ABSTRACT
This chapter provides an introduction to quasigroup module theory. Since matrix multiplication is associative, naive attempts to extend group module theory are doomed to failure. However, as described in Section 10.1, a module over a group Q yields a split extension, which may be characterized as an abelian group in the slice category of groups over Q. The most general definition of a module over a quasigroup Q is thus given in Section 10.2 as an abelian group in the slice categoryQ/Q of quasigroups over Q. An alternative characterization in terms of self-centralizing congruences is also presented. In particular, the central piques of Chapter 3 emerge as modules over the singleton quasigroup (Exercise 8). In Section 10.3, the Fundamental Theorem 10.1 of Quasigroup Representations identifies quasigroup modules as being equivalent to modules over stabilizers in the universal multiplication group. While these modules are too general to yield specific information about a quasigroup Q, they do provide a framework for the representations in varieties that are the topic of Section 10.5. For a unital commutative ring S, and for a quasigroup Q in a variety V, these representations are defined as S-modules in the slice category V/Q of V-quasigroups over Q. They are equivalent to modules over a certain quotient of the group S-algebra of stabilizers in the universal multiplication group of Q in V. The quotient is determined by a process of combinatorial partial differentiation of the quasigroup words appearing in the identities defining the variety V. This process is described in Section 10.4. Section 10.6 shows that modules over groups may be recovered as quasigroup modules in the variety of associative quasigroups.
