi ) with a highest weight (α1 . . . αm−1 ) (which is diﬀerent from (1, 0, . . . , 0), (0, 1, 0 . . . , 0), (0, . . . , 0, 1), and (0, . . . , 0, 1, 0)) is a subrepresentation of a tensor product of representations such that at least one of these representations has the highest weight of the form 1., 2., or 3. However, the dimension of each representation with the highest weight (1), (2), or (3) is greater than m2 /2. Hence, as was shown above, the dimension of our subrepresentation is also greater than m2 /2. Therefore the dimension of Wi is greater than m2 /2. This contradicts the fact that was before that n ≤ m2 /2. LEMMA 32.16

Any representation (Wi , ρi ) , i = 1 . . . s is nontrivial.

PROOF Let us assume the contrary, that is, there exists Wi such that K0 acts trivially in Wi . Next we will show that there exist m2 + 1 independent elements in K1 . Let (W1 , ρ1 ) be a nontrivial representation and (W2 , ρ2 ) be a trivial representation of K0 . Then there exists a basis for V such that K0 consists of the matrices of the form: ⎛ ⎞ X 0 ... 0 ⎜ 0 X1 0 0 ⎟ ⎜ ⎟ (32.35) ⎜ .. . ⎟, ⎝ . 0 X .. ⎠ 2

0 0 ... ∗ where X is an arbitrary matrix of the order m, with zero trace, X1 is matrix of order m(m−1) and X2 is zero matrix of order m2 > 0. 2 An element from K1 has the following form: ⎞ ⎛ 0 C1 c¯2 C3 ⎜ D1 0 . . . 0 ⎟ ⎟ ⎜ (32.36) ⎜ . . ⎟, ⎝ d¯ .. . . . .. ⎠ 2

D3 0 . . . 0 where C1 is a matrix of order m × m(m−1) , D1 is a matrix of order m(m−1) × m, c¯2 is a 2 2 ¯ column vector of dimension m, d2 is a row vector of dimension m and C3 , D3 are matrices of appropriate orders. We consider a subspace of K1 consisting of the matrices of the form (32.36) such that the ﬁrst row of c¯2 and C3 are zero, the ﬁrst column of d¯2 , D3 is zero, the ﬁrst row of C1 and the ﬁrst column of D1 are arbitrary. Let us denote this subspace as T1 . Obviously that the dimension of T1 is greater than or equal to m(m − 1). We know that ⎡⎛ ⎞ ⎛ ⎞⎤ ⎛ ⎞ X 0 0 XC1 − C1 X1 X c¯2 0 C1 c¯2 0 ⎣⎝ 0 X1 0 ⎠ , ⎝ D1 0 0 ⎠⎦ = ⎝ X1 D1 − D1 X 0 0 ⎠. 0 0 0 0 0 d¯2 0 0 d¯2 X According to this formula and since X is an arbitrary matrix from sl(m), K1 contains elements of the form (32.36) where d¯2 X is an arbitrary vector. In particular, K1 contains an element x of the form (32.36) where d¯2 = (10 . . . 0) We consider the vector c¯2 of the element x . Two cases are possible. Case 1. c¯2 = 0.

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Acting in the same manner as before, we prove that K1 contains the subspace T2 of the form (32.36) where d¯2 X is an arbitrary vector and c¯2 = 0. Clearly the dimension of T2 is greater than or equal to m and the intersection of T1 and T2 is a zero vector. By Lemma 32.11 and the fact that S = K + L, we can choose an element from K1 of the form (32.36) where the ﬁrst coeﬃcient of c¯2 is 1. Clearly this element linearly independent with elements from T1 and T2 . Thus we have found m(m − 1) + m + 1 = m2 + 1 linearly independent in K1 , which is a contradiction. Case 2. c¯2 = 0 At ﬁrst we assume that there exists i0 = 1 such that a coeﬃcient (c¯2 )i0 of c¯2 is nonzero. We consider the following commutator: ⎡⎛ ⎞ ⎞ ⎛ ⎞⎤ ⎛ 0 ∗ X c¯2 0 0 ∗ X c¯2 ∗0 ⎣⎝ ∗ 0 0 ⎠ , ⎝ ∗ 0 0 ⎠⎦ = ⎝ ∗ ∗ ⎠. 0 d¯2 X 0 0 d¯2 X 0 0 0 0 d¯2 XX c¯2 As was shown in Lemma 32.10, if X = E1k + Eki0 , then XX = E1i0 , 1 = i0 . Hence d¯2 XX c¯2 = d¯2 E1i0 c¯2 = (c¯2 )i0 = 0. On the other hand, d¯2 XX c¯2 = 0 since [K1 , K1 ] = K0 and K0 has the form(32.35) where X2 = 0. This is a contradiction. Next we assume that (c¯2 )1 is nonzero. As was shown in Lemma 32.10, if X = E11 − E22 , then XX = E11 − E22 . Hence d¯2 XX c¯2 = d¯2 (E11 − E22 )c¯2 = (c¯2 )1 = 0. On the other hand, d¯2 XX c¯2 = 0 since [K1 , K1 ] = K0 and K0 has the form (15) where X2 = 0, a contradiction. REMARK 32.6 Note that according to Lemma 32.15 and Lemma 32.16, the highest weight of a representation (Wi , ρi ), i = 1 . . . s is equal to either (010 . . . 0) or (0 . . . 010). 2 Hence dim Wi = m(m−1) . Since n ≤ m2 and m ≥ 3, it follows that s = 1. Therefore 2 n = m(m−1) . 2 ∼ sl(m, n) and K ∼ LEMMA 32.17 Let K be a subalgebra of S = = P (m−1) (or Q(m−1)). Then K contains a subalgebra K ∼ = P (m − 2) (or Q(m − 2)) such that K0 consist of the following matrices: ⎞ ⎛ 0 0 0 ⎝ 0 X 0 ⎠ , (32.37) 0 0 Y where X is a square matrix of order m − 1 and Y is a matrix of order n. PROOF Let Kst be a standard realization of P (m − 1) (resp. Q(m − 1)). We consider an automorphism ϕ of gl(2m) of the form ˜ C˜ −1 , ϕ(X) = CX where C˜ is a nonsingular matrix of the order 2m. In the case Kst ∼ = Q(m − 1), C 0 C˜ = , 0 C where C is a nonsingular matrix of order m. In the case when Kst ∼ = P (m − 1), C 0 C˜ = , 0 (C −1 )t

(32.38)

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where C is a nonsingular matrix of order m. In both cases ϕ is an automorphism of Kst . Let ρ1 be a projection of (Kst )0 ⊂ sl(m) ⊕ sl(m) into the ﬁrst ideal. Denote ρ1 ((Kst )0 ) as Kρ . Let Kπ = π1 (K0 ) where π1 was deﬁned above. Since K0 and (Kst )0 are simple Lie algebras, π1 and ρ1 are isomorphisms. Let ψ be an is the isomorphism from Kπ onto Kρ . We isomorphism from Kst onto K. Then π1 ψρ−1 1 is an consider Kπ ∼ = sl(m) and Kρ ∼ = sl(m) in their matrix realizations. Then π1 ψρ−1 1 takes the following forms: automorphism of sl(m). Hence π1 ψρ−1 1 −1 1. π1 ψρ−1 . 1 (X) = SXS t −1 2. π1 ψρ−1 where S is non-singular matrix of order m. 1 (X) = S(−X )S Since Kst is a standard realization of P (m − 1) (Q(m − 1)), it is clear that Kst contains a subalgebra K ∼ = P (m − 2) (Q(m − 2)) such that ρ1 (K ) consists of the following matrices: 0 0 , (32.39) 0 X where X is a matrix of order m − 1 with zero trace. −1 At ﬁrst we assume that π1 ψρ−1 . 1 (X) = SXS Let ϕ be an automorphism of Kst of the form (32.38) where C = S −1 . Then ψϕ is an isomorphism between Kst and K. We will prove that ψ(ϕ(K )) has the form (32.37). −1 Indeed, we consider the mapping π1 ψϕρ−1 1 from Kρ into Kπ . Hence π1 (ψ(ϕ(ρ1 (X)))) = −1 −1 S(S XS)S = X for any X ∈ sl(m). Since ρ1 (K ) has the form (32.39) and π1 ψϕρ−1 1 (X) = X, it follows that ψ(ϕ(K )) has the form (32.37). t −1 Next we assume that π1 ψρ−1 . 1 (X) = S(−X )S Let ϕ be an automorphism of Kst of the form (32.38) where C = S t . Then ψϕ is an isomorphism between Kst and K. We will prove that ψ(ϕ(K )) has the form (32.37). Indeed, which sends Kρ into Kπ . Hence π1 (ψ(ϕ(ρ−1 we consider the mapping π1 ψϕρ−1 1 1 (X)))) = t t −1 t −1 −1 t S(−(S X(S ) ) )S = −S(S X S)S −1 = −X t for any X ∈ sl(m). Since ρ1 (K ) has t the form (32.39) and π1 ψϕρ−1 1 (X) = −X , it follows that ψ(ϕ(K )) has the form (32.37).

32.4.3

Main theorem

THEOREM 32.4 A simple Lie superalgebra S ∼ = sl(m, n) cannot be decomposed into the sum of subalgebras K and L such that K is isomorphic to either P (k) or Q(k) and L is isomorphic to sl(l, q). PROOF Let us assume the contrary. Then, by the previous Lemma, K contains a subalgebra K such that K 0 has the form (32.37). We consider K 0 -module K 1 . According to [FS], if K ∼ = P (m − 2) then K 0 -module K 1 is the direct sum of two irreducible K 0 -submodules of dimensions m(m − 1)/2 and (m − 1)(m − 2)/2. If K ∼ = Q(m − 2) then the K 0 -module K 1 is an irreducible submodule 2 of dimension (m − 1) − 1. Therefore, in both cases, K 1 contains an irreducible K 0 -submodule M such that dim M ≥ m(m − 1)/2. We will prove that an arbitrary element from M has the form: 0 C , (32.40) D 0

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where C is a matrix of order m × n, and the ﬁrst row of C is zero. Indeed, if there exists at least one element of the form (32.40) then the commutator of this element and arbitrary element from K 0 also has the form (32.40) since K 0 has the form (32.37). Hence this element generates a nonzero K0 -submodule of K1 . Since K0 submodule M is irreducible, an arbitrary element from M has the form (32.40). Let us assume that M does not contain any elements of the form (32.40). According to Remark 32.6, n = m(m − 1)/2, but dim M ≥ m(m − 1)/2. Hence M contains matrices of the form 0 C , (32.41) D 0 where the ﬁrst row of matrix C is arbitrary The projection π2 induces a representation of K 0 in a vector space W of dimension n. Note that this representation is reducible. Indeed, according [FS], a restriction of an irreducible representation of K0 ∼ = sl(m − 1) to a subalgebra K 0 ∼ = sl(m − 2) is reducible. Hence there exists two invariant subspaces of W of dimensions m1 > 0 and m2 > 0, respectively. Therefore K 0 takes the form ⎛ ⎞ 0 0 ⎜ 0 X ⎟ ⎜ ⎟, ⎝ Y1 0 ⎠ 0 Y2 where X is the set of all matrices of order m − 1, Y1 and Y2 are of orders m1 and m2 , respectively. Let T be a subspace of the form (32.41) such that c¯1 c¯2 C= , C1 C2 where C1 is matrix of order (m − 1) × m1 , C2 is matrix of order (m − 1) × m2 , c¯1 is a row vector of dimension m1 , c¯2 is a row vector of dimension m2 , and c¯1 = 0, c¯2 = 0. Obviously, the commutator of an element from T and an arbitrary element from K 0 belongs to T . Hence M ⊂ T because M is an irreducible K0 - submodule, which is a contradiction. Thus M has the form (32.40). Clearly the same results hold true for the block D. So, K1 contains the subspace M of dimension m(m − 1)/2 such that the ﬁrst row and the ﬁrst column of matrices from M are zero. Hence dim K1 ≥ n + n + m(m − 1)/2 = m(m − 1)/2 + m(m − 1)/2 + m(m − 1)/2 > m2 , which is a contradiction.

32.5

Decompositions of Simple Superalgebras of the Type sl(m, n) into the Sum of Two Strange Lie Subalgebras

We consider the decomposition of S ∼ = sl(m, n) into the sum of simple subalgebras K and L such that K ∼ = P (k) (Q(k)), and L ∼ = P (l) (Q(l)). First we prove that m = n. Indeed, if m = n then S0 is isomorphic to sl(m) ⊕ sl(n) ⊕ U . Since K0 ∼ = sl(k), L0 ∼ = sl(l), S0 cannot be the sum of K0 and L0 . It is well known that the dimension of Q(m−1) is m2 −1 and the dimension of P (m−1) is m2 . Hence dim K1 ≤ k 2 , dim L1 ≤ l2 . Since dim S1 ≤ dim K1 + dim L1 and dim S0 = 2m2 , we obtain that m = k + 1 = l + 1. Moreover, K ∼ = P (m − 1), L ∼ = P (m − 1).

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Therefore there exists a basis for V such that K takes one of the following forms: X C , (32.42) D −X t or

X C DX

,

(32.43)

where X is an arbitrary matrix of order m with a zero trace, and C, D are of order m. Note that K ∼ = P (m−1) cannot be of the form (32.43). Indeed, if K ∼ = P (m−1) then K0 submodule K1 is a direct sum of two irreducible submodules of dimensions m(m − 1)/2 and m(m + 1)/2. On the other hand, by Lemma 32.13, K1 contains an irreducible submodule of the dimension m2 − 1, which is a contradiction. Then L has the form:

5 ϕY C , D −Y t where Y is an arbitrary matrix of order m with a zero trace, ϕ is an automorphism of sl(m), C and D are of order m. Let us consider the intersection of K0 and L0 . We will prove that this intersection contains a nonzero element . Any automorphism of sl(m) has the forms: 1. ϕ(X) = C(−X t )C −1 2. ϕ(X) = CXC −1 where C is arbitrary nonsingular matrix of order m. As was shown above, ϕ cannot be of form (32.43) since K ∼ = P (m − 1), L ∼ = P (m − 1). Hence we consider only the ﬁrst case. Thus L0 has the form: 5

CY C −1 0 , 0 −Y t and K0 has the form:

X 0 0 −X t

5 ,

s I = 0 belongs to sl(m). We prove that the Let s = tr C. Then the matrix F = C − m matrix G of the form F 0 0 −F t

belongs to K0 and L0 . Clearly, G belongs to K0 . We consider the following element: CF C −1 0 . 0 −F t By deﬁnition this element belongs to L0 . Hence CF C −1 0 C(C − sI)C −1 0 F 0 = = . 0 −F t 0 −F t 0 −F t This implies that G ∈ L0 . On the other hand m2 = dim S0 ≤ dim K0 + dim L0 = dim K0 + dim L0 − dim(K0 ∩ L0 ) = m2 + m2 − 1 < m2 , which is a contradiction. Therefore this proves that a simple Lie superalgebra S ∼ = sl(m, n) cannot be decomposed into the sum of two strange Lie subalgebras.

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32.6

429

Acknowledgments

This research was supported in part by NSERC.

References [BK] Bahturin Yu.A, Kegel O.H. Sums of simple subalgebras. Algebra, 11. J. Math. Sci. (New York) 93 (1999), 830-835. [BTT] Bahturin Yu, Tvalavadze M, Tvalavadze T. Sums of simple and nilpotent Lie subalgebras. Comm. in Algebra, vol. 30, 2002, 9, 4455-4471. [BN] Burbaki, N. Algebra: Modules, Rings, Forms. Izdat. Nauka, Moscow, 1966. [FS] Frappat L, Sciarrino A. Dictionary on Lie algebras and superalgebras. In: Questions of Group Theory and Homological Algebra, London, 2000. [ON] Onishchik A.L. Topology of Transitive Transformation Groups. Johann Ambrosius Barth Verlag GmbH, Leipzig, 1994. [ON2] Onishchik A.L. Decompositions of reductive Lie groups. Mat. Sbornik, 80(122), 1969, 4, 515-554. [KV] Kac V.G. Classiﬁcation of Lie superalgebras. Funct. Anal. Appl., 9, 1975. [TT] Tvalavadze M.V., Tvalavadze T. V. Decompositions of simple special Jordan algebras. Comm. in Algebra, Vol. 33, 7, 2005, 2403–2421

Chapter 33 The Structure and Classiﬁcation of Finite Division Rings G.P. Wene Department of Applied Mathematics, The University of Texas at San Antonio

33.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33.3 Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33.3.1 Power-associative algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33.3.2 Three examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

431 432 433 433 435 438 439

Abstract Little is known of the structure of ﬁnite division rings. The major results are the classiﬁcation of ﬁnite division rings quadratic over a weak nucleus by Knuth [15] and ampliﬁed in the papers by Cohen and Ganley [5] and Ganley [11], the determination of suﬃcient conditions for the existence of subrings by Zemmer [23] and the proof that all ﬂexible ﬁnite division rings of characteristic not 2 are commutative by Oehmke [17] . The traditional method of classifying ﬁnite division rings is to give the discoverer’s name, i.e., Dickson commutative division rings [10] or Sandler division rings [20]. Our examples highlight some of the diﬀerences between ﬁnite ﬁelds and ﬁnite division rings and demonstrate that isotopic division rings need not have isomorphic automorphism groups nor isomorhpic nuclei. One example is that of a division ring of order 81 that contains no subring of order 9 or 27. We will examine structure and automorphism groups in preparation to establishing a coherent theory of ﬁnite division rings. Keywords: nonassociative rings, ﬁnite division rings, identities 2000 Mathematics Subject Classiﬁcation: Primary 17A35: Secondary 17A60

33.1

Introduction

A nonassociative division ring, the octonions, was known as early as 1844 (Grave’s correspondence with Hamilton in Hamilton [12]). Finite nonassociative division rings were introduced in 1905 by L. E. Dickson [8]. Current interest is driven by the fact that the ﬁnite planes of Lenz-Barlotti type V.1 (translation planes) are precisely the planes coordinatable by division rings that are not ﬁelds (see the book by Biliotti, Jha, and Johnson [4]). A ﬁnite division ring (or semiﬁeld [15]) is a ﬁnite algebraic system containing at least two distinguished elements 0 and 1. A division ring Δ possesses two binary operations, addition and multiplication, designated in the usual notation and satisfying the following axioms:

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(i) (Δ, +) is a group with identity 0. (ii) If a, b ∈ Δ and ab = 0 then a = 0 or b = 0. (iii) If a, b, c ∈ Δ then a(b + c) = ab + ac and (a + b)c = ac + bc. (iv) The element 1 satisﬁes the relationship 1 · a = a · 1 = a for all a ∈ Δ. It is easily seen that there are unique solutions to the equations ax = b and xa = b for every nonzero a and every b in Δ. It also follows easily that addition is commutative. In fact it can be shown that Δ is an algebra over some prime ﬁeld F = GF (p) and that Δ has pn elements where n is the dimension of Δ over F [15]. Any subring of a ﬁnite division ring will be a division ring. One of the diﬃculties in attempting to ﬁnd a coherent theory of ﬁnite division rings is the vast profusion of them. In order to reduce the number of possibilities of nonassociative algebras, Albert [1] introduced the concept of isotopy. An isomorphism is an isotopy and isotopy is an equivalence relation. Thus isotopy is a reasonable generalization of isomorphism. An isotopy of a division algebra is a division algebra. Isotopic division algebras determine isomorphic projective planes (Albert [3]). Our examples demonstrate the inadequacy of isotropy as a classiﬁcation tool for ﬁnite division rings. Unless otherwise stated, division ring Δ will mean a ﬁnite division ring.

33.2

Preliminaries

Two tools used to study the associativity of ﬁnite division rings and nonassociative rings in general are the associator of elements a, b, and c: (a, b, c) = (ab)c − a(bc) and the commutator of elements a and b: [a, b] = ab − ba. An associative commutative algebra C can be described as one in which (a, b, c) = [a, b] = 0 for all a, b, c ∈ C . The ﬂexible algebras satisfy a relaxed form of commutativity, the identity (a, b, a) = 0 for all a, b ∈ A. The linerization of the ﬂexible identity gives (x, y, z) + (z, y, x) = 0. If the characteristic is not two, the linearized relation implies the ﬂexible property: (wx, y, z) − (w, xy, z) + (w, x, yz) = w(x, y, z) + (w, x.y)z. The nuclei are a collection of subrings that has proven particularly useful in the study of nonassociative algebras. Let A be an arbitrary nonassociative algebra; deﬁne the left nucleus: Nl = {x ∈ A : (x, a, b) = 0 for all a, b ∈ A} , the middle nucleus: Nm = {x ∈ A : (a, x, b) = 0 for all a, b ∈ A} ,

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the right nucleus: Nr = {x ∈ A : (a, b, x) = 0 for all a, b ∈ A} and the nucleus: N = Nl ∩ Nm ∩ Nr . The center N Z = {n ∈ N | nx = xn for all x ∈ Δ}. The commutative center Z = {z ∈ Δ | zx = xz for all x ∈ Δ}. Clearly any nucleus of a ﬁnite division ring will be a ﬁnite ﬁeld. A subﬁeld K of a division ring Δ is called a weak nucleus if (x, y, z) = 0 whenever two of x, y, z are in K. The division ring Δ is a vector space over its weak nucleus but right and left multiplication may not be linear transformations over the weak nucleus. If the dimension of Δ over its weak nucleus is two, we say that Δ is quadratic over a weak nucleus. The classiﬁcation of ﬁnite division rings quadratic over a weak nucleus began by Knuth[15] is further ampliﬁed in the papers by Cohen and Ganley [5] and Ganley [11]. For any algebra A of characteristic not 2, deﬁne the plus algebra, A+ , as the vector space A with a new multiplication ◦ deﬁned in terms of the addition and multiplication of A: a ◦ b = 12 (ab + ba). The algebra A is called Jordan-admissible if A+ is a Jordan algebra. The minus algebra, A− , is deﬁned similarly, but the multiplication, [, ], is given by [a, b] = ab − ba; if the algebra A− is a Lie algbra, A is said to Lie-admissible. The Malcev-admissible algebras are a generalization of the Lie-admissible algebras in which the minus algebras satisfy the Malcev identity.

33.3

Structure

The condition of being ﬁnite causes many of the more familiar division rings to collapse into ﬁnite ﬁelds. THEOREM 33.1 ﬁeld.

(Wedderburn[19]) Every ﬁnite associative division ring is a ﬁnite

Nicolas Lichiardopol [16] has recently published an interesting new proof of the Wedderburn theorem.

33.3.1

Power-associative algebras

Let A be an arbitrary nonassociative algebra. The right powers of x ∈ A are deﬁned by x1 = x and xn = (xn−1 )x for all positive integers n > 1. An algebra A is called powerassociative if xn xm = xn+m for all x ∈ A and all positive integers m and n. Equivalently, the powers of a single element generate an associative subalgebra of A. The associative, alternative, and Jordan algebras are all power-associative. An even stronger form of powerassociativity, strict power-associative, requires that the algebra A remain power-associative under every extension of the scalar ﬁeld. Any power-associative ring (see Albert [2]) enjoys the associativity of cubes: (x, x, x) = 0 and the associativity of fourth powers:

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(x2 , x, x) = 0. In any commutative ring with characteristic prime to 30, associativity of fourth powers is equivalent to power-associativity. Albert further extended this result to any ﬂexible ring of characteristic prime to 30. In the characteristic zero case both the associativity of cubes and fourth powers are needed to imply power-associativity. The associativity of cubes appears as a minimal assumption on the associativity of many of the classes of rings and is particularly useful when passing to the plus algebra (see, for example, Propositions 9.7 and 10.1 of Osborn [18]. We note that for each nonzero element δ of a ﬁnite division ring Δ there exists an integer n(δ) > 1 such that δ = (· · ·(((δδ)δ)δ) · · · δ) is a product of n right multiplications by itself. This provides a “right powers” generalization of the concept of a periodic ring. Osborn [18] calls a power-associative algebra A periodic if for each element a of A there exists a positive integer n(a) such that an(a) = a. With this deﬁnition he generalizes Wedderburn’s theorem. THEOREM 33.2 (Osborn) Let A be a periodic ring of characteristic p = 2, and let A be strictly power-associative if p = 3, and let A contain only one idempotent. Then A is a periodic ﬁeld. Theorem 33.2 implies that if Δ is a ﬁnite power-associative division ring of characteristic p = 2, 3 or if Δ is strictly power-associative ﬁnite division ring of characteristic p = 3, then Δ is a ﬁnite ﬁeld. Albert’s theorem then follows as a corollary. THEOREM 33.3 (Albert) Let Δ be a ﬁnite power-associative division algebra of characteristic = 2, 3, 5. Then Δ is a ﬁnite ﬁeld. Schafer [21] has a proof of Albert’s theorem that does not use Osborn’s theorem. THEOREM 33.4 Any ﬁnite Jordan division algebra is power-associative, and, if the characteristic is not two, it is a ﬁnite ﬁeld. There is no restriction on the characteristic in the alternative case. THEOREM 33.5

A ﬁnite alternative division algebra is a ﬁnite ﬁeld.

A division ring Δ is said to have the right inverse property (RIP) if for each x ∈ Δ, x = 0, there is an element x−1 ∈ Δ such that (yx)x−1 = y for all y ∈ Δ. A left inverse property (LIP) is deﬁned similarly. See the text by Hughes and Piper [13] for a proof of the following. THEOREM 33.6 (Skornyakov-San Soucie Theorem) A division algebra Δ with the right inverse property is alternative. Hence a ﬁnite division algebra with the right inverse property is a ﬁnite ﬁeld.

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THEOREM 33.7 (D. Zalinsky [186]) A division algebra of dimension n over a ﬁeld F cannot contain a subalgebra containing 1 and of dimension n − 1 over F . Zalinsky’s result tells us that a division algebra of order p3 for any prime p is generated by any nonscalar element (a scalar element refers to a scalsr multiple of the identity.) THEOREM 33.8 (Zemmer [23]) Let Δ be a division algebra of order n over a ﬁeld F = GF (q k ), where n = hq t , (h, q) = 1. Let T be an automporphism of Δ relative to F with period n. If the minimun function of T is of degree n, then for every divisor m of n, Δ contains a subalgebra Δm , of order m over F , whose automorphism group relative to F contains the cyclic group of order m.

33.3.2

Three examples

The following examples illustrate the rich structure of ﬁnite, nonassociative division algebra and highlight the inadequacy of the concept of isotopism as an algebraic, as opposed to a geometric, classiﬁcation tool. Additional examples are given in the paper by Cordero and Wene [6]. The internal structure of ﬁnite division algebras may be radically diﬀerent in the nonassociative case. Particularly, we cannot extend to the class of all ﬁnite division algebras the well-known result for ﬁnite ﬁelds: THEOREM 33.9 For each divisor d of n, GF (pn ) has a unique subﬁeld of order pd . Moreover, these are the only subﬁelds of GF (pn ). The corresponding conjecture for ﬁnite division algebras replaces the ﬁnite ﬁeld GF (pn ) with a division algebra of order pn and subﬁelds with subdivision algebras. The 16-element division algebras provide counterexamples to both parts of this conjecture. The 23 isomorphisms classes of nonassociative sixteen (24 )-element division algebras form two isotopism classes, which Kleinfeld [14] denoted by T and V. Knuth’s system V is representative of the isotopism class V and corresponds to Kleinfeld’s system V13. We give Knuth’s well-known construction of system V13. System V13 contains four distinct subdivision algebras of order four and has an automorphism group isomorphic to the group S3 . The division algebras given in Examples 33.1 and 33.2 are isotopic. Example 33.1 (Knuth’s construction of system V13) Let F be the ﬁeld GF (4) with elements 0, 1, ω and ω 2 (= ω + 1). The elements of V are of the form a + λb where a, b ∈ F . Addition and multiplication are deﬁned in terms of the addition and multiplication of F . (a + λb) + (c + λd) = (a + c) + λ(b + d) and

(a + λb)(c + λd) = (ac + b2 d) + λ(bc + a2 d + b2 d2 ).

System V13 possesses four four-element subrings: the subring consisting of all elements {a + λ0 : a ∈ GF (4)}, the subring of all elements {a + λb : a, b ∈ GF (2)}, the subring consisting of the elements {0, 1, λω, 1 + λω} and the subring consisting of the elements 0, 1, λω 2 , 1 + λω 2 . All nuclei are isomorphic to GF (2). The ﬁeld F is a weak nucleus. The six automorphisms σij are given by

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for i = 0, 1, 2

and j = 1, 2.

Example 33.1 demonstrates that a nonassoicative division algebra may have several subdivision rings of the same order. We present an alternative construction of the system V13. Let V13 denote a vector space over the ﬁeld GF (2) and {1, I, J, K} a basis of V13 over GF (2). Deﬁne a multiplication in V13 using both distributive properties and the following relations. The element 1 is the unity of V13. I2 = 1 + I IJ = K IK = J + K

JI = J + K J2 = 1 + J JK = 1 + I + J + K

KI = J + K KJ = I + J + K K2 = 1 + K

Kleinfeld’s system V12 is isotopic to his system V13 but has no subdivision algebra of order four and only a trivial automorphism group. We see that isotopic division algebras do not necessarily have isomorphic subdivision structures nor isomorphic automorphism groups. Example 33.2 Let V12 denote a vector space over the ﬁeld GF (2) and {1, I, J, K} a basis of V1. Deﬁne a multiplication in V12 using both distributive properties and the following relations. The element 1 is the unity of V12. I2 = J IJ = K IK = 1 + I + J + K

JI = I + K J2 = 1 + J JK = I + J

KI = 1 + K KJ = I + J + K K2 = 1 + J + K

These examples beg the question “What if the characteristic is not 2?” The author discovered Example 33.3 while searching for an 81-element division algebra in which all nuclei are trivial, i.e., isomorphic to the ﬁeld GF (3). Since the kernel (the left nucleus) is trivial, these are the “geometrically” uninteresting division rings. There are no commutative 81-element division rings with trivial nuclei. Various FORTRAN language programs assisted in deriving the results presented here. Example 33.3 Let Kno denote a vector space over the ﬁeld GF (3) with basis {1, I, J, K}. Deﬁne a multiplication in Kno using both distributive properties and the following relations. The element 1 is the unity of Kno. I2 = J IJ = K IK = 1 + J

JI = K J2 = 1 + K JK = 2I + 2J + 2K

KI = 1 + K KJ = I + J K 2 = 2J

THEOREM 33.10 System Kno is an 34 -element division algebra that contains no division algebra of order 32 or order 33 . Consequently, all of its nuclei will be trivial. PROOF A computer run shows that the algebraic system deﬁned above is a division algebra. Since the multiplication is not commutative, the division ring is not associative.

The structure and classiﬁcation of ﬁnite division rings

437

Call each element of Kno of the form α1, α ∈ GF (3) a scalar. A straightforward, but nontrivial, calculation veriﬁes that the elements of each of the sets 1, x, x2 , x a nonscalar, are linearly independent. By Zalinsky [186], a division algebra of dimension 4 over a ﬁeld F cannot contain a subalgebra containing 1 of dimension 3. A nucleus would be a division ring. COROLLARY 33.1 Each algebra having a unit element and isotopic to the system Kno has only trivial subalgebras. PROOF

This is an immediate consequence of Theorem 9 of Albert[1].

COROLLARY 33.2

There is no commutative isotope of the system Kno.

PROOF Apply the test in Theorem 17 of Albert[1]. A computer search demonstrated that there is no no-zero element x0 such that (x0 a)b = (x0 b)a for all a and b in the system Kno. THEOREM 33.11

The system Kno satisﬁes the identity, [[[x, y] , [x, z]] , [[y, z] , [y, w]]] = 0.

The author used a FORTRAN language program to discover this identity. The 81-element division algebras quadratic over a weak nucleus constructed in [15] do not satisfy this identity. All division rings of order pn for any prime p, satisﬁes this identity. More examples of ﬁnite division rings are needed; we present a brief discussion of a technique that may be used to construct additional examples. There are many ways in which one can construct isotopic division rings; we give a construction utilizing the concept of a t-spread set. Following the discussion in Dembowski [7] we can easily construct isotopic division rings. A collection Ψ of (t + 1, t + 1)-matrices over the ﬁeld GF (q) a t-spread set if it satisﬁes the following conditions: |Ψ| = q t+1 , 0 ∈ Ψ and I ∈ Ψ, If X, Y ∈ Ψ and X = Y, then det(X − Y ) = 0. Here 0 and I are the zero and identity (t + 1, t + 1)-matrices, respectively. Let V be a t+1-dimensional vector space over GF (q) and let Ψ be a t-spread of (t+1, t+1)matrices over GF (q). Let e be the vector (1, 0, · · ·, 0) in V . If y ∈ V , deﬁne C(y) as the matrix C in Ψ such that y = eC(y). Deﬁne a multiplication on V by xy = xC(y). The vector space V becomes a quasiﬁeld with this multiplication and the vector space addition of V . If the set Ψ is closed under addition, the vector space V becomes a division ring. If Ψ is closed under both addition and multiplication, V will be a ﬁnite ﬁeld. Denote this algebra by (V, Ψ) The set of matrices, ΨK , determining the right multiples for the system Kno as deﬁned above is a 3-spread over GF (3). Let A be an invertible (4, 4)-matrix over GF (3). Then the set of matrices

438

G.P. Wene

AΨK A−1 = {ACA−1 | C ∈ ΨK } deﬁnes a division ring isomorphic to system Kno. As the examples show, forming isotopes does great violence. We can ﬁnd a nontrivial isotopism that preserves the right nucleus. THEOREM 33.12 Let Δ be an n-dimensional division algebra over the ﬁeld GF (q) and r a generator of the multiplicative group of the right nucleus an let M be an invertible (n,n)matrix over GF (q). If C(r) is the matrix for right multiplication by r and M C(r)M −1 = C(r) then the right nucleus of Δ = (V, Ψ) is the right nucleus of (V, M ΨM −1 ). PROOF Let r be a generator of the multipicative group of the ﬁeld that is the right nucleus of (V, Ψ). Let M be an invertible (n,n)-matrix over GF (q) and M C(r)M −1 = C(r). Denote the product in (V, Ψ) by · and the product in (V, M ΨM −1 ) by then for elements a, b ∈ Δ. We have (a · b) · r = aC(b)C(r) and a · (b · r) = aC(b(C(r)). While (a b) r = aM C(b)M −1 M C(r)M −1 = aM C(b)C(r)M −1 . But C(b)C(r) = C(b(C(r)). Furthermore, a (b r) = aM (C(bM C(r))M −1 )M −1 = aM (C(bC(r)))M −1 = aM C(b)C(r)M −1 .

33.4

Conclusion

There is much to learned about ﬁnite division rings. Some particular questions arise from the above discussion. Does there exist a ﬂexible ﬁnite division algebra that is not commutative? Of course it would necessarily have characteristic two. Are there any power-associative ﬁnite division algebras that are not ﬁelds? Here the characteristic would necessarily be 2 or 3. If A is a ﬁnite division algebra and A+ is a ﬁeld, is A necessarily a ﬁeld? If A is a ﬁnite division algebra and A+ is power-associative, is A necessarily a ﬁeld? Find all ﬁnite division algebras Δ that satisfy the identity [[[x, y] , [x, z]] , [[y, z] , [y, w]]] = 0. Is there a systematic way to go about ﬁnding identities for ﬁnite division algebras? Is there a classiﬁcation tool that is less discriminating than isomorphism and more restrictive than isotopism? Can we count the number of subalgebras? More examples of ﬁnite division rings are needed.

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439

References [1] A. A. Albert, Non-associative algebras: fundamental concepts and isotopy, Annals of Math. 43 (1942), 685-707. [2] A. A. Albert, On the power-associativity of rings, Sum. Bras. Math., Vol. II, Fasc. 2, 1948, 21-32. [3] A. A. Albert, Finite division algebras and ﬁnite planes, Proc. Sympos. Appl. Math. , Vol. 10, Amer. Math. Soc., Providence, RI, 1960, pp. 53-70. [4] M. Biliotti, V. Jha, and N. L. Johnson, Foundations of Translation Planes. Pure and Applied Mathematics, Vol. 243, Marcel Dekker, New York, Basel, 2001. [5] S. D. Cohen and M. J. Ganley, Commutative semiﬁelds, two dimensional over their middle nucleui, J. Algebra 75 (1982), 373-385. [6] M. Cordero and G. P. Wene, A survey of ﬁnite semiﬁelds, Discrete Mathematics 208/209 (1999), 125-137. [7] P. Dembowski, Finite Geometries, Springer-Verlag, Berlin, 1968. [8] L. E. Dickson, On ﬁnite algebras, Nachr. Ges. Wiss. G¨ ottingen (1905), 358-393. [9] L. E. Dickson, Linear algebras in which division is always uniquely possible, Trans. Amer. Math. Soc. 7 (1906), 370-390. [10] L. E. Dickson, On commutative linear algebras in which division is always uniquely possible, Trans. Amer. Soc. 7 (1906) 514-522. [11] M. J. Ganley, Central weak nucleus semiﬁelds, Europ. J. Combinatorics 2 (1981), 339-347. [12] W. R. Hamilton, In: The Mathematical Papers of William Rowan Hamilton, Vol. III, eds. H. Halberstam and R. E. Ingram, Cambridge University Press, Cambridge, 1967. [13] D. R. Hughes and F. C. Piper, Projective Planes, Springer-Verlag, New York, 1982. [14] E. Kleinfeld, Techniques for enumerating Veblen-Wedderbyrn systems, J. Assoc. Comp. Mach. 7 (1960), 330-337. [15] D. E. Knuth, Finite semiﬁelds and projective planes, J. Algebra 2 (1965), 182-217. [16] N. Lichiardopol, A new proof of Wedderburn’s theorem, Amer. Math. Monthly 110 (2003),735-737. [17] R. H. Oehmke, On ﬁnite division rings, Proc. Amer. Math. Soc. 79 (1980), 174-176. [18] J. M. Osborn, Varieties of Algebras, Advances in Mathematics 8 (1972), 163-369. [19] J. H. M. Wedderburn, A theorem on ﬁnite algebras, Trans. Amer. Math. Soc. 6 (1905), 349-352. [20] R. Sandler, Autotopism groups of some ﬁnite non-associative algebras, Amer.J. Math.

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[21] R. D. Schafer, An Introduction to Nonassociative Algebras, Academic Press, New York, 1966. [22] D. Zalinsky, Nonassociative algebras of order three, unpublished master’s thesis, University of Chicago, 1943. [23] J. L. Zemmer, Jr., On the subalgebras of ﬁnite division algebras, Canadian J. Math. 4 (1952),391-503.

Appendix A Some Problems in the Theory of Rings That are Nearly Associative A. I. Shirshov Translated by Murray Bremner and Natalia Fomenko

A.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.1.1 Associative and nonassociative rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.1.2 Origins of the theory of nonassociative rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.1.3 General results on nonassociative rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.1.4 Free rings and algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.1.5 Identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.2 Alternative rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.2.1 The doubling process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.2.2 Identities in alternative rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.2.3 Subrings of alternative rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.2.4 Free alternative rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.2.5 Cayley-Dickson algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.2.6 Right alternative rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.3 J-rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.3.1 J-rings and special J-rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.3.2 An exceptional J-algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.3.3 Identities in J-rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.3.4 J-rings and alternative rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.4 Lie rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.4.1 Lie rings and Lie algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.4.2 Relations between Lie rings and groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.4.3 The Engel condition and the Burnside problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.4.4 Free Lie rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.4.5 Binary-Lie rings and Moufang-Lie rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.5 Some wider classes of rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.5.1 Power-associative rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.5.2 Decomposition with respect to an idempotent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.5.3 Noncommutative J-rings and power-commutative rings . . . . . . . . . . . . . . . . . . . . . . . . A.6 Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

442 442 442 443 443 444 444 444 445 446 446 447 447 448 448 448 450 450 451 451 451 452 453 453 454 454 454 455 455 455

The original Russian article, Nekotorye voprosy teorii kolets, blizkikh k assotsiativnym, appeared in Uspekhi Matematicheskikh Nauk XIII, 6 (1958) 3–20. This translation was done by Murray Bremner (Research Unit in Algebra and Logic, University of Saskatchewan, Saskatoon, Canada) and Natalia Fomenko. The subsection headings have been added by the translators. The bibliographical references have been put into English alphabetical order.

441

442

A. I. Shirshov

The words “some problems” in the title of this article mean primarily that the article considers absolutely no results about algebras of ﬁnite dimension. Among other questions that remain outside the scope of the article, we mention, for example, various theorems about decomposition of algebras (see for example [47, 70]) which are closely related to the theory of algebras of ﬁnite dimension. The author is grateful to A. G. Kurosh and L. A. Skornyakov who got acquainted with the ﬁrst draft of the manuscript and made a series of very valuable comments.

A.1 A.1.1

Introduction Associative and nonassociative rings

Until recently the theory of rings and algebras was regarded exclusively as the theory of associative rings and algebras. This was a result of the fact that the ﬁrst rings encountered in the course of the development of mathematics were associative (and commutative) rings of numbers and rings of functions, and also associative rings of endomorphisms of Abelian groups, in particular, rings of linear transformations of vector spaces. In the survey article by A. G. Kurosh [40] he persuasively argued that the contemporary theory of associative rings is only a part of a general theory of rings, although it continues to play an important role in mathematics. The present article, in contrast to the article of A. G. Kurosh, is dedicated to a survey of one part of the theory of rings: precisely, the theory of rings, which although nonassociative, are more or less connected with the theory of associative rings. More precise connections will be mentioned during the discussion of particular classes of rings. Because the classes of rings that are studied in this article were mentioned to some extent in the article by A. G. Kurosh, there is some intersection in the content of these two articles. In what follows, the author assumes that the following notions are understood: rings, algebras, ideals, quotient rings, rings with a domain Σ of operators (or Σ-operator rings). These notions and also some other main notions of the theory of rings can be found in the same article by A. G. Kurosh.

A.1.2

Origins of the theory of nonassociative rings

We brieﬂy describe the origins of the theory of nonassociative rings. Examples of such rings were known a long time ago. The nonassociativity of the vector product of 3dimensional vectors was known in mechanics. With this operation and vector addition the collection of vectors is a Lie ring. Another very beautiful example is the algebra of so-called Cayley numbers, which have been used in diﬀerent parts of mathematics. The development of the theory of continuous groups in general and Lie groups in particular contributed to the study of Lie algebras of ﬁnite dimension, which are closely connected to Lie groups. Another connection between Lie algebras and groups that appears to be very fruitful has been studied in the works of W. Magnus [45], I. N. Sanov [50], A. I. Kostrikin [35], and others. There is an interesting relationship between associative rings on the one hand and Lie rings and J-rings1 on the other hand, constructed by the introduction of a new operation on an associative ring. This relationship, in addition to giving certain information about Lie rings and J-rings, allows us to study associative rings from some new directions. 1 Now

called Jordan rings. (Translator)

Some problems in the theory of rings that are nearly associative

A.1.3

443

General results on nonassociative rings

Because there are diﬀerences between the properties of rings in diﬀerent classes, there are few results that have a universal character. We will describe some of them. Let A be an associative ring, and let a be some element of the ring A. It is possible to connect with this element a new operation of “multiplication,” which is deﬁned by x · y = axy. It is easy to check that the set of elements of the ring A form, under this operation and addition, a ring (in general, already nonassociative), which we will denote by A(a). In [48] A. I. Malcev proved that any ring is isomorphic to some subring of a ring of the form A(a). Let the additive group of an associative ring be decomposed into the direct sum of subgroups A1 and A2 . Then every element a ∈ A allows a unique representation of the form a = a1 + a2 . Under the operations of “multiplication” x · y = (xy)1 and addition the set of elements of the ring A is a (in general nonassociative) ring. We denote this ring by A . In [66] L. A. Skornyakov proved that any ring is isomorphic to some subring of a ring of the form A . The preceding results of Malcev and Skornyakov indicate the possibility of developing the entire theory of rings in terms of associative rings. However, nobody until now has been able to get any precise theorems about rings of some class based on this method. Among the reasons for this is the fact that we cannot transfer the properties of A to A(a) and A . So, for example, if A is a Lie ring, then the rings A(a) and A may not be Lie rings. The results and problems that correspond to diﬀerent classes of rings are formulated very diﬀerently and require speciﬁc methods, and because of this it is diﬃcult to imagine the development of the entire theory of rings from the theory of one speciﬁc, suﬃciently studied class.

A.1.4

Free rings and algebras

In the theory of rings, as in the theory of groups and other algebraic systems, free systems play an important role: free rings, free associative rings, free Lie rings, etc. Let ν be a cardinal number. The free ring (free associative ring, free Lie ring, etc.) on ν generators is a ring (associative ring, Lie ring, etc.) that has a system S of generators of cardinality ν such that any mapping from S onto any system of generators of any ring (associative ring, Lie ring, etc.) can be extended to a homomorphism of rings. The free ring Aν with the set S of generators of cardinality ν can be built constructively by the following steps. We will call the elements of the set S words of length 1. If α and β are words of lengths m and n (respectively) then the symbol (α)(β) will be called a word of length m + n; furthermore, we will consider two words (α)(β) and (α1 )(β 1 ) to be equal if and only if α = α1 and β = β1 . The collection of ﬁnite sums of the form s ks γs where ks is an integer and γs is a word (we assume γs = γt when s = t) becomes a ring, which we will denote by Aν , when we deﬁne the operations as follows:

ks γs +

s

s

ls γs =

s

ks γs ·

t

(ks + ls )γs ,

s

lt γt =

ks lt (γs )(γt ).

s,t

It easy to check that the ring Aν satisﬁes the above-formulated deﬁnition, and that any ring that satisﬁes that deﬁnition is isomorphic to Aν .

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If the symbols ks are allowed to come from some associative ring Σ and we deﬁne k ks γs = (kks )γs , k ∈ Σ, s

s

then the ring Aν will be a free Σ-operator ring with ν generators in the sense of Σ-operator homomorphisms. If, furthermore, Σ is a ﬁeld, then Aν is a free algebra with ν generators over the ﬁeld Σ. In the works of Kurosh [39], [41] it was proved that any subalgebra of a free algebra is again free, and some generalizations of this result to free sums of algebras were given. A. I. Zhukov [74] positively solved the problem of identities for algebras with a ﬁnite number of generators and a ﬁnite number of deﬁning relations, which is analogous to a famous problem of identities in the theory of groups.

A.1.5

Identities

With additional axioms, or so-called identities, we may deﬁne various classes of rings. The general method applied to this problem is as follows. Let Aω be the free ring with a countably inﬁnite number of generators xi (i = 1, 2, . . .). In the ring Aω we consider a subset Q. Any ring C that satisﬁes the condition, that any substitution of any elements of C into the generators xi in any element of the set Q gives zero, will be regarded as belonging to the class deﬁned by the set Q or simply to the class of Q-rings. If in some free ring Aν we take the ideal J generated by the elements obtained by substituting all the elements of Aν into the generators xi in the elements of Q, then the quotient ring D = Aν /J will be isomorphic to the free Q-ring in the sense given earlier. For example, if the set Q consists of the single element (x1 x2 )x3 − x1 (x2 x3 ), then we obtain the class of associative rings. If the set Q consists of elements qα , then it is sometimes said that the class of Q-rings is deﬁned by the identities qα = 0. The same concepts can be deﬁned in a very similar way for Σ-operator Q-rings. For the case when the set Q is ﬁnite, Yu. I. Sorkin [69] showed that the corresponding class of rings can be given with the help of one ternary operation (that is, deﬁned on ordered triples of elements) and one relation that this operation must satisfy.

A.2 A.2.1

Alternative Rings The doubling process

It is known that the ﬁeld of complex numbers can be represented as the collection of pairs of real numbers with the natural addition and the familiar deﬁnition of multiplication. If, on the Abelian group of ordered pairs (p, q) of complex numbers with coordinate-wise addition, an operation of multiplication is deﬁned by the formula (p1 , q1 ) · (p2 , q2 ) = (p1 p2 − q2 q1 , q2 p1 + q1 p2 ),

(A.1)

where p2 and q2 are the complex conjugates of the complex numbers p2 and q2 , then one can easily check that with respect to these operations the set we are considering is a ring. In this ring it happens that the equations AX = B and XC = D have a uniquely determined solution when A = 0, C = 0 and so this ring is the (associative but not commutative) division ring of real quaternions. If in Eq. (A.1) we replace the symbols pi and qi by

Some problems in the theory of rings that are nearly associative

445

real quaternions, and we understand p to be the quaternion conjugate of the quaternion p = (a, b), that is p = (a, −b), then the pairs of quaternions become a ring with respect to these operations, which in this case is a nonassociative division ring. If for every real number α and pair (p, q) we deﬁne α(p, q) = (αp, αq) then the additive groups of the above division rings become vector spaces over the ﬁeld of real numbers with corresponding dimensions 4 and 8 and the division rings become algebras over the ﬁeld of real numbers. The constructed nonassociative algebra of dimension 8 over the ﬁeld of real numbers is called the algebra of Cayley numbers. In what follows we will denote it by R8 .

A.2.2

Identities in alternative rings

The associator of the elements a, b, c in any ring is deﬁned to be the element [a, b, c] = (ab)c − a(bc). The algebra R8 satisﬁes the following identities: [x, y, y] = 0,

(A.2)

[x, x, y] = 0, [x, y, x] = 0,

(A.3) (A.4)

each of which is implied by the other two. Rings in which the identities (A.2)–(A.4) are satisﬁed are called alternative. A more general class of 8-dimensional alternative algebras was studied by Dickson. These algebras received the name Cayley-Dickson algebras. In this and the following section (if this is not stated explicitly) for simplicity of language we will assume that the additive groups of the rings do not contain elements of order 2. We next list some identities that hold in every alternative ring: [(xy)z]y = x[(yz)y], y[z(yx)] = [y(zy)]x,

(A.5) (A.6)

(xy)(zx) = x[(yz)x].

(A.7)

To prove relation (A.5) we notice that substitution of y + z for y in Eq. (A.2) leads to the equation: [x, y, z] = −[x, z, y]. (A.8) Using Eqs. (A.2) and (A.8) gives 2x[(yz)y] = x 2(yz)y + [z, y, y] − [y, z, y] − [y, y, z] = x (yz)y + (zy)y − zy 2 + y(zy) − y 2 z + y(yz) = [x(yz)]y + (xy)(yz) − [x, yz, y] − [x, y, yz] + [x(zy)]y + (xy)(zy) − [x, zy, y] − [x, y, zy] + [x, z, y 2 ] + [x, y 2 , z] − (xz)y 2 − (xy 2 )z = x(yz) + x(zy) y + (xy)(yz + zy) − [(xz)y]y − [(xy)y]z = 2[(xy)z]y.

Thus Eq. (A.5) is proved, and for its proof we used only Eq. (A.2). From this it follows that Eq. (A.5) holds in any ring which satisﬁes Eq. (A.2), that is, in any so-called right alternative ring. The proofs of Eqs. (A.6) and (A.7) are left to the reader.

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A. I. Shirshov

Subrings of alternative rings

Let us notice one property of alternative rings, which makes them close to associative rings. Let a and b be two elements of some alternative ring A, and let D be the subring of the ring A generated by the elements a and b. It happens that the ring D is associative. To prove this proposition it is enough to show that any two elements of the ring D obtained by diﬀerent parenthesizations of an associative monomial in a and b are equal. Let c be some associative monomial as described. We denote by c the nonassociative monomial obtained from the monomial c by the following parenthesization: when c = c1 a or c = c1 b we let c = (c1 )a or c = (c1 )b, respectively; and a = a, b = b. For example, a2 bab2 = ((((aa)b)a)b)b. If d is a nonassociative monomial in some parenthesization, then we will denote by d the associative monomial obtained by removing the parentheses from d. The associativity of the ring D is equivalent to the equation d = d holding where d is any nonassociative monomial in the generators a and b. The last equality, which is obvious if the degree of the monomial d in a and b is less than or equal to 3, will be proved by induction on the degree of d. Let the degree of the monomial d be greater than 3: d = d1 d2 , d1 = ad3 , and we assume that the equality to be proved holds for monomials with lower degree. Then we have the following cases: 1)

d2 = d4 a, d = (ad3 )(d4 a) = [a(d3 d4 )]a = d,

where we have used Eq. (A.7). If there is no such monomial d3 then we obtain the following using Eq. (A.4): 2)

d2 = (bd4 )b, d = (ad3 )[(bd4 )b] = [(d1 b)d4 ]b = d,

where Eq. (A.5) is used. Finally, 3)

d = (ad4 )b, d = (ad3 )[(ad4 )b] = −(ad3 )[b(ad4 )] + [(ad3 )(ad4 )]b + [(ad3 )b](ad4 ) = −d5 + d + d5 ,

where we have used Eq. (A.8) and also the above-proved identities from cases 1) and 2). Repeating (if necessary) the same transformation on d5 and so on, we come in a ﬁnite number of steps to the identity which we are proving.

A.2.4

Free alternative rings

In spite of the noted closeness of alternative rings to associative rings, up to now there is no general method that allows us to prove equations in alternative rings. Each such presently known equation requires a separate and in some cases very diﬃcult proof. This happens because up to now there has been no known method to build constructively free alternative rings, so there is no known algorithm that solves the problem of equality in free alternative rings; that is, an algorithm that allows us, for every element of this ring written in terms of the generators, to determine if it is zero or not.

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447

We mention the following interesting identity: [(ab − ba)2 , c, d] (ab − ba) = 0, which was proved by Kleinfeld (see for example [67]) and which shows that in the free alternative ring there are zero divisors.

A.2.5

Cayley-Dickson algebras

The study of alternative rings in general began with the study of alternative division rings, which in the theory of projective planes play the role of the so-called natural division rings of alternative planes (see [65]); that is, planes for which the little Desargues theorem holds. In the works of L. A. Skornyakov [62, 63] a full description is given of alternative but not associative division rings. It happens that every such division ring is an algebra of dimension 8 over some ﬁeld (a Cayley-Dickson algebra). Later and independently of Skornyakov this statement was proved by Bruck and Kleinfeld [8], but Kleinfeld [28] proved that even simplicity (that is, not having two-sided ideals) of an alternative but not associative ring implies that the ring is a Cayley-Dickson algebra. If for an element a of some ring A there exists a natural number n(a) such that an(a) = 0 (with any parenthesization of the expression an(a) ) then this element is called a nilpotent element. If all the elements in a ring (resp. ideal) are nilpotent, it is called a nilring (resp. nilideal). Recently Kleinfeld [30] strengthened his results by proving that any alternative but not associative ring, in which the intersection of all the two-sided ideals is not a nilideal, is a Cayley-Dickson algebra over some ﬁeld. Hence the class of alternative rings is much larger than the class of associative rings but only within the limits explained above.

A.2.6

Right alternative rings

Some attention has been given to right alternative rings (rings that satisfy identity (A.2)). Skornyakov [64] proved that every right alternative division ring is alternative. Kleinfeld [29] proved that for the alternativity of a right alternative ring it is suﬃcient that [x, y, z]2 = 0 implies [x, y, z] = 0. Smiley [68] analyzed the proof of Kleinfeld and noticed that it is suﬃcient to check only these cases: x = y, x = yz − zy, x = (yz − zy)y, x = [y, y, z], or z = wy and x = [y, y, w] for some w. We know about the structure of free right alternative rings as little as we know about the structure of free alternative rings. The study of these rings is one of the main tasks of the theory of alternative rings. It would be interesting to ﬁnd out whether there are identities that are not implied by Eqs. (A.2)–(A.4) and are satisﬁed in the free alternative ring with three generators as, for example, the relation (xy)z − x(yz) = 0 is satisﬁed by the free alternative ring with two generators. Because alternative rings are close relatives of associative rings, we may ask of any statement that holds for associative rings whether it also holds for alternative rings. One such problem (the Kurosh problem) will be discussed in the next section. San Soucie [51, 52] studied alternative and right alternative rings in characteristic 2 (2x = 0).

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J -Rings J -rings and special J -rings

Let A be an associative ring. If we set a ◦ b = ab + ba then with respect to addition and the operation ◦ the set of elements of the ring A becomes a ring that is in general nonassociative. We denote this ring by A(+) . For an associative algebra (or a Σ-operator ring) B it is possible in a similar way to deﬁne an algebra (or a Σ-operator ring) B (+) over the same ﬁeld; for an algebra it is more convenient to use the operation a ◦ b = 12 (ab + ba). It is easy to check that in the ring A(+) the following identities hold: a ◦ b = b ◦ a,

(A.9)

((a ◦ a) ◦ b) ◦ a = (a ◦ a) ◦ (b ◦ a).

(A.10)

Rings in which the multiplication satisﬁes (9) and (10) are called J-rings (or Jordan rings). It can happen that some subset of a ring, which is not a subring, becomes a J-ring under the operation ◦. As an example, consider the set of all real symmetric matrices of some ﬁxed degree n. A J-ring that is isomorphic to a subring of some ring of the form A(+) is called a special J-ring. Special J-algebras can be deﬁned in a similar way.

A.3.2

An exceptional J -algebra

Not every J-ring and not every J-algebra is special. The classical example, that will be discussed below, of a nonspecial (often called exceptional) J-algebra of ﬁnite dimension belongs to Albert [5]. In the algebra R8 , which was discussed at the beginning of Section A.2, for any element s = (p, q) we set s = (p, −q). In the set of all matrices of degree 3 with elements from the algebra R8 we consider the subspace C27 of self-conjugate matrices (that is, matrices that do not change when the elements are conjugated and the matrix is transposed). It is possible to check that the set C27 with respect to addition, the usual multiplication of real numbers, and the operation s ◦ t = 12 (s · t + t · s) is a J-algebra of dimension 27 over the ﬁeld of real numbers. Let x be an element of the algebra R8 . Denote by xij the matrix S from the algebra C27 in which sij = x and sji = x and all other entries are zero; by e denote the identity of the algebra R8 . Assume that there exists an associative algebra A, such that the J-algebra A(+) has a isomorphic to the algebra C27 . For simplicity in what follows we will identify subalgebra C27 with the algebra C27 . If s, t ∈ C27 then it is obvious that s · t + t · s = st + ts the algebra C27 where st is the product of the elements s and t in the algebra A. The last observation allows us to easily verify the following equations: e2ij = eij eij = eij · eij = eii + ejj ,

(A.11)

eii xij + xij eii = ejj xij + xij ejj = xij ,

(A.12)

ekk xij + xij ekk = 0 (for k = i, j), x12 y23 + y23 x12 = (x · y)13 ,

(A.13) (A.14)

x12 y13 + y13 x12 = (x · y)23 , x13 y23 + y23 x13 = (x · y)12 .

(A.15) (A.16)

From Eq. (A.13) we have ekk (ekk xij + xij ekk ) = (ekk xij + xij ekk )ekk = 0,

Some problems in the theory of rings that are nearly associative

449

and because of e2kk = ekk , it easily follows that ekk xij = xij ekk = 0 (k = i, j).

(A.17)

Setting fij = eii + ejj , from the obvious equalities fij xij + xij fij = 2xij ,

2fij xij = fij xij + fij xij fij ,

we easily obtain fij xij = fij xij fij = xij fij = xij .

(A.18)

eii yij eii = ejj yij ejj = 0,

(A.19)

Finally, because, for example, eii yij eii = eii (yij − eii yij ) = 0, (Eq. (A.12)). If x ∈ R8 then we set x = e11 x12 e12 . We show that the map x → x is a homomorphism of the algebra R8 into the algebra A. Clearly (x + y) = x + y . From Eqs. (A.14)–(A.17) it follows that (x · y) = e11 (x · y)12 e12 = e11 (x13 y 23 + y 23 x13 )e12 = e11 x13 y 23 e12 = e11 (x12 e23 + e23 x12 )y 23 e12 = e11 x12 e23 y 23 e12 = e11 x12 e23 (y12 e13 + e13 y12 )e12 . On the other hand, y12 e13 e12 = y12 e13 f13 e12 = y12 e13 e11 e12 = (y 23 − e13 y12 )e11 e12 = −e13 y12 e11 e12 = −e13 f13 y12 e11 e12 = −e13 e11 y12 e11 e12 = 0, and e23 e13 y12 = e23 e13 f12 y12 = e23 e13 e11 y12 = (e12 − e13 e23 )e11 y12 = e12 e11 y12 . Making the corresponding substitution in the expression (x · y) we get (x · y) = e11 x12 e12 e11 y12 e12 = x y . Because of the absence of proper ideals in the algebra R8 , and also because e = e11 e12 e12 = e11 f12 = e11 = 0, we conclude that the algebra R8 is isomorphic to a subalgebra of the associative algebra A, which contradicts the nonassociativity of the algebra R8 . This contradiction shows that there is no associative algebra A with the required properties.

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Identities in J -rings

It would be natural to assume that special J-algebras satisfy some system of identities that do not follow from Eqs. (A.9) and (A.10). At the present time such identities have not been found. Moreover, every attempt to characterize special J-rings with the help of any system of identities must be completely unsuccessful, because Cohn [9] gave many examples of nonspecial J-algebras which are homomorphic images of special J-algebras. It was also shown by Cohn that any homomorphic image of a special J-algebra with two generators is also a special J-algebra. Let B be some J-ring. We deﬁne by the formula {a, b, c} = (ab)c + (bc)a − (ca)b, a ternary operation on the set of elements of the ring B. It is easy to check that if B is a special Jordan ring then we have the identity {a, b, a}2 = {a, {b, a2 , b}, a}.

(A.20)

Harper [17] and Hall [15] independently proved that (A.20) holds for any J-ring. In the author’s work [58] it was proved that every J-ring on two generators is special. From this result it easily follows that any identity, which involves, like (A.20), only two variables and which holds in any special J-ring, also holds in any J-ring. This result was recently reproved by Jacobson and Paige [27]. At present it is still not known whether the identities {{a, x, a}, x, {a, x, b}} = {{{a, x, a}, x, b}, x, a},

(A.21)

{{x, b, x}, a, {x, b, x}} = {x, {b, {x, a, x}, b}, x},

(A.22)

which hold in any special J-ring, also hold in any J-ring. These identities, which were conjectured by Jacobson, were proved by him in [26] to hold in the algebra C27 . Jacobson proposed the question: Does there exist a J-algebra that is not a homomorphic image of a special J-algebra? Albert [6] proved that the algebra C27 is not a homomorphic image of any special Jalgebra of ﬁnite dimension. The above-mentioned problem is equivalent to the following: Is the free J-ring on three or more generators special? From a positive answer would follow trivially the solution of the problem of identities of a free J-ring, but from this we could still not solve the problem of ﬁnding a basis for the free J-algebra on three or more generators (see Cohn [9]).

A.3.4

J -rings and alternative rings

If, on the set of elements of a right-alternative ring T , we deﬁne the operation a ◦ b = ab + ba, then it is easy to show that in this case the ring T (+) will be a J-ring. However, it turns out that the class of all J-rings that can be obtained in this way is equal to the class of all special J-rings. Indeed, the mapping f : x → Rx of elements of the ring T , to the associative ring generated by right multiplications Rx (aRx = ax) in the ring T ∗ of all endomorphisms of the additive group of the ring T , is a homomorphism of the ring T (+) onto some subring of the special J-ring T ∗(+) . The mapping f will be an isomorphism if we initially extend the ring T by an identity element (after which the extended ring remains right alternative). The possibility of connecting every right alternative ring to an associative ring through the corresponding (special) J-ring (in general this mapping is not bijective) appears to be very good in the study of right alternative rings, and so also with alternative rings.

Some problems in the theory of rings that are nearly associative

451

Using this method, the author proved in [59, 60] that all the results obtained up to the present toward solving the Kurosh problem [38] (or its special case, the Levitzki problem) for associative algebras (or rings) also hold for alternative algebras (or rings) and for special J-algebras (J-rings). Let us formulate one of them. The alternative ring D with a ﬁnite number of generators and the identity xn = 0 is nilpotent, that is, there exists a natural number N such that any product of N elements of D is zero. The closest generalization of J-rings are the so-called noncommutative J-rings, the study of which was started by Schafer. The natural place for them in the present article is in the last section.

A.4 A.4.1

Lie Rings Lie rings and Lie algebras

A ring that satisﬁes the identities, x2 = 0, (xy)z + (yz)x + (zx)y = 0,

(A.23) (A.24)

is called a Lie ring. In this article we completely avoid the discussion of Lie algebras of ﬁnite dimension, which are more naturally related to the theory of Lie groups. If, in an associative ring A we deﬁne a new operation by the equation a · b = ab − ba, then the set of elements of A will be a Lie ring with this operation and addition. We denote this new ring by A(−) . Birkhoﬀ [7] and Witt [71] independently proved that every Lie algebra is isomorphic to a subalgebra of some algebra of the form A(−) . If we use the terminology of J-rings, then we can say that every Lie ring is special. Lazard [42] and Witt [72] studied representations of Σ-operator Lie rings in Σ-operator associative rings. The existence of such a representation was proved by them in the case of Σ-principal ideal rings and in particular for Lie rings without operators. The example constructed by the author in [56] shows that there exist nonrepresentable Σ-operator Lie rings that do not have elements of ﬁnite order in the additive group. I. D. Ado [1, 2] proved that any ﬁnite dimensional Lie algebra over the ﬁeld of complex numbers can be represented in a ﬁnite dimensional associative algebra. Later HarishChandra [16] and Iwasawa [24] proved that Ado’s theorem holds for any ﬁnite dimensional Lie algebra. We mention the cycle of works of Herstein [19]–[21] on associative rings that are dedicated to studying the rings A(−) with diﬀerent assumptions on the ring A.

A.4.2

Relations between Lie rings and groups

There are interesting relations between the theory of Lie rings and the theory of groups. Let K be the ring of formal power series with rational coeﬃcients in the noncommutative variables xi (i = 1, 2, . . .). Magnus [45] proved that the elements yi = 1 + xi of the ring K generate a free subgroup G of the multiplicative group of the ring K, and every element of the subgroup Gn (the nth commutator subgroup) has the form 1 + n + ω, where n is some homogeneous Lie polynomial (with respect to the operations a · b and a + b) of degree n in the generators xi , and ω is a formal power series in which all the terms have a degree

452

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greater than n. Then because of known criteria [11, 12, 44], which allow us to determine whether a given polynomial is a Lie polynomial, the above-mentioned representation of the free group allows us to determine whether any given element lies in one term or another of the lower central series. The elements zi = exi of the ring K also generate a free group [46] and if ex ey = et , then t is a power series, the terms of which are homogeneous Lie polynomials in x and y [18]. The relations that exist between the theory of groups and the theory of Lie rings allow us to obtain group-theoretical results from statements proved for Lie rings. For example, Higman [23] proved nilpotency (see the deﬁnition below) of any Lie ring that has an automorphism of prime order without nonzero ﬁxed points. This statement allowed him to prove the nilpotency of ﬁnite solvable groups that have an automorphism satisfying the analogous condition. Earlier Lazard [43] studied nilpotent groups using large parts of the apparatus of Lie ring theory.

A.4.3

The Engel condition and the Burnside problem

We consider one more circle of questions that are relevant to the theory of groups. A Lie ring L is called a ring satisfying the nth Engel condition if for any elements x and y we have the relation {· · · [(xy)y] · · · }y = 0 (n y’s). We introduce the following notation: L = L1 = L(1) , Lk = Lk−1 L, L(k) = L(k−1) L(k−1) . A Lie ring is called nilpotent (resp. solvable) if there exists a natural number m such that Lm = 0 (resp. L(m) = 0). With some restrictions on the additive group, Higgins [22] proved that solvable rings satisfying the nth Engel condition are nilpotent. Then Cohn [10] constructed an example of a solvable Lie ring, with additive p-group (in characteristic p) and satisfying the p th Engel condition, which is not nilpotent. For Lie rings with a ﬁnite number of generators and some restrictions on the additive group, A. I. Kostrikin [37] proved that the Engel condition implies nilpotency. This result is especially interesting because from it follows the positive solution of the group-theoretical restricted Burnside problem for p-groups with elements of prime order [35, 36]. An element a in a Lie algebra L is called algebraic if the endomorphism Ra : x → xa generates a ﬁnite dimensional subalgebra in the (associative) algebra of all endomorphisms of the additive group of the algebra L. It is not known whether there exists a Lie algebra with a ﬁnite number of generators and inﬁnite dimension in which every element is algebraic. This problem is analogous to the famous Kurosh problem [38] for associative algebras. We mention one more simply stated but unsolved problem. Let the Lie algebra L be such that any two elements belong to a subalgebra, the dimension of which does not exceed some ﬁxed number. Does it follow from this that every ﬁnite subset of the algebra L belongs to some subalgebra of ﬁnite dimension?

Some problems in the theory of rings that are nearly associative

A.4.4

453

Free Lie rings

An important role in the theory of Lie rings is played by free Lie rings. In contrast to free alternative rings and free J-rings, free Lie rings have been thoroughly studied. M. Hall [14] pointed out a method for constructing a basis of a free Lie algebra; E. Witt [71] found a formula for computing the rank of the homogeneous modules in a free Lie algebra on a ﬁnite number of generators. We brieﬂy describe one constructive method of building a free Lie ring. Let A be a free associative Σ-operator ring with some set R = {ai } (i = 1, 2, . . . , k) as a set of free generators. As shown in [61] the elements of the set R generate in the Lie ring A(−) a free Lie ring L for which they are free generators. We order the elements of the set R in some way, and then we order lexicographically every set of (associative) monomials of the same degree in the elements of the set R. Let W be the set of all monomials w such that w = w1 w2 > w2 w1 , for any representation of the monomial w as a product of two monomials w1 and w2 . Let v ∈ W with v = v1 v2 where v1 is a monomial from W of minimal degree such that v2 ∈ W . We parenthesize the monomial v in the following way: v = (v1 )(v2 ) and we repeat this method of parenthesization on the monomials v1 and v2 . The set of nonassociative monomials obtained from the elements of the set W by this method of parenthesization with the operation interpreted as a · b = ab − ba will be a basis of the ring L. The author in [57] and independently Witt in [73] proved that any subalgebra of a free Lie algebra is again free. This theorem is analogous to the theorem of Kurosh mentioned in Section A.1 for subalgebras of free algebras. Use of the above method of constructing a free Lie algebra allowed the author in [61] to prove that any Lie algebra of ﬁnite or countable dimension can be embedded in a Lie algebra with two generators. Analogous theorems about embedding of arbitrary algebras and of associative rings were proved, respectively, by A.I. Malcev [48] and A.I. Zhukov [74]. The study of Lie algebras over ﬁelds of prime characteristic has led to the discussion of so-called restricted Lie algebras. In a restricted Lie algebra over a ﬁeld of characteristic p > 0 an additional unary operation is deﬁned with some natural axioms that are typical of the usual (associative) p th power. Jacobson [25] proved a theorem for restricted Lie algebras analogous to the Birkhoﬀ-Witt theorem, which in this case includes a theorem similar to Ado’s theorem.

A.4.5

Binary-Lie rings and Moufang-Lie rings

Recently A. I. Malcev [49] considered a class of binary-Lie rings, which are related to Lie rings in a way analogous to the way alternative rings are related to associative rings. A ring is called binary-Lie if every two elements lie in some Lie subring. A. T. Gainov [13] proved that in the case of a ring for which the additive group has no elements of order two, for a ring to be binary-Lie it is suﬃcient that these identities hold: x2 = [(xy)y]x + [(yx)x]y = 0. If, on the set of elements of some alternative ring D, we deﬁne the above-described operation a · b = ab − ba, then in the ring D(−) , as was shown by A. I. Malcev [49], these identities hold: x2 = [(x · y) · z] · x + [(y · z) · x] · x + [(z · x) · x] · y − (x · y) · (x · z) = 0.

(A.25)

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Rings satisfying the identities (25) are called by A. I. Malcev Moufang-Lie rings2 , and he also showed that the class of Moufang-Lie rings without elements of additive order 6 is properly contained in the class of binary-Lie rings. Recently Kleinfeld [31] proved that a Moufang-Lie ring M without elements of additive order 2, which has an element a such that aM = M , is a Lie ring. A corresponding result can clearly be formulated in the language of alternative rings. The problem of the truth of a theorem similar to the Birkhoﬀ-Witt theorem, connecting the theory of Moufang-Lie rings with the theory of alternative rings, remains open.

A.5 A.5.1

Some Wider Classes of Rings Power-associative rings

As was shown earlier, a ring is alternative if and only if every two elements lie in some associative subring. Algebraists working in the theory of rings have been attracted for a long time to the wider class of rings with associative powers. A ring is called power-associative if every element lies in some associative subring. It is not diﬃcult to check that all the classes of rings discussed in the present article are power-associative. In the case of rings for which the additive group has no torsion, Albert [3] has shown that the identities x2 x = xx2 and (x2 x)x = x2 x2 are suﬃcient to guarantee power-associativity. This result was recently given another proof by A. T. Gainov [13]. Albert proved in [4] that if in the additive group of a ring there are no elements of order 30 then power-associativity follows from the identities (xy)x = x(yx) and

(x2 x)x = x2 x2 .

For rings of small characteristic some suﬃcient conditions for power-associativity were found by Kokoris [32, 33].

A.5.2

Decomposition with respect to an idempotent

We mention one method for studying power-associative rings that has been used extensively in the works of Albert. Let A be a commutative power-associative ring in which the equation 2x = a has a unique solution for every a ∈ A and which contains an idempotent e (e2 = e). Then it happens that every element b ∈ A has a unique representation in the form b = b0 + b1 + b1/2 where bλ e = λbλ ; that is, the ring A can be represented as the direct sum of three modules A = A0 + A1 + A1/2 , the study of which gives some information about the ring A. If the ring A is noncommutative, then we can study the commutative ring A(+) which is obtained from the ring A with the help of the new multiplication a ◦ b = 12 (ab + ba). It is obvious that the subrings generated by a single element in the rings A and A(+) are the same. Therefore the ring A(+) is again power-associative. Another very wide class of rings is the class of ﬂexible rings; that is, rings that satisfy the identity (A.4). All the rings discussed in this article, except for right alternative rings, are from this class. Important general results, which go beyond the class of algebras of ﬁnite dimension, have not been obtained for ﬂexible rings. 2 Now

called Malcev rings. (Translator)

Some problems in the theory of rings that are nearly associative

A.5.3

455

Noncommutative J-rings and power-commutative rings

It would be natural to expect deeper results for ﬂexible power-associative rings. However, comparatively recently Schafer [53] began the study of the class of so-called noncommutative J-rings, deﬁned by identities Eqs. (A.4) and (A.10), which is slightly narrower than the class of ﬂexible power-associative rings, but contains most of the rings mentioned above. The study of this class of rings at the present time is contained in the theory of algebras of ﬁnite dimension (see [34, 54, 55]); however, we can hope that in the future a suﬃciently interesting theory of this class of rings will be constructed. Finally, we mention one very wide class, the so-called power-commutative rings; that is, rings in which every element belongs to a commutative (but not necessarily associative) subring. This class includes not only the ﬂexible rings, but also the power-associative rings. Unfortunately, at this point in time, we do not even know whether this class can be deﬁned with the help of a ﬁnite system of identities.

A.6

Acknowledgments

The translators thank Alexander Pozhidaev for proofreading the translation and suggesting improvements. Murray Bremner thanks NSERC (Natural Sciences and Engineering Research Council of Canada) for ﬁnancial support.

References [1] I. D. Ado, On representations of ﬁnite continuous groups using linear substitutions, Izvestiya Kazanskogo Fiziko-Matematicheskogo Obshchestva 7 (1934–35) 1–43. [2] I. D. Ado, Representation of Lie algebras by matrices, Uspekhi Matematicheskikh Nauk II, 6 (1947) 159–173. [3] A. A. Albert, On the power-associativity of rings, Summa Brasiliensis Mathematicae 2 (1948) 21–33. [4] A. A. Albert, Power-associative rings, Transactions of the American Mathematical Society 64 (1948) 552–593. [5] A. A. Albert, A note on the exceptional Jordan algebra, Proceedings of the National Academy of Sciences of the USA 36 (1950) 372–374. [6] A. A. Albert, A property of special Jordan algebras, Proceedings of the National Academy of Sciences of the USA 42 (1956) 624–625. [7] G. Birkhoﬀ, Representability of Lie algebras and Lie groups by matrices, Annals of Mathematics 38 (1937) 526–532.

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[8] R. H. Bruck, E. Kleinfeld, The structure of alternative division rings, Proceedings of the American Mathematical Society 2 (1951) 878–890. [9] P. M. Cohn, On homomorphic images of special Jordan algebras, Canadian Journal of Mathematics 6, 2 (1954) 253–264. [10] P. M. Cohn, A non-nilpotent Lie ring satisfying the Engel condition and a nonnilpotent Engel group, Proceedings of the Cambridge Philosophical Society 51 (1955) 401–405. [11] E. B. Dynkin, Computation of the coeﬃcients in the Campbell-Hausdorﬀ formula, Doklady Akademii Nauk USSR 57, 4 (1947) 323–326. [12] K. O. Friedrichs, Mathematical aspects of the quantum theory of ﬁelds, V, Communications on Pure and Applied Mathematics 6 (1953) 1–72. [13] A. T. Gainov, Identical relations for binary-Lie rings, Uspekhi Matematicheskikh Nauk XII, 3 (1957) 141–146. [14] M. Hall, A basis for free Lie rings and higher commutators in free groups, Proceedings of the American Mathematical Society 1 (1950) 575–581. [15] M. Hall, An identity in Jordan rings, Proceedings of the American Mathematical Society 42 (1956) 990–998. [16] Harish-Chandra, Faithful representations of Lie algebras, Annals of Mathematics 50 (1949) 68–76. [17] L. R. Harper, A proof of an identity for Jordan algebras, Proceedings of the National Academy Sciences of the USA 42 (1956) 137–139. [18] F. Hausdorﬀ, Die symbolische Exponentialformel in der Gruppentheorie, Bericht der K¨ oniglichen S¨ achsischen Gesellschaft der Wissenschaften, Mathematisch-Physikalische Klasse 58 (1906) 19–48. [19] I. N. Herstein, On the Lie and Jordan rings of a simple associative ring, American Journal of Mathematics 77 (1955) 279–285. [20] I. N. Herstein, The Lie ring of a simple associative ring, Duke Mathematical Journal 22 (1955) 471–476. [21] I. N. Herstein, Lie and Jordan systems in simple rings with involution, American Journal of Mathematics 78 (1956) 629–649. [22] P. J. Higgins, Lie rings satisfying the Engel condition, Proceedings of the Cambridge Philosophical Society 50 (1954) 8–15. [23] G. Higman, Groups and rings having automorphisms without non-trivial ﬁxed elements, Journal of the London Mathematical Society 32, 3 (1957) 321–332. [24] K. Iwasawa, On the representation of Lie algebras, Japanese Journal of Mathematics 19 (1948) 405–426. [25] N. Jacobson, Restricted Lie algebras of characteristic p, Transactions of the American Mathematical Society 50 (1941) 15–25.

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[26] N. Jacobson, Jordan algebras, Report of a conference on linear algebras, June 1956, National Academy of Sciences – National Research Council, Washington, DC, Public. 502 (1957) 12–19. [27] N. Jacobson, L. J. Paige, On Jordan algebras with two generators, Journal of Mathematics and Mechanics 6 (1957) 895–906. [28] E. Kleinfeld, Simple alternative rings, Annals of Mathematics 58 (1953) 544–547. [29] E. Kleinfeld, Right alternative rings, Proceedings of the American Mathematical Society 4 (1953) 939–944. [30] E. Kleinfeld, Generalization of a theorem on simple alternative rings, Portugaliae Mathematica 14, 3–4 (1955) 91–94. [31] E. Kleinfeld, A note on Moufang-Lie rings, Proceedings of the American Mathematical Society 9 (1958) 72–74. [32] L. A. Kokoris, New results on power-associative algebras, Transactions of the American Mathematical Society 77 (1954) 363–373. [33] L. A. Kokoris, Power-associative rings of characteristic two, Proceedings of the American Mathematical Society 6 (1955) 705–710. [34] L. A. Kokoris, Some nodal noncommutative Jordan algebras, Proceedings of the American Mathematical Society 9 (1958) 164–166. [35] A. I. Kostrikin, On the relation between periodic groups and Lie rings, Izvestiya Akademii Nauk USSR 21 (1957) 289–310. [36] A. I. Kostrikin, Lie rings satisfying the Engel condition, Izvestiya Akademii Nauk USSR 21 (1957) 515–540. [37] A. I. Kostrikin, On the Burnside problem, Doklady Akademii Nauk USSR 119, 6 (1958) 1081–1084. [38] A. G. Kurosh, Problems in the theory of rings related to the problem of Burnside on periodic groups, Izvestiya Akademii Nauk USSR 5 (1941) 233–247. [39] A. G. Kurosh, Free nonassociative algebras and free products of algebras, Matematicheskii Sbornik 20, 62 (1947) 239–262. [40] A. G. Kurosh, The current state of the theory of rings and algebras, Uspekhi Matematicheskikh Nauk VI, 2 (1951) 3–15. [41] A. G. Kurosh, Nonassociative free sums of algebras, Matematicheskii Sbornik 37, 79 (1955) 251–264. [42] M. Lazard, Sur les alg`ebres enveloppantes universelles des certaines alg`ebres de Lie, Publications Scientiﬁques de l’Universit´e d’Alger, S´erie A 1, 2 (1954) 281–294. [43] M. Lazard, Sur les groupes nilpotents et les anneaux de Lie, Annales Scientiﬁques de ´ l’Ecole Normale Sup´erieure 71 (1954) 101–190. [44] R. C. Lyndon, A theorem of Friedrichs, Michigan Mathematical Journal 3, 1 (1955–56)

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¨ [45] W. Magnus, Uber Beziehungen zwischen h¨ oheren Kommutatoren, Journal f¨ ur die reine und angewandte Mathematik 177 (1937) 105–115. ¨ [46] W. Magnus, Uber Gruppen und zugeordnete Liesche Ringe, Journal f¨ ur die reine und angewandte Mathematik 182 (1940) 142–149. [47] A. I. Malcev, On the decomposition of an algebra into the direct sum of the radical and a semi-simple subalgebra, Doklady Akademii Nauk USSR 36, 1 (1942) 46–50. [48] A. I. Malcev, On a representation of nonassociative rings, Uspekhi Matematicheskikh Nauk VII, 1 (1952) 181–185. [49] A. I. Malcev, Analytic loops, Matematicheskii Sbornik 36, 78 (1955) 569–576. [50] I. N. Sanov, Investigations into the relation between periodic groups of prime period and Lie rings, Izvestiya Akademii Nauk USSR 16 (1952) 23–58. [51] R. L. San Soucie, Right alternative division rings of characteristic 2, Proceedings of the American Mathematical Society 6 (1955) 291–296. [52] R. L. San Soucie, Right alternative rings of characteristic two, Proceedings of the American Mathematical Society 6 (1955) 716–719. [53] R. D. Schafer, Non-commutative Jordan algebras of characteristic zero, Proceedings of the American Mathematical Society 6 (1955) 472–475. [54] R. D. Schafer, On non-commutative Jordan algebras, Proceedings of the American Mathematical Society 9 (1958) 110–117. [55] R. D. Schafer, Restricted non-commutative Jordan algebras of characteristic p, Proceedings of the American Mathematical Society 9 (1958) 141–144. [56] A. I. Shirshov, On the representation of Lie rings in associative rings, Uspekhi Matematicheskikh Nauk VIII, 5 (1953) 173–175. [57] A. I. Shirshov, Subalgebras of free Lie algebras, Matematicheskii Sbornik 33, 75 (1953) 441–452. [58] A. I. Shirshov, On special J-rings, Matematicheskii Sbornik 38, 80 (1956) 149–166. [59] A. I. Shirshov, On some nonassociative nilrings and algebraic algebras, Matematicheskii Sbornik 41, 83 (1957) 381–394. [60] A. I. Shirshov, On rings with identical relations, Matematicheskii Sbornik 43, 85 (1957) 277–283. [61] A. I. Shirshov, On free Lie rings, Matematicheskii Sbornik 45, 87 (1958) 113–122. [62] L. A. Skornyakov, Alternative division rings, Ukrainskii Matematichnii Zhurnal 2, 1 (1950) 70–85. [63] L. A. Skornyakov, Alternative division rings of characteristic 2 and 3, Ukrainskii Matematichnii Zhurnal 2, 3 (1950) 94–99.

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[64] L. A. Skornyakov, Right-alternative division rings, Izvestiya Akademii Nauk USSR 15 (1951) 177–184. [65] L. A. Skornyakov, Projective planes, Uspekhi Matematicheskikh Nauk VI, 6 (1951) 112–154. [66] L. A. Skornyakov, Representation of nonassociative rings in associative rings, Doklady Akademii Nauk USSR 102, 1 (1955) 33–35. [67] M. F. Smiley, Kleinfeld’s proof of the Bruck-Kleinfeld-Skornyakov theorem, Mathematische Annalen 134 (1957) 53–57. [68] M. F. Smiley, Jordan homomorphisms and right alternative rings, Proceedings of the American Mathematical Society 8 (1957) 668–671. [69] Yu. I. Sorkin, Rings as sets with one operation which satisfy only one relation, Uspekhi Matematicheskikh Nauk XII, 4 (1957) 357–362. [70] Liu-Shao Syue, On decomposition of locally ﬁnite algebras, Matematicheskii Sbornik 39, 81 (1956) 385–396. [71] E. Witt, Treue Darstellung Liescher Ringe, Journal f¨ ur die reine und angewandte Mathematik 177 (1937) 152–160. [72] E. Witt, Treue Darstellung beliebiger Liescher Ringe, Collectanea Mathematicae 6, 1 (1953) 107–114. [73] E. Witt, Die Unterringe der freien Liescher Ringe, Mathematische Zeit schrift 64 (1956) 195–216. [74] A. I. Zhukov, Complete systems of deﬁning relations in nonassociative algebras, Matematicheskii Sbornik 27, 69 (1950) 267–280.

Appendix B Dniester Notebook: Unsolved Problems in the Theory of Rings and Modules Edited by V.T. Filippov, V.K. Kharchenko, and I.P. Shestakov Mathematics Institute, Russian Academy of Sciences Siberian Branch, Novosibirsk

B.1 B.2 B.3 B.4 B.5 B.6

B.1

Translators’ introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Part one . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Part two . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Part three . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

461 462 463 481 494 504 504

Translators’ introduction

The Dniester Notebook (Dnestrovskaya Tetrad) is a collection of problems in algebra, especially the theory of rings (both associative and nonassociative) and modules, which was ﬁrst published in the Soviet Union in 1969. The second and third editions of 1976 and 1983 expanded the list of problems, and included comments on the current status of each problem together with bibliographical references, especially when a solution or a counterexample had been discovered. The fourth Russian edition of 1993 (edited by V. T. Filippov, V. K. Kharchenko, and I. P. Shestakov) was the last; this is the edition that we have translated for the present English version. The problems in the Dniester Notebook originate primarily from the Novosibirsk school of algebra and logic, which was founded by the mathematician and logician A. I. Malcev. The ring theory branch of this school was developed by the algebraist A. I. Shirshov. These problems have had a considerable inﬂuence on research in algebra in the countries of the former Soviet Union. They cover a wide range of topics, with a special emphasis on research directions that are characteristic of the “Russian School”: associative rings and algebras, nonassociative rings and algebras, polynomial identities, free algebras, varieties of rings and algebras, group rings, topological and ordered rings, embedding of rings into ﬁelds and rings of fractions, and the theory of modules. Nonassociative rings receive as much attention as Dnestrovskaya tetrad: Nereshennye problemy teorii kolets i modulei (Izdanie chetvyortoye, Novosibirsk, 1993). Compiled by V. T. Filippov, V. K. Kharchenko, and I. P. Shestakov, with the assistance of A. Z. Ananyin, L. A. Bokut, V. N. Gerasimov, A. V. Iltyakov, E. N. Kuzmin, I. V. Lvov, and V. G. Skosyrskii. Translated by Murray R. Bremner and Mikhail V. Kochetov (Research Unit in Algebra and Logic, University of Saskatchewan, Saskatoon, Canada). The remarks on progress have been placed after each problem rather than altogether in a separate section as in the Russian original. The references have been put into English alphabetical order, and include MR numbers and cross-references to problem numbers.

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associative rings, and there is a notable emphasis on problems with connections to universal algebra and mathematical logic. Since the publication of the fourth edition in 1993, many problems that were mentioned as unsolved have in fact been solved, partially or completely. However we have decided to go ahead with the publication of this translation, the ﬁrst English version of the Dniester Notebook, for three major reasons. First, there are many mathematicians working in areas related to the problems in the Notebook who do not read Russian. We hope that this English version will make it easier for them to appreciate the signiﬁcant Russian work in these areas. Second, even though some parts of the Notebook are somewhat out of date, it is still very stimulating to read as a source of research ideas. There are many contemporary areas of research, some of which did not even exist at the original publication date, which are closely related to the problems in the Notebook. We hope that reading the current version will inspire further research in those areas. Third, we plan to prepare a ﬁfth edition of the Dniester Notebook, which will be bilingual in Russian and English. We hope that the publication of the fourth edition in English will facilitate the collection on a worldwide basis of information on the current status of the problems, and of new problems to be included in the ﬁfth edition. We would appreciate it very much if readers of this translation would send any comments on old problems or suggestions for new problems to V. K. Kharchenko [email protected] I. P. Shestakov M. R. Bremner

[email protected] [email protected]

With the inﬂux of many mathematicians from the former Soviet Union to the West during the last two decades, the signiﬁcance of the Dniester Notebook to western mathematicians has never been greater. We believe that this is an opportune moment to make this important work easily accessible to the English-speaking world. Murray R. Bremner and Mikhail V. Kochetov

B.2

Preface

In September 1968 in Kishinev, at the First All-Union Symposium on the Theory of Rings and Modules, it was resolved to publish a collection of open problems in the theory of rings and modules, and as a result the Dniester Notebook appeared in 1969. Since then it has been republished twice, in 1976 and 1982. The ﬁrst and both subsequent editions were quickly sold out, and to a certain extent promoted the development of research in ring theory in the USSR. Of the 326 problems in the third edition, at present more than one-third have been solved. In the present collection we oﬀer the reader the fourth edition of the Dniester Notebook, which consists of three parts. The ﬁrst two parts are reproduced from the third edition with small editorial changes. The comments on the problems have been updated and extended. As before, the problems that have been completely solved are marked by an asterisk; a small circle indicates those problems on which progress has been made. The third part of the collection consists of new problems. The compilers thank everyone who has taken part in the preparation of this fourth edition.

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463

Part One

Problem 1.1 ◦ (A. A. Albert, reported by K. A. Zhevlakov) Let A be a ﬁnite dimensional commutative power-associative nilalgebra over a ﬁeld of characteristic = 2. Is A solvable? Remark: It is known that such an algebra is not necessarily nilpotent: there exists a solvable but not nilpotent ﬁnite dimensional commutative power-associative nilalgebra over any ﬁeld of characteristic = 2 (D. Suttles [164]). Problem 1.2 ∗ (S. Amitsur [78]) Is the Jacobson radical of a ﬁnitely generated associative algebra over any ﬁeld necessarily a nilalgebra? Remark: The answer is No (K. I. Beidar [13]). Problem 1.3 (A. Z. Ananiyn) Find necessary and suﬃcient conditions for the existence of a faithful representation of associative PI algebra of n × n matrices over an associative commutative algebra. Problem 1.4 (A. Z. Ananiyn) Is it true that the variety M of associative algebras over a ﬁeld k of characteristic 0 is a matricial variety if and only if each algebra A in M satisﬁes the identities [x1 , x2 , · · · , xn ]z1 z2 · · · zn [y1 , · · · , yn ] = 0, [z1 , z2 ][z3 , z4 ] · · · [z2n−1 , z2n ] = 0? Problem 1.5 (V. A. Andrunakievich) It is known that in any associative ring R the sum of all right nilideals Σ(R) coincides with the sum of all left nilideals. Is the quotient ring R/Σ(R) a ring without one-sided nilideals, where Σ(R) is the sum of all one-sided nilideals? Problem 1.6 (V. A. Andrunakievich) By transﬁnite induction using the ideal Σ(R) (see the previous problem) we construct the ideal N analogous to the Baer radical as the sum of all nilpotent ideals in the class of associative rings. The radical N is pronilpotent. Is the radical N special, that is, is any associative ring without one-sided nilideals homomorphic to the (ordinally) ﬁrst ring without one-sided nilideals? Problem 1.7 ∗ (V. A. Andrunakievich, submitted by L. A. Bokut) Find necessary and suﬃcient conditions to embed an associative ring in a radical ring (in the sense of Jacobson). Remark: Such conditions have been found (A. I. Valitskas [179]). In the same paper it is shown that these conditions are not equivalent to a ﬁnite system of quasi-identities. Problem 1.8 ∗ (V. A. Andrunakievich, Yu. M. Ryabukhin) Find necessary and suﬃcient conditions for an algebra over any associative commutative ring with identity to be decomposable into the direct sum of simple algebras. (The corresponding question for division algebras is solved.) Remark: A ring R is isomorphic to a direct sum of rings without proper ideals if and only if the following two conditions are both satisﬁed: (a) R satisﬁes the minimum condition on principal ideals; (b) R has no large ideals. (An ideal is called large if it has nontrivial intersection with every nonzero ideal). Indeed, let E be the ring of

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endomorphisms of the additive group of R, and let T be the subring of E generated by the identity element of E and all left and right multiplications by the elements of R. Consider R as a unital (right) module over T . It is clear that R is isomorphic to a direct sum of rings without proper ideals if and only if this module is semisimple. Note that if a unital module M is semisimple then every ﬁnitely generated submodule N is isomorphic to a ﬁnite direct sum of simple modules and therefore has ﬁnite length. In particular, every cyclic submodule N satisﬁes the minimum condition on submodules. It follows that M satisﬁes the minimum condition on cyclic submodules. Now if H is a submodule of M and M = H then H has a complement in M ; that is, there exists a submodule H of M such that H ⊕ H = M . Therefore, H is not large. Thus M satisﬁes the minimum condition on cyclic submodules and does not have proper large submodules. Conversely, let M be a module that does not have proper large submodules and satisﬁes the minimum condition on cyclic submodules. Let S be the socle of M (the sum of all simple submodules). Assume that S = M . Then S is not a large submodule and therefore there exists a nonzero submodule G of M such that G ∩ S = (0). Denote by P the minimal element of the set of all nonzero cyclic submodules of G. It is clear that P is simple and that P ⊆ G. Therefore P ∩ S = (0). This contradicts the inclusion P ⊆ S, which holds by deﬁnition of the socle. Thus a unital module is semisimple if and only if it satisﬁes the minimum condition on cyclic submodules and has no proper large submodules. From this and from the fact that the direct sum of arbitrary rings is semiprime if and only if every summand is semiprime, it follows that the ring R is isomorphic to a direct sum of simple rings if and only if R is semiprime and satisﬁes conditions (a) and (b). (I. V. Lvov). Problem 1.9 ∗ (V. I. Arnautov) An associative commutative ring R is called weakly Boolean if for any element x ∈ R there exists a natural number n(x) > 1 such that xn(x) = x. (Boolean algebras correspond to the case n(x) = 2 for all x.) Is there any weakly Boolean (or Boolean) ring on which it is possible to deﬁne a topology that makes the ring into a connected topological ring? Remark: The answer is Yes (V. I. Arnautov, M. I. Ursul [8]).

Problem 1.10 (V. I. Arnautov) Does there exist a “non-weakenable” topology on the ring Z of integers in which Z does not contain closed ideals? Problem 1.11 (V. I. Arnautov) Is it possible to embed any topological ﬁeld F into a connected ﬁeld? This is true if F is given the discrete topology. Problem 1.12 ∗ (V. I. Arnautov) The ring R is called hereditarily linearly compact if any closed subring in R is linearly compact. Is the direct product, with the Tikhonov topology, of hereditarily linearly compact rings Ri also hereditarily linearly compact? This is true if the Jacobson radical of every Ri is a bounded set. Remark: The answer is Yes (M. I. Ursul [176]). Problem 1.13 (V. I. Arnautov) Must a complete topological associative ring R, in which every closed commutative subring is compact, be compact?

Problem 1.14 (B. E. Barbaumov) Does there exist a division algebra, inﬁnite dimensional over its center, in which all proper subalgebras are PI algebras?

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Problem 1.15 (A. A. Bovdi) If the crossed product (G, K, ρ, σ) is a division ring, then G is a periodic group and K is a division ring. Is the group G locally ﬁnite? Problem 1.16 ∗ (L. A. Bokut) Is it possible to embed every solvable Lie algebra of countable dimension into a solvable Lie algebra with two generators? Remark: The answer is Yes (G. P. Kukin [98]). Problem 1.17 ◦ (L. A. Bokut) Let R be an associative algebra over the ﬁeld P , and let F be the free associative algebra over P on the countable set of generators X = {xi }. Let R ∗ F be the free product of the algebras R and F . By an equation over R we mean an expression f = 0 where f ∈ R ∗ F , f ∈ / R. We will call the algebra R algebraically closed if any equation over R has a solution in R. Do there exist algebraically closed associative algebras? Remark: A positive solution for equations in one variable is obtained by M. G. Makar-Limanov [106]. Problem 1.18 (L. A. Bokut) For which varieties M of rings (resp. algebras) is the groupoid ΓM of subvarieties free? When is ΓM a free semigroup? Problem 1.19 ◦ (L. A. Bokut) Describe (in terms of identities) varieties of rings (resp. algebras) with a distributive lattice of subvarieties. Remark: For associative algebras over a ﬁeld of characteristic 0 the description has been obtained by A. Z. Ananyin and A. R. Kemer [4], and for right alternative algebras by V. D. Martirosyan [109]. Problem 1.20 ∗ (L. A. Bokut) Is a ring, which is the sum of three nilpotent subrings, also nilpotent? Remark: Not always (L. A. Bokut [25]). The nilpotency of an associative ring that is the sum of two nilpotent subrings has been proved by O. Kegel [81]. Problem 1.21 (L. A. Bokut) Do there exist two semigroup algebras F1 (S) and F2 (S) without zero-divisors (here S is a semigroup and F1 and F2 are ﬁelds) such that one of them can be embedded in a division ring but the other cannot? Problem 1.22 ∗ (L. A. Bokut) Is it possible to embed any recursively deﬁned associative algebra (that is, ﬁnitely generated with recursively enumerable deﬁning relations) over a prime ﬁeld into a ﬁnitely deﬁned associative algebra? The same question for Lie algebras. Remark: The answer is Yes (V. Ya. Belyaev [16] for associative algebras, and G. P. Kukin [101] for Lie algebras). In G. P. Kukin [102] the following general result is obtained for Lie algebras and groups: Every recursively presented Lie algebra (resp. group) in a variety M is embeddable into a ﬁnitely presented Lie algebra (resp. group) in the variety MA2 (here A denotes the Abelian variety). Problem 1.23 ◦ (L. A. Bokut) Describe the identities that hold in all n-dimensional associative algebras (with ﬁxed n). Remark: A ﬁnite basis of identities in the variety generated by n-dimensional unital algebras (n ≤ 18) over a ﬁeld of characteristic 0 has been found by S. A. Pikhtilkov [132]. Problem 1.24

(L. A. Bokut) Describe Lie algebras for which the universal enveloping

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algebra has a classical ring of fractions. Problem 1.25 ◦ (L. A. Bokut) Describe varieties of associative (resp. Lie) algebras that are not decomposable into a product. Remark: A series of results on this problem for Lie algebras has been obtained by M. V. Zaicev [185]. Problem 1.26 ∗ (L. A. Bokut) Find the generators of the group of automorphisms of the free algebra of rank 2 in the variety Var Mn (k) where k is a ﬁeld. Remark: These have been found (G. Bergman, preprint). Problem 1.27 (A. T. Gainov) Is it possible to deﬁne by a ﬁnite number of identities the variety of power-commutative algebras over a ﬁeld of characteristic 0? Problem 1.28 ∗ (A. T. Gainov) Describe all ﬁnite dimensional simple binary-Lie algebras over an algebraically closed ﬁeld of characteristic 0. Remark: These have been described (A. N. Grishkov [63]). Problem 1.29 ∗ (N. Jacobson) We say that a Jordan ring J has no zero-divisors if for any a, b ∈ J the equation aUb ≡ 2(ab)b − ab2 = 0 implies either a = 0 or b = 0. Two elements a, b are said to have a common multiple if the quadratic ideals JUa and JUb satisfy JUa ∩ JUb = (0). Suppose that in a Jordan ring J without zero divisors, any two nonzero elements have a common multiple. Is J embeddable in a Jordan division ring? Remark: The answer is Yes (E. I. Zelmanov [189]). Problem 1.30 ∗ (N. Jacobson) Find necessary and suﬃcient conditions on a ﬁnite dimensional Lie algebra for its universal enveloping algebra to be primitive. Remark: These have been found (A. Ooms [125]). Problem 1.31 (N. Jacobson, reported by G. P. Kukin) Let L be a Lie p-algebra with a periodic p-operation. Is it true that L has zero multiplication? Problem 1.32 (V. P. Elizarov) Find necessary and suﬃcient conditions for a division ring T to be a left or right (classical) ring of quotients of a proper subring. Problem 1.33 ∗ (K. A. Zhevlakov) Let A be a ﬁnitely generated associative ring, and B a locally nilpotent ideal. Does B contain nilpotent ideals? Remark: Not necessarily (E. I. Zelmanov [190]). Problem 1.34 ∗ (K. A. Zhevlakov) Let A be a ﬁnitely generated associative algebra satisfying an identity. Is every algebraic ideal of A ﬁnite dimensional? Remark: Not necessarily (Yu. N. Malcev [107]). Problem 1.35 (K. A. Zhevlakov) If an associative algebra contains a nonzero algebraic right ideal, must it also contain a nonzero algebraic two-sided ideal?

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Problem 1.36 ∗ (K. A. Zhevlakov) Let A be an associative ring and let A+ be the special Jordan ring generated by some set of generators of A. Suppose that A+ satisﬁes some (associative) identity. Must A satisfy some identity? The same question if A is ﬁnitely generated. Remark: If A is ﬁnitely generated then the answer is Yes; in general, No (I. P. Shestakov [149]). Problem 1.37 ∗ (K. A. Zhevlakov) Let A be an associative ring and let A+ be the special Jordan ring generated by some set of generators of A. Let J(X) denote the quasiregular radical of the ring X. Is it true that J(A+ ) = A+ ∩ J(A)? Remark: The answer is Yes (E. I. Zelmanov [198]). Problem 1.38 ∗ (K. A. Zhevlakov) Is it true that every minimal ideal of a Jordan ring either is a simple ring or has zero multiplication? Remark: The answer is Yes (V. G. Skosyrskii [158]). Problem 1.39 ∗ (K. A. Zhevlakov) Let I be a locally nilpotent ideal in a Jordan ring J, and suppose that J satisﬁes the minimum condition on ideals contained in I. Is it true that I is nilpotent?1 Remark: The answer is Yes (V. G. Skosyrskii [158]). Problem 1.40 (K. A. Zhevlakov) In a Jordan algebra J, the least ideal for which the quotient is a special Jordan algebra will be called the specializer of J. Describe generators of the specializer of the free Jordan algebra on three generators. Problem 1.41 ∗ (K. A. Zhevlakov) Is it always possible (at least over a ﬁeld of characteristic 0) to express a Jordan algebra as a direct sum (of vector spaces) of its specializer and a special Jordan algebra? Remark: Not always. As an example we can take the free nilpotent Jordan algebra A of index 9 on 3 generators. Let S(A) be the specializer of A. The quotient algebra A/S(A) is isomorphic to the free nilpotent special Jordan algebra of index 9. Suppose that A contains a subalgebra B isomorphic to A/S(A) such that B ∩S(A) = (0). If x, y, z are the generators of B then x, y, z are linearly independent modulo A2 and by nilpotency of A they generate A. Therefore B = A and S(A) = (0). But A is not special [61] so S(A) = (0). This contradiction shows that A cannot be decomposed into the sum of the specializer and a special algebra. (I. P. Shestakov). Problem 1.42 ∗ (K. A. Zhevlakov) Is the locally nilpotent radical of a Jordan ring always ideally hereditary? Remark: The answer is Yes (A. M. Slinko [160]). Problem 1.43 ∗ (K. A. Zhevlakov) Let J be an algebraic Jordan algebra with the maximal condition on subalgebras. Must J be ﬁnite dimensional? Remark: The answer is Yes (E. I. Zelmanov, unpublished, and A. V. Chekhonadskikh [32]). Problem 1.44 ∗ (K. A. Zhevlakov) Do there exist solvable prime Jordan rings? Remark: The answer is No (E. I. Zelmanov and Yu. A. Medvedev [121]). 1 This problem is incomplete in the original text. The translation is from the third edition; A and B have been changed to J and I, respectively. (Translators)

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Problem 1.45 ∗ (K. A. Zhevlakov) Do there exist nil elements in the free Jordan algebra with n generators (n ≥ 3)? Remark: The answer is Yes (Yu. A. Medvedev [117]). Problem 1.46 ∗ (K. A. Zhevlakov) Describe nil elements in free alternative rings. Remark: These have been completely described (I. P. Shestakov [146]). Problem 1.47 ∗ (K. A. Zhevlakov) Is it true that the additive group of a free alternative ring on any number of generators is torsion free? Remark: The answer is No (S. V. Pchelintsev [128]). Problem 1.48 (K. A. Zhevlakov) (a) Describe trivial ideals of the free alternative ring on n generators. (b) ◦ Is the free alternative ring on 3 generators semiprime? Remark: (b) The free alternative algebra on 3 generators over a ﬁeld of characteristic = 2, 3 is semiprime (A. V. Iltyakov [70]). Problem 1.49 (K. A. Zhevlakov) Find elements that generate the quasiregular radical of a free alternative ring as a T -ideal. Problem 1.50 ∗ (K. A. Zhevlakov) Describe identities satisﬁed by the quasiregular radical of a free alternative ring. In particular, is it nilpotent or solvable? It is known to be locally nilpotent. Remark: The nilpotency has been proved in the case of ﬁnitely many generators, and also for the free alternative algebra over a ﬁeld of characteristic 0 with any number of generators (I. P. Shestakov [149], E. I. Zelmanov and I. P. Shestakov [199]). In the general case the quasiregular radical is not solvable (S. V. Pchelintsev [128]). Problem 1.51 ◦ (K. A. Zhevlakov) Does a free alternative ring have nonzero ideals contained in its commutative center? Remark: The answer is Yes for the free alternative algebra of characteristic = 2, 3 with a ﬁnite number k ≥ 5 of free generators (V. T. Filippov [48]). Problem 1.52 ∗ (K. A. Zhevlakov) Let A be an alternative ring. Let Z(A), N (A) and D(A), respectively, be the commutative center, the associative center, and the ideal generated by all the associators. It is known (G. V. Dorofeev [37]) that (N (A) ∩ D2 (A))2 ⊂ Z(A). Is it true that N (A) ∩ D2 (A) ⊂ Z(A)? Remark: The answer is No (E. Kleinfeld [88]). Problem 1.53 ∗ (K. A. Zhevlakov) Let A be an alternative ring, I an ideal of A, and H an ideal of I such that in A the ideal I is generated by H. Is the quotient ring B = I/H nilpotent or solvable? Remark: If the ring of operators contains 1/6 or if A is ﬁnitely generated, then B is nilpotent (S. V. Pchelintsev [127], I. P. Shestakov [147]); in the general case the answer is No (S. V. Pchelintsev [129]). Problem 1.54 ◦ (K. A. Zhevlakov) Is every nil subring of a Noetherian alternative ring nilpotent? Remark: If the ring of operators contains 1/3 then the answer is Yes (Yu. A. Medvedev [116]).

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Problem 1.55 (K. A. Zhevlakov) Find a basis of identities for the Cayley-Dickson algebra (a) ∗ over a ﬁnite ﬁeld; (b) over a ﬁeld of characteristic 0; (c) over an inﬁnite ﬁeld of prime characteristic. Remark: (a) A basis was found by I. M. Isaev [74]. Problem 1.56 ∗ (K. A. Zhevlakov) Let σ be an arbitrary radical in the class of alternative rings. Are the following statements equivalent: a ∈ σ(A), and Ra ∈ σ(A∗ ) (where A∗ is the ring of right multiplications of A)? Remark: The equivalence has been proved by A. M. Slinko and I. P. Shestakov [162] for the quasiregular radical, by V. G. Skosyrskii [158] for the locally nilpotent and locally ﬁnite radicals. In general these inclusions are not equivalent. Indeed, let s1 and s2 be the upper radicals for the class of alternative algebras deﬁned respectively by the class of all simple associative rings and the class containing only the Cayley-Dickson algebra C. Then s1 (C) = C, s1 (C∗ ) = (0), s2 (C) = (0), s2 (C∗ ) = C∗ , and so for every a = 0 in C we have a ∈ s1 (C), Ra ∈ / s1 (C∗ ),

a∈ / s2 (C), Ra ∈ s2 (C∗ ).

(I. P. Shestakov). Problem 1.57 ◦ (K. A. Zhevlakov) Describe simple nonalternative right alternative rings. Is it true that every simple right alternative ring with a nontrivial idempotent is alternative? Remark: An example of a simple nonalternative right alternative ring has been constructed by I. M. Mikheev [122]. Any simple right alternative ring that is not nil (in particular, a ring with a nonzero idempotent) is alternative (V. G. Skosyrskii [156], Ts. Dashdorzh [35]). Problem 1.58 ∗ (K. A. Zhevlakov) Can every ﬁnite dimensional right alternative algebra over a “good” ﬁeld be expressed as a direct sum (of vector spaces) of its nil radical and a semisimple subalgebra? Remark: The answer is No (A. Thedy [169], I. P. Shestakov, unpublished, see [169], p. 428). Problem 1.59 (K. A. Zhevlakov) For right alternative rings, do there exist polynomials that take on values only in the right (resp. left) associative center? In the alternative center? Problem 1.60 (K. A. Zhevlakov) Does every right alternative ring on two generators have a ﬁnite normal series with associative quotients? Problem 1.61 (K. A. Zhevlakov) Let A be an Engel Lie algebra, A∗ its multiplication algebra, and L(X) the locally nilpotent radical of the algebra X. Are the statements a ∈ L(A) and Ra ∈ L(A∗ ) equivalent? Problem 1.62 (A. E. Zalessky) Let G = SL(n, Z) where Z is the ring of integers and n ≥ 3. Let P (G) be the group algebra of G over a ﬁeld P . Does the maximal condition on two-sided ideals hold in P (G)? Problem 1.63

∗ (A. E. Zalessky) Let P (A) be the group algebra over a ﬁeld P of a

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ﬁnitely generated torsion-free Abelian group A. Let G be the group of automorphisms of A; G acts on P (A) in a natural way. Let J be an ideal of P (G) of inﬁnite index, and H ⊆ G the subgroup stabilizing this ideal: H = { h ∈ G | h(J) ⊆ J }. Is it true that in this case H has a subgroup H0 of ﬁnite index such that A has a subgroup A0 of inﬁnite index that is invariant under H0 ? (This is equivalent to the statement that H0 can be block-triangularized in GL(n, Z) where Z is the ring of integers and n is the rank of A.) Remark: The answer is Yes (G. Bergman [18]). Problem 1.64 ∗ (A. E. Zalessky) Do there exist nonisomorphic ﬁnitely generated nilpotent groups whose group algebras over some ﬁeld have isomorphic division rings of quotients? Remark: The answer is No (D. Farkas, A. Schoﬁeld, R. Snider, J. Staﬀord [40]). Problem 1.65 (I. Kaplansky) Does there exist a torsion-free group whose group algebra has zero-divisors? Problem 1.66 (I. Kaplansky) Must the group algebra of an arbitrary group G over a ﬁeld k of characteristic 0 be semisimple in the sense of Jacobson? Problem 1.67 (I. Kaplansky, reported by A. A. Bovdi) If the augmentation ideal of the group algebra KG is a nilideal then K is a ﬁeld of characteristic p and G is a p-group. Must G also be locally ﬁnite? Problem 1.68 (H. K¨ othe) Is it true that in any associative ring a sum of two left nilideals is a left nilideal? Problem 1.69 (A. I. Kokorin) Develop a theory of totally ordered skew ﬁelds analogous to the Artin-Schreier theory of totally ordered ﬁelds. Problem 1.70 (A. I. Kokorin) Is it always possible to embed a totally ordered skew ﬁeld into another totally ordered skew ﬁeld whose set of positive elements is a divisible multiplicative group? Problem 1.71 ∗ (A. T. Kolotov) Let d be a derivation of the free associative algebra kX. Must the kernel of d be a free algebra? Remark: The answer is No. Let F = kx, y, z be the free associative algebra and d the derivation of F deﬁned on generators by d(x) = xyx + x, d(y) = −yxy − y, d(z) = −x. Then ker d coincides with the subalgebra G = algp, q, r, s that has a single deﬁning relation pq = rs where p = xyz + x + z, q = yx + 1, r = xy + 1, s = zyx + x + z. The subalgebra is not free. (G. Bergman). Problem 1.72 ∗ (A. T. Kolotov) Is it true that the union of any increasing chain of free subalgebras in an arbitrary free associative algebra is also free? Remark: The answer is No. Let F = ks, xw , yw , zw be the free associative algebra with generators indexed by w ranging over the free semigroup p, q, b, d. Deﬁne inductively the elements sw ∈ F by setting 1) s1 = s, 2) swp = sw (xw yw + 1), swb = zw yw + 1, swq = sw (xw yw zw + xw + zw ),

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swd = yw . For every n ≥ 0 set Sn = { sw : |w| = n }. Then Sn is a family of free generators of a free subalgebra Gn and G1 ⊂ G2 ⊂ · · · , but the algebra G = n Gn is not free. (G. Bergman). Problem 1.73 ∗ (A. T. Kolotov) Is it true that the intersection of two free subalgebras of an arbitrary free associative algebra is again free? Remark: The answer is No. Let F = kx, y1 , y2 , y3 , z, G1 = algx, y1 , y3 , y2 z, z, G2 = algx, xy1 y2 −y3 , y2 , z. Then G1 and G2 are free subalgebras but G1 ∩ G2 is not free. (G. Bergman). Problem 1.74 (P. M. Cohn, reported by L. A. Bokut) Is every automorphism of an arbitrary free associative algebra tame (that is, a product of elementary automorphisms)? Problem 1.75 (V. M. Kopytov) Describe the real Lie algebras that admit a topology in which the Campbell-Hausdorﬀ series converges for any two elements from some neighborhood of zero. Are all such Lie algebras residually ﬁnite dimensional? Problem 1.76 ∗ (V. M. Kopytov) Is the free Lie product of ordered Lie algebras again ordered? Remark: The answer is Yes (S. A. Agalakov, L. S. Shtern [2]). Problem 1.77 ◦ (E. G. Koshevoy) Describe complete subalgebras of free unital associative algebras. A subalgebra A ⊂ kX is called complete if f (a) ∈ A implies a ∈ A for any nonconstant polynomial f (t) ∈ k[t]. Remark: An example of a complete subalgebra in the free algebra on 3 generators can be found in the work of E. G. Koshevoy [90]. Problem 1.78 ◦ (E. N. Kuzmin) Is every Malcev algebra that satisﬁes the nth Engel condition locally nilpotent? Remark: The answer is Yes, if the characteristic of the ground ﬁeld is p = 2 (V. T. Filippov [43], E. I. Zelmanov [197]). Problem 1.79 ∗ (E. N. Kuzmin) Is every Malcev algebra over a ﬁeld of characteristic 0 that satisﬁes the nth Engel condition solvable? Remark: The answer is Yes (V. T. Filippov [42], E. I. Zelmanov [195]). Problem 1.80 ∗ (E. N. Kuzmin) Can every ﬁnite dimensional Malcev algebra A over a ﬁeld of characteristic 0 be expressed as a direct sum (of vector spaces) of its radical and a semisimple subalgebra? Are the semisimple components of this decomposition conjugate by automorphisms of A? (Analog of Levi-Malcev theorem for Lie algebras.) Remark: The answer is Yes. This decomposition has been obtained independently by A. N. Grishkov [62], R. Carlsson [29] and E. N. Kuzmin [104]. The conjugacy of semisimple factors is proved by R. Carlsson [28]. Problem 1.81 ◦ (E. N. Kuzmin) Does an arbitrary Malcev algebra over a ﬁeld of characteristic = 2, 3 have a representation as a subalgebra of A(−) where A(−) is the minus algebra of some alternative algebra A? (Analog of the Poincar´e-Birkhoﬀ-Witt theorem for Lie algebras.) Remark: There exists a representation for the ideal (in an arbitrary Malcev algebra) generated by all Jacobians (V. T. Filippov [49]).

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Problem 1.82 ∗ (E. N. Kuzmin) Let G be a local analytic Moufang loop. Is G locally isomorphic to an analytic Moufang loop? Remark: The answer is Yes (F. S. Kerdman [84]). Problem 1.83 (E. N. Kuzmin) Does there exist an invariant integral (that is, a Haar integral) on compact Moufang loops? Problem 1.84 ∗ (E. N. Kuzmin) Let G be a simply connected Moufang loop whose tangent algebra is a solvable Malcev algebra. Is the topological space G homeomorphic to a Euclidean space Rn ? Remark: The answer is Yes (F. S. Kerdman [84]). Problem 1.85 (G. P. Kukin) Is it true that the minimal number of generators of the free product of two algebras is equal to the sum of the minimal numbers of generators of the factors? Problem 1.86 ∗ (G. P. Kukin) Is the membership problem for the free product of Lie algebras decidable if it is decidable for both factors? Remark: The answer is No (U. U. Umirbaev [174]). Problem 1.87 (G. P. Kukin) Is the isomorphism problem decidable for Lie algebras with one deﬁning relation? Problem 1.88 (G. P. Kukin) Is it true that every two decompositions of a Lie algebra into a free product have isomorphic reﬁnements? Problem 1.89 ∗ (V. N. Latyshev) Must the Jacobson radical of a ﬁnitely generated PI algebra of characteristic 0 be nilpotent? Remark: The answer is Yes (Yu. P. Razmyslov [141, 142], A. R. Kemer [82]). Later, A. Braun proved the nilpotency of the radical in any ﬁnitely generated PI algebra over a Noetherian ring [27]. Problem 1.90 ∗ (V. N. Latyshev) If a PI algebra has a classical ring of quotients must this ring of quotients also be a PI algebra? Remark: The answer is Yes (K. I. Beidar [12]). Problem 1.91 (V. N. Latyshev) What are the necessary and suﬃcient conditions that a semigroup must satisfy so that its semigroup algebra will be a PI-algebra? Problem 1.92 (V. N. Latyshev) Let A be an associative algebra with a ﬁnite number of generators and relations. If A is a nilalgebra must it be nilpotent? Problem 1.93 ◦ (I. V. Lvov) Does there exist an inﬁnite critical associative (resp. nonassociative) ring? A ring is called critical if it does not lie in the variety generated by its proper quotient rings. Remark: The answer is Yes in the nonassociative case (Yu. M. Ryabukhin, R. S. Florya [143]). Problem 1.94

(I. V. Lvov) Find all critical ﬁnite associative commutative rings.

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Problem 1.95 ◦ (I. V. Lvov) Must the variety generated by a ﬁnite right alternative (resp. Jordan, Malcev, binary-Lie) ring have a ﬁnite basis of identities? Remark: The answer is No in the right alternative case (I. M. Isaev [76]), and Yes in the Jordan and Malcev cases (Yu. A. Medvedev [113, 115]). Problem 1.96 ∗ (I. V. Lvov) Is it true that every minimal variety of rings is generated by a ﬁnite ring? Remark: The answer is No (Yu. M. Ryabukhin, R. S. Florya [143]). Problem 1.97 (I. V. Lvov) Is it true that an associative algebra of dimension greater than one over the ﬁeld of rational numbers, all of whose proper subalgebras are nilpotent, is also nilpotent? This is true for algebras over ﬁelds satisfying the Brauer condition; for instance, over ﬁnite or algebraically closed ﬁelds. Problem 1.98 (I. V. Lvov) Let f be a multilinear polynomial over a ﬁeld k. Is the set of values of f on the matrix algebra Mn (k) a vector space? Problem 1.99 ∗ (I. V. Lvov, V. A. Parfyonov) Is every radical (in the sense of Kurosh) on the class of Lie algebras characteristic? A Lie subalgebra is called characteristic if it is invariant under all derivations. Remark: The answer is No (Yu. A. Kuczynski [95]). Problem 1.100 ∗ (K. McCrimmon, reported by K. A. Zhevlakov) Is it true that the quasiregular radical of a Jordan ring is equal to the intersection of the maximal modular quadratic ideals? Remark: The answer is Yes (L. Hogben, K. McCrimmon [68]). Problem 1.101 ∗ (K. McCrimmon, reported by K. A. Zhevlakov) Is it true that in a Jordan ring A with minimum condition on quadratic ideals the quasiregular radical J(A) is nilpotent? Remark: The answer is Yes (E. I. Zelmanov [187]). Problem 1.102 (A. I. Malcev, reported by A. A. Bovdi, L. A. Bokut and D. M. Smirnov) Is it possible to embed the group algebra of a right-ordered group into a division ring? Problem 1.103 ∗ (A. I. Malcev, reported by L. A. Bokut) Find necessary and suﬃcient conditions for the embeddability of an associative ring into a division ring. Remark: Such conditions have been found by P. M. Cohn [34]. Problem 1.104 (A. I. Malcev, reported by L. A. Bokut) Do there exist two associative rings with isomorphic multiplicative semigroups one of which is embeddable in a division ring and the other is not? Problem 1.105 ∗ (A. I. Malcev) Do there exist varieties of Lie algebras that are not ﬁnitely axiomatizable? Remark: They exist over a ﬁeld of characteristic p > 0 (V. S. Drensky [38], M. Vaughan-Lee [183]). Problem 1.106 (A. I. Malcev) Does there exist a ﬁnitely axiomatizable variety of rings whose set of identities is not recursive?

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Problem 1.107 (A. I. Malcev) What is the structure of the groupoid of the following quasivarieties: (a) all rings; (b) all associative rings? Problem 1.108 ∗ (A. I. Malcev, reported by E. N. Kuzmin) Is every ﬁnite dimensional real Malcev algebra the tangent algebra of some locally analytic Moufang loop? Remark: The answer is Yes (E. N. Kuzmin [103]). Problem 1.109 ∗ (Yu. N. Malcev) Find a basis of identities for the algebra of upper triangular matrices over a ﬁeld of characteristic p > 0. Remark: A basis has been found by S. V. Polin [135] and P. I. Siderov [152]. Problem 1.110 ∗ (Yu. N. Malcev) Let R be an associative algebra with no nilideals, which is a radical extension of some PI subalgebra A (that is, for any x ∈ R there exists n(x) such that xn(x) ∈ A). Must R be a PI algebra? Remark: The answer is Yes (E. I. Zelmanov [188]). Problem 1.111 ∗ (Yu. N. Malcev) Let R be an associative algebra, which is an Hextension of some PI subalgebra A (that is, for any x ∈ R there exists n(x) > 1 such that xn(x) − x ∈ A). Must R be a PI algebra? Remark: The answer is Yes (M. Chacron [30]). Problem 1.112 (R. E. Roomeldi) Describe minimal ideals of right alternative rings. Is it true that they are either simple as rings or solvable (resp. right nilpotent)? Problem 1.113 (Yu. M. Ryabukhin) Find necessary and suﬃcient conditions for an algebra R over an arbitrary associative commutative unital ring k to be decomposable into a subdirect product of algebras with unique left and right division. Problem 1.114 (Yu. M. Ryabukhin) Let F be an arbitrary ﬁeld. Do there exist (a) an associative nilalgebra A of at most countable dimension such that every countable dimensional nilalgebra is a homomorphic image of A; (b) ◦ an algebraic algebra with the analogous property with respect to algebraic algebras? Remark: (b) The answer is No if the ground ﬁeld is uncountable (G. P. Chekanu [31]). Problem 1.115 ∗ (Yu. M. Ryabukhin, I. V. Lvov) Let S be a class of algebras over a ﬁxed ﬁeld F closed under homomorphic images. If S is not radical in the sense of Kurosh then is it true that the chain of Kurosh classes S = S0 ⊆ S1 ⊆ S2 ⊆ · · · ⊆ Sα ⊆ · · · formed by constructing the lower radical does not stabilize? This is true if the class S is closed not only under homomorphic images but also under ideals. Remark: The answer is Yes (K. I. Beidar [14]). Problem 1.116 ∗ (L. A. Skornyakov) Over which rings is every left module (resp. every ﬁnitely generated left module) decomposable into a direct sum of distributive modules (that

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is, modules with distributive lattices of submodules)? Does there exist a non-Artinian ring with this property? Remark: For the ﬁrst question, descriptions of such rings are given in the works of A. A. Tuganbaev [171] and K. R. Fuller [57]. For the second question, the answer is No. Problem 1.117 ◦ (L. A. Skornyakov) Over which rings is every ﬁnitely generated left module decomposable into a direct sum of uniserial modules? Remark: Such rings have been described by G. Ivanov [77]. Problem 1.118 ◦ (L. A. Skornyakov) Which rings (resp. algebras) are projective in the category of rings (resp. algebras over a ﬁxed ﬁeld)? Do there exist projective rings other than free rings? Remark: The ring P is projective if and only if P ∼ = S with S ⊕ K ∼ =F (direct sum of Abelian groups) where F is a free ring, K is an ideal, and S is a subring. The solution of this problem probably depends on the bicategory in which we work. Therefore it is useful to take into account the fact that the collection of such bicategories is not a set (S. V. Polin [134]). (L. A. Skornyakov). Problem 1.119 ∗ (L. A. Skornyakov) Does there exist a ring A that is not left Noetherian and such that every module, which is injective in the category of ﬁnitely generated left Amodules, is injective? Remark: The negative answer can be extracted from the results of V. S. Ramamurthi and K. N. Rangaswamy [138]. Indeed, let Q be a module that is injective in the category of ﬁnitely generated left modules over an arbitrary ring A. Then Q is ﬁnitely generated and injective with respect to natural embeddings of ﬁnitely generated left ideals of A, which implies the injectivity of Q (Theorem 3.1 and Corollary 3.4(i) in [138]). Therefore every non-Noetherian ring, whose only ﬁnitely generated injective module is zero, gives the desired example. From results of E. Matlis [110] (Theorem 4) and D. Gill [60] it follows that every non-Noetherian almost maximal commutative uniserial ring is an example. More or less the same considerations were articulated by C. U. Jensen (private correspondence, 1969). (L. A. Skornyakov). Problem 1.120 ◦ (L. A. Skornyakov) Describe all the rings whose left ideals are homomorphic images of injective modules. Remark: For the commutative case the answer is known (L. A. Skornyakov [154]). Problem 1.121 ∗ (L. A. Skornyakov) Must a ring, over which every module has a decomposition complementing direct summands, be a generalized uniserial ring? Remark: The answer is No (K. R. Fuller [56]). Problem 1.122 ∗ (L. A. Skornyakov, reported by L. A. Bokut) Do there exist free (with respect to T -homomorphisms) associative division rings? Remark: The answer is Yes. Let R be an arbitrary semiﬁr. Then there exists a universal R-division ring U that contains R. (Every R-division ring is a specialization, or a T -homomorphic image, of the R-division ring U .) In particular, if R = kX, the free algebra on an inﬁnite set X of generators, over the prime ﬁeld of characteristic p ≥ 0, then the universal R-division ring U is a “free” division ring in the class of division rings of characteristic p and cardinality ≤ |X|. (See P. M. Cohn [34]). The ﬁrst proof of the existence of a universal division ring was given by S. Amitsur [3]. J. Lewin [105] proved that the division subring generated by kX in the division ring of Malcev-Neumann (containing kX) is the universal kX-division ring. (L. A. Bokut).

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Problem 1.123 ∗ (M. Slater, reported by K. A. Zhevlakov) Does there exist a prime alternative ring that is neither associative nor Cayley-Dickson? Remark: The answer is Yes (S. V. Pchelintsev [129]). Problem 1.124 ∗ (M. Slater, reported by K. A. Zhevlakov) Let A be a free alternative ring, D the ideal generated by the associators of A, and U a maximal ideal of A lying in the associative center. Must U ∩ D be nonzero? Is it true that in a free alternative ring every trivial ideal is contained in U ∩ D? Remark: The answer to the ﬁrst question is Yes, to the second No (V. T. Filippov [47, 45]). Problem 1.125 (M. Slater, reported by K. A. Zhevlakov) Let M be the ideal of a free alternative ring A generated by the set [N, A] where N is the associative center of A. Is it true that M ⊆ N ? (This is equivalent to the statement [n, t](x, y, z) = 0 for all x, y, z, t ∈ A and n ∈ N .) This statement is true for rings with three generators. Problem 1.126 Jordan algebra?

(A. M. Slinko) What is the minimal possible dimension of a nonspecial

Problem 1.127 (A. M. Slinko) Is every ideal of a semiprime Jordan ring itself semiprime? This condition is necessary and suﬃcient for the class of Jordan rings to have the lower nilradical. Problem 1.128 ∗ (A. M. Slinko) It is known that in a special Jordan algebra J every absolute zero divisor (that is, an element b such that aUb = 2(ab)b − ab2 = 0 for all a ∈ J) generates a locally nilpotent ideal (A. M. Slinko [161]). Is this true for arbitrary Jordan algebras? Remark: The answer is Yes (E. I. Zelmanov [191]). Problem 1.129 ◦ (A. M. Slinko) Does every variety of solvable alternative (resp. Jordan) algebras have a ﬁnite basis of identities? Remark: In the case of alternative algebras the answer is Yes if the characteristic is not 2 or 3 (U. U. Umirbaev [172]), and No over a ﬁeld of characteristic 2 (Yu. A. Medvedev [114]). In the case of Jordan algebras the answer is Yes for algebras of solvability index 2 (Yu. A. Medvedev [112]). Problem 1.130 (A. M. Slinko, I. P. Shestakov) Find a system of relations that deﬁnes right representations of alternative algebras. Does there exist a ﬁnite system of relations? Problem 1.131 ◦ (A. M. Slinko, I. P. Shestakov) Let A be an alternative PI algebra. Is the universal associative algebra R(A) for alternative representations of A also PI? Remark: The answer is Yes for ﬁnitely generated algebras (I. P. Shestakov [149]). Problem 1.132 (A. M. Slinko, I. P. Shestakov) Let C be a Cayley-Dickson algebra. It is known (A. M. Slinko, I. P. Shestakov [162]) that the map ρ : x → Lx is a right-alternative right representation of C. Is ρ an alternative right representation of C? Problem 1.133 ∗ (D. M. Smirnov) What is the cardinality of the set of minimal varieties of rings? Remark: The cardinality is that of the continuum (Yu. M. Ryabukhin, R. S. Florya

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[143]). Problem 1.134 be Hopf?

(D. M. Smirnov) If a group G is Hopf must the group ring Z(G) also

Problem 1.135 (D. M. Smirnov, A. A. Bovdi) Can the group ring Z(G) of a torsion-free group contain invertible elements other than ±g, g ∈ G? Problem 1.136 (E. A. Sumenkov) Does the universal enveloping algebra of an arbitrary PI Lie algebra satisfy the Ore condition? Problem 1.137 ∗ (V. T. Filippov) Does a free Malcev algebra have trivial ideals? Remark: The answer is Yes (I. P. Shestakov [148]). Problem 1.138 ∗ (V. T. Filippov) Does the simple 7-dimensional non-Lie Malcev algebra over a ﬁeld of characteristic 0 have a ﬁnite basis of identities? Remark: The answer is Yes (A. V. Iltyakov [71]). Problem 1.139 (V. T. Filippov) Let A be a free Malcev algebra and J(A) the ideal generated by the Jacobians. Does the variety generated by J(A) have a ﬁnite basis of identities? Problem 1.140 (I. Fleischer, reported by V. I. Arnautov) Does there exist a topological ﬁeld that is not locally bounded whose topology cannot be weakened? Problem 1.141 ◦ (P. A. Freidman) Describe right Hamiltonian rings (that is, rings such that every subring is a right ideal). Remark: Right Hamiltonian rings have been described (P. A. Freidman [50], V. I. Andriyanov, P. A. Freidman [5]). A description of periodic rings and torsion-free rings has been announced (O. D. Artemovich [9]). Problem 1.142 ◦ (P. A. Freidman) Describe the rings whose lattice of subrings is modular. Remark: These rings have been described in the case of prime characteristic (P. A. Freidman, Yu. G. Shmalakov [52, 53]) and in the case of torsion-free nil rings (P. A. Freidman [51]). Problem 1.143 ∗ (P. A. Freidman) Must a ring in which all proper subrings are nilpotent also be nilpotent? Remark: The answer is No. An obvious example is the ﬁeld with p elements (p prime). The answer is Yes if we assume that the ring is nil (I. L. Khmelnitsky [86]). Problem 1.144 ∗ (P. A. Freidman) A subring Q of a ring K is called a meta-ideal of ﬁnite index if Q is a member a ﬁnite normal series Q = A0 A1 · · · An = K, where Ai is a two-sided ideal in Ai+1 (Baer). Must a nilpotent p-nilring K, in which all subrings are meta-ideals of ﬁnite index with uniformly bounded indices, be nilpotent?

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Remark: The answer is Yes, even without the assumption that the indices are uniformly bounded (I. L. Khmelnitsky [87]). Problem 1.145 ∗ (V. K. Kharchenko) Let L be a Lie algebra that admits an automorphism of ﬁnite order such that the ﬁxed elements are in the center of L. Must L be solvable? Remark: The answer is No (A. I. Belov, A. G. Gein [15]). Problem 1.146 (I. Herstein) Must the Jacobson radical of a left and right Noetherian associative ring be generalized nilpotent? Problem 1.147 (I. Herstein, reported by Yu. N. Malcev) Let R be an associative ring, without nilideals, that satisﬁes the condition ∀x, y ∈ R, ∃n = n(x, y) such that [x, y]n = 0. Must R be commutative? Problem 1.148 (P. Hall, reported by A. A. Bovdi) If the group ring K(G) satisﬁes the maximal condition on right ideals then G is Noetherian and K satisﬁes the maximal condition on right ideals. Is the converse true? It is true for solvable groups (P. Hall [66]). Problem 1.149 ◦ (I. P. Shestakov) Is it true that the center of the free alternative ring on three generators is equal to the intersection of the associative center and the associator ideal? If not, the free alternative ring on three generators is not semiprime. Remark: For the free alternative algebra on 3 generators over a ﬁeld of characteristic = 2, 3 the answer is Yes (A. V. Iltyakov [70]). Problem 1.150 ∗ (I. P. Shestakov) Is it true that every simple exceptional Jordan algebra is ﬁnite dimensional over its center? The answer is not known even in the case of Jordan division algebras (N. Jacobson [79]). Remark: The answer is Yes (E. I. Zelmanov [193]). Problem 1.151 ∗ (I. P. Shestakov) Is every solvable subring of a ﬁnitely generated alternative (resp. Jordan) ring nilpotent? Remark: For alternative algebras over a ﬁeld of characteristic = 2, 3 the answer is Yes (I. P. Shestakov [148], V. T. Filippov [46]). For Jordan algebras the answer is No (I. P. Shestakov [148]). Problem 1.152 (I. P. Shestakov) Must a right alternative nilalgebra over an associative commutative ring Φ with the maximal condition on Φ-subalgebras be right nilpotent? Problem 1.153 ∗ (A. I. Shirshov) Describe subalgebras of a free product of Lie algebras. Remark: These have been described (G. P. Kukin [96, 97]). Problem 1.154 ∗ (A. I. Shirshov) Is the word problem decidable in the class of all Lie algebras? Remark: The answer is No (L. A. Bokut [24]). An explicit example has been constructed by G. P. Kukin [99]. Problem 1.155

∗ (A. I. Shirshov) Is the word problem decidable in the class of all Lie

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algebras that are solvable of a ﬁxed index? Remark: The answer is No for solvability index ≥ 3 (G. P. Kukin [100], see also O. G. Kharlampovich [85]). Problem 1.156 ∗ (A. I. Shirshov) Must a Jordan nil ring of bounded index be locally nilpotent? Remark: The answer is Yes (E. I. Zelmanov [191]). Problem 1.157 ◦ (A. I. Shirshov) Must a Jordan nil ring of index n in characteristic 0 or p > n be solvable? Remark: The answer is Yes for algebras over a ﬁeld of characteristic 0 (E. I. Zelmanov [196]). Problem 1.158 ∗ (A. I. Shirshov) Does there exist a natural number n such that every Jordan algebra of at most countable dimension embeds in a Jordan algebra with n generators? Remark: For special Jordan algebras n = 2 (A. I. Shirshov [150]). In the general case there is no such number n (E. I. Zelmanov [192]). Problem 1.159 ◦ (A. I. Shirshov) Let Altn be the variety of alternative rings generated by the free alternative ring on n generators. Does the chain Alt1 ⊆ Alt2 ⊆ Alt3 ⊆ · · · stabilize? The same question applies to Jordan, right alternative, Malcev and binary-Lie rings. It is known that in the class of (−1, 1) rings this chain does not stabilize (S. V. Pchelintsev [126]). Remark: For alternative and Malcev rings it does not stabilize (I. P. Shestakov [148]). Problem 1.160 (A. I. Shirshov) Construct a basis of the free alternative (resp. right alternative, Jordan, Malcev, binary-Lie) algebra on n generators. Problem 1.161 ∗ (A. I. Shirshov) Must a right alternative nil ring of bounded index be locally nilpotent? Remark: The answer is No (G. V. Dorofeev [36]). Problem 1.162 (A. I. Shirshov) Is it true that every ﬁnitely generated right alternative nil ring of bounded index is solvable? Problem 1.163 ∗ (A. I. Shirshov, A. T. Gainov) Does the variety of binary-Lie algebras of characteristic 2 have a ﬁnite basis of identities? Remark: The answer is Yes if the ground ﬁeld has more than 3 elements (A. T. Gainov [58]). Problem 1.164 ∗ (W. Specht [163]) Is it true that every variety of associative (unital) algebras over a ﬁeld of characteristic 0 has a ﬁnite basis of identities? Remark: The answer is Yes (A. R. Kemer [83]). Problem 1.165 ∗ (Reported by V. I. Arnautov) Does there exist an inﬁnite ring that admits only the discrete topology? Remark: The answer is Yes (V. I. Arnautov [7]). An associative commutative ring always admits a nondiscrete topology (V. I. Arnautov [6]).

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Problem 1.166 ◦ (Reported by L. A. Bokut) Is the freeness theorem true for associative algebras with one relation? Remark: References and some partial results on this problem and 1.168 can be found in V. N. Gerasimov [59]. Problem 1.167 (Problem of Keller, reported by L. A. Bokut) Let f : xi → fi (1 ≤ i ≤ n) be an endomorphism of the polynomial algebra F [x1 , x2 , . . . , xn ] where n ≥ 2 and F is a ﬁeld of characteristic 0. Suppose that the Jacobian ∂fi det ∂xj is equal to 1. Must f be an automorphism? Problem 1.168 ◦ (Reported by L. A. Bokut) Is the word problem decidable for associative algebras with a single relation? Remark: References and some partial results on this problem and 1.166 can be found in V. N. Gerasimov [59]. Problem 1.169 (Reported by L. A. Bokut) Does there exist a group such that its group ring does not have zero divisors but is not embeddable into a division ring? Problem 1.170 (Reported by L. A. Bokut and A. R. Kemer) Let R be an associative ring without nilideals that satisﬁes the condition ∀x, y ∈ R, ∃n = n(x, y) ≥ 2 such that (xy)n = xn y n . Must R be commutative? Problem 1.171 (Reported by L. A. Bokut) Does there exist an inﬁnite associative division ring that is ﬁnitely generated as a ring? Problem 1.172 nil ring?

(Reported by K. A. Zhevlakov) Does there exist a simple associative

Problem 1.173 (Reported by K. A. Zhevlakov and V. N. Latyshev) Does there exist an algebraic, but not locally ﬁnite, associative division algebra? Problem 1.174 ∗ (Reported by E. N. Kuzmin) Is it true that every algebraic Lie algebra of bounded degree over a ﬁeld of characteristic 0 must be locally ﬁnite? Remark: The answer is Yes (E. I. Zelmanov [194]). Problem 1.175 ◦ (Reported by E. N. Kuzmin and A. I. Shirshov) Must a Lie ring of characteristic 0 or p > n satisfying the n-th Engel condition be nilpotent? Remark: The answer is Yes in the case of characteristic 0 (E. I. Zelmanov [195]). If p = n + 2 then the answer is No (Yu. P. Razmyslov [139]). Problem 1.176 (Reported by G. P. Kukin) Is the membership problem decidable in a Lie algebra with a single deﬁning relation? Problem 1.177 ◦ (Reported by I. V. Lvov and Yu. N. Malcev) Is the variety of associative algebras generated by a full matrix algebra ﬁnitely based or Specht

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(a) ∗ over a ﬁeld of characteristic 0; (b) over a ﬁeld of characteristic p? Remark: (a) The answer is Yes. Moreover, every variety of associative algebras over a ﬁeld of characteristic 0 is Specht (A. R. Kemer [83]). Problem 1.178 ∗ (Reported by Yu. N. Malcev) An algebra A over a ﬁeld F is said to have type Mk if A satisﬁes all identities of the matrix algebra Mk (F ) and only those. Let the matrix algebra Mn (R) over an algebra R have type Mk . Does it follow that R has type Mt for some t? Remark: The answer is No. It is easy to see that this is not true for a free algebra in the variety deﬁned by the identities [x1 , x2 ]x3 = x1 [x2 , x3 ] = 0 over an inﬁnite ﬁeld (I. I. Benediktovich). Problem 1.179 (Reported by V. A. Parfyonov) Describe all Schreier varieties of nonassociative algebras. Do there exist Schreier varieties other than the known ones: the variety of all algebras, ε-algebras, Lie algebras, and algebras with zero multiplication? Problem 1.180 ∗ (Reported by A. A. Nikitin and S. V. Pchelintsev) Do there exist nonassociative prime (−1, 1)-rings without elements of order 6 in the additive group? Remark: The answer is Yes (S. V. Pchelintsev [129]). Problem 1.181 (Reported by A. I. Shirshov) Is the isomorphism problem decidable in the class of nonassociative algebras over a “good” ﬁeld, for instance, over the ﬁeld of rational numbers?

B.4

Part Two

Problem 2.1 (S. Amitsur) Find the conditions for embeddability of an algebra over a ﬁeld into an algebra that is a ﬁnitely generated module over a commutative ring. Problem 2.2 (A. Z. Ananyin, L. A. Bokut, I. V. Lvov) A variety is called locally residually ﬁnite if every ﬁnitely generated ring (resp. algebra) can be approximated by ﬁnite rings (resp. ﬁnite dimensional algebras). Describe (in terms of identities) locally residually ﬁnite varieties of (a) associative rings; (b) associative algebras over a ﬁnite ﬁeld. Problem 2.3 (V. I. Arnautov) Does there exist an inﬁnite associative ring that admits only the discrete topology? Problem 2.4 (V. A. Artamonov) Let k be a principal ideal domain and V a variety of linear k-algebras deﬁned by multilinear identities. Is it true that every retract of a V -free algebra is again a V -free algebra?

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Problem 2.5 (V. A. Artamonov) Let k be a commutative associative unital ring and G an almost polycyclic group without torsion. (a) Compute the kernel of the natural epimorphism of K0 (kG) onto K0 (k). (b) If all projective modules over the group algebra kG are free, must G be commutative? Problem 2.6 (Yu. A. Bahturin) Prove that for multilinear monomials [x1 , · · · , xn ]σ = wσ and [x1 , · · · , xn ]τ = wτ (with bracket structures σ and τ ), the Lie algebra identities wσ = 0 and wτ = 0 are equivalent if and only if wσ and wτ are equal as elements of the free commutative nonassociative groupoid on x1 , · · · , xn . Problem 2.7 (Yu. A. Bahturin) Find examples of varieties U , V of Lie algebras over a ﬁnite ﬁeld that have a ﬁnite basis of identities such that one of the varieties U V , U ∪ V , [U, V ] does not have a ﬁnite basis of identities. Problem 2.8 (Yu. A. Bahturin) Find an example of a variety V of Lie algebras that has a ﬁnite basis of identities but for some natural number n the variety V (n) (deﬁned by all the identities of V in n variables) does not have a ﬁnite basis of identities. Problem 2.9 (Yu. A. Bahturin) Find a basis of identities for the Lie algebra Wn of derivations of the ring of polynomials in n variables over a ﬁeld of characteristic 0. Problem 2.10 ∗ (Yu. A. Bahturin) Find a basis of identities for the full matrix Lie algebra gl(2, k) over a ﬁnite ﬁeld k of characteristic = 2. Remark: A basis has been found (K. N. Semenov [145]). For an inﬁnite ﬁeld of positive characteristic see S. Yu. Vasilovsky [182]. Problem 2.11 ∗ (Yu. A. Bahturin) Prove that a Lie algebra L whose derived algebra L = [L, L] is nilpotent of index c (c < p) over a ﬁeld of characteristic p > 0 has a ﬁnite basis of identities. Remark: This has been proved by A. N. Krasilnikov [94]. Problem 2.12 (Yu. A. Bahturin) Describe solvable special varieties of Lie algebras (that is, varieties generated by a special Lie algebra) over a ﬁeld of characteristic 0. Problem 2.13 ∗ (Yu. A. Bahturin) Is it true that a central extension of a special Lie algebra is again special (that is, embeddable into an associative PI algebra)? In the case of characteristic 0 this question is equivalent to Latyshev’s problem 2.64 (S. A. Pikhtilkov [133]). Remark: The answer is No (Yu. V. Billig [20], see also Yu. A. Bahturin, A. I. Kostrikin [11]). Problem 2.14 (Yu. A. Bahturin) Is it true that free algebras of ﬁnite rank in an arbitrary variety over a ﬁnite ﬁeld are residually ﬁnite or at least Hopf? Problem 2.15 (Yu. A. Bahturin, G. P. Kukin) Describe Hopf (resp. locally Hopf) varieties of Lie algebras.

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Problem 2.16 ◦ (Yu. A. Bahturin, L. A. Bokut) Describe in terms of identities locally residually ﬁnite varieties of Lie algebras (a) ∗ over a ﬁeld of characteristic 0; (b) over a ﬁnite ﬁeld. Remark: (a) They have been described by M. V. Zaicev [186]. Problem 2.17 (K. I. Beidar) Must a ﬁnitely generated domain be semisimple in the sense of Jacobson? Problem 2.18 (L. A. Bokut) A variety is called Higman if every recursively presented algebra of this variety is embeddable into a ﬁnitely presented algebra. Are the following varieties of rings (algebras over a prime ﬁeld) Higman: (a) alternative; (b) Jordan; (c) Malcev; (d) binary-Lie; (e) solvable Lie algebras (resp. groups) of index n ≥ 3? Problem 2.19 (L. A. Bokut) Is the problem of existence of a solution for an equation in a free associative (resp. Lie) algebra over an algebraically closed ﬁeld decidable? Problem 2.20 (L. A. Bokut) Find the axiomatic rank of the class of associative rings that are embeddable into division rings. Problem 2.21 (L. A. Bokut) Is the class of associative rings that are embeddable into division rings deﬁnable by an independent system of quasi-identities? Problem 2.22 (L. A. Bokut) For a given p ≥ 0 construct a noninvertible ring of characteristic p whose multiplicative semigroup of nonzero elements is embeddable into a group. An associative ring is called invertible if all nonzero elements are invertible in some ring extension. So far the only known example is in the case p = 2 (L. A. Bokut [22, 23]). Problem 2.23 (L. A. Bokut) Is an arbitrary ﬁnitely generated associative (resp. Lie) algebra with a recursive basis over a prime ﬁeld embeddable into a simple ﬁnitely presented associative (resp. Lie) algebra? Problem 2.24 (L. A. Bokut) Is the problem of equality decidable in the following classes of rings? In a class of rings, the problem of equality is the question of the existence of an algorithm to decide the truth of a quasi-identity in that class: (a) ﬁnite Lie; (b) ﬁnite alternative; (c) ﬁnite Jordan; (d) ﬁnite binary-Lie; (e) free associative algebras; (f) free Lie algebras.

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Problem 2.25 (L. A. Bokut) A variety is called Magnus if the word problem is decidable for algebras with one relation. Determine whether the following varieties of algebras are Magnus: (a) the variety M (n) generated by the full matrix algebra Mn of order n over a ﬁeld of characteristic 0; (b) the variety S(n) deﬁned by the standard identity

(−1)σ xσ(1) xσ(2) · · · xσ(n) = 0.

σ∈Sn

Problem 2.26 and S(n)?

(L. A. Bokut) Is the problem of equality decidable in the varieties M (n)

Problem 2.27 Magnus?

(L. A. Bokut, I. P. Shestakov) Is the variety of alternative algebras

Problem 2.28 (L. A. Bokut. I. V. Lvov) Must every relatively free algebra in a variety of associative algebras over a ﬁnite ﬁeld be residually ﬁnite? [Compare with Problem 2.14] Problem 2.29 ∗ (Bj¨ ork, reported by V. N. Gerasimov) Suppose that a division ring is ﬁnitely generated as a right module over a subring. Must this subring also be a division ring? Remark: The answer is No (G. Bergman [19]). Problem 2.30 (P. Gabriel, reported by Yu. A. Drozd) Prove (or disprove) that for any natural number n there exist only ﬁnitely many (up to isomorphism) associative algebras of dimension n over an algebraically closed ﬁeld K that have only ﬁnitely many nonisomorphic indecomposable representations. Problem 2.31 ∗ (V. N. Gerasimov) Suppose that R is a radical ring that satisﬁes a nontrivial identity with the signature +, ·, where is the quasi-inverse. Must R satisfy a polynomial identity? Remark: The answer is Yes (A. I. Valitskas [180]). Problem 2.32 (A. G. Gein, A. Yu. Olshanski) Do there exist inﬁnite dimensional simple Lie algebras over a ﬁeld such that every proper subalgebra is one-dimensional? Problem 2.33 (A. N. Grishkov) Describe semisimple ﬁnite dimensional binary-Lie algebras over a ﬁeld of characteristic p > 3. Problem 2.34 (A. N. Grishkov) Assume that the annihilator of every noncentral element of a Lie algebra, which is nilpotent of index 2 over an algebraically closed ﬁeld, is ﬁnite dimensional modulo the center. Prove that the algebra is residually ﬁnite dimensional. Problem 2.35 ∗ (A. N. Grishkov) Must a ﬁnite dimensional solvable binary-Lie algebra over a ﬁeld of characteristic p > 3 have an Abelian ideal? Remark: The answer is Yes (A. N. Grishkov [65]).

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Problem 2.36 (K. R. Goodearl) A ring R is called invertibly regular if for every a ∈ R the equation axa = a has an invertible solution. Must a regular ring whose homomorphic images are directly ﬁnite (see Problem 2.141) be invertibly regular? Problem 2.37 (K. R. Goodearl) Let A and B be ﬁnitely generated projective right modules over an invertibly regular ring. If An is isomorphic to B n must A and B be isomorphic? If An is isomorphic to a direct summand of B n then must A be isomorphic to a direct summand of B? Problem 2.38 (A. Jategaonkar, reported by A. A. Tuganbaev) Must every ideal of a prime ring, all of whose right ideals are principal, be a product of prime ideals? Problem 2.39

(V. P. Elizarov) For a prime p describe nilpotent rings of order p4 .

Problem 2.40 (K. A. Zhevlakov, reported by I. P. Shestakov) Must the locally nilpotent (or even antisimple) radical of a weakly Noetherian associative (resp. alternative, Jordan) algebra be nilpotent? Problem 2.41 (V. N. Zhelyabin) Must any two inertial subalgebras of a ﬁnite dimensional Jordan algebra over a local Hensel ring be conjugate? Problem 2.42 (A. E. Zalesski) Is it true that the left annihilator of every element of a group algebra over a ﬁeld is ﬁnitely generated as a left ideal? Problem 2.43 ∗ (A. E. Zalesski) Is it true that every idempotent of a group algebra over a ﬁeld is conjugate by an automorphism to an idempotent whose support subgroup is ﬁnite? Remark: The answer is No (D. P. Farkas, Z. S. Marciniak [39]). Problem 2.44 (A. E. Zalesski, D. Passman) Find necessary and suﬃcient conditions for the group algebra of a locally ﬁnite group (over a ﬁeld of nonzero characteristic) to be semisimple. Problem 2.45 (I. Kaplansky, reported by A. A. Tuganbaev) Describe the rings in which every one-sided ideal is two-sided and over which every ﬁnitely generated module can be decomposed as a direct sum of cyclic modules. Problem 2.46 (O. V. Kaptsov) Let R be the ﬁeld of real numbers. Consider the commutative diﬀerential ring R[ui ] in inﬁnitely many variables ui , i ≥ 0. The derivation d acts on the ui as follows: d(ui ) = ui+1 . Deﬁne a Lie algebra structure on R[ui ] by [f, g] =

∞ i=0

(fi di g − gi di f ), where fi =

∂f ∂g , gi = , for any f, g ∈ R[ui ]. ∂ui ∂ui

Is it true that if [f, g] = 0 (f = λg, λ ∈ R) and fk = 0, gm = 0 for some k, m > 1, then the centralizer of g is inﬁnite dimensional? For instance, if g = u3 + u0 u1 (the right side of the Korteweg-deVries equation ut = uxxx − uux ), this conjecture holds. A positive answer

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would allow us to approach a solution of the following well-known problem: Describe the set of elements g that have an inﬁnite dimensional centralizer. Problem 2.47 (A. V. Kaptsov) Let R[ui ] be the ring deﬁned in the previous problem. Deﬁne a new multiplication by f ∗g =

∞

fi di g + (−d)i (f gi ) .

i=0

Is it true that if f ∗ g = 0 where f = f (u0 , . . . , un ), g = g(u0 , · · · , um ) and fn = 0, gm = 0 for some n > m > 1, then the subspace H of all h such that h∗g = 0 is inﬁnite dimensional? Is it possible to prove that if H is inﬁnite dimensional then so is the centralizer of g (in the sense of the previous problem)? Problem 2.48 (H. K¨ othe, reported by A. A. Tuganbaev) Describe the rings over which every right and left module is a direct sum of cyclic modules. Problem 2.49 (L. A. Koifman) Let R be a left hereditary ring and P (R) its prime radical. Is it true that P (R) is nilpotent and the quotient ring R/P (R) is also left hereditary? If R does not have an inﬁnite set of orthogonal idempotents then this is true (Yu. A. Drozd). Problem 2.50 (A. T. Kolotov) Let F be a free associative algebra of ﬁnite rank, and A a ﬁnitely generated subalgebra of F , and I an ideal of F such that I ⊂ A and F/I is a nilalgebra. Is it true that codim A < ∞? Problem 2.51 ∗ (A. T. Kolotov) Does there exist an algorithm that decides, for any ﬁnite family of elements of a free associative algebra, whether this family is algebraically dependent? Remark: The answer is No (U. U. Umirbaev [175]). Problem 2.52 (A. T. Kolotov, I. V. Lvov) Let k be a ﬁeld, and let D consist of the pairs (F, A) where F is a free associative k-algebra and A is a subalgebra of F . Let D0 ⊂ D consist of those pairs in which A is free. Can D0 be deﬁned axiomatically in D if we add to the signature the predicate that deﬁnes the subalgebra? Problem 2.53 (P. M. Cohn) Must a retract of a free associative algebra also be free? (This is a special case of Problem 2.4.) Problem 2.54 (A. I. Kostrikin) Can every ﬁnite dimensional complex simple Lie algebra be decomposed into a direct sum of Cartan subalgebras, which are pairwise orthogonal with respect to the Killing form? One of the conjectured negative examples is the Lie algebra of type A5 . (See A. I. Kostrikin, I. A. Kostrikin, V. A. Ufnarovskii [92].) Problem 2.55 ∗ (A. I. Kostrikin) Do there exist ﬁnite dimensional simple Lie algebras over a ﬁeld of characteristic p > 5 such that (ad x)p−1 = 0 for all x = 0? The conjectured answer is negative. Remark: The answer is No (A. A. Premet [136]).

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Problem 2.56 ∗ (A. I. Kostrikin, I. R. Shafarevich) Prove that every ﬁnite dimensional simple Lie p-algebra over an algebraically closed ﬁeld of characteristic p > 5 is isomorphic to one of the algebras of classical or Cartan type. (See A. I. Kostrikin, I. R. Shafarevich [93].) Remark: The conjecture is true for characteristic > 7 (R. E. Block, R. L. Wilson [21]). Problem 2.57 (A. I. Kostrikin) What are the maximal subalgebras of simple Lie algebras of classical type over an algebraically closed ﬁeld of characteristic p > 0? Problem 2.58 (E. N. Kuzmin) Must a binary-Lie algebra that has a regular automorphism of ﬁnite order be solvable? Problem 2.59 ∗ (E. N. Kuzmin) Is there a connection between Moufang loops of prime exponent p and Malcev algebras of characteristic p analogous to the connection between groups and Lie algebras? Remark: The answer is Yes (A. N. Grishkov [64]). Problem 2.60 (G. P. Kukin) Describe the varieties of Lie algebras in which every ﬁnitely presented algebra (resp. ﬁnitely presented algebra with decidable word problem) is residually ﬁnite dimensional. Here ﬁnite presentability can be understood in the absolute or relative sense. Problem 2.61 ∗ (G. P. Kukin) It can be shown that a free Lie algebra of characteristic p > 0 is residually ﬁnite with respect to inclusion into a ﬁnitely generated subalgebra. Is this true for Lie algebras of characteristic 0? Remark: The answer is Yes (U. U. Umirbaev [173]). Problem 2.62 (G. P. Kukin) Is the problem of conjugacy by an automorphism for ﬁnitely generated subalgebras of a free algebra (resp. free Lie algebra) decidable? Problem 2.63 ∗ (G. P. Kukin) Must every ﬁnitely generated subalgebra of a free solvable Lie algebra be ﬁnitely separated? Remark: The answer is No (S. A. Agalakov [1]). Problem 2.64 ∗ (V. N. Latyshev) Is it true that a homomorphic image of a special Lie algebra is again a special Lie algebra (that is, embeddable into an associative PI algebra)? Remark: The answer is No (Yu. V. Billig [20]). Problem 2.65 (I. V. Lvov) Is it true that every PI ring is a homomorphic image of a PI ring with torsion-free additive group? Problem 2.66 (I. V. Lvov) Does there exist a nonzero PI ring that coincides with its derived Lie algebra? Problem 2.67 (I. V. Lvov) Suppose that a ring satisﬁes an identity of degree d with coprime coeﬃcients. Is it true that this ring satisﬁes a multilinear identity of degree d with some coeﬃcient equal to 1?

488

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Problem 2.68 (I. V. Lvov) (a) Does there exist a simple inﬁnite dimensional ﬁnitely presented algebra R over an arbitrary ﬁeld k of positive characteristic? (b) The same question with additional assumptions: R is Noetherian without zero divisors and has ﬁnite Gelfand-Kirillov dimension (see W. Borho, H. Kraft [26]). In the case of a ﬁeld k of characteristic 0, Weyl algebras are examples. Problem 2.69 (I. V. Lvov) Does every ﬁnitely generated algebra (over a ﬁeld) with ﬁnite Gelfand-Kirillov dimension have a greatest nilpotent ideal? Problem 2.70 (I. V. Lvov) Is it true that every (right) primitive ring is a subdirect product of subdirectly indecomposable (right) primitive rings? Problem 2.71 (I. V. Lvov) Is the class of residually ﬁnite rings (resp. groups) axiomatizable in the language L∞,∞ ? Problem 2.72 (I. V. Lvov) Is it true that the class of free associative algebras over a ﬁxed ﬁeld k is not axiomatizable in the language L∞,∞ ? Problem 2.73 tizable?

(I. V. Lvov) Is the class of subdirect products of division rings axioma-

Problem 2.74 (I. V. Lvov) Does every algebra (over a ﬁeld) without zero divisors and with the maximal condition on subalgebras satisfy a polynomial identity? Problem 2.75 (I. V. Lvov) Must every ﬁnitely generated nilalgebra (over a ﬁeld) with ﬁnite Gelfand-Kirillov dimension be nilpotent? Problem 2.76 (I. V. Lvov) Let A be a Noetherian alternative algebra. Is the algebra of formal power series A[[x]] also Noetherian? Problem 2.77 (I. V. Lvov) (a) Must the Gelfand-Kirillov dimension of a ﬁnitely generated Noetherian PI algebra be an integer? (It is ﬁnite by Shirshov’s height theorem.) The same question for reduced-free (not necessarily Noetherian) algebras. (b) Describe varieties of algebras over an inﬁnite ﬁeld in which all ﬁnitely generated algebras have integral Gelfand-Kirillov dimension. Problem 2.78 (I. V. Lvov) Is it true that two free associative algebras (over a ﬁeld) of ﬁnite ranks m, n (m > n ≥ 2) are elementarily equivalent? Problem 2.79 (A. I. Malcev, reported by A. N. Grishkov) Prove that every analytic alternative local loop is locally isomorphic to an analytic alternative loop. Problem 2.80 ∗ (Yu. N. Malcev) Let R be a critical unital ring. Is it true that the matrix ring Mn (R) is also critical? Remark: The answer is Yes (Yu. N. Malcev [108]).

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Problem 2.81 ◦ (Yu. N. Malcev) Describe varieties of rings whose lattice (of subvarieties) is not distributive but the lattice of every proper subvariety is distributive. Remark: A complete description has not yet been obtained. Signiﬁcant progress on this problem has been announced by M. V. Volkov [184]. Problem 2.82 (Yu. N. Malcev) Let M be the variety of associative rings satisfying the identity x3 = xn for some n ≥ 4. Does M satisfy the minimum condition on subvarieties? Problem 2.83 (Yu. N. Malcev) Describe the critical rings in the variety of rings satisfying the identity x3 = xn for some n ≥ 4. Problem 2.84 (E. Matlis, reported by A. A. Tuganbaev) Must a direct summand of a direct sum of indecomposable injective modules also be a direct sum of indecomposable injective modules? Problem 2.85 (Yu. A. Medvedev) Must a variety M of alternative algebras be solvable if every associative algebra in M is nilpotent? Problem 2.86 ◦ (Yu. A. Medvedev) Let A be an alternative (resp. Jordan) ring, G a ﬁnite group of automorphisms, AG the subalgebra of ﬁxed elements. Must A be solvable if AG is solvable and A has no |G|-torsion? Remark: If A is an algebra over a ﬁeld of characteristic 0 then the answer is Yes (A. P. Semenov [144]). Problem 2.87 (Yu. A. Medvedev) Does every ﬁnite alternative (resp. Jordan) ring have a ﬁnite basis of quasi-identities? Problem 2.88 ∗ (S. Montgomery, V. K. Kharchenko) Consider the free associative algebra F of rank n over a ﬁeld k as the tensor algebra of the n-dimensional space V . For which linear groups G ⊆ GL(V ) is the subalgebra of invariants of F with respect to G ﬁnitely generated? Remark: A description of such groups has been obtained (A. I. Koryukin [89]). Problem 2.89 (Yu. A. Ryabukhin, R. A. Florya) Does there exist in some variety a simple free ring with characteristic p ≥ 3? Problem 2.90 (L. A. Skornyakov) Describe the rings over which all ﬁnitely presented modules are injective. Problem 2.91 (A. M. Slinko) What is the minimal possible dimension of a nilpotent exceptional Jordan algebra? Problem 2.92 ◦ (A. M. Slinko) Does every variety of solvable alternative algebras over a ﬁeld of characteristic = 2 have a ﬁnite basis of identities? (This is a more precise version of Problem 1.129.) Remark: The answer is Yes over a ﬁeld of characteristic = 2, 3 (U. U. Umirbaev [172]).

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Problem 2.93 ◦ (A. M. Slinko) Must every nilideal of a Jordan algebra with the minimum condition on annihilators be nilpotent? Remark: The solvability of nil subalgebras of Jordan algebras with minimum condition on annihilators has been proved by A. V. Chekhonadskikh [33]. Problem 2.94

(A. M. Slinko) Describe maximal special varieties of Jordan algebras.

Problem 2.95 ◦ (A. M. Slinko) Must the variety generated by the Jordan algebra of a bilinear form be special? Remark: The answer is Yes in the case of a ﬁeld of characteristic 0 (S. R. Sverchkov [165]) and in the case of a ﬁnite ﬁeld (I. M. Isaev [75]). Problem 2.96 (A. M. Slinko) Find a basis of weak identities of the pair (F2 , H(F2 )). Do they all follow from the standard identity S4 (x1 , x2 , x3 , x4 )? Problem 2.97 (A. M. Slinko) If a homogeneous variety of algebras has a locally nilpotent radical, must it also have a locally ﬁnite dimensional radical? Problem 2.98 ∗ (G. F. Smit) Must a right alternative nilalgebra with the minimum condition on right ideals be right nilpotent? Remark: The answer is Yes (V. G. Skosyrskii [157]). Problem 2.99 nilpotent?

(G. F. Smit) Must a one-sided nilideal of a Noetherian (−1, 1) ring be

Problem 2.100 (D. A. Suprunenko) Must a torsion group of matrices over a division ring be locally ﬁnite? The modular case is especially interesting. Problem 2.101 (A. Thedy, reported by I. P. Shestakov) Is it true that every ﬁnite dimensional right alternative algebra has an isotope that splits over its radical? Problem 2.102 (A. A. Tuganbaev) A module is called weakly injective if every endomorphism of every submodule can be extended to an endomorphism of the whole module. Describe the rings over which all cyclic modules are weakly injective. Problem 2.103 (A. A. Tuganbaev) Must a weakly injective module with an essential socle be quasi-injective? Problem 2.104 (A. A. Tuganbaev) Can every right Noetherian ring with a distributive lattice of right ideals be decomposed as a direct sum of a right Artinian ring and a semiprime ring? Problem 2.105 (V. T. Filippov) Let A be the free Malcev algebra of countable rank, and let M[n] = Var(An ). Does the chain of varieties M[1] ⊂ M[2] ⊆ M[3] ⊆ · · · ⊆ M[n] ⊆ · · ·

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stabilize after a ﬁnite number of steps? Problem 2.106 nilpotent ideals?

(V. T. Filippov) Does the free binary-Lie algebra contain nonzero

Problem 2.107 trivial?

(V. T. Filippov) Is the associative center of a free Moufang loop non-

Problem 2.108 (V. T. Filippov) Let A be a free Malcev algebra over a ﬁeld F of characteristic 0, and C7 the simple 7-dimensional non-Lie Malcev algebra over F . Is the ideal of identities of C7 a Lie ideal in A? Problem 2.109 (V. T. Filippov) An algebra is called assocyclic if it satisﬁes the identity (x, y, z) = (z, x, y) where (x, y, z) = (xy)z − x(yz). It is easy to show that the minus algebra of such an algebra is binary-Lie. Is every binary-Lie algebra over a ﬁeld of characteristic = 2, 3 embeddable into the minus algebra of a suitable assocyclic algebra? Problem 2.110 (V. T. Filippov) Are the varieties generated by the free Malcev algebras (over a ﬁeld of characteristic = 2, 3) of ranks 3 and 4 distinct? Problem 2.111 (V. T. Filippov) Is the ideal of the free alternative algebra over a ﬁeld F of characteristic 0, generated by the identities of the split Cayley-Dickson algebra over F , associative? Problem 2.112 (V. T. Filippov) Describe the class of ﬁnite dimensional Malcev algebras over a ﬁeld of characteristic 0 that have a faithful (not necessarily ﬁnite dimensional) representation. Problem 2.113 (V. T. Filippov) Describe the center of the algebra of right multiplications of the free Malcev algebra. Problem 2.114 ∗ (Fischer, reported by L. A. Skornyakov) Is the ring of matrices over an invertibly regular ring also invertibly regular (that is, a ring in which for every a the equation axa = a has an invertible solution)? Remark: The answer is Yes (M. Henriksen [67]). Problem 2.115 (V. K. Kharchenko) Is the subalgebra of constants (that is, invariants) for a ﬁnite dimensional Lie p-algebra of derivations of the free associative algebra (over a ﬁeld of characteristic p) also free? Problem 2.116 (V. K. Kharchenko) Is the subalgebra of invariants for a ﬁnite group of automorphisms of the free associative algebra also free? The answer is unknown also for inﬁnite groups. Problem 2.117

(V. K. Kharchenko) Let L be a Lie algebra that admits an automor-

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phism of order 2 all of whose ﬁxed elements are in the center of L. Must L be solvable? Problem 2.118 ∗ (V. K. Kharchenko) Must the restricted enveloping algebra of a Lie palgebra with a polynomial identity also satisfy a polynomial identity? Remark: The answer is No (V. M. Petrogradsky [131]). Problem 2.119 ∗ (V. K. Kharchenko) Can every associative algebra over a ﬁeld of characteristic 0 that satisﬁes the identity xn = 0 be represented by n × n matrices over a commutative ring? Remark: The answer is Yes (C. Procesi [137]). Problem 2.120 ◦ (I. P. Shestakov) Describe the ideal of identities of the free alternative algebra on 3 generators. Remark: This ideal coincides with the radical over a ﬁeld of characteristic = 2, 3 (A. V. Iltyakov [70]). Over a ﬁeld of characteristic 0 it is ﬁnitely generated as a T-ideal (A. V. Iltyakov [72]) and nilpotent (E. I. Zelmanov and I. P. Shestakov [199]). Problem 2.121 (I. P. Shestakov) Describe the center and the associative center of a free alternative algebra as completely characteristic subalgebras. Are they ﬁnitely generated? Problem 2.122 (I. P. Shestakov) Does the antisimple radical of an associative (resp. Jordan) algebra coincide with the intersection of the kernels of all irreducible birepresentations of this algebra? Problem 2.123 (I. P. Shestakov) Is every ﬁnitely generated associative (resp. special Jordan) PI algebra embeddable into a 2-generated PI algebra? Problem 2.124 ◦ (I. P. Shestakov) Describe the following varieties of alternative and Jordan algebras (resp. rings): (a) almost nilpotent; (b) almost Cross; (c) ◦ locally residually ﬁnite; (d) ◦ locally Noetherian (resp. weakly Noetherian); (e) Hopf; (f) alternative almost associative; (g) Jordan almost special; (h) Jordan distributive. Remark: (c,d) Locally residually ﬁnite and locally (weakly) Noetherian varieties of alternative algebras have been described (S. V. Pchelintsev [130]). Problem 2.125 ∗ (I. P. Shestakov) Does the free Jordan algebra on three or more generators contain Albert subrings? Remark: The answer is Yes (Yu. A. Medvedev [120]). Problem 2.126 ∗ (I. P. Shestakov) Find a basis of identities of the Jordan algebra of a bilinear form over an inﬁnite ﬁeld. Does this algebra generate a Specht variety? Remark: A ﬁnite basis of identities has been found by S. Yu. Vasilovsky [181]. Over a ﬁeld of characteristic 0 the unitary Specht property has been proved (S. Yu. Vasilovsky [181], A.

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V. Iltaykov [71], P. E. Koshlukov [91]). Problem 2.127 ◦ (I. P. Shestakov) Are the varieties of Jordan algebras generated by (a) ∗ the algebra Fn+ ; (b) ∗ the algebra H(Fn ); (c) ∗ the algebra H3 (C); ﬁnitely based or Specht? Describe the trace identities that hold in these algebras. Do they have a ﬁnite basis? Remark: It has been shown that every ﬁnitely generated Jordan PI algebra over a ﬁeld of characteristic 0 is Specht (A. Ya. Vais, E. I. Zelmanov [178]). Problem 2.128 ∗ (I. P. Shestakov) Suppose that a special Jordan algebra J satisﬁes an identity that does not hold in a Jordan algebra of a bilinear form on an inﬁnite dimensional space. Must J have an enveloping associative PI algebra? Remark: The answer is No (S. V. Pchelintsev [129]). Problem 2.129 ∗ (I. P. Shestakov) Let J be a ﬁnitely generated Jordan PI algebra. Is its universal multiplicative enveloping algebra R(J) also a PI algebra? Remark: The answer is Yes (Yu. A. Medvedev [119]). Problem 2.130 ∗ (I. P. Shestakov) Let J be a Jordan algebra, I J, H I. Suppose that the ideal I is generated in J by the set H. Must the quotient algebra I/H be solvable or nilpotent? Remark: The answer is No (Yu. A. Medvedev [118], S. V. Pchelintsev [129]). Problem 2.131 ∗ (I. P. Shestakov) Let A be a simple right alternative ring such that A(+) is a simple Jordan ring. Must A be alternative? Remark: The answer is Yes (V. G. Skosyrskii [156]). Problem 2.132 (I. P. Shestakov) Describe ﬁnite dimensional irreducible right alternative bimodules over the matrix algebra M2 (F ). Is their number ﬁnite (up to isomorphism)? Problem 2.133 ∗ (I. P. Shestakov) Describe noncommutative Jordan division algebras, at least in the ﬁnite dimensional case. Remark: Strictly prime algebras of characteristic = 2, 3 have been described by V. G. Skosyrskii [159]. Problem 2.134 (I. P. Shestakov) Does there exist a simple inﬁnite dimensional noncommutative Jordan algebra, with the identity ([x, y], y, y) = 0, that is neither alternative nor Jordan? Problem 2.135 ∗ (A. I. Shirshov) Must the variety of algebras generated by a ﬁnite dimensional associative (resp. Lie) algebra over a ﬁeld of characteristic 0 have a ﬁnite basis of identities? Remark: The answer is Yes (A. R. Kemer [83], A. V. Iltyakov [73]). Problem 2.136 (A. L. Shmelkin) Do there exist inﬁnite dimensional Noetherian Lie algebras (that is, satisfying the maximal condition on subalgebras) that can be approximated by nilpotent Lie algebras? The analogous question for groups: Does there exist a nonnilpotent group that is approximable by nilpotent torsion-free groups?

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Problem 2.137 (Reported by V. A. Artamonov) Let k be a principal ideal domain and G an almost polycyclic group without torsion. Is it true that every projective module over the group algebra kG is a direct sum of a free module and a one-sided ideal? Problem 2.138 ◦ (Reported by Yu. A. Bahturin) Is it true that a variety of Lie algebras over a ﬁeld k of characteristic 0 that does not contain the algebra sl(2, k) is (locally) solvable? Remark: The answer is Yes for special Lie algebras (A. Ya. Vais [177]) and also for some other varieties (S. P. Mishchenko [123]). Problem 2.139 (Reported by Yu. A. Bahturin) Describe ﬁnite dimensional simple Lie algebras (over an arbitrary ﬁeld) such that all proper subalgebras are nilpotent (or even Abelian). Problem 2.140 (Reported by A. L. Voronov) Let G be a polycyclic group. Is it true that the algebra kG is primitive if and only if the ﬁeld k is not absolute and Δ(G) = 1? Problem 2.141 (Reported by K. R. Goodearl and L. A. Skornyakov) A ring is called directly ﬁnite if xy = 1 implies yx = 1. Is the ring of matrices over a regular directly ﬁnite ring also directly ﬁnite? Problem 2.142 (Reported by Yu. A. Drozd) Let A be a ﬁnite dimensional Lie algebra, U its universal enveloping algebra, and P a ﬁnitely generated projective U -module. Must P be a free module? Problem 2.143 (Reported by I. V. Lvov) Does every ﬁnitely generated PI ring satisfy all the identities of a ring of n × n matrices over the integers? Problem 2.144 (Reported by Yu. M. Ryabukhin) Describe in terms of identities the varieties of commutative associative algebras over a ﬁnite ﬁeld. Problem 2.145 (Reported by A. I. Kostrikin) Find a formula for the dimensions of the irreducible p-modules of the classical Lie algebras over a ﬁeld of characteristic p > 0.

B.5

Part Three

Problem 3.1 (T. Anderson) Let M be a variety of power-associative algebras whose ﬁnite dimensional solvable algebras are nilpotent. Must the nilalgebras of M be solvable? Problem 3.2 (V. I. Arnautov) Is every ring topology of a ring (resp. division ring) R a greatest lower bound of some family of maximal ring topologies of R in the lattice of all topologies?

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Problem 3.3 (V. I. Arnautov) Is there a ring in which one can build maximal ring topologies, without assuming the Continuum Hypothesis, such that the corresponding topological ring is not complete? Problem 3.4 (V. I. Arnautov) Is there a ring that is complete with respect to any maximal ring topology? Problem 3.5 (V. I. Arnautov, A. V. Mikhalev) Is it possible to embed an arbitrary topological group into the multiplicative semigroup of a topological ring? Problem 3.6 (V. I. Arnautov, A. V. Mikhalev) Is it true that for any topological ring (R, τ0 ) and any discrete monoid G the topology τ0 can be extended to a ring topology on the semigroup ring RG? Problem 3.7 (V. I. Arnautov, I. V. Protasov) Is it true that for an arbitrary ring there exists a ring topology for which all endomorphisms of the ring are continuous? Problem 3.8 (A well-known problem reported by V. I. Arnautov) Can any ring topology of a division ring be weakened to a ring topology in which the inverse operation is a continuous function? Problem 3.9 (V. A. Artamonov) Let B be an associative left Noetherian algebra of Krull dimension d, let H be a commutative and cocommutative Hopf algebra, and let A = B#t H be a crossed product. Suppose that P is a ﬁnitely generated projective module of rank > d. If P is stably extended from B, then is P extended from B? Problem 3.10 (L. A. Bokut) Is an arbitrary ﬁnitely generated associative (resp. Lie) algebra with a recursive basis embeddable in a ﬁnitely deﬁnable associative (resp. Lie) algebra? Problem 3.11 (L. A. Bokut) How many nonisomorphic algebraically closed Lie algebras of a given cardinality are there? Problem 3.12 (L. A. Bokut, V. N. Gerasimov) Is an arbitrary free associative algebra embeddable in an algebraically closed associative algebra (that is, an algebra in which any nontrivial generalized polynomial in one variable has a root)? Problem 3.13 (L. A. Bokut, V. N. Gerasimov) Is it true that the class of associative rings embeddable into division rings cannot be deﬁned by an independent system of quasiidentities? Problem 3.14 (L. A. Bokut, V. N. Gerasimov) Is it true that the class of semigroups embeddable into groups cannot be deﬁned by an independent system of axioms? Problem 3.15

(L. A. Bokut, M. V. Sapir) Describe all varieties of algebras over a ﬁeld

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of characteristic 0 in which for every ﬁnitely deﬁnable algebra the word problem is solvable. Problem 3.16 (L. A. Bokut, M. V. Sapir) Describe all varieties of algebras over a ﬁeld of characteristic 0 in which every ﬁnitely deﬁnable algebra is residually ﬁnite dimensional. Problem 3.17 (L. A. Bokut, M. V. Sapir) Describe all varieties of algebras over a ﬁeld of characteristic 0 in which every ﬁnitely deﬁnable algebra is representable. Problem 3.18 (N. A. Vavilov, A. V. Mikhalev) This and the following problem are related to the attempt to extend some results in algebraic K-theory from commutative rings to PI rings. Positive answers are known in the case when the ring is a ﬁnitely generated module over its center (A. A. Suslin, M. S. Tulenbaev). Let R be a unital PI ring. Consider, in the group GLn (R) of all invertible n × n matrices over R, the subgroup of elementary matrices En (R) generated by the transvections tij (r) = I + rEij ,

i = j,

1 ≤ i, j ≤ n,

r ∈ R.

Is En (R) a normal subgroup of GLn (R)? Problem 3.19 (N. A. Vavilov, A. V. Mikhalev) Let R be a unital PI ring, and let Stn (R) be its nth Steinberg group, that is, the group generated by formal transvections over R: the elements uij (a) with deﬁning relations: [uij (a), ujk (b)] = uik (ab), i = j, j = k, k = i, [uij (a), ukl (b)] = 1, j = k, i = l, uij (a)uij (b) = uij (a + b). There exists a group homomorphism of Stn (R) onto En (R) (see 3.18) sending uij (r) to tij (r). Let K2,n (R) be its kernel: 1 −→ K2,n (R) −→ Stn (R) −→ En (R) −→ 1. Is it true that K2,n (R) is contained in the center of the group Stn (R) for suﬃciently large n (for instance, for n ≥ 5)? Problem 3.20 (Yu. M. Vazhenin) What are the SA-critical theories of a free associative ring? The lists of all SA-critical theories of the ring of integers and of the absolutely free (nonassociative) ring are known. 2 Problem 3.21 (Yu. M. Vazhenin) Of the rings deﬁned by one relation in the following varieties: (a) alternative rings; (b) Jordan rings; (c) associative rings; which have a decidable elementary theory? 2 For the deﬁnition of SA-critical theories, see Yu. M. Vazhenin, Algorithmic problems and hierarchies of ﬁrst-order languages, Algebra i Logika 26 (1987) 419-525, MR963095 (90e:03054). [Translators]

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Problem 3.22 (Yu. M. Vazhenin, I. P. Shestakov) What are the SA-critical theories of a free Jordan ring? Problem 3.23 (Yu. M. Vazhenin, I. P. Shestakov) What are the SA-critical theories of the variety of all Jordan rings? Problem 3.24 (A. T. Gainov) Let Mn (Φ) be the matrix algebra over a ﬁeld Φ of characteristic not 2. We call a subspace V of the algebra Mn (Φ) a space of anticommuting matrices (SAM for short) if x2 = 0 for all x ∈ V . Two SAMs V and W in Mn (Φ) are called equivalent if φ(V ) = W for some automorphism or antiautomorphism φ of Mn (Φ). Find all inequivalent maximal (with respect to inclusion) SAMs of the algebra Mn (Φ). Problem 3.25 (A. T. Gainov) Let Φ be a ﬁeld of characteristic not 2, and assume Φ2 = Φ. We will call a subspace V of the algebra Mn (Φ) a space of anticommuting antisymmetric (resp. symmetric) matrices (SAAM for short, resp. SASM) if x2 = 0 and xt = −x (resp. xt = x) for all x ∈ V . Two SAAMs (resp. SASMs) V and W in Mn (Φ) are called equivalent if W = qV q t for some orthogonal matrix q ∈ Mn (Φ). Find all inequivalent maximal (with respect to inclusion) SAAMs (resp. SASMs) of the algebra Mn (Φ). Problem 3.26 (A. T. Gainov) Describe all ﬁnite dimensional simple anticommutative algebras A over an inﬁnite ﬁeld of characteristic not 2 that satisfy the condition that any element a ∈ A lies in some two-dimensional subalgebra. Problem 3.27 (A. G. Gein) An element a of a Lie algebra L is called ad-pure if any ﬁnite dimensional ad a-invariant subspace of the algebra L lies in the kernel of the operator ad a. Is there a simple Lie algebra all of whose elements are ad-pure? Problem 3.28 (A. G. Gein) Does there exist (a) an inﬁnite dimensional Lie algebra all of whose proper subalgebras are ﬁnite dimensional; (b) an inﬁnitely generated Lie algebra all of whose proper subalgebras are ﬁnite dimensional; (c) an inﬁnitely generated Lie algebra all of whose proper subalgebras are ﬁnitely generated? Problem 3.29 (A. V. Grishin) Let F be a countably generated free algebra over a ﬁeld of characteristic 0 from a variety of ﬁnite base rank, F(d) a d-generated subalgebra of F. We say that a subspace V of F(d) is a T -space if V = V ∩ F(d) where V is the subspace of F spanned by all possible substitutions into the polynomials in V of elements in F. Does any T -space have a ﬁnite base? A positive answer is known in the case of the variety of associative algebras. In particular, this result would imply that the variety is Specht. It is interesting to consider also the cases of alternative, Jordan and (−1, 1) algebras. Problem 3.30 (A. V. Grishin) Find an upper bound for the nilpotency index of the radical of the free (associative) algebra satisfying the standard identity of degree n. Problem 3.31 (A. V. Grishin) Find an upper bound for the dimension of the (ﬁnite dimensional) algebra of least dimension that generates the variety of associative algebras deﬁned by the standard identity of degree n. (Such an algebra exists by the results of the

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author and A. R. Kemer.) Problem 3.32 (A. V. Grishin) Let M(n) be the variety generated by all n-dimensional algebras, and A the variety of all associative algebras. Is it true that M(n) ∩ A can be deﬁned by the Capelli identity of order n + 1: y0 xσ(1) y1 xσ(2) · · · yn xσ(n+1) yn+1 = 0 ? σ∈Sn+1

Problem 3.33 (A. V. Grishin) If a variety is Specht then it is a sum of indecomposable subvarieties. Investigate the question of the uniqueness of such a decomposition in the associative or nearly associative case. Problem 3.34 (A. N. Grishkov) Describe ﬁnite dimensional Malcev algebras (resp. binary Lie algebras) to which there correspond algebraic Moufang loops (resp. alternative loops). Problem 3.35 Problem 3.36 nilring solvable?

(A. N. Grishkov) Describe all simple algebraic Bol loops. (V. N. Zhelyabin) Is a countably categorical alternative (resp. Jordan)

Problem 3.37 (A. E. Zalessky) Describe the two-sided ideals of the group ring of the ﬁnitary symmetric group over a ﬁeld of prime characteristic. (The ﬁnitary symmetric group consists of all permutations of an inﬁnite set which only move a ﬁnite number of elements.) The description is known over a ﬁeld of characteristic 0. Problem 3.38 (A. E. Zalessky) Let P be a ﬁeld of characteristic p > 0, and A an associative algebra over P graded by a ﬁnite Abelian group of order k. Assume that the zero component A0 is commutative. Is it true that A satisﬁes the standard identity of degree kp? This is the case for the matrix algebra Mk (P ). Problem 3.39 (E. I. Zelmanov) Let F2,m be the free 2-generated associative ring with identity xm = 0. Is it true that the nilpotency index of F2,m grows exponentially as a function of m? Problem 3.40 (E. I. Zelmanov) Is it true that the nilpotency index of the m-generated (p − 1)-Engel Lie algebra over a ﬁeld of characteristic p > 0 grows linearly as a function of m and exponentially as a function of p? Problem 3.41 (E. I. Zelmanov) Let L be a (p − 1)-Engel Lie algebra over a ﬁeld of characteristic p > 0. Is it true that an arbitrary element of L generates a nilpotent ideal? Problem 3.42 (A. V. Iltyakov) Let A be a ﬁnitely generated alternative (resp. Jordan) PI algebra. Does there exist a ﬁnite dimensional alternative (resp. Jordan) algebra B whose ideal of identities T (B) is contained the ideal of identities T (A) of the algebra A?

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Problem 3.43 (I. Kaplansky, reported by A. E. Zalessky) Let H be a group, P a ﬁeld, and A(P H) the augmentation ideal of the group ring P H. Describe all the groups H for which A(P H) is a simple ring (at least for P = C, the ﬁeld of complex numbers). Problem 3.44 (I. Kaplansky, M. Henriksen, reported by A. A. Tuganbaev) Let M be a 2 × 2 matrix with entries from a commutative B´ezout domain A. Is it true that there always exist invertible 2 × 2 matrices C and D such that CM D is a diagonal matrix? Problem 3.45 (I. Kaplansky, reported by K. A. Pavlov) Is it true that there are only ﬁnitely many (up to isomorphism) Hopf algebras of a given dimension? Problem 3.46 (A. R. Kemer) Does the algebra of 2 × 2 matrices over an inﬁnite ﬁeld of positive characteristic have a ﬁnite basis of identities? Problem 3.47 (G. P. Kukin) Prove that a Lie algebra has cohomological dimension ≤ 2 if and only if its module of relations is free. Problem 3.48 (G. P. Kukin) Prove that the elementary (resp. universal) theory of a free Lie algebra over a ﬁeld F is decidable if and only if the elementary (resp. univeral) theory of F is decidable. Problem 3.49 (I. V. Lvov) At the present time there is no reasonable conjecture about the structure of the automorphism group of a free PI algebra. A question in the negative direction: Let M be a variety of PI algebras strictly containing the variety of commutative algebras, and A a free algebra (in countably many generators). Is it true that the automorphism group of the algebra A is not generated by tame automorphisms? If A has a nontrivial center, then the answer is Yes (G. Bergman). Problem 3.50 (I. V. Lvov) Let A be a free PI ring. Does there always exist an epimorphism B → A where B is a free PI ring without additive torsion? If yes (or in those cases when the answer is yes) then what is the “minimal” B with this property? The analogous question under the assumption that A has prime characteristic p > 0. Problem 3.51 (I. V. Lvov, Yu. N. Malcev) Is a free PI ring residually ﬁnite? Equivalently, is every variety of PI rings generated by its ﬁnite rings? Problem 3.52 representable?

(Yu. N. Malcev) Is a ﬁnite local (associative) unital ring necessarily

Problem 3.53 (Yu. N. Malcev) Describe the minimal non-Engel varieties of associative rings. In particular, are they Cross varieties? Problem 3.54 (S. P. Mishchenko) Describe the nonsolvable varieties of Lie algebras that have almost polynomial growth. (An example of such a variety is Var(sl2 ).)

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Problem 3.55 (S. P. Mishchenko) Describe the solvable varieties of Lie algebras that have exponential growth. (An example is AN2 .) Problem 3.56 (S. P. Mishchenko) Does the identity x0 (¯ x1 yˆ1 ) · · · (¯ xm yˆm ) = 0 (for some m) follow from the standard Lie identity? Here bars and hats denote skew-symmetrization in the corresponding groups of variables. Problem 3.57 (S. P. Mishchenko) Is there a variety of Lie algebras over a ﬁeld of characteristic 0 with a distributive lattice of subvarieties and whose basis of identities is not limited to degree 6? Problem 3.58 (S. Montgomery) Let R be an associative ring with a derivation d, and let S = R[x; d] be the Ore extension. Is it true that if R has no nonzero nilideals then S is semisimple? If d = 0 then it is true by a well-known theorem of Amitsur. See some partial results in J. Bergen, S. Montgomery, D. S. Passman [17]. Problem 3.59 (V. M. Petrogradsky) Suppose that a Lie p-algebra has no elements algebraic with respect to the p-mapping. Is it true that its restricted enveloping algebra has no zero-divisors? Problem 3.60 (V. M. Petrogradsky) Let R be a PI subalgebra of the restricted enveloping algebra of a Lie p-algebra, and n the minimal number such that R satisﬁes a power of the standard identity S2n . Is it true that n = pk ? Problem 3.61 (S. V. Pchelintsev) Is the ideal of a ﬁnitely generated binary-(−1, 1) algebra generated by the alternators nilpotent or solvable? Problem 3.62 (S. V. Pchelintsev) Do there exist simple nonalternative right alternative Malcev-admissible algebras? Problem 3.63 (S. V. Pchelintsev) Is it true that the additive group of the free alternative ring on three generators is torsion-free? Problem 3.64 (S. V. Pchelintsev) Is it true that every prime nonassociative (−1, 1) algebra over a ﬁeld of characteristic 0 generates the variety of all strictly (−1, 1) algebras? Problem 3.65 (S. V. Pchelintsev) Is the variety of alternative algebras over a ﬁeld of characteristic 0 decomposable into a union of proper subvarieties? Problem 3.66 (G. E. Puninsky) Let R be a uniserial ring without zero divisors. Is it true that every purely injective module over R contains an indecomposable direct summand? Problem 3.67 (Yu. P. Razmyslov, reported by S. P. Mishchenko) Prove that the variety of algebras with the standard identity has exponential growth.

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Problem 3.68 (D. A. Rumynin) Let k be an absolute algebraic ﬁeld, and H a Hopf algebra over k. Is it true that every irreducible H-module is ﬁnite dimensional? Problem 3.69 bras.

(D. A. Rumynin) Describe all ﬁnite dimensional semisimple Hopf alge-

Problem 3.70 (Yu. M. Ryabukhin) Is it true that every reduced-free quasiregular algebra (that is, an algebra with an additional unary operation x → x∗ that provides the adjoint group) is generalized nilpotent? Problem 3.71 (L. V. Sabinin) Develop the structure theory of ﬁnite dimensional Bol algebras of characteristic 0. Problem 3.72 (M. V. Sapir) Let k be a ﬁeld of characteristic 0, and let R be a k-algebra. Do there exist an extension ﬁeld F (R) ⊃ k, and an algebra A(R) ⊃ R ﬁnite dimensional over F (R), such that every family of elements of R, which becomes linearly dependent upon some embedding of the k-algebra R into an algebra ﬁnite dimensional over some ﬁeld extension of k, is linearly dependent in A(R)? It is not even clear whether, for any two ﬁnite subsets U1 , U2 ⊂ R, which are linearly dependent in algebras A1 , A2 ⊃ R ﬁnite dimensional over extension ﬁelds F1 , F2 ⊃ k, it is possible to make U1 , U2 simultaneously linearly dependent in some algebra A ⊃ R ﬁnite dimensional over some extension ﬁeld F ⊃ k. Problem 3.73 (M. V. Sapir) Is it true that in a variety of associative algebras (over a constructive ﬁeld of characteristic 0) the word problem is decidable if and only if the variety does not contain the variety deﬁned by the identities x[y, z][t, u]v = 0,

x[y, z, t]u = 0?

Problem 3.74 (A. I. Sozutov) Describe all ﬁnite dimensional simple Lie algebras with a monomial basis. Problem 3.75 (A. A. Tuganbaev) Describe all rings over which every right module is a distributive left module over its endomorphism ring. (A module is called distributive if its submodule lattice is distributive.) Problem 3.76

(A. A. Tuganbaev) Describe all right distributive monoid rings.

Problem 3.77 (A. A. Tuganbaev) Does every left and right distributive ring have a classical ring of quotients? Problem 3.78 (A. A. Tuganbaev) Is every right distributive ring, which is integral over its center, also left distributive? Problem 3.79 (A. A. Tuganbaev) Describe all the rings over which every left module is isomorphic to a submodule of a direct sum of uniserial modules.

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Problem 3.80 (A. A. Tuganbaev) Let M be a maximal right ideal of a right distributive ring A, and let T = A \ M . Does there always exist a ring Q and a ring homomorphism f : A → Q such that the elements f (T ) are invertible in Q, and such that ker f = { a ∈ A | ∃t ∈ T, at = 0 },

Q = { f (a)f (t)−1 | a ∈ A, t ∈ T }?

Problem 3.81 (A. A. Tuganbaev) Let A be a right distributive ring without nonzero nilpotent elements. Are all right ideals of A ﬂat? Problem 3.82 hereditary ring?

(A. A. Tuganbaev) Is every left and right distributive domain a semi-

Problem 3.83 (A. A. Tuganbaev) Describe all the rings over which the ring of formal power series in one variable has weak global dimension one. Problem 3.84 (V. T. Filippov) Let A be the free Malcev algebra over a ﬁeld of characteristic 0, let Var(A2 ) be the variety generated by the square of A, and let M3 be the variety generated by the free Malcev algebra on three generators. Is it true that Var(A2 ) = M3 ? Problem 3.85 (V. T. Filippov) Does there exist a trivial characteristic ideal, not lying in the Lie center, in the free countably generated Malcev algebra of characteristic = 2, 3? Problem 3.86 (V. T. Filippov) Does there exist a trivial characteristic ideal, not lying in the associative center, in the free countably generated alternative algebra of characteristic = 2, 3? Problem 3.87 (V. T. Filippov) Does there exist a simple non-Malcev binary-Lie algebra of characteristic 0? Problem 3.88 (V. T. Filippov) Classify simple ﬁnite dimensional n-Lie algebras over an algebraically closed ﬁeld of characteristic 0. Problem 3.89 (V. T. Filippov) Is it true that in any nonsolvable ﬁnite dimensional n-Lie algebra over an algebraically closed ﬁeld of characteristic 0 there exists an (n + 1)dimensional simple subalgebra? Problem 3.90 (V. T. Filippov) Do there exist non-Lie simple ﬁnite dimensional Sagle algebras over a ﬁeld of characteristic 0? A Sagle algebra is an anticommutative algebra satisfying the identity J(x, y, z)t = J(xt, y, z) + J(x, yt, z) + J(x, y, zt), where J(x, y, z) = (xy)z + (zx)y + (yz)x. Problem 3.91 (J. Faulkner) An Abelian group A together with mappings ja (deﬁned for each 0 = a ∈ A) from the set A ∪ {∞} to itself is called a Hua system if the following

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conditions are satisﬁed: (HS1) ja2 = id, ja (a) = a, ja (−a) = −a, ja (0) = ∞ (HS2) (sa ja )3 = id where sa (b) = a − b, sa (∞) = ∞ (HS3) ja jb ∈ End A Every quadratic Jordan division ring is a Hua system if we set ja (b) = Ua (b−1 ). Can every Hua system be obtained from a quadratic Jordan division ring in this way? (See J. R. Faulkner [41] for references and a survey of known results.) Problem 3.92 (P. A. Freidman) Is it true that every associative nil ring all of whose proper subrings have an annihilator series also has an annihilator series? Problem 3.93 (V. K. Kharchenko) Let an associative unital ring satisfy an essential polynomial identity with automorphisms and skew derivations. Will it be a PI ring? (An identity is called essential if the two-sided ideal generated by all values of its generalized monomials contains the unit element.) Problem 3.94 (V. K. Kharchenko) Let R be a prime ring with generalized centroid C, and B a quasi-Frobenius ﬁnite dimensional C-subalgebra of RC. Is R necessarily a PI ring if the centralizer of B in R is a PI ring? Problem 3.95 (V. K. Kharchenko) Describe the identities with skew derivations and automorphisms of an arbitrary prime ring. Problem 3.96 (V. K. Kharchenko) Develop Galois theory in the class of prime rings for reduced ﬁnite groups that have a quasi-Frobenius group algebra. At the present time such a theory has been developed for groups that have a semisimple group algebra, and it is also known that a reduced ﬁnite group with a quasi-Frobenius group algebra is a Galois group. Problem 3.97 (V. K. Kharchenko) Let a Hopf algebra H act on an associative unital algebra R, and suppose that R satisﬁes an essential multilinear generalized identity with operators from H. Is R necessarily a PI algebra? A multilinear generalized identity is called essential if the two-sided ideal generated by the values of all generalized monomials contains the unit element. A generalized monomial is the sum of all the monomials having a ﬁxed order of variables. Problem 3.98 (I. P. Shestakov) Do there exist exceptional prime noncommutative alternative algebras (that is, algebras other than associative or Cayley-Dickson rings)? Problem 3.99 (I. P. Shestakov) Compute (or at least ﬁnd an upper bound for) the nilpotency index of the radical of the free alternative algebra over a ﬁeld of characteristic 0. Problem 3.100 (I. P. Shestakov) Describe all simple ﬁnite dimensional superalgebras for the following classes of algebras:

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noncommutative Jordan (that are not super-anticommutative); right alternative; structurable; binary-Lie.

Problem 3.101 (I. P. Shestakov) Describe all ﬁnite dimensional irreducible superbimodules for the following classes of algebras: (a) alternative; (b) Jordan; (c) Malcev; (d) structurable. Problem 3.102 (I. P. Shestakov) Describe all simple ﬁnite dimensional Jordan superpairs and triple supersystems. Problem 3.103 (I. P. Shestakov) Do there exist ﬁnite dimensional central simple algebras over a ﬁeld of characteristic 0 that do not have a ﬁnite basis of identities? Problem 3.104 (I. P. Shestakov) Let A be a ﬁnite dimensional central simple algebra over a ﬁeld F , let Fk (A) be the free algebra of rank k in the variety generated by A, and let Γk be the ﬁeld of quotients of the centroid of Fk (A). Is Γk always a purely transcendental extension of F ? If A = Mn (F ) then this is the well-known problem on the center of the ring of generic matrices, which has been solved positively only for n ≤ 4. Problem 3.105 (I. P. Shestakov) Is it true that every nilpotent (not necessarily associative) algebra is representable (that is, embeddable in a ﬁnite dimensional algebra over some extension of the ground ﬁeld)?

B.6

Acknowledgments

The translators thank the editors of the Proceedings of NONAA-V for their support of this project, and NSERC (Natural Sciences and Engineering Research Council) for ﬁnancial assistance.

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