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

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.

The trascendence of Kac-Moody algebras

“…Indeed, R.M. Navarro classified filiform Lie algebras of order 3, [195], Falc ón, Falc ón, N ú ñez, Pacheco and Villar the filiform Lie algebras up to dimension 7 over finite fields [86], and I. Darijani and H. Usefi [74] the 5-dimensional p-nilpotent restricted Lie algebras over perfect fields.…”

On Properties of Five-dimensional Nonstandard Filiform Lie algebra

“…Some researchers classified the Filiform Lie algebra up to the dimension 7( [1], [2]). The 3-dimensional Filiform Lie algebras are known as the Heisenberg Lie algebra 𝔥 3 as mentioned above and it has been widely studied as the best model or the best example of representation theory of Lie algebras and their Lie groups, especially the Schrödinger representation ([3]- [8]).…”

“…Furthermore, the unitary irreducible representation of Lie groups of the five-dimensional standard Filiform Lie algebra has also been obtained by means of the orbit method [10]. We emphasize that the classification of Filiform Lie algebra up to dimension six has been classified by [2] and classification of Filiform Lie algebra up to dimension seven has been obtained by ( [1], [11], [12]). To the best of author's knowledge, there is none of researcher has yet classified the Filiform Lie algebra with dimension more than seven.…”

Classification of types A and A_+ from low dimensional standard and non-standard filiform Lie Algebras

“…Jesus & Schneider, 2023;Falcón et al, 2016;Gómez et al, 1998). The studies of filiform Lie algebras for examples focus on their classification, index of graded filiform, center and invariants.…”