Skip to main content
Log in

About A-properties of some elastic three-webs

  • Published:
Russian Mathematics Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Moufang, R. “Zur Struktur von Alternativkörpern”, Math. Ann. 110, 416–430 (1935).

    Article  MathSciNet  MATH  Google Scholar 

  2. Maltsev, A. I. “Analytic Loops”, Mat. Sb. 36, No. 3, 569–575 (1955) [in Russian].

    MathSciNet  Google Scholar 

  3. Sabinin, L. V. Analytic Quasigroups and Geometry (Peoples Friendship Univ., Moscow, 1991) [in Russian].

    MATH  Google Scholar 

  4. Mikheev, P. O., Sabinin, L. V. “Smooth Quasigroups and Geometry”, J. Sov. Math. 51, No. 6, 2642–2666 (1990).

    Article  MathSciNet  MATH  Google Scholar 

  5. Akivis, M. A., Shelekhov, A. M. Multidimensional Three-Webs and Their Applications (Tver Gos. Univ., 2010) [in Russian].

    MATH  Google Scholar 

  6. Belousov, V. D. Fundamentals of the Theory of Quasigroups and Loops (Nauka, Moscow, 1967) [in Russian].

    Google Scholar 

  7. Shelekhov, A. M. “On Analytic Solutions of the Functional Equation x(yx) = (xy)x”, Mat. Zametki 50, No. 4, 132–140 (1991) [in Russian].

    MathSciNet  MATH  Google Scholar 

  8. Balandina, G. A., Shelekhov, A. M. “On General Theory of ElasticWebs”, in Webs and Quasigroups (Tver State Univ., 1995), pp. 622–74 [in Russian].

    Google Scholar 

  9. Dzhukashev, K. R. “About Three-Webs with Flexible Coordinate Loops”, in Trudy Mezhdunar. Geom. Tsentra (Odessa, 2013), 52–80 [in Russian].

    Google Scholar 

  10. Khasina, V. I. “Multidimensional Three-Webs with ElasticW-Algebras”, Sib.Mat. Zh. 17, 945–949 (1976).

    Google Scholar 

  11. Fedorova, V. I. “Three-Webs with Partially Skew-Symmetric Curvature Tensions”, Soviet Mathematics (Iz. VUZ) 20, No. 11, 114–117 (1976).

    MathSciNet  Google Scholar 

  12. 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].

    MathSciNet  Google Scholar 

  13. Fedorova, V. I. “A Condition DefiningMultidimensional Bol Three-Webs”, Sib.Mat.Zh. 19, 922–928 (1978) [in Russian].

    MathSciNet  Google Scholar 

  14. 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].

    Google Scholar 

  15. 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].

    Google Scholar 

  16. Antipova, M. V., Shelekhov, A. M. “Eight-Dimensional Bol Webs with Almost Zero Curvature Tensor”, Russian Mathematics (Iz. VUZ) 57, 2, 1–12 (2013).

    MathSciNet  MATH  Google Scholar 

  17. Murdoch, D. G. “Quasigroups Which Satisfy Certain Generalized Associative Laws”, Amer. J.Math. 61, No. 2, 509–522 (1939).

    Article  MathSciNet  MATH  Google Scholar 

  18. Osborn, M. J. “A Theorem on A-Loops”, Proc. Amer.Math. Soc. 9, No. 3, 347–349 (1958).

    MathSciNet  MATH  Google Scholar 

  19. Phillips, J. D. “On Moufang A-Loops”, Comment.Math. Univ. Carolinae 41, No. 2, 371–349 (2000).

    MathSciNet  MATH  Google Scholar 

  20. Drápal, A. “A-Loops Close to Code Loops are Groups”, Comment. Math. Univ. Carolinae 41, No. 2, 245–249 (2000).

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to K. R. Dzhukashev.

Additional information

Original Russian Text © K.R. Dzhukashev, 2016, published in Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, 2016, No. 7, pp. 23–33.

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Dzhukashev, K.R. About A-properties of some elastic three-webs. Russ Math. 60, 19–28 (2016). https://doi.org/10.3103/S1066369X16070045

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.3103/S1066369X16070045

Keywords

Navigation