
Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki, 2018, Volume 28, Issue 3, Pages 305–327
(Mi vuu641)




MATHEMATICS
Embedding of an additive twodimensional phenomenologically symmetric geometry of two sets of rank $(2,2)$ into twodimensional phenomenologically symmetric geometries of two sets of rank $(3,2)$
V. A. Kyrov^{}, G. G. Mikhailichenko^{} ^{} GornoAltaisk State University, ul. Lenkina, 1, GornoAltaisk, 649000, Russia
Abstract:
In this paper, the method of embedding is used to construct the classification of twodimensional phenomenologically symmetric geometries of two sets (PS GTS) of rank $(3,2)$ from the previously known additive twodimensional 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 twodimensional PS GTS of rank $(3,2)$ admit groups of transformations of dimension 4, and the twodimensional 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 twodimensional 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 twodimensional PS GTS of rank $(3,2)$.
Keywords:
phenomenologically symmetric geometry of two sets, system of differential equations, Lie algebra, Lie transformation group.
DOI:
https://doi.org/10.20537/vm180304
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UDC:
517.912, 514.1
MSC: 39A05, 39B05 Received: 07.06.2018
Citation:
V. A. Kyrov, G. G. Mikhailichenko, “Embedding of an additive twodimensional phenomenologically symmetric geometry of two sets of rank $(2,2)$ into twodimensional phenomenologically symmetric geometries of two sets of rank $(3,2)$”, Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki, 28:3 (2018), 305–327
Citation in format AMSBIB
\Bibitem{KyrMik18}
\by V.~A.~Kyrov, G.~G.~Mikhailichenko
\paper Embedding of an additive twodimensional phenomenologically symmetric geometry of two sets of rank $(2,2)$ into twodimensional phenomenologically symmetric geometries of two sets of rank~$(3,2)$
\jour Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki
\yr 2018
\vol 28
\issue 3
\pages 305327
\mathnet{http://mi.mathnet.ru/vuu641}
\crossref{https://doi.org/10.20537/vm180304}
\elib{http://elibrary.ru/item.asp?id=35645984}
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