Bulletin of the American Mathematical Society

Author(s): Maks A. Akivis; Vladislav V. Goldberg
Journal: Bull. Amer. Math. Soc. 43 (2006), 207-226.
MSC (2000): Primary 20N05; Secondary 53A60, 17D99
Posted: February 15, 2006
Errata: Bull. Amer. Math. Soc. 43 (2006), 397
Retrieve article in: PDF DVI PostScript

Abstract: The authors provide a summary of results in the theory of differential quasigroups and their local algebras and indicate the relationship of these results to recent work on this subject.

1.: M. A. Akivis, The canonical expansions of the equations of a local analytic quasigroup (Russian), Dokl. Akad. Nauk SSSR 188 (1969), no. 5, 967-970. English translation: Soviet Math. Dokl. 10 (1969), no. 5, 1200-1203. MR 0262413 (41:7021); Zbl 205:02601.
2.: M. A. Akivis, The local differentiable quasigroups and three-webs that are determined by a triple of hypersurfaces (Russian), Sibirsk. Mat. Zh. 14 (1973), no. 3, 467-474. English translation: Siberian Math. J. 14 (1973), no. 3, 319-324. MR 0324559 (48:2911); Zbl 267:53005 & 281:53002.
3.: M. A. Akivis, Closed -structures on a differentiable manifold, Problems in Geometry, 7, 69-79 (Russian), Akad. Nauk SSSR Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1975. MR 0478055 (57:17549); Zbl 549:53032.
4.: M. A. Akivis, The local algebras of a multidimensional three-web (Russian), Sibirsk. Mat. Zh. 17 (1976), no. 1, 5-11. English translation: Siberian Math. J. 17 (1976), no. 1, 3-8. MR 0405261 (53:9055); Zbl 337:53018.
5.: M. A. Akivis, Geodesic loops and local triple systems in a space with an affine connection, Sibirsk. Mat. Zh. 19 (1978), no. 2, 243-253 (Russian). English translation: Siberian Math. J. 19 (1978), no. 2, 171-178. MR 0487840 (58:7438); Zbl 388:53007 & 409:53008.
6.: M. A. Akivis, Bol three-webs and non-Euclidean geometries, Webs and Quasigroups, 16-26, Tver. Gos. Univ., Tver, 1994. MR 1413333 (97f:53024); Zbl 896:53015.
7.: M. A. Akivis and V. V. Goldberg, Projective Differential Geometry of Submanifolds, North-Holland, Amsterdam, 1993, xii+362 pp. MR 1234487 (94i:53001); Zbl 865:53013.
8.: M. A. Akivis and V. V. Goldberg, Differential geometry of webs, Chapter 1 in Handbook of Differential Geometry, pp. 1-152, Elsevier Science B. V., 2000. MR 1736852 (2001f:53036); Zbl 968:53001.
9.: M. A. Akivis and V. V. Goldberg, Algebraic aspects of web geometry, Comment. Math. Univ. Carolin. 41 (2000), no. 2, 205-236. MR 1780866 (2001i:63026); Zbl 1042:53007.
10.: M. A. Akivis and A. M. Shelekhov, Foundations of the Theory of Webs (Russian), Kalinin. Gos. Univ., Kalinin, 1981, 88 pp. MR 0639895 (83h:53001); Zbl 475:53016.
11.: M. A. Akivis and A. M. Shelekhov, Fourth-order alternators of a local analytic loop and three-webs of multidimensional surfaces (Russian), Izv. Vyssh. Uchebn. Zaved. Mat. 1989, no. 4 (334), 12-16. English translation: Soviet Math. (Iz. VUZ) 33 (1989), no. 4, 13-18. MR 1008142 (90f:53032); Zbl 701:53027.
12.: M. A. Akivis and A. M. Shelekhov, Geometry and Algebra of Multidimensional Three-Webs, translated from Russian by V. V. Goldberg, Kluwer Academic Publishers, Dordrecht, 1992, xvii+358 pp. MR 1196908 (93k:53021); Zbl 771:53001.
13.: V. D. Belousov, Foundations of the Theory of Quasigroups and Loops, Izdat. ``Nauka'', Moscow, 1967, 223 pp. (Russian). MR 0218483 (36:1569); Zbl 163:01801.
14.: W. Blaschke Thomsens Sechseckgewebe. Zueinander diagonale Netze, Math. Z. 28 (1927), 150-157. JFM 54:074404.
15.: W. Blaschke and G. Bol, Geometrie der Gewebe. Springer-Verlag, Berlin, 1938, viii+339 pp. Zbl 20:06701.
16.: G. Bol, Über Drei-Gewebe im vierdimensionalen Raum. Math. Ann. 110 (1935), 431-463. Zbl 10:22203.
17.: R. H. Bruck, A survey of binary systems, Springer-Verlag, Berlin, 1971, 3rd printing, viii + 185 pp. MR0093552 (20:76); Zbl 206:30301.
18.: S. S. Chern, Eine Invariantentheorie der Dreigewebe aus -dimensionalen Mannigfaltigkeiten in ${\bf R}_{2r}$ , Abh. Math. Sem. Univ. Hamburg 11 (1936), no. 1-2, 333-358. Zbl 13:41802.
19.: A. Einstein, Zur Elektrodynamik bewegter Körper, Ann. Phys. (4) 17 (1905), 891-921. JFM 36:092002.
20.: L. Euler, Recherches sur une espéce de carrés magiques, in Commentationes Arithmeticae Collectae, vol. II, St. Petersburg Acad. Sci., St. Petersburg, 1849, pp. 302-361; see also reprint made by Kraus Reprint, Nendeln/Liechtenstein, 1969, Vol. I, 1xxxix+584 pp.; Vol. II: ix+651 pp. MR 0250831 (40:4063).
21.: L. Euler, Recherches sur une nouvelle espèce de quarrés magiques, Verh. Zeeuwsch. Genootsch. Wetensch. Vlissengen Vol. 9 (1782), pp. 85-239; see also Euler, L., Opera Omnia., Series I: Opera mathematica. Vol. VII: algebraicae ad theoriam combinationum et probabilitatum pertinentes (Latin), Teubner, Leipzig and Berlin, 1923, pp. 291-392. JFM 49:0007.
22.: V. I. Fedorova, A condition that defines multidimensional Bol three-webs, Sibirsk. Mat. Zh. 19 (1978), no. 4, 922-928 (Russian). English translation: Siberian Math. J. 19 (1978), no. 4, 657-661. MR 514146 (58:24036); Zbl 398:53007 & 409:53007.
23.: Ya. Friedman, Physical Applications of Homogenous Balls. With the assistance of T. Scarr, Progress in Mathematical Physics 40, Birkhäuser Boston, Inc., Boston, MA, 2005, xxiv+279 pp. MR 2104434 (2005m:32044).
24.: V. V. Goldberg, Local differentiable quasigroups and webs, Chapter X in the book Quasigroups and Loops: Theory and Applications, 263-311, Heldermann-Verlag, Berlin, 1990. MR 1125816 (93g:20133); Zbl 737:53015.
25.: K. H. Hofmann and K. Strambach, The Akivis algebra of a homogeneous loop, Mathematika 33 (1986), no. 1, 87-95. MR 0859501 (88d:17003); Zbl 601:22002.
26.: K. H. Hofmann and K. Strambach, Topological and analytic loops, Chapter IX in the book Quasigroups and Loops: Theory and Applications, 205-262, Heldermann-Verlag, Berlin, 1990. MR 0859501 (93g:20133); Zbl 737:53015.
27.: M. Kikkawa, On local loops in affine manifolds, J. Sci. Hiroshima Univ. Ser A-I Math. 28 (1964), 199-207. MR 0187173 (32:4627); Zbl 141:19603.
28.: E. N. Kuz'min, On relation between Mal'cev algebras and analytic Moufang loops, Algebra i Logika 10 (1971), no. 1, 3-22 (Russian). English translation: Algebra and Logic 10 (1971), no. 1, 1-14. MR 0297916 (45:6968); Zbl 244:17019 & 248:17001.
29.: O. Loos, Symmetric Spaces, vol. 1 (viii+198 pp.), vol. 2 (viii+193 pp.), Benjamin, New York, 1969. MR 0239005 (39:365a,b); Zbl 175:48601.
30.: J. Lòhmus, E. Paal and L. Sorgsepp, Nonassociative Algebras in Physics, Hadronic Press, Palm Harbor, FL, xiv+271 pp. MR 1320709 (97d:81095); Zbl 840:17001.
31.: A. I. Mal'cev, Analytical loops, Mat. Sb. 36 (1955), 569-576 (Russian). MR 0069190 (16:997g); Zbl 65:00702.
32.: P. O. Mikheev, On a problem of Chern-Akivis-Shelekhov on hexagonal tree-webs, Aequationes Math. 51 (1996), no. 1-2, 1-11. MR 1372779 (97a:53019); Zbl 845:53011.
33.: P. O. Mikheev and L. V. Sabinin, Quasigroups and differential geometry, Chapter XII in the book Quasigroups and Loops: Theory and Applications, 357-430, Heldermann-Verlag, Berlin, 1990. MR 1125816 (93g:20133); Zbl 737:53015.
34.: R. Moufang, Zur Struktur von Alternativkörpern, Math. Ann. 110 (1935), 416-430. Zbl 10:00403.
35.: G. P. Nagy, The Campbell-Hausdorff series of local analytic Bruck loops, Abh. Math. Semin. Univ. Hamb. 72 (2002), 79-87. MR 1941548 (2003m:20097); Zbl 1015:20050.
36.: P. T. Nagy and K. Strambach, Loops in Group Theory and Lie Theory, de Gruyter Expositions in Mathematics, 35, Walter de Gruyter & Co., Berlin, 2002, xii+361 pp. MR 1899331 (2003d:20110); Zbl 1050:22001.
37.: H. O. Pflugfelder, Quasigroups and Loops: Introduction, Heldermann-Verlag, Berlin, 1990, iii+147 pp. MR 1125767 (93g:20132); Zbl 715:20043.
38.: M. Reiss, Mémoire sur les propriétés générales des courbes algébriques, Correspondance Mathématique et Physique de Quêtelet 9 (1837), 249-308.
39.: L. V. Sabinin, The geometry of loops, Mat. Zametki 12 (1972), no. 5, 605-616 (Russian). English translation: Math. Notes 12 (1972), no. 5, 799-805. MR 0340461 (49:5216); Zbl 258:20006.
40.: L. V. Sabinin, Homogeneous spaces and quasigroups, Izv. Vyssh. Uchebn. Zaved. Mat. 1996, no. 7 (410), 77-84 (Russian). English translation: Soviet Math. (Iz. VUZ) 40 (1996), no. 7, 74-81. MR 1430701 (97k:53053); Zbl 874:53037.
41.: L. V. Sabinin, Smooth Quasigroups and Loops, Kluwer Academic Publishers, Dordrecht, 1999, xvi+249 pp. MR 1727714 (2001e:20069); Zbl 939:20064.
42.: L. V. Sabinin and P. O. Mikheev, Analytic Bol loops, Webs and Quasigroups, 102-109 (Russian), Kalinin. Gos. Univ., Kalinin, 1982. MR 0674312 (84c:22007); Zbl 499:20044.
43.: L. V. Sabinin and P. O. Mikheev, A symmetric connection in the space of an analytic Moufang loop, Dokl. Akad. Nauk SSSR 262 (1982), no. 4, 807-809 (Russian). English translation: Soviet Math. Dokl. 25 (1982), no. 1, 136-138. MR 0644875 (84e:53027); Zbl 495:53048.
44.: L. V. Sabinin and P. O. Mikheev, On the geometry of smooth Bol loops, Webs and Quasigroups, 144-154 (Russian), Kalinin. Gos. Univ., Kalinin, 1984. MR 0896092; Zbl 568:53009.
45.: L. V. Sabinin, and P. O. Mikheev, The differential geometry of Bol loops, Dokl. Akad. Nauk SSSR 281 (1985), 1055-1057 (Russian). English translation: J. Sov. Math. Dokl. 31 (1985), no. 2, 389-391. MR 0787287 (86k:53022); Zbl 587:53021.
46.: L. V. Sabinin, and P. O. Mikheev, The Theory of Smooth Bol Loops, Dual Russian/English text, Univ. Druzhby Narodov, Moscow, 1985, 81 pp. MR 0831661 (87j:22030); Zbl 584:53001.
47.: L. V. Sabinin, and P. O. Mikheev, Infinitesimal theory of local analytic loops, Dokl. Akad. Nauk SSSR 297 (1987), no. 4, 801-804 (Russian). English translation: Soviet Math. Dokl. 36 (1988), no. 3, 545-548. MR 0924255 (89g:22003); Zbl 659:53018.
48.: A. A. Sagle, Mal'cev algebras, Trans. Amer. Math. Soc. 101 (1961), no. 3, 426-458. MR 0143791 (26:1343); Zbl 101:02302.
49.: A. M. Shelekhov, Classification of multidimensional three-webs according to closure conditions (Russian), Probl. Geom. 21, 109-154, Itogi Nauki i Tekhniki, Akad. Nauk SSSR Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1989. English translation: Soviet Math. 55 (1991), no. 6, 2140-2168. MR 1027856 (91b:53018); Zbl 705:53014.
50.: I. P. Shestakov, Linear representability of Akivis algebras (Russian), Dokl. Akad. Nauk 368 (1999), no. 1, 21-23. English translation: Dokl. Math. 60 (1999), No.2, 164-166. MR 1726261 (2001e:17003); Zbl 1034:17500.
51.: I. P. Shestakov, Every Akivis algebra is linear, Geom. Dedicata 77 (1999), no. 2, 215-223. MR 1713296 (2000f:17004); Zbl 1043:17002.
52.: I. P. Shestakov, Speciality and deformations of algebras, Bahturin, Yu. (ed.), Algebra. Proceedings of the international algebraic conference on the occasion of the 90th birthday of A. G. Kurosh, Moscow, Russia, May 25-30, 1998, 345-356, de Gruyter, Berlin, 2000. MR 1754680 (2001b:17005); Zbl 1027:17001.
53.: I. P. Shestakov and U. U. Umirbaev, Free Akivis algebras, primitive elements, and hyperalgebras, IX Algebra Meeting USP/UNICAMP/UNESP (Portuguese) (São Pedro, 2001), Resenhas 5 (2001), no. 2, 149-158. MR 1945299 (2003m:17002); Zbl 1034:17002.
54.: I. P. Shestakov and U. U. Umirbaev, Free Akivis algebras, primitive elements and hyperalgebras, J. Algebra 250 (2002), no. 2, 533-548. MR 1899864 (2003c:17005); Zbl 993:17002.
55.: A. K. Suschkewitsch, On a generalization of the associative law, Trans. Amer. Math. Soc. 31 (1929), 204-214. JFM 55:008501.
56.: A. A. Ungar, Beyond the Einstein Addition Law and its Gyroscopic Thomas Precession. The Theory of Gyrogroups and Gyrovector Spaces, Fundamental Theories of Physics, 117, Kluwer Academic Publishers Group, Dordrecht, 2001, xlii+413 pp. MR 1978122 (2004c:83006); Zbl 972:83002.
57.: K. Yamaguti, On algebras of totally geodesic space Lie triple systems, J. Sci. Hiroshima Univ. Ser A 21 (1957), no. 2, 107-113. MR 0100046 (20:6482); Zbl 84:18405.
58.: K. Yamaguti, On the Lie triple system and its generalization, J. Sci. Hiroshima Univ. Ser A 21 (1958), 155-160. MR 0100047 (20:6483); Zbl 84:18404.

Retrieve articles in Bulletin of the American Mathematical Society with MSC (2000): 20N05, 53A60, 17D99

Maks A. Akivis
Affiliation: Department of Mathematics, Jerusalem College of Technology--Mahon Lev, Havaad Haleumi St., P. O. Box 16031, Jerusalem 91160, Israel
Email: akivis@mail.jct.ac.il

Vladislav V. Goldberg
Affiliation: Department of Mathematical Sciences, New Jersey Institute of Technology, University Heights, Newark, New Jersey 07102
Email: vlgold@oak.njit.edu

DOI: 10.1090/S0273-0979-06-01094-9
PII: S 0273-0979(06)01094-9
Keywords: Quasigroup, web, local algebra, Akivis algebra
Received by editor(s): January 2005
Received by editor(s) in revised form: August 10, 2005
Posted: February 15, 2006
Additional Notes: A preliminary version of this article was presented at the Mile High Conference on Quasigroups, Loops and Nonassociative Systems (University of Denver, Denver, Colorado, July 2005).
Copyright of article: Copyright 2006, American Mathematical Society

			Journals Home Search Author Info Subscribe Tech Support Help
ISSN 1088-9485(e) ISSN 0273-0979(p)

Previous issue \| Table of contents \| Next issue Articles in press \| Recently posted articles \| Most recent issue \| All issues: 1891 - Present Previous Article \| Next Article


	Comments: webmaster@ams.org © Copyright 2008, American Mathematical Society Privacy Statement		Search the AMS