zbMATH — the first resource for mathematics

A class of nilpotent Lie algebras. (English) Zbl 0970.17007
The classification of the nilpotent complex Lie algebras of finite dimension is still an open problem. One would like to classify at least the \(p\)-filiform Lie algebras, i.e., nilpotent Lie algebras \({\mathfrak g}\) such that the sequence \(({\mathfrak g}-p,1,1,\dots,1)\) is lexicographically the biggest among the sequences \(c(X)\) for \(X\in{\mathfrak g}\), where \(c(X)\) stands for the decreasing sequence of dimensions of the cells occurring in a Jordan form of the nilpotent map \(\text{ ad} X:{\mathfrak g}\to{\mathfrak g}\). For \(p=1\) one has the so-called filiform Lie algebras and their classification is still out of reach. On the other hand, the case \(p=\dim{\mathfrak g}-1\) is that of the Abelian Lie algebras, which are easily classified (by means of their dimension).
Trying to reach the case \(p=1\), one has first classified the \(p\)-filiform Lie algebras \({\mathfrak g}\) for \(\dim{\mathfrak g}-3\leq p\leq\dim{\mathfrak g}-2\) [J. M. Cabezas, J. R. Gómez and A. Jimenez-Merchán, Algebra and operator theory. Proc. Colloq., Tashkent, 1997, 93-102 (1998; Zbl 0924.17005)] and then for \(p=\dim{\mathfrak g}-4\) [J. M. Cabezas and J. R. Gómez, Commun. Algebra 27, 4803-4819 (1999; Zbl 0934.17003)]. In the case \(p=\dim{\mathfrak g}-5\), one could solve the classification problem for \(\dim{\mathfrak g}= 8\) [L. M. Camacho, J. R. Gómez and R. M. Navarro, Ann. Math. Blaise Pascal 6, No. 2, 1-13 (1999; Zbl 0942.17003)].
The paper under review also deals with the case \(p=\dim{\mathfrak g}-5\), but for filiform Lie algebras \({\mathfrak g}\) of arbitrary dimension such that the dimension of \([{\mathfrak g},{\mathfrak g}]\) is maximal. One first proves a theorem which shows in particular that, in this case, \(\dim[{\mathfrak g},{\mathfrak g}]=6\). An explicit classification of these algebras is then exhibited. Finally, the result is used to find out which of these algebras is isomorphic to the corresponding graded algebra associated to the filtration provided by the descending central series.

17B30 Solvable, nilpotent (super)algebras
17B05 Structure theory for Lie algebras and superalgebras
Full Text: DOI
[1] Ancochea J.M., Archiv. Math 52 (2) pp 175– (1989) · Zbl 0672.17005 · doi:10.1007/BF01191272
[2] Ancochea J.M., Archiv. Math 50 (2) pp 511– (1988) · Zbl 0628.17005 · doi:10.1007/BF01193621
[3] Cabezas J.M., PhD thesis (1996)
[4] Cabezas, J.M., Camacho, L.M., Gómez, J.R., Navarro, R.M. and Pastor, E. 1999. Low-dimensional naturally graded 3-filiform Lie algebra. Meeting León. 1999, Junio. SAGA V
[5] Cabezas, J.M. and Gómez, J.R. Generación de lafamilia de leyes de algebras de Lie (n-5)-filiformes en dimensión cualquiera. Meeting on Matrix Analysis and its Applications. pp.109–116. Sevilla
[6] Cabezas J.M., Algebra and Operator Theory pp 93– (1998) · doi:10.1007/978-94-011-5072-9_7
[7] Gómez, J.R. and Echarte, F.J. 1991.Classification of complex filiform Lie algebras of dimension 9, Vol. 61, 21–29. Cagliari: Rend. Sem. Fac. Sc. Univ. 1 · Zbl 0790.17004
[8] Gómez J.R., Communications in Algebra 25 (2) pp 431– (1997) · Zbl 0866.17009 · doi:10.1080/00927879708825864
[9] Gómez J.R., Journal of Pure and Applied Algebra 130 (2) pp 133– (1998) · Zbl 0929.17004 · doi:10.1016/S0022-4049(97)00096-0
[10] Goze M., Nilpotent Lie Algebras (1996) · Zbl 0845.17012 · doi:10.1007/978-94-017-2432-6
[11] Vergne M., Bull. Soc. Math 98 pp 81– (1970)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.