Publication: Classification des algèbres de Lie filiformes de dimension 8
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1988-06-02
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Birkhäuser Verlag
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In this work the authors classify the filiform Lie algebras (i.e., Lie algebras that are nilpotent with an adjoint derivation of maximal order) of dimension m=8 over the field of complex numbers. These algebras, introduced by M. Vergne in her thesis ["Variétés des algèbres de Lie nilpotentes'', Thèse de 3 ème cycle, Univ. Paris, Paris, 1966; BullSig(110) 1967:299], form a Zariski-open set in the variety N m of nilpotent Lie algebra laws of C n . For each m≤6 there exists a filiform algebra which gives the only rigid law (i.e., the orbit is open under the natural action of the linear group GL m (C) ) in N m . In N 7 there exists a one-parameter family of filiform algebras but there is no rigid law.
For m=8 the authors enumerate six continuous families with one complex parameter and fourteen algebras, one of which is rigid in N 8 ; this is the most remarkable fact. The methods use perturbations, which are the analogue of deformations in the language of nonstandard analysis, and similarity invariants for a nilpotent matrix. Notice that we do not have for the moment an example of a Lie algebra which is nilpotent and rigid in the variety of all the Lie algebras in dimension m ; such an algebra cannot be filiform.
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J. M. Ancochéa-Bermudez et M. Goze, Sur la classification des algèbres de Lie nilpotentes de dimension 7. C.R.A.S. Paris t 302. I.17, 611–613 (1986).
A.Cerezo, Les algèbres de Lie nilpotentes réelles et complexes de dimension 6. Publi. Univ. Nice27 (1983).
J. Dixmier, Sur les représentations unitaires des groupes de Lie nilpotents. Bull. Soc. Math. France85, 325–388 (1957).
M. Goze et J. M. Ancochéa-Bermudez, Classification des algèbres de Lie nilpotentes de dimension7. I.R.M.A. Strasbourg 1985.
M.Goze, Thèse Mulhouse 1982.
L.Magnin, Sur les algèbres de Lie nilpotentes de dimension ≦7. J. Geom. and Physics 1986.
V. V.Morozov, Classification des algèbres de Lie nilpotentes de dimension6. Isv Vyss. Ucebn. Zaved., Math. A190 (1958).
M.Vergne, Sur la variété des lois nilpotentes. Thèse 3è cycle Paris, 1966.
G.Vranceanu, Leçons de géométrie différentielle. 4. Edition de l'académie roumaine 1975.