Abstract
A survey of the methods of the theory of quasigroups and loops in algebra and geometry is presented in order to attract the attention of mathematicians and physicists to promising applications of this new branch of mathematics in applied sciences.
Similar content being viewed by others
References
Afanasiev, I. L.: Algebraic structures of affine connection, PhD Thesis, Friendship of Nations Univ., Moscow, 1984.
Akivis, M. A.: Differential geometry of webs, in Probl Geom. 15, VINITI, Moscow, 1983, pp. 187-213.
Aleksandrov, A. D., Berestovski, V. N., and Nicolaev, I. G.: Generalized Riemannian spaces, Adv. Math. Sci. 41 (1986), 3-44.
Al-Houjeiri, M. J.: Bol algebras for involutive pairs of rank one, PhD Thesis, Friendship of Nations Univ., Moscow, 1987.
Batalin, I. A.: Quasigroup construction and first class constraints, J. Math. Phys. 22 (1981), 1837-1850.
Fedenko, A. S.: Regular spaces with symmetries, Matem. Zametki 14 (1973), 113-120.
Fedenko, A. S.: Spaces with Symmetries, Belarus Univ. Press, Minsk, 1977.
Karanda, H. M.: On geometry of symmetric loops, PhD Thesis, Friendship of Nations Univ., Moscow, 1972.
Kikkawa, M.: Geometry of homogeneous Lie loops, Hiroshima Math. J. 5 (1975), 141-179.
Kikkawa, M.: A note on subloops of a homogeneous Lie loop and subsystems of its triple Lie algebra, Hiroshima Math. J. 5 (1975), 439-446.
Kowalski, O.: Generalized Symmetric Spaces, Lecture Notes in Math. 805, Springer-Verlag, Berlin, Heidelberg, New York, 1980.
Ledger, A. J.: Espaces de Riemann symetriques generalises, C. R. Acad. Sci. Paris, Ser. A 264 (1967), 947-948.
Lõhmus, J., Paal, E., and Sorgsepp L.: On currents and symmetries associated with Mal'tsev algebras, Preprint F-50, Inst. of Phys., Estonian Acad. Sci., Tartu, 1989
Matveev, O. A.: On manifolds with geodesics, in Webs and Quasigroups, Kalinin Univ., 1986, 44-49.
Nesterov, A. I. and Stepanenko, V. A.: On methods of nonassociative algebra in geometry and physics, Siberian Branch of Acad. Sci. USSR, Krasnojarsk, 1986.
Nono, T.: On geodesic subspaces of group spaces, J. Sci. Hiroshima Univ. Ser. A 21 (1958), 167-176.
Nono, T.: Sur les familles triples locales de transformations locales de Lie, J. Sci. Hiroshima Univ., Ser. A-1 (Math.) 25 (1961), 357-366.
Paal, E.: Analytic Moufang-transformations, Preprint F-46 (1988), Inst. of Phys., Estonian Acad. Sci., Tartu, 1988.
Sabinin, L. V.: On the equivalence of categories of loops and homogeneous spaces, Dokl. Akad. Nauk USSR 205 (1972), 533-536 (in Russian; translation: Soviet Math. Dokl. 13 (1972), 970-974).
Sabinin, L. V.: On geometry of loops, Matem. Zametki 12 (1972), 605-616 (in Russian; translation: Math. Notes 12 (1973), 799-805).
Sabinin, L. V.: On geometry of loops, in The 5th Conf. on Contemp. Problems in Differential Geom. (Samarkand 1972), Abstracts, p. 192.
Sabinin, L. V.: Odules as a new approach to a geometry with connection, Dokl. Acad. Nauk USSR 233 (1977), 800-803 (in Russian; translation: Soviet Math. Dokl. 18 (1977), 515-518).
Sabinin, L. V.: The methods of nonassociative algebra in differential geometry, Suppl. to the Russian translation of S. Kobayashi and K. Nomizu: Foundations of Differential Geometry, vol. 1, Nauka, Moscow, 1981, pp. 293-339 (in Russian).
Sabinin, L. V.: On tangent connections of loopuscular structures, in Webs and Quasigroups, Kalinin Univ. Press, Kalinin, 1986, pp. 86-89 (in Russian).
Sabinin, L. V.: Geometric odules, in Webs and Quasigroups, Kalinin Univ. Press, Kalinin, 1987, pp. 88-98 (in Russian).
Sabinin, L. V.: Differential equations of smooth loops, in Proc. Sem. on Vector and Tensor Analysis, vol. 23, Moscow University, Moscow, 1988, pp. 133-146.
Sabinin, L. V.: Differential geometry and quasigroups, in Contemporary Problems of Geometry and Analysis, Trans. Inst. Math., Siberian Branch of Acad. Sci. USSR, Nauka, Novosibirsk, 1989, pp. 208-221 (in Russian).
Sabinin, L. V.: Smooth hyporeductive loops, in Variational Methods in Contemporary Geometry, Friendship of Nations Univ., Moscow, 1990, pp. 50-69.
Sabinin, L. V.: Analytic Quasigroups and Geometry, Friendship of Nations Univ., Moscow, 1991.
Sabinin, L. V.: Smooth hyporeductive loops, in: Webs and Quasigroups, Tver Univ. Press, Tver, 1991, pp. 129-137 (in Russian).
Sabinin, L. V.: On smooth hyporeductive loops, Soviet Math. Dokl. 42 (1991), 524-526.
Sabinin, L. V. and Mikheev, P. O.: On a symmetric connection in the space of an analytic Moufang loop, Dokl. Acad. Nauk USSR 262 (1982), 807-809 (in Russian; translation: Soviet Math. Dokl. 25 (1982), 136-138).
Sabinin, L. V. and Mikheev, P. O.: On analytic Bol loops, in Webs and Quasigroups, Kalinin Univ. Press, Kalinin, 1982, pp. 102-109 (in Russian).
Sabinin, L. V. and Mikheev, P. O.: On the geometry of smooth Bol loops, in Webs and Quasigroups, Kalinin Univ. Press, Kalinin, 1984, pp. 144-154 (in Russian).
Sabinin, L. V. and Mikheev, P. O.: On differential geometry of Bol loops, Dokl. Acad. Nauk USSR 281 (1985), 1055-1057 (in Russian; translation: Soviet Math. Dokl. 31 (1985), 389-391).
Sabinin, L. V. and Mikheev, P. O.: Lectures on the Theory of Smooth Bol Loops, Friendship of Nations Univ., Moscow, 1985.
Sabinin, L. V. and Mikheev, P. O.: On locally analytic loops with the identity of right alternativity, in Webs and Quasigroups, Kalinin Univ. Press, Kalinin, 1985, pp. 72-75 (in Russian).
Sabinin, L. V. and Mikheev, P. O.: On the infinitesimal theory of local analytical loops, Dokl. Acad. Nauk USSR 297 (1987), 801-804 (in Russian; translation: Soviet Math. Dokl. 36 (1988), 545-548).
Sabinin, L. V. and Mikheev, P. O.: Smooth quasigroups and geometry, in Probl. Geom. 20, VINITI, Moscow, 1988, p. 75-110.
Sabinin, L. V. and Mikheev, P. O.: Quasigroups and differential geometry, Ch. 12 in: Quasigroups and Loops: Theory and Applications, Heldermann-Verlag, Berlin, 1990, pp. 357-430.
Sabinin, L. V. and Sabinina, L. L.: On geometry of transsymmetric spaces, in Webs and Quasigroups, Tver Univ. Press, Tver, 1991, pp. 117-122.
Sabinin, L. V. and Sbitneva, L. V.: Reductive spaces and left F-quasigroups, in Webs and Quasigroups, Tver Univ. Press, Tver, 1991, pp. 123-128.
Sabinin, L. V. and Sharma, B.: On the existence of half-Bol loops, An. Stiint. Univ. 'Al Cusa' (Iasi), Sec. 1 (Math.) 22 (1976), 147-148.
Sbitneva. L. V.: Perfect s-structures, in Differential Geometry of Manifolds of Figures, Kaliningrad Univ., 1979, pp. 97-103.
Sbitneva. L. V.: On Lie algebras of perfect s-spaces, in Webs and Quasigroups, Kalinin Univ. Press, Kalinin, 1982, pp. 128-133.
Sbitneva. L. V.:Perfect s-spaces, PhD Thesis, Friendship of Nations Univ., Moscow, 1984.
Yamaguti, K.: On the Lie triple system and its generalization, J. Sci. Hiroshima Univ. Ser. A 21 (1957/58), 155-160.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Sabinin, L.V. Quasigroups, Geometry and Nonlinear Geometric Algebra. Acta Applicandae Mathematicae 50, 45–66 (1998). https://doi.org/10.1023/A:1005858932289
Issue Date:
DOI: https://doi.org/10.1023/A:1005858932289