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## Archive of Issues

Russia Gorno-Altaisk
Year
2018
Volume
28
Issue
3
Pages
305-327
 Section Mathematics Title Embedding of an additive two-dimensional phenomenologically symmetric geometry of two sets of rank (2,2) into two-dimensional phenomenologically symmetric geometries of two sets of rank (3,2) Author(-s) Kyrov V.A.a, Mikhailichenko G.G.a Affiliations Gorno-Altaisk State Universitya Abstract In this paper, the method of embedding is used to construct the classification of two-dimensional phenomenologically symmetric geometries of two sets (PS GTS) of rank $(3,2)$ from the previously known additive two-dimensional PS GTS of rank $(2,2)$ defined by a pair of functions $g^1=x+\xi$ and $g^2 = y+\eta$. The essence of this method consists in finding the functions defining the PS GTS of rank $(3,2)$ with respect to the functions $g^1=x+\xi$ and $g^2 = y+\eta$. In solving this problem, we use the fact that the two-dimensional PS GTS of rank $(3,2)$ admit groups of transformations of dimension 4, and the two-dimensional PS GTS of rank $(2,2)$ is of dimension 2. It follows that the components of the operators of the Lie algebra of the transformation group of the two-dimensional PS GTS of rank $(3,2)$ are solutions of a system of eight linear differential equations of the first order in two variables. Investigating this system of equations, we arrive at possible expressions for systems of operators. Then, from the systems of operators, we select the operators that form Lie algebras. Then, applying the exponential mapping, we recover the actions of the Lie groups from the Lie algebras found. It is precisely these actions that specify the two-dimensional PS GTS of rank $(3,2)$. Keywords phenomenologically symmetric geometry of two sets, system of differential equations, Lie algebra, Lie transformation group UDC 517.912, 514.1 MSC 39A05, 39B05 DOI 10.20537/vm180304 Received 7 June 2018 Language Russian Citation Kyrov V.A., Mikhailichenko G.G. Embedding of an additive two-dimensional phenomenologically symmetric geometry of two sets of rank (2,2) into two-dimensional phenomenologically symmetric geometries of two sets of rank (3,2), Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2018, vol. 28, issue 3, pp. 305-327. References Mikhailichenko G.G. Matematicheskie osnovy i rezul'taty teorii fizicheskikh struktur (The mathemetical basics and results of the theory of physical structures), Gorno-Altaisk: Gorno-Altaisk State University, 2016, 297 p. Kyrov V.A., Mikhailichenko G.G. An analytic method for the embedding of the Euclidean and pseudo-Euclidean geometries, Trudy Inst. Mat. Mekh. Ural. Otd. Ross. Akad. Nauk, 2017, vol. 23, no. 2, pp. 167-181 (in Russian). DOI: 10.21538/0134-4889-2017-23-2-167-181 Kyrov V.A., Mikhailichenko G.G. An analytic method for the embedding of the symplectic geometry, Siberian Electronic Mathematical Reports, 2017, vol. 14, pp. 657-672 (in Russian). DOI: 10.17377/semi.2017.14.057 Kyrov V.A. Embedding of phenomenologically symmetric geometries of two sets of the rank $(N,2)$ into phenomenologically symmetric geometries of two sets of the rank $(N+1,2)$, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp’yuternye Nauki, 2016, vol. 26, issue 3, pp. 312-323 (in Russian). DOI: 10.20537/vm160302 Kyrov V.A. Embedding of phenomenologically symmetric geometries of two sets of the rank $(N,M)$ into phenomenologically symmetric geometries of two sets of the rank $(N+1,M)$, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp’yuternye Nauki, 2017, vol. 27, issue 1, pp. 42-53 (in Russian). DOI: 10.20537/vm170104 Simonov A.A. Pseudomatrix groups and physical structures, Siberian Mathematical Journal, 2015, vol. 56, no. 1, pp. 177-190. DOI: 10.1134/S0037446615010176 Mikhailichenko G.G. Bimetric physical structures of rank $(n+1,2)$, Siberian Mathematical Journal, 1993, vol. 34, no. 3, pp. 513-522. DOI: 10.1007/BF00971227 Kyrov V.A. Phenomenologically symmetric local Lie groups of transformations of the space $R$$s$ , Russian Mathematics, 2009, vol. 53, no. 7, pp. 7-16. DOI: 10.3103/S1066369X09070020 Bredon G. Introduction of compact transformation groups, New York-London: Academic Press, 1972. xiii+459 p. Kostrikin A.I. Vvedenie v algebru (Introduction to Algebra), Moscow: Nauka, 1977, 496 p. El'sgol'ts L.E. Differentsial'nye uravneniya i variatsionnoe ischislenie (Differential equations and the calculus of variations), Moscow: Nauka, 1969, 424 p. Ovsyannikov L.V. Gruppovoi analiz differentsial'nykh uravnenii (Group analysis of differential equations), Moscow: Nauka, 1978, 400 p. Lie S., Engel F. Theorie der transformationsgruppen. Bd. 3, Leipzig: B.G. Teubner, 1893. 324 p. Full text