Person: Ancochea Bermúdez, José María
Universidad Complutense de Madrid
Faculty / Institute
Álgebra, Geometría y Topología
Geometría y Topología
Now showing 1 - 10 of 45
PublicationOn the cohomology of Frobenius model Lie algebras(WALTER DE GRUYTER, 2004) Ancochea Bermúdez, José María; Campoamor-Stursberg, RutwigLie algebra model theory studies the closure O() of the Gln(C)-orbit in the variety Ln of Lie algebra laws of a law on Cn using nonstandard analysis. In this context, given a Lie algebra law , a contraction of is a law μ with μ 2 O(). On the other hand, a perturbation of in Ln is a μ 2 Ln such that the absolute value of the difference of the structure constants of and μ over a standard basis is smaller than any strictly positive real standard. Keeping this in mind, a Lie algebra g0 = (Cn, μ0) is called a model relative to a property (P) if any Lie algebra law μ satisfying (P)contracts to μ0 and any perturbation of μ0 satisfies (P). The property (P) studied in the article under review is the one to be Frobenius, i.e. the property that there exists a linear form ! 2 g on the 2n-dimensional Lie algebra g whose differential is symplectic, i.e. !n 6= 0. A family F of Lie algebras satisfying a property (P) is called a multiple model relative to (P) if any Lie algebra satisfying (P) contracts to a member of F and any perturbation of a member of F satisfies (P). M. Goze found in [C. R. Acad. Sci., Paris, S´er. I 293, 425–427 a multiple model for the above stated property (P). The authors of the present article compute first and second cohomology space of the Lie algebras in this family with respect to the adjoint representation. Furthermore, they compute the first obstruction for infinitesimal deformations to be prolonged which turns out to be zero. PublicationSur la classification des algèbres de Lie nilpotentes de dimension 7(Elsevier, 1986) Ancochea Bermúdez, José María; Goze, MichelThe 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 the similitude exponents of the nilpotent operators ad X with X in the complementary of the first derived algebra. The classification described in this paper is given by considering a fixed characteristic sequence. To illustrate this approach, one gives the classification of 7-dimensional complex nilpotent Lie algebras corresponding to the following sequences: (6,1) (filiforme case) and (5,1,1). Other cases and the complete proofs are given in a preprint edited by I.R.M.A Strasbourg (France). PublicationOn the classification of rigid Lie algebras(Academic Press, 2001-11-01) Goze, Michel; Ancochea Bermúdez, José MaríaAfter 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. PublicationClassification of Lie algebras with naturally graded quasi-filiform nilradicals(Elsevier, 2011) Ancochea Bermúdez, José María; Campoamor Stursberg, Otto Ruttwig; García Vergnolle, LucíaThe whole class of complex Lie algebras g having a naturally graded nilradical with characteristic sequence c(g) = (dim g − 2, 1, 1) is classified. It is shown that up to one exception, such Lie algebras are solvable. PublicationOn the nonrationality of rigid Lie algebras(America Mathematical Society, 1999) Ancochea Bermúdez, José María; Goze, MichelIn 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 defined over Q. The goal of this note is to provide a large number of examples of rigid but nonrational and nonreal Lie algebras. PublicationContractions of Low-Dimensional Nilpotent Jordan Algebras(Taylor & Francis, 2011) Ancochea Bermúdez, José María; Fresán, Javier; Margalef Bentabol, JuanIn this article, we classify the laws of three-dimensional and four-dimensional nilpotent Jordan algebras over the field of complex numbers. We describe the irreducible components of their algebraic varieties and extend contractions and deformations among their isomorphism classes. In particular, we prove that 2 and 3 are irreducible and that 4 is the union of the Zariski closures of the orbits of two rigid Jordan algebras PublicationTwo-step solvable Lie algebras and weight graphs(Birkhäuser, 2002) Ancochea Bermúdez, José María; Campoamor-Stursberg, RutwigIn this paper the authors propose a new approach to the study of weight systems. Instead of considering graphs whose vertices correspond to the generators of a Lie algebra (as for Cartan subalgebras in the semisimple case), the authors consider the whole weight system. The purpose is to extract information about the weight system from the geometry of the weights. The considerations are restricted to the case where a torus of derivations induces a decomposition of a nilpotent Lie algebra g into one-dimensional weight spaces, none of which is associated with the zero weight. The paper is structured as follows: In Section 2 the most important facts of weight systems of nilpotent Lie algebras and the root system associated to solvable Lie algebras are recalled. In Section 3 the authors formulate their conditions on the weight systems and analyze the consequences of these conditions on the structure of the weight system. They also define associated weight graphs and deduce their elementary geometrical properties. This provides a characterization of the three-dimensional Heisenberg Lie algebra in terms of trees. Section 4 is devoted to the study of certain subgraphs of a weight graph which can be used to reconstruct the weight system from the weight graph. If r is a semidirect product of g and a torus T these subgraphs determine bounds for the solvability class of r . In Section 5 these results are applied to obtain a geometrical proof of the nonexistence of two-step solvable rigid Lie algebras. PublicationOn the product by generators of characteristically nilpotent Lie S-algebras(Elsevier, 2003-11-01) Ancochea Bermúdez, José María; Campoamor-Stursberg, RutwigWe show that the product by generators preserves the characteristic nilpotence of Lie algebras, provided that the multiplied algebras belongs to the class of S-algebras. In particular, this shows the existence of nonsplit characteristically nilpotent Lie algebras h such that the quotient dim h−dim Z(h)=dim Z(h) is as small as wanted. PublicationSome remarks on the classification of graded nilpotent Lie algebras(Editorial Complutense, 2004) Ancochea Bermúdez, José María; Campoamor-Stursberg, RutwigWe present the classification of naturally graded nilpotent Lie algebras g with linear characteristic sequence in arbitrary dimension modulo a finite set of particular solutions corresponding to solutions with very small nilindex with respect to the nilindex of g.” PublicationSur les composantes irréductibles de la varieté des lois d'algèbres de Lie nilpotentes(Elsevier Science, 1996-01-15) Ancochea Bermúdez, José María; Gómez-Martín, José Ramón; Valeiras, Gerardo; Goze, MichelIn 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 complexity of the calculus. Thus we have developed a formal calculus for use in computers. The case of the dimension 8 is interesting because it is the first case where many nonfiliform components appear.