Quasi-Associative Algebras on the Frobenius Lie Algebra M_3 (R)⊕gl_3 (R)

https://doi.org/10.24042/ajpm.v12i1.8485

Henti Henti, Edi Kurniadi, Ema Carnia

Abstract


In this paper, we study the quasi-associative algebra property for the real Frobenius  Lie algebra  of dimension 18. The work aims  to prove that  is a quasi-associative algebra and to compute its formulas explicitly. To achieve this aim, we apply the literature reviews method corresponding to Frobenius Lie algebras, Frobenius functionals, and the structures of quasi-associative algebras. In the first step, we choose a Frobenius functional determined by direct computations of a bracket matrix of  and in the second step, using an induced symplectic structure, we obtain the explicit formulas of quasi-associative algebras for . As the results, we proved that  has the quasi-associative algebras property, and we gave their formulas explicitly. For future research, the case of the quasi-associative algebras on   is still an open problem to be investigated. Our result can motivate to solve this problem. 

 


Keywords


Frobenius Lie Algebras; Frobenius Functionals; Quasi- Associative Algebras; Symplectic Structures.

Full Text:

PDF

References


Alvarez, M. A., & et al. (2018). Contact and Frobenius solvable Lie algebras with abelian nilradical. Comm. Algebra, 46, 4344–4354.

Burde, D. (2015). Left-symmetric algebras, or pre-Lie algebras in geometry and physics. Central European Journal of Mathematics, 4(3), 323–357. https://doi.org/10.2478/s11533-006-0014-9.

Csikós, B., & Verhóczki, L. (2007). Classification of frobenius Lie algebras of dimension ≤ 6. Publicationes Mathematicae, 70(3–4), 427–451.

Diatta, A., Manga, B., & Mbaye, A. (2020). On Systems of Commuting Matrices, Frobenius Lie Algebras and Gerstenhaber’s Theorem. ArXiv:2002.08737.

Diatta, A., & Manga, B. (2014). On properties of principal elements of frobenius lie algebras. J. 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. (February), 0–12.

Graaf, W. A. De. (2007). Classification of 6-dimensional nilpotent Lie algebras over fields of characteristic not 2. 309, 640–653. https://doi.org/10.1016/j.jalgebra.2006.08.006

Hadjer, A., & Makhlouf, A. (2012). Index of Graded Filiform and Quasi Filiform Lie Algebras. (May 2014). https://doi.org/10.2298/FIL1303467A.

Hendrawan, R. (2020). Aljabar Simetrik Kiri Pada Aljabar Lie Frobenius Riil Berdimensi 6. Unpad.

Henti, Kurniadi,Edi, & Carnia, E. (2021). On Frobenius functionals of the Lie algebra M_3 (R)oplus gl_3 (R). Journal of Physics : Conference Series Accepted.

Hilgert, J., & Neeb, K.-H. (2012). Structure and Geometry of Lie Groups. New York: Springer Monographs in Mathematics, Springer.

Kurniadi, E., Gusriani, N., & Subartini, B. (2020). Struktur Affine Aljabar Lie Real dari Grup Lie Similitude Berdimeni 5. Jurnal Teorema, Universitas Galuh.

Ooms, A. I. (1980). On frobenius lie algebras. In Communications in Algebra (Vol. 8). https://doi.org/10.1080/00927878008822445.

Pham, D. N. (2016). G-Quasi-Frobenius Lie Algebras. Archivum Mathematicum, 52(4), 233–262. https://doi.org/10.5817/AM2016-4-233.

Rais, M. (1978). La representation du groupe affine. Ann.Inst.Fourier,Grenoble, 26, 207--237.




DOI: https://doi.org/10.24042/ajpm.v12i1.8485

Article Metrics

Abstract views : 138 | PDF downloads : 55

Refbacks

  • There are currently no refbacks.


Copyright (c) 2021 Al-Jabar : Jurnal Pendidikan Matematika

Creative Commons License
This work is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License.

 

Creative Commons License
Al-Jabar : Jurnal Pendidikan Matematika is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License.