On the Local Extension of the Group of Parallel Translations in Three-Dimensional Space. II

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Home Editorial board Publication ethics Peer review guidelines Latest issue All issues Rules for authors Online submission system's guidelines Submit manuscript Contacts Address: Vatutina st. 53, Vladikavkaz, 362025, RNO-A, Russia Phone: (8672)23-00-54 E-mail: rio@smath.ru	DOI: 10.46698/a9077-8757-4946-m On the Local Extension of the Group of Parallel Translations in Three-Dimensional Space. II Kyrov, V. A. Vladikavkaz Mathematical Journal 2024. Vol. 26. Issue 2. Abstract: This article solves the problem of local extension of the group of parallel translations of a three-dimensional space to a locally bounded exactly doubly transitive group of Lie transformations of the same space. Locally bounded exactly twice transitivity means the existence of a unique transformation that takes an arbitrary pair of non-coinciding points from some open neighborhood to almost any pair of points from the same neighborhood. The problem posed is solved for four cases related to Jordan forms of third-order matrices. Using these Jordan matrices, systems of linear differential equations are written, the solutions of which lead to the basis operators of a six-dimensional linear space. Requiring that the commutators of the basis operators be closed, we find Lie algebras. By checking the condition of bounded exactly twice transitivity, we obtain the Lie algebras of the required Lie transformation groups. At the end of the paper it is proved that these Lie algebras are either solvable or representable as a direct sum of a solvable ideal and a subalgebra isomorphic to \(sl(2,R)\). In this case, solvable Lie algebras are decomposed into the direct sum of a nilpotent ideal and a solvable subalgebra. Finally, the isomorphism of some above found Lie algebras is established. Keywords: Lie group of transformations, locally bounded exactly doubly transitive Lie group of transformations, Lie algebra, Jordan form of a matrix Language: Russian Download the full text For citation: Kyrov, V. A. On the Local Extension of the Group of Parallel Translations in Three-Dimensional Space. II, Vladikavkaz Math. J., 2024, vol. 26, no. 2, pp. 54-69 (in Russian). DOI 10.46698/a9077-8757-4946-m + References 1. Kyrov, V. A. On the Local Extension of the Group of Parallel Translations in Three-Dimensional Space, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Kompyuternye Nauki, 2022, vol. 32, no. 1, pp. 62-80 (in Russian). DOI: 10.35634/vm220105. 2. Kyrov, V. A. On the Question of Local Extension of the Group of Parallel Translations of Three-Dimensional Space, Vladikavkaz Mathematical Journal, 2021, vol. 23, no. 1, pp. 32-42 (in Russian). DOI: 10.46698/q6524-1245-2359-m. 3. Gorbatsevich, V. V. Extension of Transitive Actions of Lie Groups, Izvestiya: Mathematics, 2017, vol. 81, no. 6, pp. 38-51. DOI: 10.1070/im8506. 4. Mikhailichenko, G. G. Group Symmetry of Physical Structures, Barnaul, BGPU, 2003, 203 p. (in Russian). 5. Kyrov, V. A. and Mikhailichenko, G. G. Embedding of the Additive Two-Metric Phenomenologically Symmetric Geometry of Two Sets of Rank (2,2) into the Two-Metric Phenomenologically Symmetric Geometries of Two Sets of Rank (3,2), Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Kompyuternye Nauki, 2018, vol. 28, no. 1, pp. 305-327 (in Russian). DOI: 10.20537/vm180304. 6. Ovsyannikov, L. V. Group Analysis of Differential Equation, Moscow: Nauka, 1978, 400 p. (in Russian). 7. Morozov, V. V. Classification of Nilpotent Lie Algebras of the Sixth Order, Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, 1958, no. 4, pp. 161-171 (in Russian). 8. Mubarakzyanov, G. M. On Solvable Lie Algebras, Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, 1963, no. 1, pp. 114-123 (in Russian). 9. Mubarakzyanov, G. M. Classification of Sixth-Order Solvable Lie Algebras with one Non-Nilpotent Basis Element, Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, 1963, no. 4, pp. 104-116 (in Russian). 10. Turkowski, P. Solvable Lie Algebras of Dimension Six, Journal of Mathematical Physics, 1990, vol. 31, no. 6, pp. 1344-1350 (in Russian). DOI: 10.1063/1.528721. 11. Turkowski P. Lowdimensional Real Lie Algebras, Journal of Mathematical Physics, 1988, vol. 29, no. 10, pp. 2139-2144 (in Russian). DOI: 10.1063/1.528140. The sole copyright holder of the published work is the Founder of the Vladikavkaz Mathematical Journal. The terms of use of this work are governed by an open license (Creative Commons Attribution-NonCommercial 4.0 International). The use of metadata of the scientific article, including the title, abstract, author information, references, identifiers, and other bibliographic description elements for subsequent unrestricted use, is carried out under the terms of the CC BY or CC0 open licenses. ← Contents of issue


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