Classification of types A and A_+ from low dimensional standard and non-standard filiform Lie Algebras
DOI:
https://doi.org/10.21580/jnsmr.2023.9.2.18083Keywords:
Types A and A_ , Filiform Lie algebras, Heisenberg Lie algebras.Abstract
In this paper, we study low-dimensional Filiform Lie algebras. Specifically, three-dimensional standard Filiform Lie algebras and five-dimensional non-standard Filiform Lie algebras. The classification method was given in the following stage. For given a low-dimensional Filiform Lie algebra, we compute its second centre. We showed that three-dimensional Filiform Lie algebra-called Heisenberg Lie algebra-is type ???? and ????+ as well. On the other hand, for ????≥3, the standard Filiform Lie algebras are type ???? but not type ????+. In this case, we give a concrete example of case five-dimensional Heisenberg Lie algebra. Moreover, we proved that five-dimensional non-standard Filiform Lie algebra is type ???? but not type ????+. It is still an open problem to classify types ???? and ????+ for the general case of non-standard Filiform Lie algebra of dimension ≥6.
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Adimi, H., & Makhlouf, A. (2013). Index of graded filiform and quasi filiform Lie algebras. Filomat, 27(3), 467–483. https://doi.org/10.2298/FIL1303467A
Alvarez, M. A., Rodríguez-Vallarte, M. C., & Salgado, G. (2018a). Contact and Frobenius solvable Lie algebras with abelian nilradical. Communications in Algebra, 46(10), 4344–4354. https://doi.org/10.1080/00927872.2018.1439048
Alvarez, M. A., Rodríguez-Vallarte, M. C., & Salgado, G. (2018b). Contact and Frobenius solvable Lie algebras with abelian nilradical. Communications in Algebra, 46(10), 4344–4354.
https://doi.org/10.1080/00927872.2018.1439048
Alvarez, M. A., Rodríguez-Vallarte, M., & Salgado, G. (2016). Contact nilpotent Lie algebras. Proceedings of the American Mathematical Society, 145(4), 1467–1474. https://doi.org/10.1090/proc/13341
Ayala, V., Kizil, E., & de Azevedo Tribuzy, I. (2012). On an algorithm for finding derivations of Lie algebras. Proyecciones (Antofagasta), 31(1), 81–90. https://doi.org/10.4067/S0716-09172012000100008
Beltiţă, I., & Beltiţă, D. (2015). On Kirillov’s lemma for nilpotent Lie algebras. Journal of Algebra, 427, 85–103.. https://doi.org/10.1016/j.jalgebra.2014.12.026
Csikós, B., & Verhóczki, L. (2007). Classification of Frobenius Lie algebras of dimension $leq 6$. Publicationes Mathematicae Debrecen, 70(3–4),427–451.https://doi.org/10.5486/PMD.2007.3556
de Graaf, W. A. (2007). Classification of 6-dimensional nilpotent Lie algebras over fields of characteristic not 2. Journal of Algebra, 309(2), 640–653. https://doi.org/10.1016/j.jalgebra.2006.08.006
de Jesus, V. L., & Schneider, C. (2023). The center and invariants of standard filiform Lie algebras. Journal of Algebra, 628, 584–612. https://doi.org/10.1016/j.jalgebra.2023.04.002
Diatta, A., & Manga, B. (2014). On Properties of Principal Elements of Frobenius Lie Algebras . Journal of Lie Theory, 24(3), 849–864.
Diatta, A., Manga, B., & Mbaye, A. (2020). On systems of commuting matrices, Frobenius Lie algebras and Gerstenhaber’s Theorem. http://arxiv.org/abs/2002.08737
Falcón, Ó. J., Falcón, R. M., Núñez, J., Pacheco, A. M., & Villar, M. T. (2016). Classification of Filiform Lie Algebras up to dimension 7 Over Finite Fields. Analele Universitatii “Ovidius” Constanta - Seria Matematica, 24(2), 185–204. https://doi.org/10.1515/auom-2016-0036
Gerstenhaber, M., & Giaquinto, A. (2009). The Principal Element of a Frobenius Lie Algebra. Letters in Mathematical Physics, 88(1–3), 333–341. https://doi.org/10.1007/s11005-009-0321-8
Gómez, J. R., Jimenéz-Merchán, A., & Khakimdjanov, Y. (1998). Low-dimensional filiform Lie algebras. Journal of Pure and Applied Algebra,
(2), 133–158. https://doi.org/10.1016/S0022-4049(97)00096-0
Henti, H., Kurniadi, E., & Carnia, E. (2021). Quasi-Associative Algebras on the Frobenius Lie Algebra M_3 (R)⊕gl_3 (R). Al-Jabar : Jurnal Pendidikan Matematika, 12(1), 59–69. https://doi.org/10.24042/ajpm.v12i1.8485
Kirillov, A. A. (1962). Unitary Representations Of Nilpotent Lie Groups. Russian Mathematical Surveys, 17(4), 53–104. https://doi.org/10.1070/RM1962v017n04ABEH004118
Kruglikov, B. (1997). Symplectic and contact Lie algebras with an application to Monge-Ampére equations.
Kurniadi, E., Carnia, E., & Supriatna, A. K. (2021). The construction of real Frobenius Lie algebras from non-commutative nilpotent Lie algebras of dimension. Journal of Physics: Conference Series, 1722(1), 012025. https://doi.org/10.1088/1742-6596/1722/1/012025
Ooms, A. I. (2009). Computing invariants and semi-invariants by means of Frobenius Lie algebras. Journal of Algebra, 321(4), 1293–1312. https://doi.org/10.1016/j.jalgebra.2008.10.026
Pham, D. N. (2016). $mathfrak{g}$-quasi-Frobenius Lie algebras. Archivum Mathematicum, 4, 233–262. https://doi.org/10.5817/AM2016-4-233
Rodríguez-Vallarte, M. C., & Salgado, G. (2016). 5-dimensional indecomposable contact Lie algebras as double extensions. Journal of Geometry and Physics, 100, 20–32.
https://doi.org/10.1016/j.geomphys.2015.10.014
Salgado-González, G. (2019). Invariants of contact Lie algebras. Journal of Geometry and Physics, 144, 388–396. https://doi.org/10.1016/j.geomphys.2019.06.014
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