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Natural Sciences Issue (English)
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03/05/2026
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Nguyen, T. M. T., Pham, Q. T., Pham, P. H., & Thieu, P. M. T. (2026). CLASSIFICATION OF 8-DIMENSIONAL SOLVABLE LIE ALGEBRAS HAVING 7-DIMENSIONAL HEISENBERG NILRADICAL. Dong Thap University Journal of Science, Online First. https://doi.org/10.52714/dthu.ns.2406.1895
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CLASSIFICATION OF 8-DIMENSIONAL SOLVABLE LIE ALGEBRAS HAVING 7-DIMENSIONAL HEISENBERG NILRADICAL
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Abstract
In this paper, we classify all indecomposable 8-dimensional solvable Lie algebras whose nilradical is the 7-dimensional Heisenberg algebra. Our method is based on the observation that any solvable Lie algebra can be viewed as an extension of its nilradical , namely, the maximal nilpotent ideal of . Using standard techniques from Lie theory, we begin with the 7-dimensional Heisenberg algebra and classify all 8-dimensional algebras that admit it as their nilradical.
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References
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https://doi.org/10.1080/00927872.2022.2145300
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Parry, A. R. (2007). A classification of real indecomposable solvable Lie algebras of small dimension with codimension one nilradicals [Master thesis]. Logan, UT: Utah State University. https://digitalcommons.usu.edu/etd/7145
Rubin, J. L., & Winternitz, P. (1993), Solvable Lie algebras with Heisenberg ideals. Journal of Physics A: Mathematical and General, 26, 1123-1138.
https://iopscience.iop.org/article/10.1088/0305-4470/26/5/031
Šnobl, L., & Winternitz, P. (2014). Classification and Identification of Lie Algebras, Vol. 33 of CRM Monograph Series. Providence, RI: American Mathematical Society.
Turkowski, P. (1990). Solvable Lie algebras of dimension six. Journal of Mathematical Physics, 31(6), 1344–1350. https://doi.org/10.1063/1.528721
Cartan, E. (1894). Sur la structure des groupes de transformations finis et continus. Faculty of Science, University of Paris, Academy of Paris.
Gantmacher, F. R. (1939). On the classification of real simple Lie groups. Sbornik Mathematics, 5, 217–250.
Gong, M. P. (1998). Classification of nilpotent Lie algebras of dimension 7 (Over algebraically closed fields and R) [PhD dissertation]. Waterloo, Ontario, CA: University of Waterloo. http://hdl.handle.net/10012/1148.
Hindeleh, F., Thompson, G. (2008). Seven-dimensional Lie algebras with a four-dimensional nilradical. Algebras Groups and Geometries. 25(3):243–265.
Kruchkovich, G. I. (1954). Classification of three-dimensional Riemannian spaces according to groups of motions. Uspekhi Matematicheskikh Nauk, 9(1), 3–40.
Le, A. V., Nguyen, A. T., Nguyen, T. C. T., Nguyen, T. M. T., & Vo, N. T. (2023). Classification of 7-dimensional solvable Lie algebras having 5-dimensional nilradicals. Communications in Algebra, 51(5), 1885–1899.
https://doi.org/10.1080/00927872.2022.2145300
Levi, E. E. (1905). Sulla struttura dei gruppi finiti e continui. Atti della Accademia delle Scienze di Torino. Classe di Scienze Fisiche, Matematiche e Naturali, 40, 551–565.
Lie, M. S., & Engel, F. (1893). Theorie der Transformationsgruppen III. Leipzig: B. G. Teubner.
Malcev, A. I. (1945). On solvable Lie algebras. Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya, 9(5), 329–356.
Mubarakzyanov, G. M. (1963). Classification of real structures of Lie algebras of fifth order. Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, 3, 99–106.
Mubarakzyanov, G. M. (1963). Classification of solvable Lie algebras of sixth order with a non-nilpotent basis element. Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, 4, 104–116.
Mubarakzyanov, G. M. (1963). On solvable Lie algebras. Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, 1, 114–123.
Parry, A. R. (2007). A classification of real indecomposable solvable Lie algebras of small dimension with codimension one nilradicals [Master thesis]. Logan, UT: Utah State University. https://digitalcommons.usu.edu/etd/7145
Rubin, J. L., & Winternitz, P. (1993), Solvable Lie algebras with Heisenberg ideals. Journal of Physics A: Mathematical and General, 26, 1123-1138.
https://iopscience.iop.org/article/10.1088/0305-4470/26/5/031
Šnobl, L., & Winternitz, P. (2014). Classification and Identification of Lie Algebras, Vol. 33 of CRM Monograph Series. Providence, RI: American Mathematical Society.
Turkowski, P. (1990). Solvable Lie algebras of dimension six. Journal of Mathematical Physics, 31(6), 1344–1350. https://doi.org/10.1063/1.528721
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