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Russia Gorno-Altaisk
Year
2022
Volume
32
Issue
1
Pages
62-80
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Section Mathematics
Title On local extension of the group of parallel translations in three-dimensional space
Author(-s) Kyrov V.A.a
Affiliations Gorno-Altaisk State Universitya
Abstract In this paper, we solve the problem of extending the group of parallel translations of a three-dimensional space to a locally boundedly sharply doubly transitive Lie group of transformations of the same space. Local bounded sharply double transitivity means that there is a single transformation that takes an arbitrary pair of non-coincident points from some open neighborhood to almost any pair of points from the same neighborhood. In this article, the problem posed is solved for two cases related to Jordan forms of third-order matrices. These matrices are used to write systems of linear differential equations, whose solutions lead to the basic operators of a six-dimensional linear space. Requiring the closedness of the commutators of these operators, we select the Lie algebras. Checking also the condition of local bounded sharply double transitivity, we obtain the Lie algebras of locally boundedly sharply doubly transitive Lie groups of transformations of a three-dimensional space with a subgroup of parallel translations. As a result, three Lie algebras are obtained, two of which can be represented as a half-line sum of a commutative three-dimensional ideal and a three-dimensional Lie subalgebra, and the third one decomposes into a half-line sum of a commutative three-dimensional ideal and a subalgebra isomorphic to $sl(2,R)$.
Keywords Lie group of transformations, locally boundedly sharply doubly transitive Lie group of transformations, Lie algebra, Jordan form of a matrix
UDC 512.816.3
MSC 22E99
DOI 10.35634/vm220105
Received 19 September 2021
Language Russian
Citation Kyrov V.A. On local extension of the group of parallel translations in three-dimensional space, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2022, vol. 32, issue 1, pp. 62-80.
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