An Introduction to Quasigroups and Their Representations: 1st Edition (Hardback) book cover

An Introduction to Quasigroups and Their Representations

1st Edition

By Jonathan D. H. Smith

Chapman and Hall/CRC

352 pages | 14 B/W Illus.

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pub: 2006-11-15
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Description

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.

Table of Contents

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

About the Series

Studies in Advanced Mathematics

Learn more…

Subject Categories

BISAC Subject Codes/Headings:
MAT002000
MATHEMATICS / Algebra / General
MAT022000
MATHEMATICS / Number Theory
MAT036000
MATHEMATICS / Combinatorics