On quasi-filiform Lie algebras admitting a torus of derivations. (Sur les algèbres de Lie quasi-filiformes admettant un tore de dérivations.) (English) Zbl 1172.17006

The main theorem of the paper under review provides a classification for the quasi-filiform Lie algebras that allow for nontrivial diagonalizable derivations. We recall that a nilpotent Lie algebra of dimension \(n\) is said to be quasi-filiform if its nilpotency index is equal to \(n-2\). The present classification is based on the classification of the naturally graded quasi-filiform algebras given by J. R. Gómez and A. Jiménez-Merchán [J. Algebra 256, No. 1, 211–228 (2002; Zbl 1030.17007)].


17B30 Solvable, nilpotent (super)algebras
17B70 Graded Lie (super)algebras


Zbl 1030.17007
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[1] Favre G. (1973). Système des poids sur une algèbre de Lie nilpotente. Manuscripta Math. 9: 53–90 · Zbl 0253.17011
[2] Gómez J.R. and Jiménez-Merchán A. (2002). Naturally graded quasi-filiform Lie algebras. J. Algebra 256: 221–228 · Zbl 1030.17007
[3] Goze M. and Ancochea J.M. (2001). On the classification of rigid Lie algebras. J. Algebra 245: 68–91 · Zbl 0998.17010
[4] Goze M. and Hakimjanov Y. (1994). Sur les algèbres de Lie nilpotentes admettant un tore de dérivations. Manuscripta Math. 84: 115–124 · Zbl 0823.17009
[5] Goze M. and Remm E. (2004). Valued Deformations of Algebras. J. Alg. Appl. 3: 345–365 · Zbl 1062.17010
[6] Mal’cev A.I. (1945). Solvable Lie algebras. Izv. Akad. Nauk SSSR 9: 329–356 · Zbl 0061.05303
[7] Vergne M. (1970). Cohomologies des algèbres de Lie nilpotentes. Application à l’étude de la variété des algèbres de Lie Nilpotentes. Bull. Soc. Math. France 98: 81–116 · Zbl 0244.17011
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