
José María Ancochea BermudezComplutense University of Madrid | UCM · Departamento de Geometría y Topología
José María Ancochea Bermudez
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Publications
Publications (49)
The class of rank one solvable Lie algebras possessing a maximal torus t with eigenvalue spectrum spec(t)=(1,4,5,…,n+2) is studied in the context of rigidity. It is shown that from the value n ≥ 18, three isomorphism classes of rigid Lie algebras exist, two of them being algebraically rigid, and the third being geometrically rigid with a two-dimens...
It is shown that the two-known series of rank one
and rank two
finite-dimensional solvable rigid Lie algebras with non-vanishing second cohomology can be extended to solvable rigid Lie algebras of arbitrary rank
such that the cohomology is preserved exactly. For the second series, it is further proved that an extension decreasing the cohomology ex...
It is shown that for a finite-dimensional solvable rigid Lie algebra r, its rank is upper bounded by the length of the characteristic sequence c(n) of its nilradical n. For any characteristic sequence c=(n1,⋯,nk,1), it is proved that there exists at least a solvable Lie algebra rc the nilradical of which has this characteristic sequence and that sa...
The complete classification of real solvable rigid Lie algebras possessing a nilradical of dimension at most six is given. Eleven new isomorphism classes of indecomposable algebras are obtained. It is further shown that the resulting solvable Lie algebras have a vanishing second Chevalley cohomology group, thus correspond to algebraically rigid Lie...
We show the rigidity of a parameterized family of solvable Leibniz non-Lie algebras in arbitrary dimension, obtaining an irreducible component in the variety [Inline formula] that does not intersect the variety of Lie algebras non-trivially. Moreover it is shown that for any [Inline formula] the Abelian Lie algebra [Inline formula] appears as the a...
We show that the only indecomposable solvable Leibniz non-Lie algebra L0L0 with nilradical of maximal nilpotence index is rigid in any dimension, and moreover that it is complete, i.e., only possesses inner derivations. The possible contractions of L0L0 onto Lie algebras are obtained.
It is proved that for any quasi-filiform of non-zero rank the solvable Lie algebra obtained by adjoining a maximal torus of outer derivations is complete. Further, for any positive integer m, it is shown that there exist solvable complete Lie algebras with the second Chevalley–Eilenberg cohomology group of arbitrary dimension.
The whole class of complex Lie algebras g having a naturally graded nilradical with characteristic sequence c(g)=(dimg−2,1,1) is classified. It is shown that up to one exception, such Lie algebras are solvable.
In this paper 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 them. In particular, we prove that J2 and J3 are irreducible
and that J4 is the union of the Zari...
On détermine les classes d'isomorphisme des algèbres de Jordan en dimension deux sur le corps des nombres réels. En utilisant des techniques d'Analyse Non Standard, on étudie les propriétés de la variété des lois d'algèbres de Jordan, et aussi les contractions parmi ces algèbres.
R�sum�. We present all real solvable algebraically rigid Lie algebras of dimension lower or equal than eight. We point out the differences
that distinguish the real and complex classification of solvable rigid Lie algebras.
This paper has been withdrawn by the author due to a crucial error in
example.
This paper consists of a description of the variety of two dimensional
associative algebras within the framework of Nonstandard Analysis. By
decomposing each algebra in A^2 as sum of a Jordan algebra and a Lie algebra,
we calculate the isomorphism classes of two dimensional real associative
algebras over the field of real numbers and determine the...
We determine the isomorphism classes of Jordan algebras in dimension two over the field of real numbers. Using techniques of non-standard analysis we study the properties of the variety of Jordan algebras, and also the contractions among these algebras.
On montre qu’une algèbre de Lie résoluble rigide réelle n’est pas nécessairement complètement résoluble. On construit un exemple n⊕tn⊕t de dimension minimale dont le tore extérieur tt n’est pas formé par des dérivations ad-semi-simples sur RR. Nous étudions les formes réelles des nilradicaux des algébres de résolubles rigides en dimension n ⩽ 7 et...
We prove first that every (n-p)-filiform Lie algebra, p≤3, is the nilradical of a solvable, nonnilpotent rigid Lie algebra. We also analyze how this result extends to (n-4)-filiform Lie algebras. For this purpose, we give a classification of these algebras and then determine which of the obtained classes appear as the nilradical of a rigid algebra....
We show that a solvable real rigid Lie algebra is not completelt rigid, by constructing an example of minimal dimension where the external torus is not spanned by $ad$-semisimple derivations over $\mathbb{R}$. We analyze the real forms of nilradicals of solvable rigid Lie algebras in dimensions $n\leq 7$ and give the real classification for dimensi...
In the variety of finite dimensional nilpotent Lie algebras over the field of complex numbers, the set of characteristically nilpotent Lie algebras is not closed. In this Note we show that it is also not open. To cite this article: J.M. Ancochea Bermúdez et al., C. R. Acad. Sci. Paris, Ser. I 337 (2003). © 2003 Académie des sciences. Publié par Els...
We give examples of a family of nonsolvable complete Lie algebras isomorphic to the Lie algebra of derivations of a proper ideal.
By using the concept of weight graph associated to nonsplit complex nilpotent Lie algebras
\mathfrakg\mathfrak{g}, we find necessary and sufficient conditions for a semidirect product
\mathfrakgÅ® Ti\mathfrak{g}\overrightarrow{\oplus } T_{i} to be two-step solvable, where $T_{i}<T$T_{i}<T is a subalgebra of a maximal torus of derivations TT ove...
We introduce the product by generators of two nilpotent Lie algebras as a central extension of the direct sum and analyze symplectic structures on them. We show that, up to few exceptions, these products do not admit symplectic forms. Besides a general criterion, we indicate a procedure to construct symplectic forms in natural manner on quotient Li...
In [6] and [7] the author introduces the notion of filiform Lie superalgebras, generalizing the filiform Lie algebras studied by Vergne in the sixties. In these appers, the superalgebras whose even part is isomorphic to the model filiform Lie algebra Ln are studied and classified in low dimensions. Here we consider a class of superalgebras whose ev...
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.
We construct characteristically nilpotent Lie algebras with characteristic sequence (2m,1,1) in any even dimension 2m≥8 which are central extensions of nilradicals of complete rigid laws.
We review the known results about characteristically nilpotent complex Lie algebras, as well as we comment recent developements in the theory.
We construct large families of characteristically nilpotent Lie algebras by considering deformations of the Lie algebra g_{m,m-1}^{4} of type Q_{n},and which arises as a central extension fo the filiform Lie algebra L_{n}. By studying the graded cohomology spaces we obtain that the sill algebras are isomorphic to the nilradicals of solvable, comple...
In his thesis, Carles made the following conjecture: Every rigid Lie algebra is dened on the eld Q. This was quite an interesting question because a positive answer would give a nice explanation of the fact that simple Lie algebras are dened over Q. The goal of this note is to provide a large number of examples of rigid but nonrational and nonreal...
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 is rigid then T is a maximal torus on n . The authors then study algebras of this form where n is a filiform...
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 symbols of the nilpotent linear maps ad x with x in the complement of the derived subalgebra. This invariant w...
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 ["Varietes des algebres de Lie nilpotentes'', These de 3 eme cycle, Univ. Paris, Paris, 1966; BullSi...
Soit \( {{\mathcal{L}}^{n}} \)
la variété des lois d’algèbres de Lie de dimension n sur un corps K algébriquement clos et de caractéristique nulle. La structure de variété algébrique de \( {{\mathcal{L}}^{n}} \)
est définie comme suit: soient \( \mu \in {{\mathcal{L}}^{n}} \)
et (e
1, e
2, …, e
n
) une base fixée de K
n
. Les constantes de structur...