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On the infinitesimal theory of local analytic loops. (English. Russian original) Zbl 0659.53018
Sov. Math., Dokl. 36, 545-548 (1988); translation from Dokl. Akad. Nauk SSSR 297, 801-804 (1987).
E. Cartan [J. de Math. (9) 6, 1–119 (1927; JFM 53.0388.01)] has shown that any Lie group may be considered as a torsion-free affinely connected space with covariantly constant torsion. An analogous construction is carried out in the paper under review for the local analytic loops (Q,$$\times,e)$$ of general kind. For this the authors first consider the case of local analytic loops with the right mono-alternatively law, i.e. when for arbitrary integers k and $$\ell$$ and any $$a,b\in Q$$, sufficiently close to e, we have $$(a\times b^ k)\times b^{\ell}=a\times b^{k+\ell}$$, and then they treat the general situation as a “perturbation” of the right mono-alternative case. As result they obtain in the tangent space at the unit of the loop variety a hyperalgebra structure with some multi-operators whose specification uniquely defines a local analytic loop.
The results obtained in [A. I. Mal’tsev, Mat. Sb., Nov. Ser. 36(78), 569–576 (1955; Zbl 0065.00702); E. N. Kuz’min, Algebra Logika 10, 3–22 (1971; Zbl 0244.17019); O. Loos, “Symmetric spaces”, Vol. I, II. New York etc.: W. A. Benjamin (1969; Zbl 0175.48601); the first author, “Methods of nonassociative algebras in differential geometry”, Appendix to the Russian translation of S. Kobayashi, K. Nomizu, “Foundations of differential geometry”, Vol. I. Moskva: Nauka, 293–339 (1981; Zbl 0508.53002) the authors in: Webs and Quasigroups, Kalinin 1984, 144–154 (1984; Zbl 0568.53009); the authors, “The theory of smooth Bol loops” (Lecture Notes, Moskva 1985; Zbl 0584.53001); M. Kikkawa, Hiroshima Math. J. 5, 141-179 (1975; Zbl 0304.53037)] are special cases of this construction.

##### MSC:
 53A60 Differential geometry of webs 22A30 Other topological algebraic systems and their representations 53B05 Linear and affine connections