Relations among invariants of complex filiform Lie algebras.
In the paper under review, the authors pursue their previous investigation [Algebras Groups Geom. 16, 297--317 (1999; Zbl 1001.17010)] on the classification of complex filiform \(n\)-di\-men\-sion\-al Lie algebras in terms of the invariants \(i\) and \(j\) defined by \[ i=\max\{k\in{\mathbb N}\mid[{\mathcal L}^{n-k+2},{\mathcal L}^2]=\{0\}\} \] and \[ j=\max\{k\in{\mathbb N}\mid[{\mathcal L}^{n-k+1},{\mathcal L}^{n-k+1}]=\{0\}\}, \] where \({\mathcal L}={\mathcal L}^1\supseteq{\mathcal L}^2\supseteq\cdots\) is the descending central sequence of a nilpotent \(n\)-dimensional Lie algebra \({\mathcal L}\). The main results of the present paper give new inequalities that relate the numbers \(i\), \(j\) and \(n\). Thus, Theorem~8 says that \(n\leq2j-3\) provided \(i\neq4\), while Theorem~7 gives a lower bound for \(i+j\). The last section of the paper discusses the classification of filiform Lie algebras with \(j=n-1\).
scientific article; zbMATH DE number 2009108