Abstract
The main goal of this paper is to show an application of Graph Theory to classifying Lie algebras over finite fields. It is rooted in the representation of each Lie algebra by a certain pseudo-graph. As partial results, it is deduced that there exist, up to isomorphism, four, six, fourteen and thirty-four 2-, 3-, 4-, and 5-dimensional algebras of the studied family, respectively, over the field ℤ/2ℤ. Over ℤ/3ℤ, eight and twenty-two 2- and 3-dimensional Lie algebras, respectively, are also found. Finally, some ideas for future research are presented.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
L. Boza, E. M. Fedriani, J. Núñez: The relation between oriented pseudo-graphs with multiple edges and some Lie algebras. Actas del IV Encuentro Andaluz de Matemática Discreta (2005), 99–104. (In Spanish.)
A. Carriazo, L. M. Fernández, J. Núñez: Combinatorial structures associated with Lie algebras of finite dimension. Linear Algebra Appl. 389 (2004), 43–61.
M. Ceballos, J. Núñez, Á. F. Tenorio: Complete triangular structures and Lie algebras. Int. J. Comput. Math. 88 (2011), 1839–1851.
M. Ceballos, J. Núñez, Á. F. Tenorio: Study of Lie algebras by using combinatorial structures. Linear Algebra Appl. 436 (2012), 349–363.
M. Ceballos, J. Núñez, A. F. Tenorio: Combinatorial structures and Lie algebras of upper triangular matrices. Appl. Math. Lett. 25 (2012), 514–519.
W. A. de Graaf: Classification of solvable Lie algebras. Exp. Math. 14 (2005), 15–25.
L. M. Fernández, L. Martín-Martínez: Lie algebras associated with triangular configurations. Linear Algebra Appl. 407 (2005), 43–63.
J. L. Gross, J. Yellen: Handbook of Graph Theory. Discrete Mathematics and its Applications, CRC Press, Boca Raton, 2004.
R. C. Hamelink: Graph theory and Lie algebra. Many Facets of Graph Theory, Proc. Conf. Western Michigan Univ., Kalamazoo/Mi. 1968. Lect. Notes Math. 110, Springer, Berlin, 1969, pp. 149–153.
J. Núñez, A. Pacheco, M. T. Villar: Discrete mathematics applied to the treatment of some Lie theory problems. Sixth Conference on Discrete Mathematics and Computer Science. Univ. Lleida, Lleida, 2008, pp. 485–492. (In Spanish.)
J. Núñez, A. M. Pacheco, M. T. Villar: Study of a family of Lie algebra over ℤ/3ℤ. Int. J. Math. Stat. 7 (2010), 40–45.
J. Patera, H. Zassenhaus: Solvable Lie algebras of dimension ⩽ 4 over perfect fields. Linear Algebra Appl. 142 (1990), 1–17.
V. S. Varadarajan: Lie Groups, Lie Algebras and Their Representations (Reprint of the 1974 edition). Springer, New York, 1984.
Author information
Authors and Affiliations
Corresponding author
Additional information
The research has been partially supported by PAI-FQM-164.
Rights and permissions
About this article
Cite this article
Boza, L., Fedriani, E.M., Núñez, J. et al. Directed pseudo-graphs and lie algebras over finite fields. Czech Math J 64, 229–239 (2014). https://doi.org/10.1007/s10587-014-0096-7
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10587-014-0096-7