%0 Journal Article
%A Ancochea Bermúdez , José María
%A Goze, Michel
%T Classification des algèbres de Lie filiformes de dimension 8
%D 1988
%@ 0003-889X
%U https://hdl.handle.net/20.500.14352/58451
%X 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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