| Title:
|
Directed pseudo-graphs and Lie algebras over finite fields (English) |
| Author:
|
Boza, Luis |
| Author:
|
Fedriani, Eugenio Manuel |
| Author:
|
Núñez, Juan |
| Author:
|
Pacheco, Ana María |
| Author:
|
Villar, María Trinidad |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
64 |
| Issue:
|
1 |
| Year:
|
2014 |
| Pages:
|
229-239 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
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. (English) |
| Keyword:
|
directed pseudo-graph |
| Keyword:
|
adjacency matrix |
| Keyword:
|
Lie algebra |
| MSC:
|
05C99 |
| MSC:
|
17B30 |
| MSC:
|
17B45 |
| MSC:
|
17B50 |
| MSC:
|
17B60 |
| idZBL:
|
Zbl 06391489 |
| idMR:
|
MR3247457 |
| DOI:
|
10.1007/s10587-014-0096-7 |
| . |
| Date available:
|
2014-09-29T09:59:40Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/143962 |
| . |
| Reference:
|
[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. |
| Reference:
|
[2] Carriazo, A., Fernández, L. M., Núñez, J.: Combinatorial structures associated with Lie algebras of finite dimension.Linear Algebra Appl. 389 (2004), 43-61. Zbl 1053.05059, MR 2080394 |
| Reference:
|
[3] Ceballos, M., Núñez, J., Tenorio, Á. F.: Complete triangular structures and Lie algebras.Int. J. Comput. Math. 88 (2011), 1839-1851. Zbl 1271.17015, MR 2810866, 10.1080/00207161003767994 |
| Reference:
|
[4] Ceballos, M., Núñez, J., Tenorio, Á. F.: Study of Lie algebras by using combinatorial structures.Linear Algebra Appl. 436 (2012), 349-363. Zbl 1276.17010, MR 2854876 |
| Reference:
|
[5] Ceballos, M., Núñez, J., Tenorio, A. F.: Combinatorial structures and Lie algebras of upper triangular matrices.Appl. Math. Lett. 25 (2012), 514-519. MR 2856025, 10.1016/j.aml.2011.09.049 |
| Reference:
|
[6] Graaf, W. A. de: Classification of solvable Lie algebras.Exp. Math. 14 (2005), 15-25. Zbl 1173.17300, MR 2146516, 10.1080/10586458.2005.10128911 |
| Reference:
|
[7] Fernández, L. M., Martín-Martínez, L.: Lie algebras associated with triangular configurations.Linear Algebra Appl. 407 (2005), 43-63. Zbl 1159.17302, MR 2161914 |
| Reference:
|
[8] Gross, J. L., Yellen, J.: Handbook of Graph Theory.Discrete Mathematics and its Applications CRC Press, Boca Raton (2004). Zbl 1036.05001, MR 2035186 |
| Reference:
|
[9] Hamelink, R. C.: Graph theory and Lie algebra.Many Facets of Graph Theory, Proc. Conf. Western Michigan Univ., Kalamazoo/Mi. 1968 Lect. Notes Math. 110 149-153 Springer, Berlin (1969). Zbl 0187.45504, MR 0256910, 10.1007/BFb0060113 |
| Reference:
|
[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. MR 2523385 |
| Reference:
|
[11] Núñez, J., Pacheco, A. M., Villar, M. T.: Study of a family of Lie algebra over $\mathbb Z/3\mathbb Z$.Int. J. Math. Stat. 7 (2010), 40-45. MR 2755406 |
| Reference:
|
[12] Patera, J., Zassenhaus, H.: Solvable Lie algebras of dimension $\leq 4$ over perfect fields.Linear Algebra Appl. 142 (1990), 1-17. MR 1077969 |
| Reference:
|
[13] Varadarajan, V. S.: Lie Groups, Lie Algebras and Their Representations (Reprint of the 1974 edition).Graduate Texts in Mathematics 102 Springer, New York (1984). Zbl 0955.22500, MR 0746308, 10.1007/978-1-4612-1126-6 |
| . |
