An introduction to quasigroups and their representations

342 Pages·2007·1.578 MB·English

by  Jonathan D. H. Smith

An Introduction
to Quasigroups
and Their
Representations

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An Introduction
to Quasigroups
and Their
Representations
Jonathan D. H. Smith
Boca Raton London New York
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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.
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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 gen-
eralizations 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 sym-
metry, probabilisticinnature, thatmayincludecertainformsofapproximate
symmetry. Chapter 4 describes this symmetry, and the quasigroup homoge-
neous 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 alge-
bra 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 quasi-
group 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 quasi-
groups,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 vari-
ous 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