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Smooth loops. I. (English) Zbl 1001.22003
This paper is the first part of a survey paper related to smooth loops theory, the development of which is motivated by physics and mechanics as well as by certain interior problems of differential geometry. So, this theory has been elaborated and successfully applied to the geometry of homogeneous spaces, mostly by the author’s contributions [see L. V. Sabinin, Tr. Semin. Vektorn. Tenzorn. Anal. 23, 133-146 (1988; Zbl 0814.22001), Analytic quasigroups and geometry (Friendship of Nations University Press, Moscow, 1991), and L. V. Sabinin and P. O. Miheev, Quasigroups and differential geometry (in: Quasigroups and loops: theory and applications, Sigma Ser. Pure Math. 8, 357-430, 1990; Zbl 0721.53018)].
In this part the fundamentals of the general infinitesimal theory of local smooth loops are introduced. So, an approach, which is similar to the original approach of S. Lie’s Lie groups theory is used. The differential equations of smooth loops are introduced and their infinitesimal theory is constructed generalizing again Lie groups theory. It is shown that any geometric odule is a geometric odule for some affine connection.
Differential equations for geometric and holonomial odules are obtained. On the tangent space of a local smooth loop at the neutral element the structures of a $$\nu$$-hyperalgebra with two binary operations which determine the initial local analytic loop uniquely up to isomorphism are introduced.
##### MSC:
 22A30 Other topological algebraic systems and their representations 22A22 Topological groupoids (including differentiable and Lie groupoids) 20N05 Loops, quasigroups