ABSTRACT
In the first four sections of this chapter, properties of the permutation action of the multiplication group of a finite quasigroup are used to describe some of the algebra structure associated with a homogeneous space for that quasigroup. The remaining sections introduce the characters of a finite quasigroup that are associated with permutation actions of the quasigroup. These give a direct generalization of the permutation characters of a group. The fundamental tool for the chapter is the linear map (9.1). Associated with each quasigroup homogeneous space is an algebra known
as the enveloping algebra. This algebra is defined in Section 9.1 as a subspace of the domain of (9.1), equipped with an algebra structure that makes the restriction of (9.1) an algebra homomorphism, the so-called canonical representation of the enveloping algebra (Definition 9.1). Section 9.2 describes the structure of the enveloping algebra (Theorem 9.1), while Section 9.3 analyses the canonical representation. As an application of the enveloping algebra, Section 9.4 presents sufficient conditions for the commuting of the action matrices of a homogeneous space, as observed in the example of Section 4.2. A different restriction of (9.1), namely (9.21), furnishes a representation
of the centralizer ring of the quasigroup (Theorem 9.3). Definition 9.3 then defines a homogeneous space to be faithful when this restriction injects. The definition is consistent with the classical terminology in the group case. In Section 9.6, the restriction (9.21) is used to define quasigroup permutation characters for quasigroup homogeneous spaces, and the definition is extended to general permutation representations in Section 9.7. As an illustration, Section 9.8 computes the permutation characters of the quasigroup of integers modulo 4 under subtraction.
