Abstract
We consider two families of multidimensional three-webs and prove that these webs are webs E, i.e., in their coordinate loops the elasticity identity holds true. We also show that these webs have A-properties.
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References
Moufang, R. “Zur Struktur von Alternativkörpern”, Math. Ann. 110, 416–430 (1935).
Maltsev, A. I. “Analytic Loops”, Mat. Sb. 36, No. 3, 569–575 (1955) [in Russian].
Sabinin, L. V. Analytic Quasigroups and Geometry (Peoples Friendship Univ., Moscow, 1991) [in Russian].
Mikheev, P. O., Sabinin, L. V. “Smooth Quasigroups and Geometry”, J. Sov. Math. 51, No. 6, 2642–2666 (1990).
Akivis, M. A., Shelekhov, A. M. Multidimensional Three-Webs and Their Applications (Tver Gos. Univ., 2010) [in Russian].
Belousov, V. D. Fundamentals of the Theory of Quasigroups and Loops (Nauka, Moscow, 1967) [in Russian].
Shelekhov, A. M. “On Analytic Solutions of the Functional Equation x(yx) = (xy)x”, Mat. Zametki 50, No. 4, 132–140 (1991) [in Russian].
Balandina, G. A., Shelekhov, A. M. “On General Theory of ElasticWebs”, in Webs and Quasigroups (Tver State Univ., 1995), pp. 622–74 [in Russian].
Dzhukashev, K. R. “About Three-Webs with Flexible Coordinate Loops”, in Trudy Mezhdunar. Geom. Tsentra (Odessa, 2013), 52–80 [in Russian].
Khasina, V. I. “Multidimensional Three-Webs with ElasticW-Algebras”, Sib.Mat. Zh. 17, 945–949 (1976).
Fedorova, V. I. “Three-Webs with Partially Skew-Symmetric Curvature Tensions”, Soviet Mathematics (Iz. VUZ) 20, No. 11, 114–117 (1976).
Fedorova, V. I. “On a Class W6 of Three-Webs with a Partially Skew-Symmetric Curvature Tensor”, Ukr. Geom. Sb. 20, 115–124 (1977) [in Russian].
Fedorova, V. I. “A Condition DefiningMultidimensional Bol Three-Webs”, Sib.Mat.Zh. 19, 922–928 (1978) [in Russian].
Fedorova, V. I. “Six-Dimensional Bol Three-Webs with Symmetric Tensor a ij”, in Webs and quasigroups (Kalininsk. Gos. Univ., 1981), pp. 1102–123 [in Russian].
Antipova, M. V. “About Three-Webs of Bol with Almost Null Curvature Tensor”, Izv. Penz. Gos. Pedagog. Univ., Fiz.-Math. i Tekhn. Nauki, No. 26, 28–34 (2011) [in Russian].
Antipova, M. V., Shelekhov, A. M. “Eight-Dimensional Bol Webs with Almost Zero Curvature Tensor”, Russian Mathematics (Iz. VUZ) 57, 2, 1–12 (2013).
Murdoch, D. G. “Quasigroups Which Satisfy Certain Generalized Associative Laws”, Amer. J.Math. 61, No. 2, 509–522 (1939).
Osborn, M. J. “A Theorem on A-Loops”, Proc. Amer.Math. Soc. 9, No. 3, 347–349 (1958).
Phillips, J. D. “On Moufang A-Loops”, Comment.Math. Univ. Carolinae 41, No. 2, 371–349 (2000).
Drápal, A. “A-Loops Close to Code Loops are Groups”, Comment. Math. Univ. Carolinae 41, No. 2, 245–249 (2000).
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Original Russian Text © K.R. Dzhukashev, 2016, published in Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, 2016, No. 7, pp. 23–33.
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Dzhukashev, K.R. About A-properties of some elastic three-webs. Russ Math. 60, 19–28 (2016). https://doi.org/10.3103/S1066369X16070045
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DOI: https://doi.org/10.3103/S1066369X16070045