Articlr 3316

Title of the article

THE CORE OF THE GROUP THREE-WEB

Authors

Mikheeva Anna Aleksandrovna, Postgraduate student, Tver State University, (33 Zhelyabova street, Tver Russia), heathjensen@yandex.ru

Index UDK

514.763.7

DOI

10.21685/2072-3040-2016-3-3

Abstract

Background. Wilhelm Blaschke began to study differential geometry of multidimensional three-webs in 1920s; S. Chern, M. A. Akivis and others studied the theory of multidimensional three-webs later. A special place in this theory is occupied by the Bol three-webs, on which the structure of symmetric space arise naturally. The structure is determined on the basis of one of the foliations of the Bol three-web using a binary operation, which is called the core of this three-web. In particular, this operation arises on group three-web W(G) , which is generated by Lie group G . The core of the Bol three-web has been investigated in several studies, but the number of important questions have not been studied, in particular, the canonical decomposition for the core has not been found, the Bol three-webs with the same core have not been described. The aims of this work are to find the canonical decomposition for the core, to describe the properties of the core of group three-webs, to find the conditions characterizing group three-webs with isomorphic cores.
Materials and methods. Methods of classical differential geometry, tensor calculus, the modified method of exterior differential forms and the moving frame of Elie Cartan were applied to study the core of the Bol three webs; the theory of Lie groups and the results of the previous works were also used in the study.
Results. The canonical form of the expansion in the Taylor series for the core of the left Bol three-web has been found, the corresponding commutator and associator were calculated. It is shown how the core of group three web CW(G) is expressed through the group operation in G ; by means of the Campbell-Hausdorff's formula of group G we have found a series expansion for the core. It is proved that 1) core CW(G) is equivalent to the core of original group three-web W(G) ; 2) the symmetric connection determined by the core on the basis of the first foliation of the web coincides with the Elie Cartan's third connection on the Lie group; 3) the right translations in the group are automorphisms of its core. The conditions, under which two group three-webs have the general core, have been received . It is proved that group G is a nilpotent of height 1 if and only if the three-web, determined by core, is parallelizable. Cores of group three-webs generated by the group of affine transformations on a straight line and the Heisenberg group have been considered.
Conclusions. It turned out that the torsion tensor of the core is equal to zero in a unit, but the curvature tensor, generally speaking, isn't equal to zero in a unit. The conditions, under which two group webs have the general core, have been found. It allows to carry out a more amply study of the question of properties of group threewebs with the general core in future works.

Key words

three-web, group three-web, Bol three-web, core of the Bol three-web.

Download PDF
References

1. Belousov V. D. Issledovaniya po obshchey algebra [Research in general algebra]. Kishinev, 1965, pp. 53–65.
2. Belousov V. D. Osnovy teorii kvazigrupp i lup [Basic theories of quasigroups and loops]. Moscow: Nauka, 1967, 223 p.
3. Tolstikhina G. A., Shelekhov A. M. Izvestiya vuzov. Matematika [University proceedings. Mathematics]. 2014, no. 12, pp. 83–88.
4. Shelekhov A. M., Tolstikhina G. A. Izvestiya vuzov. Matematika [University proceedings. Mathematics]. 2015, no. 8, pp. 75–84.
5. Gegamyan G. D. Levye tri-tkani Bola s trivial'noy serdtsevinoy: dis. kand. fiz.-mat. nauk: 01.01.04 [Left Bol three-webs with trivial cores: dissertation to apply for the degree of the candidate of physical and mathematical sciences]. Gos. inzhener. un-t Armenii (Politekhnik). Erevan, 2013, 180 p.
6. Akivis M. A., Shelekhov A. M. Mnogomernye tri-tkani i ikh prilozheniya: monogr. [Multidimensional three-webs and applications thereof: onograph]. Tver: Izd-vo TvGU, 2010, 308 p.
7. Tolstikhina G. A. Vestnik Tverskogo gosudarstvennogo universiteta. Ser. Prikladnaya matematika [Bulletin of Tver State University. Series: Applied mathematics]. 2011, iss. 2 (21), pp. 117–134.
8. Tolstikhina G. A. Geometriya, topologiya ta ikh zastosuvannya: zb. prats' In-tu matematiki NAN Ukraini [Geometry, topology and applications thereof: proceedings of the Institute of Mathematics of NAS of Ukraine]. 2009, vol. 6, no. 2, pp. 247–255.
9. Loos O. Simmetricheskie prostranstva [Symmetric spaces]. Moscow: Nauka, 1985, 207 p.
10. Sabinin L. V., Mikheev P. O. Teoriya gladkikh lup Bola [Lectures on the theory of smooth Bol loops]. Moscow: Izd-vo RUDN, 1985, 80 p.
11. Kartan E. Geometriya grupp Li i simmetricheskie prostranstva [Geometry of Lie groups and symmetric spaces]. Moscow: IIL, 1949, 119 p.
12. Dynkin E. B. Doklady Akademii nauk SSSR [Proceedings of the USSR Academy of Sciences]. 1947, vol. 51, no. 4, pp. 323–326.
13. Mikheeva A. A., Tolstikhina G. A. Izvestiya vuzov. Matematika [University proceedings. Mathematics]. 2014, no. 10, pp. 1–12.

 

Дата создания: 19.12.2016 11:15
Дата обновления: 19.12.2016 15:53