1st Edition

An Introduction to Quasigroups and Their Representations

By Jonathan D. H. Smith Copyright 2007
    352 Pages 14 B/W Illustrations
    by Chapman & Hall

    Collecting results scattered throughout the literature into one source, An Introduction to Quasigroups and Their Representations shows how representation theories for groups are capable of extending to general quasigroups and illustrates the added depth and richness that result from this extension.

    To fully understand representation theory, the first three chapters provide a foundation in the theory of quasigroups and loops, covering special classes, the combinatorial multiplication group, universal stabilizers, and quasigroup analogues of abelian groups. Subsequent chapters deal with the three main branches of representation theory-permutation representations of quasigroups, combinatorial character theory, and quasigroup module theory. Each chapter includes exercises and examples to demonstrate how the theories discussed relate to practical applications. The book concludes with appendices that summarize some essential topics from category theory, universal algebra, and coalgebras.

    Long overshadowed by general group theory, quasigroups have become increasingly important in combinatorics, cryptography, algebra, and physics. Covering key research problems, An Introduction to Quasigroups and Their Representations proves that you can apply group representation theories to quasigroups as well.

    QUASIGROUPS AND LOOPS
    Latin squares
    Equational quasigroups
    Conjugates
    Semisymmetry and homotopy
    Loops and piques
    Steiner triple systems I
    Moufang loops and octonions
    Triality
    Normal forms
    Exercises
    Notes

    MULTIPLICATION GROUPS
    Combinatorial multiplication groups
    Surjections
    The diagonal action
    Inner multiplication groups of piques
    Loop transversals and right quasigroups
    Loop transversal codes
    Universal multiplication groups
    Universal stabilizers
    Exercises
    Notes

    CENTRAL QUASIGROUPS
    Quasigroup congruences
    Centrality
    Nilpotence
    Central isotopy
    Central piques
    Central quasigroups
    Quasigroups of prime order
    Stability congruences
    No-go theorems
    Exercises
    Notes

    HOMOGENEOUS SPACES
    Quasigroup homogeneous spaces
    Approximate symmetry
    Macroscopic symmetry
    Regularity
    Lagrangean properties
    Exercises
    Notes

    PERMUTATION REPRESENTATIONS
    The category IFSQ
    Actions as coalgebras
    Irreducibility
    The covariety of Q-sets
    The Burnside algebra
    An example
    Idempotents
    Burnside's lemma
    Exercises
    Problems
    Notes

    CHARACTER TABLES
    Conjugacy classes
    Class functions
    The centralizer ring
    Convolution of class functions
    Bose-Mesner and Hecke algebras
    Quasigroup character tables
    Orthogonality relations
    Rank two quasigroups
    Entropy
    Exercises
    Problems
    Notes

    COMBINATORIAL CHARACTER THEORY
    Congruence lattices
    Quotients
    Fusion
    Induction
    Linear characters
    Exercises
    Problems
    Notes

    SCHEMES AND SUPERSCHEMES
    Sharp transitivity
    More no-go theorems
    Superschemes
    Superalgebras
    Tensor squares
    Relation algebras
    The reconstruction theorem
    Exercises
    Problems
    Notes

    PERMUTATION CHARACTERS
    Enveloping algebras
    Structure of enveloping algebras
    The canonical representation
    Commutative actions
    Faithful homogeneous spaces
    Characters of homogeneous spaces
    General permutation characters
    The Ising model
    Exercises
    Problems
    Notes

    MODULES
    Abelian groups and slice categories
    Quasigroup modules
    The fundamental theorem
    Differential calculus
    Representations in varieties
    Group representations
    Exercises
    Problems
    Notes

    APPLICATIONS OF MODULE THEORY
    Nonassociative powers
    Exponents
    Steiner triple systems II
    The Burnside problem
    A free commutative Moufang loop
    Extensions and cohomology
    Exercises
    Problems
    Notes

    ANALYTICAL CHARACTER THEORY
    Functions on finite quasigroups
    Periodic functions on groups
    Analytical character theory
    Almost periodic functions
    Twisted translation operators
    Proof of the existence theorem
    Exercises
    Problems
    Notes

    APPENDIX A: CATEGORICAL CONCEPTS
    Graphs and categories
    Natural transformations and functors
    Limits and colimits

    APPENDIX B: UNIVERSAL ALGEBRA
    Combinatorial universal algebra
    Categorical universal algebra

    APPENDIX C: COALGEBRAS
    Coalgebras and covarieties
    Set functors

    REFERENCES
    INDEX

    Biography

    Jonathan D. H. Smith