Ashay Dharwadker and Jonathan D.H. Smith
Communications in Algebra 23(11), 42454255, 1995
In a paper published by E. K. Loginov the concept of linear representation for Moufang loops is introduced, based on an idea of S. Eilenberg [published in Ann. Soc. Pol. Math. 21, 125134, 1948] generalizing the concept of split extension of a group module by the group. After this, a concept of representation in a variety of quasigroups was introduced by J. D. H. Smith. This paper is concerned with investigating the relationship between these two approaches to a representation theory for Moufang loops. In the process, the general theory of representations in a variety of quasigroups is specialized explicitly to the case of Moufang loops for the first time. It is shown that, while EilenbergLoginov split extensions and representations are equivalent for groups, they are no longer equivalent for Moufang loops. The seventh section gives an example of a representation in the variety of Moufang loops that cannot be described as an EilenbergLoginov module.

