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

Abstract

This paper tries to develop a recent research which consists in us- ing Discrete Mathematics as a tool in the study of the problem of the classi cation of Lie algebras in general, dealing in this case with liform Lie algebras up to dimension 7 over nite elds. 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 nd out that there exist, up to isomorphism, six, ve and ve 6-dimensional liform Lie algebras and fteen, eleven and fteen 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 nd out properties of Lie algebras and to exemplify a suitable technique to be used in classi cations for larger dimensions.