Sur les algèbres de Lie quasi-filiformes admettant un tore de dérivations

Abstract

Dans ce travail on décrit les algèbres de Lie quasi-filiformes de rang non nul. De plus, on rappelle et corrige la classification des algèbres de Lie filiformes admettant un tore de dérivations, ainsi que la liste des algèbres graduées naturellement et quasi-filiformes. *

Naturally graded Lie algebras of slow growth

“…Suppose also that there exists an isomorphism f : g′ → g. An equality holds on f g′k = gk for the ideals of the lower central series. Whence follows that f (g ′ k ) = gk , and we have a commutative diagram (10)…”

“…Such generalizations were usually based on the concept of the length of the lower central series, since for the finite-dimensional filiform Lie algebra g the length s(g) of its lower central series is maximal for a given dimension of Lie algebras: s(g) = dim g − 1. The class of so-called quasifiliform Lie algebras (s(g) = dim g − 2) has not greatly extended the supply of examples of Lie algebras close in properties to filiform [14,10]. In this paper we propose a generalization of the class of filiform Lie algebras from the point of view of the growth of Lie algebras.…”

Solvable extensions of negative Ricci curvature of filiform Lie groups

“…As usual, the relations between the basis elements for and for which we do not list are zero (unless they follow from the given ones by the skew‐symmetry). Note that in there is one other class of algebras of rank one, , but as it is shown in [, Remarque 1] all the algebras of class are isomorphic to .…”

Classification of the Quasifiliform Nilpotent Lie Algebras of Dimension 9

“…. , 16 ), which is summarized within the cases shown in the following subsections: Proof. For the family described by (3a)-(3o) and (4a)-(4q), its descending central series is Proof.…”

“…Subsequent works about quasifiliform Lie algebras classification were centered on specific types of families or subclasses, obtaining results applicable to higher dimensions. For instance, the classifications of naturally graded [15] and graded by derivations [16] quasifiliform Lie algebras. These works extended to other algebras, with a high nilindex, the classification of graded filiform Lie algebras, studied initially by Vergne [17,18], obtained from the gradation related to the filtration produced in a natural way by the descending central sequence.…”