Quasigroup Representation Theory

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Overview

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.

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Product Details

  • ISBN-13: 9781584885375
  • Publisher: Taylor & Francis
  • Publication date: 12/15/2006
  • Series: Studies in Advanced Mathematics Series , #48
  • Edition description: 1ST
  • Pages: 352
  • Product dimensions: 6.20 (w) x 9.20 (h) x 1.00 (d)

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

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