Introduction To Quasigroups And Representation Theory, An

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