Naturally graded quasi-filiform Lie algebras (Q1858216)

A finite-dimensional nilpotent Lie algebra \(\mathfrak{g}\) is naturally filtered by the descending central sequence \(C^1 \mathfrak{ g} = \mathfrak{ g},\dots,C^i{\mathfrak{ g}}=[\mathfrak{ g},C^{i{-}1}\mathfrak{ g}]\), which induces a gradation \[ \text{ gr} \mathfrak{ g}=\bigoplus_i {\mathfrak{ g}}_i=\bigoplus_iC^i \mathfrak{ g} /C^{i-1}\mathfrak{ g}. \] If \(\mathfrak{g}\) is isomorphic to the associated graded algebra, then it is said to be naturally graded. Starting from the classification of naturally graded filiform Lie algebras (i.e., nilpotent algebras of maximal nilindex), the authors classify the naturally graded quasi-filiform algebras, i.e., nilpotent algebras \(\mathfrak{g}\) whose nilindex is \(k=\dim \mathfrak{g}-2\). The decomposable and indecomposable cases are distinguished. As a result of this classification, various families of algebras are obtained, to which some special cases appearing in concrete dimensions must be added. The paper finishes with a short discussion of the usefulness of symbolic calculus packages adapted to this kind of problem.