Directed pseudo-graphs and Lie algebras over finite fields

Luis B. Boza; Eugenio Manuel Fedriani; Juan Núñez; Ana María Pacheco; María Trinidad Villar

Czechoslovak Mathematical Journal (2014)

  • Volume: 64, Issue: 1, page 229-239
  • ISSN: 0011-4642

Abstract

top
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 / 2 ℤ . Over ℤ / 3 ℤ , eight and twenty-two 2 - and 3 -dimensional Lie algebras, respectively, are also found. Finally, some ideas for future research are presented.

How to cite

top

Boza, Luis B., et al. "Directed pseudo-graphs and Lie algebras over finite fields." Czechoslovak Mathematical Journal 64.1 (2014): 229-239. <http://eudml.org/doc/261973>.

@article{Boza2014,
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 $\mathbb \{Z\}/2\mathbb \{Z\}$. Over $\mathbb \{Z\}/3\mathbb \{Z\}$, eight and twenty-two $2$- and $3$-dimensional Lie algebras, respectively, are also found. Finally, some ideas for future research are presented.},
author = {Boza, Luis B., Fedriani, Eugenio Manuel, Núñez, Juan, Pacheco, Ana María, Villar, María Trinidad},
journal = {Czechoslovak Mathematical Journal},
keywords = {directed pseudo-graph; adjacency matrix; Lie algebra; directed pseudo-graph; Lie algebra; classification using graphs; finite fields; adjacency matrix},
language = {eng},
number = {1},
pages = {229-239},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Directed pseudo-graphs and Lie algebras over finite fields},
url = {http://eudml.org/doc/261973},
volume = {64},
year = {2014},
}

TY - JOUR
AU - Boza, Luis B.
AU - Fedriani, Eugenio Manuel
AU - Núñez, Juan
AU - Pacheco, Ana María
AU - Villar, María Trinidad
TI - Directed pseudo-graphs and Lie algebras over finite fields
JO - Czechoslovak Mathematical Journal
PY - 2014
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 64
IS - 1
SP - 229
EP - 239
AB - 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 $\mathbb {Z}/2\mathbb {Z}$. Over $\mathbb {Z}/3\mathbb {Z}$, eight and twenty-two $2$- and $3$-dimensional Lie algebras, respectively, are also found. Finally, some ideas for future research are presented.
LA - eng
KW - directed pseudo-graph; adjacency matrix; Lie algebra; directed pseudo-graph; Lie algebra; classification using graphs; finite fields; adjacency matrix
UR - http://eudml.org/doc/261973
ER -

References

top
  1. Boza, L., Fedriani, E. M., Núñez, J., The relation between oriented pseudo-graphs with multiple edges and some Lie algebras, Actas del IV Encuentro Andaluz de Matemática Discreta (2005), 99-104 Spanish. (2005) 
  2. Carriazo, A., Fernández, L. M., Núñez, J., Combinatorial structures associated with Lie algebras of finite dimension, Linear Algebra Appl. 389 (2004), 43-61. (2004) Zbl1053.05059MR2080394
  3. Ceballos, M., Núñez, J., Tenorio, Á. F., 10.1080/00207161003767994, Int. J. Comput. Math. 88 (2011), 1839-1851. (2011) Zbl1271.17015MR2810866DOI10.1080/00207161003767994
  4. Ceballos, M., Núñez, J., Tenorio, Á. F., Study of Lie algebras by using combinatorial structures, Linear Algebra Appl. 436 (2012), 349-363. (2012) Zbl1276.17010MR2854876
  5. Ceballos, M., Núñez, J., Tenorio, A. F., 10.1016/j.aml.2011.09.049, Appl. Math. Lett. 25 (2012), 514-519. (2012) MR2856025DOI10.1016/j.aml.2011.09.049
  6. Graaf, W. A. de, 10.1080/10586458.2005.10128911, Exp. Math. 14 (2005), 15-25. (2005) Zbl1173.17300MR2146516DOI10.1080/10586458.2005.10128911
  7. Fernández, L. M., Martín-Martínez, L., Lie algebras associated with triangular configurations, Linear Algebra Appl. 407 (2005), 43-63. (2005) Zbl1159.17302MR2161914
  8. Gross, J. L., Yellen, J., Handbook of Graph Theory, Discrete Mathematics and its Applications CRC Press, Boca Raton (2004). (2004) Zbl1036.05001MR2035186
  9. Hamelink, R. C., 10.1007/BFb0060113, Many Facets of Graph Theory, Proc. Conf. Western Michigan Univ., Kalamazoo/Mi. 1968 Lect. Notes Math. 110 149-153 Springer, Berlin (1969). (1969) Zbl0187.45504MR0256910DOI10.1007/BFb0060113
  10. Núñez, J., Pacheco, A., Villar, M. T., Discrete mathematics applied to the treatment of some Lie theory problems, Sixth Conference on Discrete Mathematics and Computer Science Univ. Lleida, Lleida (2008), 485-492 Spanish (2008), 485-492. (2008) MR2523385
  11. Núñez, J., Pacheco, A. M., Villar, M. T., Study of a family of Lie algebra over ℤ / 3 ℤ , Int. J. Math. Stat. 7 (2010), 40-45. (2010) MR2755406
  12. Patera, J., Zassenhaus, H., Solvable Lie algebras of dimension ≤ 4 over perfect fields, Linear Algebra Appl. 142 (1990), 1-17. (1990) MR1077969
  13. Varadarajan, V. S., 10.1007/978-1-4612-1126-6, Graduate Texts in Mathematics 102 Springer, New York (1984). (1984) Zbl0955.22500MR0746308DOI10.1007/978-1-4612-1126-6

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.