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Published **1999**
by Springer Netherlands in Dordrecht .

Written in English

- Group theory,
- Mathematics,
- Global differential geometry,
- Geometry,
- Number theory

This monograph presents the complete theory of smooth quasigroups and loops, as well as its geometric and algebraic applications. Based on a generalisation of the Lie-group theory, it establishes new backgrounds for differential geometry in the form of nonlinear geometric algebra and `loopuscular" geometry. It will prove useful in applications in such diverse fields as mathematical physics, relativity, Poisson and symplectic mechanics, quantum gravity, dislocation theory, etc. Audience: This volume will be of interest to researchers, lecturers and postgraduate students whose work involves geometry, group theory, nonassociative rings and algebras, and mathematical and theoretical physics.

**Edition Notes**

Statement | by Lev V. Sabinin |

Series | Mathematics and Its Applications -- 492, Mathematics and Its Applications -- 492 |

Classifications | |
---|---|

LC Classifications | QA174-183 |

The Physical Object | |

Format | [electronic resource] / |

Pagination | 1 online resource (xvi, 249 p.) |

Number of Pages | 249 |

ID Numbers | |

Open Library | OL27087917M |

ISBN 10 | 9401059217, 9401144915 |

ISBN 10 | 9789401059213, 9789401144919 |

OCLC/WorldCa | 851370091 |

Such exotic structures of algebra as quasigroups and loops were obtained from purely geometric structures such as affinely connected spaces. The notion ofodule was introduced as a fundamental algebraic invariant of differential geometry. For any space with an affine connection loopuscular. Smooth Quasigroups and Loops | During the last twenty-five years quite remarkable relations between nonas- sociative algebra and differential geometry have been discovered in our work. Such exotic structures of algebra as quasigroups and loops were obtained from purely geometric structures such as affinely connected spaces. Such exotic structures of algebra as quasigroups and loops were obtained from purely geometric structures such as affinely connected spaces. The notion ofodule was introduced as a fundamental algebraic invariant of differential geometry. Smooth quasigroups and loops. [Lev V Sabinin] -- "This monograph presents the complete theory of smooth quasigroups and loops, as well as its geometric and algebraic applications. Based on a generalisation of the Lie-group theory, it establishes Your Web browser is not enabled for JavaScript.

Smooth quasigroups and loops: forty-ﬁve years of incredible growth such that, for any ﬁxed a ∈ M, hM, a, a,(ta)t∈Ri is a smooth local left odule, which means that hM, a,ai is a loop with two-sided neutral a ∈ M and (t +u)ax = tax a (12) uax (monoassociativity), (13) ta(uax) = (tu)ax (pseudoassociativity), (14) 1ax = x (unitarity). Also the theory of smooth quasi-groups and loops started to find interesting applications in geometry and physics. We refer to the books [3, 2, 21, 22] and the survey articles [4, 23, 24] if terms. ordered loops and quasigroups. Richard Hubert Bruck () made a survey of binary systems. In the recent past Hala () made a description on quasigroups and loops. A quasigroup is a generalization of a group without associative law or identity element. Groups can be reached in another way from groupoids, namely through quasi groups. abelian group automorphism autotopism autotopism of Q bijection binary operation Bol loop Bruck called carrier set Cayley table commutative Moufang loop Complete the proof Corollary decomposition modulo defined Definition denote di-associative distributive quasigroup element for G entropic quasigroup equivalence relation Example Exercise G-loop.

and the numbers of isomorphism classes of quasigroups and loops, up to order The best previous results were for Latin squares of order 8 (Kolesova, Lam and Thiel, ), quasigroups of order 6 (Bower, ) and loops of order 7 (Brant and Mullen, ). The loops of order 8 have been independently found by “QSCGZ”. Download PDF: Sorry, we are unable to provide the full text but you may find it at the following location(s): (external link)Cited by: Abstract. In this chapter we develop the theory of smooth Bol loops. See [L.V. Sabinin 91b], [L.V. Sabinin, P.O. Miheev 85b, 90]. Our interest in Bol loops was stimulated by their applications to differential geometry and the theory of homogeneous : Lev V. Sabinin. Quasigroups and loops pdf And the numbers of isomorphism classes of quasigroups and loops, up to order Thiel, , quasigroups of order 6 Bower, and loops of order 7. The group Ih α MQ, αh h is called inner mapping group of a quasigroup. 7 Number of small quasigroups and loops 8 See also 9 Notes 10 References 11 External links.

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