Split extensions and representations of Moufang loops.

*(English)*Zbl 0834.20067In 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.

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.

Reviewer: C.Pereira da Silva (Curitiba)

##### Keywords:

linear representation; Moufang loops; variety of quasigroups; split extensions; variety of Moufang loops##### Citations:

Zbl 0031.34303
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\textit{A. Dharwadker} and \textit{J. D. H. Smith}, Commun. Algebra 23, No. 11, 4245--4255 (1995; Zbl 0834.20067)

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##### References:

[1] | Bruck R.H., Contributions to the theory of loops 60 pp 245– (1946) · Zbl 0061.02201 |

[2] | Bruck R.H., A Survey of Binary Systems (1958) · Zbl 0081.01704 |

[3] | Doro S., Simple Moufang Loops 83 pp 377– (1978) · Zbl 0381.20054 |

[4] | Eilenberg S., Ann. Soc. Polon. Math 21 pp 125– (1948) |

[5] | Fox R.H., Free differential calculus 57 pp 547– (1953) · Zbl 0142.22303 |

[6] | Fuad T. S. R., Quasigroups, right quasigroups and category coverings · Zbl 0863.20033 |

[7] | Herrlich H., Category Theory (1973) |

[8] | Locinov E.K., On linear representations of Moufang Loops 21 pp 2527– (1993) |

[9] | Smith J. D. H., Representation Theory of Infinite Groups and Finite Quasigroups Les Presses de L’Universite de Montreal (1986) |

[10] | Smith, J.D.H. 1992.Quasigroup representation theoryEdited by: Romanwska, A. and Smith, J.D.H. 195–207. Heldermann, Berlin Universal Algebra and Quasi-group Theory · Zbl 0772.20023 |

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