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Split extensions and representations of Moufang loops. (English) Zbl 0834.20067
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; Zbl 0031.34303)] 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.

MSC:
20N05 Loops, quasigroups
20E22 Extensions, wreath products, and other compositions of groups
Citations:
Zbl 0031.34303
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References:
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