Table Of Content

An Introduction to Quasigroups and Their Representations C5378_C000.indd 1 10/06/2006 2:07:27 PM Studies in Advanced Mathematics Titles Included in the Series John P. D’Angelo, Several Complex Variables and the Geometry of Real Hypersurfaces Steven R. Bell, The Cauchy Transform, Potential Theory, and Conformal Mapping John J. Benedetto, Harmonic Analysis and Applications John J. Benedetto and Michael W. Frazier, Wavelets: Mathematics and Applications Albert Boggess, CR Manifolds and the Tangential Cauchy–Riemann Complex Keith Burns and Marian Gidea, Differential Geometry and Topology: With a View to Dynamical Systems George Cain and Gunter H. Meyer, Separation of Variables for Partial Differential Equations: An Eigenfunction Approach Goong Chen and Jianxin Zhou, Vibration and Damping in Distributed Systems Vol. 1: Analysis, Estimation, Attenuation, and Design Vol. 2: WKB and Wave Methods, Visualization, and Experimentation Carl C. Cowen and Barbara D. MacCluer, Composition Operators on Spaces of Analytic Functions Jewgeni H. Dshalalow, Real Analysis: An Introduction to the Theory of Real Functions and Integration Dean G. Duffy, Advanced Engineering Mathematics with MATLAB®, 2nd Edition Dean G. Duffy, Green’s Functions with Applications Lawrence C. Evans and Ronald F. Gariepy, Measure Theory and Fine Properties of Functions Gerald B. Folland, A Course in Abstract Harmonic Analysis José García-Cuerva, Eugenio Hernández, Fernando Soria, and José-Luis Torrea, Fourier Analysis and Partial Differential Equations Peter B. Gilkey, Invariance Theory, the Heat Equation, and the Atiyah-Singer Index Theorem, 2nd Edition Peter B. Gilkey, John V. Leahy, and Jeonghueong Park, Spectral Geometry, Riemannian Submersions, and the Gromov-Lawson Conjecture Alfred Gray, Elsa Abbena, and Simon Salamon Modern Differential Geometry of Curves and Surfaces with Mathematica, Third Edition Eugenio Hernández and Guido Weiss, A First Course on Wavelets Kenneth B. Howell, Principles of Fourier Analysis Steven G. Krantz, The Elements of Advanced Mathematics, Second Edition Steven G. Krantz, Partial Differential Equations and Complex Analysis Steven G. Krantz, Real Analysis and Foundations, Second Edition Kenneth L. Kuttler, Modern Analysis Michael Pedersen, Functional Analysis in Applied Mathematics and Engineering Clark Robinson, Dynamical Systems: Stability, Symbolic Dynamics, and Chaos, 2nd Edition John Ryan, Clifford Algebras in Analysis and Related Topics John Scherk, Algebra: A Computational Introduction Jonathan D. H. Smith, An Introductin to Quasigroups and Their Representations PavelSolín, Karel Segeth, and Ivo Dolezel, High-Order Finite Element Method André Unterberger and Harald Upmeier, Pseudodifferential Analysis on Symmetric Cones James S. Walker, Fast Fourier Transforms, 2nd Edition James S. Walker, A Primer on Wavelets and Their Scientific Applications Gilbert G. Walter and Xiaoping Shen, Wavelets and Other Orthogonal Systems, Second Edition Nik Weaver, Mathematical Quantization Kehe Zhu, An Introduction to Operator Algebras C5378_C000.indd 2 10/06/2006 2:07:28 PM An Introduction to Quasigroups and Their Representations Jonathan D. H. Smith Boca Raton London New York Chapman & Hall/CRC is an imprint of the Taylor & Francis Group, an informa business C5378_C000.indd 3 10/06/2006 2:07:28 PM Chapman & Hall/CRC Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487‑2742 © 2007 by Taylor & Francis Group, LLC Chapman & Hall/CRC is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S. Government works Printed in the United States of America on acid‑free paper 10 9 8 7 6 5 4 3 2 1 International Standard Book Number‑10: 1‑58488‑537‑8 (Hardcover) International Standard Book Number‑13: 978‑1‑58488‑537‑5 (Hardcover) 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. 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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 Smith, Jonathan D. H. 1949‑ An introduction to quasigroups and their representations / Jonathan D.H. Smith. p. cm. ‑‑ (Studies in advance mathematics) Includes bibliographical references and index. ISBN 1‑58488‑537‑8 (alk. paper) 1. Quasigroups. 2. Nonassociative algebras. 3. Representations of groups. I. Title. QA181.5.S65 2006 512’.22‑‑dc22 2006049290 Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com C5378_C000.indd 4 10/06/2006 2:07:28 PM Preface The theory of quasigroups (“nonassociative groups”) is one of the oldest branches of algebra and combinatorics. In the guise of Latin squares, it dates back at least to Euler [54]. Nevertheless, throughout the twentieth century it was overshadowed by its subset, the theory of groups, to such an extent that Mathematical Reviews classified loops and quasigroups merely as “other generalizations of groups.” Apart from the fashions of the day, the main reason forthepredominanceofgrouptheorywasthefactthatabstractgroupsadmit representations, either linearly by matrices and modules, or as symmetries in the form of permutation representations. The aim of the present book is to show how these representations for groups are fully capable of extension to general quasigroups, and to illustrate the added depth and richness that result from such an extension. Thelineartheoryforquasigroupsseparatestwotopics,charactertheoryand moduletheory,thatareusuallyconflatedinthegroupcase. Permutationrep- resentations take on two striking new aspects when extended to quasigroups. The first is probabilistic: permutation matrices of groups are replaced by Markov matrices for quasigroups. The second is the fact that quasigroup actions are naturally described as coalgebras rather than as algebras. The book divides into three parts: • Thefirstthreechapterscoverelementsofthetheoryofquasigroupsand loops, including certain keyexamples and construction techniques, that are needed for a full appreciation of the representation theory. • The bulk of the book is devoted to the three main branches of the representation theory itself: permutation representations, characters, and modules. • Finally, three brief appendices summarize some essential topics from category theory, universal algebra, and coalgebras. Chapter 1 provides a quick elementary introduction to quasigroups and loops, as well as some of the most important special classes such as semisym- metric quasigroups, Steiner triple systems, and Moufang loops. Chapter 2 discusses the group actions on the underlying set of a quasigroup that result from the quasigroup structure. These actions are the key tools of quasigroup theory. Inparticular,theactionofthecombinatorialmultiplicationgroupona quasigroupyieldsthecombinatorialcharacters, whiletheuniversalstabilizers discussedinSection2.8formthebasisformuchofquasigroupmoduletheory. v vi Chapter3looksatthequasigroupanaloguesofabeliangroups,namelycentral quasigroupsandpiques. Italsotouchesbrieflyonaconverseinterpretationof “quasigroup representation theory,” namely the representation of groups as multiplication groups of quasigroups. Chapters 4 and 5 are devoted to the theory of permutation representations of quasigroups. With permutation representations of groups being regarded as the embodiment of symmetry, one may view permutation representations of quasigroups as the expression of a newer and more general kind of symmetry, probabilisticinnature, thatmayincludecertainformsofapproximate symmetry. Chapter 4 describes this symmetry, and the quasigroup homogeneous spaces that underlie it. Homogeneous space concepts are also used to study issues related to the breakdown of Lagrange’s Theorem in quasigroups. The slightly more advanced Chapter 5 provides the general definition of a quasigroup permutation action, as a sum of images of homogeneous spaces or as an element of a certain covariety of coalgebras. The isomorphism classes of the permutation representations of a quasigroup form a Burnside algebra, justasforgroups,andageneralformofBurnside’sLemma,linearlyalgebraic in nature, counts the number of orbits. Chapters 6 through 9 treat the oldest branch of quasigroup representation theory, the combinatorial character theory. This theory extends the ordinary character theory of finite groups. In fact, selected material from the first two ofthesechaptersmightevenbeusedasaquickintroductiontothattheoryin a more general setting. In Chapter 6, the combinatorial characters of a finite quasigroup are obtained from the action of the combinatorial multiplication group on the quasigroup. The complex incidence matrices of the orbitals in this transitive action span a commutative algebra, and the characters emerge as normalized coefficients expressing the orthogonal idempotents of the algebra as linear combinations of the incidence matrices. Chapter 7 develops those parts of quasigroup character theory that form natural generalizations of group character theory: induced characters, fusion, lifting characters from quotients, and the determination of the structure of a quasigroup from its character table. By contrast, the topics of Chapter 8 do not have direct counterparts in group character theory. The motivating question is the extent to which a combinatorial multiplication group action on a quasigroup may be recovered from character-theoretical data. Chapter9,onpermutationcharacters,servesatwofoldpurpose. Ontheone hand, itusespropertiesofthepermutationactionofthemultiplicationgroup of a quasigroup to describe some of the algebra structure associated with a homogeneousspaceforthatquasigroup. Ontheotherhand,italsointroduces the characters of a quasigroup that are associated with permutation actions of the quasigroup. These give a direct generalization of the permutation characters of a group. The final chapters study quasigroup module theory. Since the composition of matrices or module endomorphisms is associative, module representations of quasigroups require a more sophisticated definition than for groups, using vii analgebraicanalogueofthetopologicalconceptofafiberbundleorthephys- ical concept of a gauge theory (Sections 10.2 and 10.5). The fundamental theorems of Chapter 10 then show that categories of modules over a quasigroup are equivalent to categories of modules over certain rings, quotients of group algebras of universal stabilizers. In particular, there is a differential calculus for quasigroup words (Section 10.4). Various applications of quasigroup module theory are given in Chapter 11. The topics discussed include the indexing of nonassociative powers (by their derivatives), the exponent of a quasigroup, Burnside’s problem for quasigroups,constructionoffreecommutativeMoufangloops,andaquicksynopsis of cohomology and extension theory for quasigroups. Chapter12introducesanalyticalcharactersofafinitequasigroup,ascertain almost-periodicfunctions. Althoughthefinite-dimensionalcomplexrepresen- tations of a finite group are determined up to equivalence by its ordinary characters, the corresponding combinatorial characters of a finite quasigroup, astreatedinChapters6through9,areinadequateforthetaskofclassifyingall the finite-dimensional modules over the complex numbers. This classification is achieved by the analytical characters. Appendix A covers the main constructions of category theory used at various points throughout the text. Appendix B provides a quick introduction to universalalgebraicconceptssuchascongruences,freealgebras,andidentities. Although these two appendices might conceivably serve as synopses for mini- courses in their respective topics, readers are referred to [165] for more detail on categories and general algebraic methods. Appendix C summarizes the basicfactsaboutcoalgebrasthatareneededforthetreatmentofpermutation representations in Chapter 5. The structure of the book is summarized in the following chart. 1 Permutation Funda- representations mentals (cid:63) 5 (cid:190) 4 (cid:190) 2 (cid:45) 3 (cid:45)10 (cid:45)11 (cid:63) (cid:63) (cid:63) (cid:63) 9.6 (cid:190) 9 (cid:190) 7 (cid:190) 6 (cid:45)12 Combinatorial Module (cid:63) characters theory 8 Arrowsshowtheapproximatedependenciesbetweenthechaptersorsections. In case of doubt, the index may be helpful in locating keywords or symbols. viii Much of the work on this book was completed while I was on a Faculty Professional Development Assignment from Iowa State University during the academic year 2005–2006. The four-month period from October 2005 to February 2006 which I spent as a guest of the Faculty of Mathematics and InformationSciencesatWarsawUniversityofTechnologyprovedparticularly fruitful. Many thanks are also due to Bob Stern and the staff of Taylor & Francis. Contents 1 QUASIGROUPS AND LOOPS 1 1.1 Latin squares . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 Equational quasigroups . . . . . . . . . . . . . . . . . . . . . 3 1.3 Conjugates . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.4 Semisymmetry and homotopy . . . . . . . . . . . . . . . . . 7 1.5 Loops and piques . . . . . . . . . . . . . . . . . . . . . . . . 9 1.6 Steiner triple systems I . . . . . . . . . . . . . . . . . . . . . 12 1.7 Moufang loops and octonions . . . . . . . . . . . . . . . . . . 13 1.8 Triality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 1.9 Normal forms . . . . . . . . . . . . . . . . . . . . . . . . . . 19 1.10 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 1.11 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 2 MULTIPLICATION GROUPS 35 2.1 Combinatorial multiplication groups . . . . . . . . . . . . . . 35 2.2 Surjections . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 2.3 The diagonal action . . . . . . . . . . . . . . . . . . . . . . . 38 2.4 Inner multiplication groups of piques . . . . . . . . . . . . . 40 2.5 Loop transversals and right quasigroups . . . . . . . . . . . . 41 2.6 Loop transversal codes . . . . . . . . . . . . . . . . . . . . . 46 2.7 Universal multiplication groups . . . . . . . . . . . . . . . . . 50 2.8 Universal stabilizers . . . . . . . . . . . . . . . . . . . . . . . 54 2.9 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 2.10 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 3 CENTRAL QUASIGROUPS 61 3.1 Quasigroup congruences . . . . . . . . . . . . . . . . . . . . . 62 3.2 Centrality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 3.3 Nilpotence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 3.4 Central isotopy . . . . . . . . . . . . . . . . . . . . . . . . . . 69 3.5 Central piques . . . . . . . . . . . . . . . . . . . . . . . . . . 75 3.6 Central quasigroups . . . . . . . . . . . . . . . . . . . . . . . 77 3.7 Quasigroups of prime order . . . . . . . . . . . . . . . . . . . 79 3.8 Stability congruences . . . . . . . . . . . . . . . . . . . . . . 81 3.9 No-go theorems . . . . . . . . . . . . . . . . . . . . . . . . . 86 3.10 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 3.11 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 ix x 4 HOMOGENEOUS SPACES 93 4.1 Quasigroup homogeneous spaces . . . . . . . . . . . . . . . . 93 4.2 Approximate symmetry . . . . . . . . . . . . . . . . . . . . . 98 4.3 Macroscopic symmetry . . . . . . . . . . . . . . . . . . . . . 101 4.4 Regularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 4.5 Lagrangean properties . . . . . . . . . . . . . . . . . . . . . . 106 4.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 4.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 5 PERMUTATION REPRESENTATIONS 113 5.1 The category IFS . . . . . . . . . . . . . . . . . . . . . . . 113 Q 5.2 Actions as coalgebras . . . . . . . . . . . . . . . . . . . . . . 116 5.3 Irreducibility . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 5.4 The covariety of Q-sets . . . . . . . . . . . . . . . . . . . . . 121 5.5 The Burnside algebra . . . . . . . . . . . . . . . . . . . . . . 123 5.6 An example . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 5.7 Idempotents . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 5.8 Burnside’s Lemma . . . . . . . . . . . . . . . . . . . . . . . . 133 5.9 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 5.10 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 5.11 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 6 CHARACTER TABLES 139 6.1 Conjugacy classes . . . . . . . . . . . . . . . . . . . . . . . . 139 6.2 Class functions . . . . . . . . . . . . . . . . . . . . . . . . . . 140 6.3 The centralizer ring . . . . . . . . . . . . . . . . . . . . . . . 142 6.4 Convolution of class functions . . . . . . . . . . . . . . . . . 145 6.5 Bose-Mesner and Hecke algebras . . . . . . . . . . . . . . . . 147 6.6 Quasigroup character tables . . . . . . . . . . . . . . . . . . 150 6.7 Orthogonality relations . . . . . . . . . . . . . . . . . . . . . 155 6.8 Rank two quasigroups . . . . . . . . . . . . . . . . . . . . . . 158 6.9 Entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 6.10 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166 6.11 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 6.12 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 7 COMBINATORIAL CHARACTER THEORY 169 7.1 Congruence lattices . . . . . . . . . . . . . . . . . . . . . . . 169 7.2 Quotients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172 7.3 Fusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176 7.4 Induction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182 7.5 Linear characters . . . . . . . . . . . . . . . . . . . . . . . . 187 7.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 7.7 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198 7.8 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198