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Quasigroups, geometry and physics. (English) Zbl 0821.20061
Lómus, J. (ed.) et al., Quasigroups and nonassociative algebras in physics. Papers from the seminar held in Tartu (USSR), June 12-17, 1989. Tartu: Institut Fiziki AN Ehstonii, Eesti Teaduste Akadeemia Füüsika Instituudi Uurimused. 66, 24-53 (1990).
This paper is a survey on the theory of smooth nonassociative structures, intended for physicists, who use these structures in their research. Local loops and quasigroups, odules, loopuscular and odular structures, geoodular structures of affinely connected spaces, idempotent and distributive local loops and quasigroups, Bol loops and algebras are considered in this survey. Questions about inclusion of loops in groups, about representation of loops, about ties to the quasigroup and web theory and many others are considered as well. Many notions, mentioned above were introduced and studied by the author and his collaborators. Many of their results are described in detail in the author’s appendix to the book [S. Kobayashi, K. Nomizu, “Foundations of differential geometry”, Vol. 1 (Moscow, 1981; Zbl 0508.53002), p. 293].
For the entire collection see [Zbl 0788.00016].
20N05 Loops, quasigroups
22A30 Other topological algebraic systems and their representations
22E70 Applications of Lie groups to the sciences; explicit representations