Álgebras de Leibniz de longitud máxima

Abstract

Non associative algebras appear at the beginning of the twentieth century as a consequence of the development of quantum mechanics. Pascual Jordan, John von Neumann and EugeneWigner were the first researchers in introducing theses kinds of algebras in 1934 and then, Jean-Louis Loday introduced the Leibniz algebras in 1993 in his cyclic homology study [30]. Thanks to Levi’s theorem, the nilpo tent Lie algebras have played an important role in mathematics over the last years, either in the classification theory or in geometrical and analytical applications. In Leibniz algebras, Levi’s theorem does not hold, although nilpotent algebras still play a central role. The aim of this work is to continuing the study of nilpotent Leibniz algebras. Since the description of these algebras seen to be unsolvable, we reduce our discussion to the nilpotent family with some restriction on their characteristic sequences and their gradation. More precisely, we study the p-filiform Leibniz algebras whose gradation is of maximum length. This family is very interesting since it has a close relationship with their physical applications, making easier the cohomology study for instance, the space of derivation or the first space of cohomology. The n-dimensional p-filiform Leibniz algebras of maximum length have already been studied with 0 ≤ p ≤ 2. For Lie algebras whose nilindex is equal to n − 2 there is only one characteristic sequence, (n − 2, 1, 1), while in Leibniz theory we obtain two possibilities: (n−2, 1, 1) and (n−2, 2). The first case- ‘the 2-filiformcase’- is already studied in [4] and [37]. The current work appears as a result of studying the case (n − 2, 2). Therefore, we present in the second chapter of this work, the study of the quasifiliform non Lie Leibniz algebras of maximum length, with characteristic sequence (n − 2, 2), completing the study of maximum length of Leibniz algebras with nilindex n − p with 0 ≤ p ≤ 2. The following chapter deals with the classification of the 3-filiform algebras, whereas the fourth one shows the generic case partially 1: the study of p-filiform Leibniz algebras of maximum length, with p generic. In the last chapter we carry out the cohomological study of the obtained algebras. Finally, we have to mention that the software Mathematica has been very useful throughout all this work. We present two algorithms to compute the space of derivations and to establish whether or not two algebras are isomorphic.