Smooth quasigroups and loops: Forty-five years of incredible growth.

*(English)*Zbl 1038.20051This is a short but nicely written survey about the topic in the title. 13 pages are devoted to references. The stress is laid not upon the pure algebra but on those parts of the theory which were inspired by problems in differential geometry, especially by the theory of symmetric spaces. The most frequent relationship is that between an algebraic structure on a manifold and a “canonical” affine connection on the same manifold. An interesting topic of the recent research (still far from the final solution) is the generalization of the fifth Hilbert problem from groups to loops.

Here are some other keywords proposed by the author: odules, loopuscular and odular algebras, Bol loops, Moufang loops, Bol algebras, Mal’cev algebras, nonlinear geometric algebra, nonassociative geometry, \(F\)-quasigroups, hyperalgebra, hyporeductivity, pseudoreductivity. The web geometry is not treated here but in a special article by M. A. Akivis and V. V. Goldberg in the same volume [ibid. 41, No. 2, 205-236 (2000; Zbl 1042.53007)].

Here are some other keywords proposed by the author: odules, loopuscular and odular algebras, Bol loops, Moufang loops, Bol algebras, Mal’cev algebras, nonlinear geometric algebra, nonassociative geometry, \(F\)-quasigroups, hyperalgebra, hyporeductivity, pseudoreductivity. The web geometry is not treated here but in a special article by M. A. Akivis and V. V. Goldberg in the same volume [ibid. 41, No. 2, 205-236 (2000; Zbl 1042.53007)].

Reviewer: Oldřich Kowalski (Praha)

##### MSC:

20N05 | Loops, quasigroups |

22A30 | Other topological algebraic systems and their representations |

53A60 | Differential geometry of webs |

53C22 | Geodesics in global differential geometry |

53C35 | Differential geometry of symmetric spaces |

53C05 | Connections, general theory |