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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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##### References:
 [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 [5] Gómez J.R., Rend. Sem. Fac. Sc 61 pp 21– (1991) [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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