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.


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