An Introduction to Quasigroups and Their Representations
This book is based on a research program that spans over thirty years. During those years the author was consistently trying to develop tools for the theory of quasigroups that would possess a sound categorical basis and would involve the classical representation theory of groups as a special case. The earliest fruits of this effort were the notion of the centralizer ring of a quasigroup and the notion of the universal multiplication group. The former gave rise to the theory of quasigroup characters to which more than one third of the book is devoted. However, this is preceded by the more recent concepts of homogeneous spaces and permutation representations that are based on the usual definition of the multiplication group as the permutation group generated by left and right translations.
Consider the orbits of the permutation group generated by left translations supplied by elements of a subquasigroup. The key notion is that of Markov matrices indexed by these orbits, where for each element of the quasigroup one constructs a matrix in which each row describes the probability distribution of the elements in the orbit indexing the row into all orbits when the right translation is applied. This connection of quasigroup theory and probability is novel and might lead in future to surprising and interesting consequences. While the book certainly introduces the notion of the quasigroup, it cannot be regarded as a general source of knowledge about the development of the theory of quasigroups and loops in recent decades, despite many interesting exercises that often overcome the limits set by the exposition goals.