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**Smooth quasigroups and loops.**
*(English)*
Zbl 0939.20064

Mathematics and its Applications (Dordrecht). 492. Dordrecht: Kluwer Academic Publishers. xvi, 249 p. (1999).

The book is a collection of results obtained by the author and his numerous students. The main objects of investigation are the general theory of smooth loops, special classes of loops (monoalternative, Moufang and Bol loops and some of their generalizations, namely hyporeductive and pseudoreductive loops), the so-called odules introduced by the author. Also, the theory of the corresponding infinitesimal objects is considered, that is Bol algebras, hyporeductive algebras, etc. Structures of this kind naturally arise on smooth manifolds. In the book, the so-called loopuscular and geoodular structures are investigated in detail. The book is equipped with a rather complete bibliography.

Reviewer: A.M.Shelekhov (Tver’)

### MSC:

20N05 | Loops, quasigroups |

20-02 | Research exposition (monographs, survey articles) pertaining to group theory |

53A60 | Differential geometry of webs |

22A30 | Other topological algebraic systems and their representations |

22A22 | Topological groupoids (including differentiable and Lie groupoids) |

17D10 | Mal’tsev rings and algebras |