On nilpotent Lie algebras having a torus of derivations. (Sur les algèbres de Lie nilpotentes admettant un tore de dérivations.) (French) Zbl 0823.17009

Goze, Michel; Hakimjanov, You

Manuscr. Math. 84, No. 2, 115-124 (1994).

Let \({\mathfrak g}\) be a complex nilpotent Lie algebra of dimension \(n\). For \(x \in {\mathfrak g} - [{\mathfrak g}, {\mathfrak g}]\) let \(c(x)\) be the set of dimensions of Jordan blocks of ad \(x\) in decreasing order. Let \(c({\mathfrak g}) = \text{Sup} (c(x),x \in {\mathfrak g} - [{\mathfrak g}, {\mathfrak g}])\) (relative to the lexicographical order). Then \({\mathfrak g}\) is said to be filiform if \(c ({\mathfrak g})\) is maximal, i.e., \((n - 1,1)\).
The authors classify the filiform Lie algebras admitting a nonzero diagonalizable derivation. Then, in a section on characteristically nilpotent algebras (i.e., algebras with all derivations nilpotent), they prove that for dimensions greater than 8 any algebraic irreducible component of the variety of complex nilpotent filiform laws of Lie algebras contains an open set whose elements are characteristically nilpotent laws.

Reviewer: G.Brown (Boulder)

MSC:

17B30	Solvable, nilpotent (super)algebras
17B40	Automorphisms, derivations, other operators for Lie algebras and super algebras

Keywords:

complex nilpotent Lie algebra; filiform Lie algebras; derivation; characteristically nilpotent algebras

References:

[1]	[A-G] Ancochea Bermudez, J.M., Goze, M.: Le rang du système linéaire des racines d’une algèbre de Lie résoluble rigide. Comm. in Algebra20 (3), (1992), 875–887 · Zbl 0748.17005
[2]	[B] Benoist, Y.: Une nilvariété non affine. Preprint Paris (1992)
[3]	[C] Carles, R.: Sur les algèbres de Lie caractéristiquement nilpotentes. Preprint Univ. Poitiers (1984)
[4]	[Di] Dixmier, J.: Sur les représentations unitaires des groupes de Lie nilpotents III. Canad. J. Math.10 (1958) 321–348 · Zbl 0100.32401
[5]	[D-L] Dixmier, J., Lister, W.G.: Derivations of nilpotent Lie algebras. Proc. Amer. Math. Soc.8 (1957) 155–158 · Zbl 0079.04802
[6]	[Dy] Dyer, J.L.: A nilpotent Lie algebra with nilpotent automorphism group. Bull. Amer. Math. Soc.76 (1970) 52–56 · Zbl 0198.05402
[7]	[F] Favre, G.: Une algèbre de Lie caractéristiquement nilpotente de dimension 7. C.R.A.Sc. Paris, Série A,274 (1972) 1338–1339 · Zbl 0236.17003
[8]	[H1] Hakimjanov, Y.: Algèbres de Lie caractéristiquement nilpotentes. Math. Sb.181 (1990) 642–655 et Math. USSR.70, 1 (1991)
[9]	[H2] Hakimjanov, Y.: Variétés des lois d’algèbres de Lie nilpotentes. Geom. Dedicata40 (1991) 269–295
[10]	[H-R] Kraft, H.P., Riedtmann, Ch.: Geometry of representations of quivers. In Representations of algebras (Durham 1985). London Math. Soc. Lecture Notes116. Cambridge Univ. Press, Cambridge-New-York 1986. · Zbl 0632.16019
[11]	[L-T] Leger, G., Tôgô, S.: Caracteristically nilpotent Lie algebras. Duke Math. J.26 (1959) 623–628 · Zbl 0101.02402
[12]	[L] Luks, M.: What is the typical nilpotent Lie algebra? in Computers in non associative rings and algebras. Acad. Press, New-York p 189–207, 1977.
[13]	[M] Morozov, V.: Classification of nilpotent Lie algebras of sixth order. Izv. Vyssh. U.Zaved. Mat.4(5) (1958) 161–171 · Zbl 0198.05501
[14]	[Mo] Mostow, G.D.: Fully reductible subgroups of algebraic groups. Amer. J. Math.78 (1956) · Zbl 0073.01603
[15]	[V] Vergne, M.: Cohomologie des algèbres de Lie nilpotentes. Application à l’étude de la variété des algèbres de Lie nilpotentes. Bull. Soc. Math. France98 (1970) 81–116 · Zbl 0244.17011