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Matematicheskie Zametki, 1972, Volume 12, Issue 5, Pages 605–616 (Mi mzm9923)  
This article is cited in 10 scientific papers (total in 11 papers)

Loop geometries

L. V. Sabinin

Patrice Lumumba University
Full-text PDF (1286 kB) Citations (11)
Abstract: We introduce the construction of the semidirect product of a loop and its associate (or quasigroup) — the group uniquely generated by the loop. For a (left or right) loop the semidirect product is a group acting transitively on the loop so that the loop is provided with the structure of a homogeneous space, the stationary subgroup being its associate. The construction is reversible, viz.: any homogeneous space can be provided with the structure of a loop so that the semidirect product of it with the transassociate is isomorphic with the fundamental group of the homogeneous space and the transassociate is isomorphic with the stationarity group.
Received: 03.12.1971
English version:
Mathematical Notes, 1972, Volume 12, Issue 5, Pages 799–805
DOI: https://doi.org/10.1007/BF01099069
Bibliographic databases:
Document Type: Article
UDC: 519.4
Language: Russian
Citation: L. V. Sabinin, “Loop geometries”, Mat. Zametki, 12:5 (1972), 605–616; Math. Notes, 12:5 (1972), 799–805
Citation in format AMSBIB
\Bibitem{Sab72}
\by L.~V.~Sabinin
\paper Loop geometries
\jour Mat. Zametki
\yr 1972
\vol 12
\issue 5
\pages 605--616
\mathnet{http://mi.mathnet.ru/mzm9923}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=340461}
\zmath{https://zbmath.org/?q=an:0258.20066}
\transl
\jour Math. Notes
\yr 1972
\vol 12
\issue 5
\pages 799--805
\crossref{https://doi.org/10.1007/BF01099069}
Linking options:
  • https://www.mathnet.ru/eng/mzm9923
  • https://www.mathnet.ru/eng/mzm/v12/i5/p605
    • This publication is cited in the following 11 articles:
    1. Alexander I. Nesterov, Héctor Mata, “How Nonassociative Geometry Describes a Discrete Spacetime”, Front. Phys., 7 (2019)  crossref
    2. Vipul Kakkar, “Boolean loops with compact left inner mapping groups are profinite”, Topology and its Applications, 244 (2018), 51  crossref
    3. Yu. G. Nikonorov, “Double exponential map on symmetric spaces”, Siberian Adv. Math., 23:3 (2013), 210–218  mathnet  crossref  mathscinet  elib
    4. Yu. E. Borovskii, V. A. Kudryavtsev, Yu. G. Reshetnyak, “Lev Vasil'evich Sabinin (obituary)”, Russian Math. Surveys, 61:5 (2006), 975–980  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    5. ALEXANDER I. NESTEROV, LEV V. SABININ, “NONASSOCIATIVE GEOMETRY: FRIEDMANN–ROBERTSON–WALKER SPACETIME”, Int. J. Geom. Methods Mod. Phys., 03:08 (2006), 1481  crossref
    6. Maks A. Akivis, Vladislav V. Goldberg, Handbook of Differential Geometry, 1, 2000, 1  crossref
    7. Alexander I. Nesterov, Lev V. Sabinin, “Nonassociative geometry: Towards discrete structure of spacetime”, Phys. Rev. D, 62:8 (2000)  crossref
    8. V. V. Kisil, International Society for Analysis, Applications and Computation, 6, Complex Methods for Partial Differential Equations, 1999, 215  crossref
    9. Alexander I. Nesterov, Lev V. Sabinin, “Smooth loops, generalized coherent states, and geometric phases”, Int J Theor Phys, 36:9 (1997), 1981  crossref
    10. L. V. Sabinin, “On the gyrogroups of Hungar”, Russian Math. Surveys, 50:5 (1995), 1095–1096  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    11. Abraham A. Ungar, “The abstract complex Lorentz transformation group with real metric. I. Special relativity formalism to deal with the holomorphic automorphism group of the unit ball in any complex Hilbert space”, Journal of Mathematical Physics, 35:3 (1994), 1408  crossref
    Citing articles in Google Scholar: Russian citations, English citations
    Related articles in Google Scholar: Russian articles, English articles
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