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Classification des algèbres de Lie filiformes de dimension 8



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Ancochea Bermúdez , José María and Goze, Michel (1988) Classification des algèbres de Lie filiformes de dimension 8. Archiv der Mathematik, 50 (6). pp. 511-525. ISSN 0003-889X

Official URL: http://link.springer.com/article/10.1007%2FBF01193621



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.

Item Type:Article
Uncontrolled Keywords:classification; complex nilpotent filiform Lie algebras in dimension 8
Subjects:Sciences > Mathematics > Algebra
ID Code:21197
Deposited On:30 Apr 2013 17:26
Last Modified:12 Dec 2018 15:13

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