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Relations among invariants of complex filiform Lie algebras. (English) Zbl 1047.17005
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 $n$-dimensional Lie algebras in terms of the invariants $i$ and $j$ defined by $$i=\max\{k\in{\Bbb N}\mid[{\Cal L}^{n-k+2},{\Cal L}^2]=\{0\}\}$$ and $$j=\max\{k\in{\Bbb N}\mid[{\Cal L}^{n-k+1},{\Cal L}^{n-k+1}]=\{0\}\},$$ where ${\Cal L}={\Cal L}^1\supseteq{\Cal L}^2\supseteq\cdots$ is the descending central sequence of a nilpotent $n$-dimensional Lie algebra ${\Cal L}$. The main results of the present paper give new inequalities that relate the numbers $i$, $j$ and $n$. Thus, Theorem 8 says that $n\le2j-3$ provided $i\ne4$, while Theorem 7 gives a lower bound for $i+j$. The last section of the paper discusses the classification of filiform Lie algebras with $j=n-1$.

MSC:
17B30Solvable, nilpotent Lie (super)algebras
17B05Structure theory of Lie algebras
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References:
[1] Ancochea, J. M.; Goze, M.: Classification des algèbres de Lie nilpotentes complexes de dimension 7. Arch. math. 52, 175-185 (1989) · Zbl 0672.17005
[2] Reula, F. J. Echarte; Martıń, J. R. Gómez; Valdés, J. Núñez: LES algébres de Lie filiformes complexes derivées d’autres algébres de Lie. Collection travaux en cours: lois d’algèbres et variétés algébriques 50, 45-55 (1996)
[3] Echarte, F. J.; Núñez, J.; Ramıŕez, F.: Study of two invariants in complex filiform Lie algebras. Algebras groups geom. 13, 55-70 (1996) · Zbl 0868.17003
[4] Echarte, F. J.; Núñez, J.; Ramıŕez, F.: Classification of complex filiform Lie algebras depending on the invariants i, j and n. Algebras groups geom. 16, 297-317 (1999) · Zbl 1001.17010
[5] Vergne, M.: Cohomologie des algèbres de Lie nilpotentes, application à l’étude de la variété des algebres de Lie nilpotentes. Bull. soc. Math. France 98, 81-116 (1970) · Zbl 0244.17011