Jonathan D.H. Smith

Department of Mathematics
396 Carver Hall
Iowa State University
411 Morrill Road
Iowa 50011-2104



Split Extensions and Representations of Moufang Loops

Ashay Dharwadker and Jonathan D.H. Smith

Communications in Algebra 23(11), 4245-4255, 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, 125-134, 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 Eilenberg-Loginov 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 Eilenberg-Loginov module.