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Goze, Michel ; Ancochea Bermúdez , José María | Koninklijke Nederlandse Akademie van Wetenschappen | 1985The goal in this article is to give a constructive method describing the n-dimensional rigid Lie algebras ?, with "rigid'' meaning, in the simplest sense, that every Lie algebra law sufficiently close to ? is isomorphic to it. The authors use Li[...]texto impreso
Let Ln Kn3 be the affine variety of all n-dimensional Lie algebras over an algebraically closed field of characteristic zero. An element g 2 Ln can be identified with the bilinear mapping ? defining the multiplication in g. If f is an element o[...]texto impreso
In this article the authors study filiform nilpotent Lie algebras n which possess a given torus T of semisimple derivations. The solvable Lie algebras obtained by a semidirect product T?n depend, up to isomorphism, on one or many parameters (con[...]texto impreso
Ancochea Bermúdez , José María ; Goze, Michel | Université de Paris VII, U.E.R. de Mathématiques | 1989An n -dimensional complex Lie algebra is rigid if its orbit under the canonical action of the full linear group is open in the variety defined by the Jacobi identities. The authors have perfected a method for obtaining solvable rigid Lie algebra[...]texto impreso
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 t[...]texto impreso
The authors give a complete list of the 7-dimensional complex nilpotent Lie algebras. This classification is obtained by using an invariant of nilpotent Lie algebras, called a characteristic sequence and defined by the maximum of the Segre symbo[...]texto impreso
This work synthesizes a number of articles about varieties of Lie algebras. The methods employed lie in the framework of nonstandard analysis. The notion of deformation for the variation of Lie algebra laws is replaced here by a perturbation sat[...]texto impreso
After having given the classification of solvable rigid Lie algebras of low dimensions, we study the general case concerning rigid Lie algebras whose nilradical is filiform and present their classification.texto impreso
In his thesis, Carles made the following conjecture: Every rigid Lie algebra is defined on the field Q. This was quite an interesting question because a positive answer would give a nice explanation of the fact that simple Lie algebras are defin[...]texto impreso
Let Nn be the variety of n-dimensional complex nilpotent Lie algebras. We know that this algebraic variety is reducible for n?11 and irreducible for n?6. In this work we prove that N7 is composed of two algebraic components and that N8 is also r[...]texto impreso
One knows that a solvable rigid Lie algebra is algebraic and can be written as a semidirect product of the form g=T?n if n is the maximal nilpotent ideal and T a torus on n . The main result of the paper is equivalent to the following: If g [...]texto impreso
The authors give an approach to the classification of 7-dimensional nilpotent algebras. They illustrate the use of the invariant of nilpotent Lie algebras, called characteristic sequence, and defined by the maximum of the ordered sequences of th[...]texto impreso
Ancochea Bermúdez, José María ; Gómez-Martín, José Ramón ; Valeiras, Gerardo ; Goze, Michel | Elsevier Science | 1996-01-15In this paper we determine all the components fo the variety of complex nilpotent Lie algebras of dimension 8. The technique is similar to that used for the smaller dimensions. But in this case big difficulties appear resulting from the complexi[...]texto impreso
The scheme of the Lie algebras of dimension n is reducible for n?2 and the number of its components is bounded asymptotically by exp(n/4) . The same problem is studied by the authors for the subvariety N n of the nilpotent Lie algebras of di[...]texto impreso
Let Nn be the variety of nilpotent Lie algebra laws of a given complex vector space Cn. M. Vergne showed ["Variété des algèbres de Lie nilpotentes'', Thèse de 3ème cycle, Spéc. Math., Paris, 1966; BullSig(110) 1967:299; Bull. Soc. Math. France 9[...]
