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Oct 2, 2023
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Apr 5, 2024
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Abstract

In this paper, we study symplectic form on low dimensional real Lie algebra. A symplectic form is very important in classifying of Lie algebra types. Based on their dimension and certain conditions, there are two types of Lie algebras. A lie algebra with odd dimension endowed with one-form  such that   is called a contact Lie algebra, while a Lie algebra whose dimension is even and it is endowed with zero index is called a Frobenius Lie algebra. The research aimed to give explicit formula of a symplectic form  of  low dimensional contact Lie algebras and Frobenius Lie algebras. We established that a one-form associated to simplectic form determine a type of a Lie algebra whether a contact or a Frobenius Lie algebras.To clearer the main results, we give some examples of one-form and symplectic form of Frobenius and contact Lie algebras.  

Keywords

bilinear form, contact Lie algebra, Frobenius Lie algebra, one-form, symplectic form.

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References

  1. [1] G. Salgado-González, “Invariants of contact Lie algebras,” Journal of Geometry and Physics, vol. 144, pp. 388–396, Oct. 2019, doi: 10.1016/j.geomphys.2019.06.014.
  2. [2] A. I. Ooms, “Computing invariants and semi-invariants by means of Frobenius Lie algebras,” J Algebra, vol. 321, no. 4, pp. 1293–1312, Feb. 2009, doi: 10.1016/j.jalgebra.2008.10.026.
  3. [3] M. A. Alvarez, M. Rodríguez-Vallarte, and G. Salgado, “Contact nilpotent Lie algebras,” Proceedings of the American Mathematical Society, vol. 145, no. 4, pp. 1467–1474, Oct. 2016, doi: 10.1090/proc/13341.
  4. [4] M. A. Alvarez, M. C. Rodríguez-Vallarte, and G. Salgado, “Contact and Frobenius solvable Lie algebras with abelian nilradical,” Commun Algebra, vol. 46, no. 10, pp. 4344–4354, Oct. 2018, doi: 10.1080/00927872.2018.1439048.
  5. [5] B. Kruglikov, “Symplectic and contact Lie algebras with an application to Monge-Ampére equations,” 1997.
  6. [6] M. C. Rodríguez-Vallarte and G. Salgado, “5-dimensional indecomposable contact Lie algebras as double extensions,” Journal of Geometry and Physics, vol. 100, pp. 20–32, Feb. 2016, doi: 10.1016/j.geomphys.2015.10.014.
  7. [7] D. N. Pham, “$\mathfrak{g}$-quasi-Frobenius Lie algebras,” Archivum Mathematicum, no. 4, pp. 233–262, 2016, doi: 10.5817/AM2016-4-233.
  8. [8] M. Gerstenhaber and A. Giaquinto, “The Principal Element of a Frobenius Lie Algebra,” Lett Math Phys, vol. 88, no. 1–3, pp. 333–341, Jun. 2009, doi: 10.1007/s11005-009-0321-8.
  9. [9] A. Diatta and B. Manga, “On Properties of Principal Elements of Frobenius Lie Algebras ,” Journal of Lie Theory, vol. 24, no. 3, pp. 849–864, 2014.
  10. [10] A. Diatta, B. Manga, and A. Mbaye, “On systems of commuting matrices, Frobenius Lie algebras and Gerstenhaber’s Theorem,” Feb. 2020, [Online]. Available: http://arxiv.org/abs/2002.08737