Classification of Filiform Lie Algebras up to dimension 7 Over Finite Fields

Falcón, Óscar J.; Falcón, Raúl M.; Núñez, Juan; Pacheco, Ana M.; Villar, M. Trinidad

doi:10.1515/auom-2016-0036

Abstract

This paper tries to develop a recent research which consists in using Discrete Mathematics as a tool in the study of the problem of the classification of Lie algebras in general, dealing in this case with filiform Lie algebras up to dimension 7 over finite fields. The idea lies in the representation of each Lie algebra by a certain type of graphs. Then, some properties on Graph Theory make easier to classify the algebras. As main results, we find out that there exist, up to isomorphism, six, five and five 6-dimensional filiform Lie algebras and fifteen, eleven and fifteen 7-dimensional ones, respectively, over ℤ/pℤ, for p = 2, 3, 5. In any case, the main interest of the paper is not the computations itself but both to provide new strategies to find out properties of Lie algebras and to exemplify a suitable technique to be used in classifications for larger dimensions.

References

[1] L. Boza Prieto, F.J. Echarte Reula and J. Núñez Valdés, Classification of complex filiform Lie algebras of dimension 10, Algebras, Groups and Geometries 11 (1994), 253-276.
Search in Google Scholar Back to article
[2] L. Boza Prieto, E. M. Fedriani Martel and J. Núñez Valdés, A new method for classifying complex filiform Lie algebras, Applied Mathematics and Computation 121: 2-3 (2001), 169-175.10.1016/S0096-3003(99)00270-2
Search in Google Scholar Back to article
[3] L. Boza, E.M. Fedriani y J. Núñez, Una relacifion entre los pseudo-grafos dirigidos sin aristas repetidas y algunas fialgebras de Lie, Actas del IV Encuentro Andaluz de Matemfiatica Discreta (2005), 99-104.
Search in Google Scholar Back to article
[4] A. Carriazo, L.M. Fernfiandez and J. Núñez, Combinatorial structures associated with Lie algebras of finite dimension, Linear Algebra and its Applications 389 (2004), 43-61.10.1016/j.laa.2004.02.030
Search in Google Scholar Back to article
[5] S. Cicalò, W. de Graaf and C. Schneider, Six-dimensional nilpotent Lie algebras, Linear Algebra Appl. 436:1 (2012), 163-189.10.1016/j.laa.2011.06.037
Search in Google Scholar Back to article
[6] O. J. Falcón, R. M. Falcón and J. Núñez, Isomorphism and isotopism classes of filiform Lie algebras of dimension up to seven, Submitted, 2015.10.1007/s00025-015-0502-y
Search in Google Scholar Back to article
[7] E. M. Fedriani, Clasificacifion de las fialgebras de Lie filiformes complejas de dimensifion 12. Tesis de Licenciatura. Universidad de Sevilla (Espa~na), 1997.
Search in Google Scholar Back to article
[8] P. Frè, V. Gili, F. Gargiulo, A. Sorin, H. Rulik, M. Trigiante, Cosmological backgrounds of superstring theory and solvable algebras: Oxidation and branes, Nucl. Phys. B 685 (2004) 3-64.10.1016/j.nuclphysb.2004.02.031
Search in Google Scholar Back to article
[9] F. Harary, Graph Theory, Addison Wesley, Reading, Mass., 1969. 10.21236/AD0705364
Search in Google Scholar Back to article
[10] J. Núñez, A. Pacheco y M.T. Villar, Matemfiatica Discreta aplicada al tratamiento de algunos problemas de la Teorfifia de Lie, Actas de las VI Jornadas de Matemfiatica Discreta y Algorfifitmica (VIJMDA), (2008) 485-492.
Search in Google Scholar Back to article
[11] J. Núñez, A. Pacheco y M. T. Villar, Study of a family of Lie algebras over Z=3Z, International Journal of Applied Mathematics and Statistics, Special volume 7:W10 (2010), 40-45.
Search in Google Scholar Back to article
[12] C. Schneider, A computer-based approach to the classification of nilpotent Lie algebras, Experiment. Math. 14:2 (2005), 153-160.10.1080/10586458.2005.10128917
Search in Google Scholar Back to article
13] V.S. Varadarajan. Lie Groups, Lie Algebras and their Representations. Springer, 1984. 10.1007/978-1-4612-1126-6
Search in Google Scholar Back to article
[14] M. Vergne, Cohomologie des algébres de Lie nilpotentes. Application fia l'étude de la variété des algebres de Lie nilpotentes, Bull. Soc. Math. France 98 (1970), 81-116.10.24033/bsmf.1695
Search in Google Scholar Back to article

Classification of Filiform Lie Algebras up to dimension 7 Over Finite Fields

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