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\((n-4)\)-filiform Lie algebras. (English) Zbl 0934.17003
A complex \(n\)-dimensional Lie algebra is called \(p\)-filiform for some \(p\in\{1,\dots,n\}\), whenever its characteristic sequence [introduced by J. M. Ancochea-Bermudez and M. Goze, C. R. Acad. Sci., Paris, Sér. I 302, 611-613 (1986; Zbl 0591.17008)] has its first term \(n-p\) and the remaining terms are all 1. The \((n-1)\) filiform Lie algebras of dimension \(n\) are exactly the abelian ones and the classification of \(p\)-filiform Lie algebras becomes more difficult as \(p\) decreases. The \((n-3)\)- and \((n-2)\)-filiform Lie algebras of dimension \(n\) were classified by J. M. Cabezas, J. R. Gómez and A. Jiménez-Mérchan [cf. “Family of \(p\)-filiform Lie algebras”. Khakimdjanov, Yusupdjan (ed.) et al., Algebra and operator theory. Proceedings of the colloquium, Tashkent 1997, 93-102 (1998; Zbl 0924.17005)].
The aim of the paper under review is to classify the \((n-4)\)-filiform Lie algebras of dimension \(n\). One finds exactly \(6n-29\) such pairwise non-isomorphic algebras for each \(n\geq 8\). (The 7-dimensional nilpotent Lie algebras were classified by J. M. Ancochea-Bermudez and M. Goze [cf. Arch. Math. 52, No. 2, 175-185 (1989; Zbl 0672.17005)].) The method of proof for this fact consists of considering the \(n\)-dimensional nilpotent Lie algebras as central extensions of \((n-1)\)-dimensional algebras.

MSC:
17B30 Solvable, nilpotent (super)algebras
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[1] DOI: 10.1007/BF01193621 · Zbl 0628.17005 · doi:10.1007/BF01193621
[2] DOI: 10.1007/BF01191272 · Zbl 0672.17005 · doi:10.1007/BF01191272
[3] Cabezas J.M., PhD thesis (1997)
[4] Cabezas, J.M., Gómez, J.R. and Jiménez-Merchán, A. Family of p-filiform Lie algebras, Algebra and Operator Theory. Proceedings of the Colloquium in Tachkent. pp.93–102. Kluwer. · Zbl 0924.17005
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[6] DOI: 10.1016/S0022-4049(97)00096-0 · Zbl 0929.17004 · doi:10.1016/S0022-4049(97)00096-0
[7] Goze M., Nilpotent Lie Algebras (1996) · Zbl 0845.17012
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