Algebraic aspects of web geometry

Maks A. Akivis; Vladislav V. Goldberg

Commentationes Mathematicae Universitatis Carolinae (2000)

  • Volume: 41, Issue: 2, page 205-236
  • ISSN: 0010-2628

Abstract

top
Algebraic aspects of web geometry, namely its connections with the quasigroup and loop theory, the theory of local differential quasigroups and loops, and the theory of local algebras are discussed.

How to cite

top

Akivis, Maks A., and Goldberg, Vladislav V.. "Algebraic aspects of web geometry." Commentationes Mathematicae Universitatis Carolinae 41.2 (2000): 205-236. <http://eudml.org/doc/248614>.

@article{Akivis2000,
abstract = {Algebraic aspects of web geometry, namely its connections with the quasigroup and loop theory, the theory of local differential quasigroups and loops, and the theory of local algebras are discussed.},
author = {Akivis, Maks A., Goldberg, Vladislav V.},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {quasigroup; loop; web; group; local quasigroup; local loop; Akivis algebra; $n$-quasigroup; quasigroup; loop; web; group; local quasigroup; local loop; Akivis algebra; -quasigroup},
language = {eng},
number = {2},
pages = {205-236},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Algebraic aspects of web geometry},
url = {http://eudml.org/doc/248614},
volume = {41},
year = {2000},
}

TY - JOUR
AU - Akivis, Maks A.
AU - Goldberg, Vladislav V.
TI - Algebraic aspects of web geometry
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2000
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 41
IS - 2
SP - 205
EP - 236
AB - Algebraic aspects of web geometry, namely its connections with the quasigroup and loop theory, the theory of local differential quasigroups and loops, and the theory of local algebras are discussed.
LA - eng
KW - quasigroup; loop; web; group; local quasigroup; local loop; Akivis algebra; $n$-quasigroup; quasigroup; loop; web; group; local quasigroup; local loop; Akivis algebra; -quasigroup
UR - http://eudml.org/doc/248614
ER -

References

top
  1. Aczel J., Quasigroups, nets and nomograms, Adv. Math. 1 (1965), 3 383-450. (1965) Zbl0135.03601MR0193174
  2. Akivis M.A., The canonical expansions of the equations of a local analytic quasigroup, Dokl. Akad. Nauk SSSR 188 (1969), no. 5, 967-970 (Russian); English transl.: Soviet Math. Dokl. 10 (1969), no. 5, 1200-1203. MR0262413
  3. Akivis M.A., Three-webs of multidimensional surfaces, Trudy Geometr. Sem. 2 (1969), 7-31 (Russian). (1969) MR0254760
  4. Akivis M.A., The local differentiable quasigroups and three-webs that are determined by a triple of hypersurfaces, Sibirsk. Mat. Zh. 14 (1973), no. 3, 467-474 (Russian); English transl.: Siberian Math. J. 14 (1973), no. 3, 319-324. MR0324559
  5. Akivis M.A., Local differentiable quasigroups and three-webs of multidimensional surfaces, in ``Studies in the Theory of Quasigroups and Loops'', pp.3-12 (Russian), Izdat. ``Shtiintsa'', Kishinev, 1973. MR0370391
  6. Akivis M.A., Closed G -structures on a differentiable manifold, Problems in Geometry, Vol. 7, 69-79 (Russian), Akad. Nauk SSSR Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1975. Zbl0549.53032MR0478055
  7. Akivis M.A., The local algebras of a multidimensional three-web, Sibirsk. Mat. Zh. 17 (1976), no. 1, 5-11 (Russian); English transl.: Siberian Math. J. 17 (1976), no. 1, 3-8. MR0405261
  8. Akivis M.A., Geodesic loops and local triple systems in a space with an affine connection, Sibirsk. Mat. Zh. 19 (1978), no. 2, 243-253 (Russian); English transl.: Siberian Math. J. 19 (1978), no. 2, 171-178. MR0487840
  9. Akivis M.A., Goldberg V.V., The 4 -web and the local differentiable ternary quasigroup that are determined by a quadruple of surfaces of codimension two, Izv. Vyssh. Uchebn. Zaved. Mat. 1974, no. 5(144), 12-24 (Russian); English transl.: Soviet Math. (Iz. VUZ) 18 (1974), no. 5, 9-19. MR0355847
  10. Akivis M.A., Goldberg V.V., Differential geometry of webs, Chapter 1 in ``Handbook of Differential Geometry'', Vol. 1, Elsevier, 2000, pp.1-152. Zbl0968.53001MR1736852
  11. Akivis M.A., Shelekhov A.M., Local differentiable quasigroups and connections that are associated with a three-web of multidimensional surfaces, Sibirsk. Mat. Zh. 12 (1971), no. 6, 1181-1191 (Russian); English transl.: Siberian Math. J. 12 (1971), no. 6, 845-852. MR0288681
  12. Akivis M.A., Shelekhov A.M., Geometry and Algebra of Multidimensional Three-Webs, translated from the Russian by V.V. Goldberg, Kluwer Academic Publishers, Dordrecht, 1992, xvii+358 pp. Zbl0771.53001MR1196908
  13. Barlotti A., Geometry of quasigroups, Chapter VIII in ``Quasigroups and Loops: Theory and Applications'', O. Chein, H.O. Pflugfelder, and J.D.H. Smith, Eds., Heldermann-Verlag, Berlin, 1990, pp.197-203. Zbl0723.20041MR1125814
  14. Barlotti A., Strambach K., The geometry of binary systems, Adv. Math. 49 (1983), 1 1-105. (1983) Zbl0518.20064MR0715128
  15. Belousov V.D., Foundations of the Theory of Quasigroups and Loops, Izdat. ``Nauka'', Moscow, 1967, 223 pp. (Russian). MR0218483
  16. Belousov V.D., Algebraic Nets and Quasigroups, Izdat. ``Shtiintsa'', Kishinev, 1971, 165 pp. (Russian). MR0340459
  17. Belousov V.D., n -ary Quasigroups, Izdat. ``Shtiintsa'', Kishinev, 1972, 227 pp. (Russian). MR0354919
  18. Belousov V.D., Sandik M.D., n -ary quasigroups and loops, Sibirsk. Mat. Zh. 7 (1966), no. 1, 31-54 (Russian); English transl.: Siberian Math. J. 7 (1966), no. 1, 24-42. MR0204564
  19. Blaschke W., Thomsens Sechseckgewebe. Zueinander diagonale Netze, Math. Z. 28 (1928), 150-157. (1928) MR1544947
  20. Blaschke W., Projective Geometrie, Wolfenbüttel, Hannover, 1947, 160 pp.; 2nd ed., Wolfenbüttel, Hannover, 1948, 160 pp.; 3rd ed., Birkhäuser, Basel-Stuttgart, 1954, 197 pp. 
  21. Blaschke W., Einführung in die Geometrie der Waben, Birkhäuser-Verlag, Basel-Stuttgart, 1955, 108 pp.; Russian transl.: GITTL, Moskva, 1959, 144 pp. MR0075630
  22. Blaschke W., Bol G., Geometrie der Gewebe, Springer-Verlag, Berlin, 1938, viii+339 pp. Zbl0020.06701
  23. Bol G., Über 3-Gewebe in vierdimensionalen Raum, Math. Ann. 110 (1935), 431-463. (1935) MR1512949
  24. Bol G., Gewebe und Gruppen, Math. Ann. 114 (1937), 414-431. (1937) Zbl0016.22603MR1513147
  25. Chern S.S., Eine Invariantentheorie der Dreigewebe aus r -dimensionalen Mannigfaltigkeiten in 𝐑 2 r , Abh. Math. Sem. Univ. Hamburg 11 (1936), 1-2 333-358. (1936) 
  26. Fedorova V.I., A condition defining multidimensional Bol's three-webs, Sibirsk. Mat. Zh. 19 (1978), no. 4, 922-928 (Russian); English transl.: Siberian Math. J. 19 (1978), no. 4, 657-661. MR0514146
  27. Goldberg V.V., ( n + 1 ) -webs of multidimensional surfaces, Dokl. Akad. Nauk SSSR 210 (1973), no. 4, 756-759 (Russian); English transl.: Soviet Math. Dokl. 14 (1973), no. 3, 795-799. MR0324567
  28. Goldberg V.V., An invariant characterization of certain closure conditions in ternary quasigroups, Sibirsk. Mat. Zh. 16 (1975), no. 1, 29-43 (Russian); English transl.: Siberian Math. J. 16 (1975), no. 1, 23-34. MR0370392
  29. Goldberg V.V., Local ternary quasigroups that are connected with a four-web of multidimensional surfaces, Sibirsk. Mat. Zh. 16 (1975), no. 2, 247-263 (Russian); English transl.: Siberian Math. J. 16 (1975), no. 2, 190-202. MR0372101
  30. Goldberg V.V., The ( n + 1 ) -webs defined by n + 1 surfaces of codimension n - 1 , Probl. Geom., Vol. 7 (Russian), Akad. Nauk SSSR Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1975, pp.173-195. Zbl0547.53007MR0478042
  31. Goldberg V.V., Reducible, group, and ( 2 n + 2 ) -hedral ( n + 1 ) -webs of multidimensional surfaces, Sibirsk. Mat. Zh. 17 (1976), no. 1, 44-57 (Russian); English transl.: Siberian Math. J. 17 (1976), no. 1, 34-44. MR0417940
  32. Goldberg V.V., On the theory of four-webs of multidimensional surfaces on a differentiable manifold X 2 r , Izv. Vyssh. Uchebn. Zaved. Mat. 1977, no. 11(186), 15-22 (Russian); English transl.: Soviet Math. (Iz. VUZ) 21 (1977), no. 11, 97-100. 
  33. Goldberg V.V., On the theory of four-webs of multidimensional surfaces on a differentiable manifold X 2 r , Serdica 6 (1980), 2 105-119 (Russian). (1980) MR0601349
  34. Goldberg V.V., Multidimensional four-webs on which the Desargues and triangle figures are closed, Geom. Dedicata 12 (1982), 3 267-285. (1982) Zbl0488.53009MR0661533
  35. Goldberg V.V., Grassmann and algebraic four-webs in a projective space, Tensor (N.S.) 38 (1982), 179-197. (1982) Zbl0513.53009MR0832646
  36. Goldberg V.V., Theory of Multicodimensional ( n + 1 ) -Webs, Kluwer Academic Publishers, Dordrecht, 1988, xxii+466 pp. Zbl0668.53001MR0998774
  37. Goldberg V.V., Local differentiable quasigroups and webs, Chapter X in ``Quasigroups and Loops: Theory and Applications'' O. Chein, H.O. Pflugfelder, and J.D.H. Smith, Eds., Heldermann-Verlag, Berlin, 1990, pp.263-311. Zbl0737.53015MR1125816
  38. Goursat E., Sur les équations du second ordre à n variables, analogues à l’équation de Monge-Ampère, Bull. Soc. Math. France 27 (1899), 1-34. (1899) MR1504329
  39. Hofmann K.H., Strambach K., Lie's fundamental theorems for local analytical loops, Pacific J. Math. 123 (1986), 2 301-327. (1986) Zbl0596.22002MR0840846
  40. Hofmann K.H., Strambach K., The Akivis algebra of a homogeneous loop, Mathematika 33 (1986), 1 87-95. (1986) Zbl0601.22002MR0859501
  41. Hofmann K.H., Strambach K., Topological and analytic loops, Chapter IX in ``Quasigroups and Loops: Theory and Applications'', O. Chein, H.O. Pflugfelder, and J.D.H. Smith, Eds., Heldermann-Verlag, Berlin, 1990, pp.205-262. Zbl0747.22004MR1125815
  42. Kneser H., Gewebe und Gruppen, Abh. Math. Sem. Univ. Hamburg 9 (1932), 147-151. (1932) Zbl0006.03302
  43. Kuz'min E.N., La relation entre les algebras de Malcev et les boucles de Moufang analytiques, C.R. Acad. Sci. Paris Sér. A 271 (1970), 23 1152-1155. (1970) MR0269766
  44. Kuz'min E.N., On relation between Mal'tsev algebras and analytic Moufang loops, Algebra i Logika 10 (1971), 1 3-22 (Russian) English transl.: Algebra and Logic 10 (1971), 1 1-14. (1971) MR0297916
  45. Lie S., Vorlesungen über kontinuierliche Gruppen mit geometrischen und anderen Anwendungen, Teubner, Leipzig, 1893, xii+810 pp.; reprint Chelsea Publishing Co., Bronx, N.Y., 1971, xii+810 pp. MR0392458
  46. Mal'cev A.I., Analytical loops, Mat. Sb. 36 (1955), 569-576 (Russian). (1955) MR0069190
  47. Mikheev P.O., On a problem of Chern-Akivis-Shelekhov on hexagonal three-webs, Aequationes Math. 51 (1996), 1-2 1-11. (1996) Zbl0845.53011MR1372779
  48. Mikheev P.O., Sabinin L.V., Quasigroups and differential geometry, Chapter XII in ``Quasigroups and Loops: Theory and Applications'', O. Chein, H.O. Pflugfelder, and J.D.H. Smith, Eds., Heldermann-Verlag, Berlin, 1990, pp.357-430. Zbl0721.53018MR1125818
  49. Moufang R., Zur Struktur von Alternativ Körpern, Math. Ann. 110 (1935), 416-430. (1935) MR1512948
  50. Nagy P., Strambach K., Loops as invariant sections in groups, and their geometry, Canad. J. Math. 46 (1994), 5 1027-1056. (1994) Zbl0814.20055MR1295130
  51. Pickert G., Projective Ebenen, 2nd edition, Springer-Verlag, Berlin, 1975, ix+371 pp. MR0370350
  52. Rado F., Generalisation of space webs for certain algebraic structures, Studia Univ. Babeş-Bolyai Ser. Math. Phys. 1960 1, pp.41-55 (Roumanian). MR0182913
  53. Reidemeister K., Gewebe und Gruppen, Math. Z. 29 (1928), 427-435. (1928) 
  54. Reidemeister K., Vorlesungen Über Grundlagen der Geometrie, Springer-Verlag, Berlin-New York, 1968, x+148 pp. Zbl0167.18701MR0223957
  55. Sabinin L.V., Quasigroups, geometry and nonlinear geometric algebra, Acta Appl. Math. 50 (1998), 1-2 45-66. (1998) Zbl0903.20038MR1608635
  56. Sabinin L.V., Mikheev P.O., Analytical Bol loops, Webs and Quasigroups (Russian), Kalinin. Gos. Univ., Kalinin, 1982, pp.102-109. Zbl0499.20044MR0674312
  57. Sabinin L.V., Mikheev P.O., The Theory of Smooth Bol Loops, Univ. Druzhby Narodov, Moscow, 1985, 81 pp. MR0831661
  58. Sagle A.A., Mal'tsev algebras, Trans. Amer. Math. Soc. 101 (1961), 3 426-458. (1961) MR0143791
  59. Sandik M.D., Completely reducible n -quasigroups, Izv. Akad. Nauk Moldav. SSR Ser. Fiz.-Tekhn. Mat. Nauk 7 (1965), 55-67 (Russian). (1965) MR0194542
  60. Shelekhov A.M., Classification of multidimensional three-webs according to closure conditions, Probl. Geom. 21, 109-154, Itogi Nauki i Tekhniki, Akad. Nauk SSSR Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1989 (Russian); English transl.: Soviet Math. 55 (1991), no. 6, 2140-2168. Zbl0705.53014MR1027856
  61. Shelekhov A.M., The G -structures associated to a hexagonal 3 -web is closed, J. Geom. 35 (1989), 169-176. (1989) MR1002653
  62. Shelekhov A.M., On the closed G -structure associated to a hexagonal 3 -web, in ``Differential Geometry and Its Applications'', Dubrovnik, 1988, Univ. Novi Sad, Novi Sad, 1989, pp.323-326. Zbl0699.53024MR1040080
  63. Shestakov I.P., Any Akivis algebra is linear, Geom. Dedicata 77 (1999), 2 215-223. (1999) MR1713296
  64. Shestakov I.P., Linear representability of Akivis algebras, Dokl. Akad. Nauk 368 (1999), 1 21-23. (1999) Zbl1034.17500MR1726261
  65. Shestakov I.P., Speciality and deformation of algebras, Proceedings of the Intern. Algebraic Conf. dedicated to the memory of A.G. Kurosh, Moscow, 1998, to appear. MR1754680
  66. Sushkevich A.K., Theory of Generalized Groups, GNTI, Kharkiv-Kiev, 1937, 176 pp. 
  67. Smith J.D.H., Multilinear algebras and Lie’s theorem for formal n -loops, Arch. Math. (Basel) 51 (1988), 169-177. (1988) Zbl0627.22003MR0959394
  68. Thomsen G., Un teoreme topologico sulle schiere di curve e una caratterizzazione geometrica delle superficie isotermo-asintotiche, Boll. Un. Mat. Ital. Bologna 6 (1927), 80-85. (1927) 

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.