José María Ancochea Bermúdez

Classification des algèbres de Lie nilpotentes complexes de dimension 7

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 was introduced by them in an earlier paper [C. R. Acad. Sci. Paris Ser. I Math. 302 (1986), no. 17, 611–613; in which they determined the nilpotent complex Lie algebras corresponding to the characteristic sequences (6, 1) and (5, 1, 1). The paper under review contains no proofs; for details the authors refer to another article [the authors, “Classification des algebres de Lie nilpotentes de dimension 7”, Univ. Louis Pasteur, Strasbourg, 1986; per bibl.].

On the varieties of nilpotent Lie algebras of dimension 7 and 8

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 reducible.

ON LIE ALGEBRAS WHOSE NILRADICAL IS (n — p)-FILIFORM

We prove first that every (n — p)-filiform Lie algebra, p ≤ 3, is the nilradical of a solvable, nonnilpotent rigid Lie algebra. We also analize how this result extends to (n — 4)-filiform Lie algebras. For this purpose, we give a classificaction of these algebras and then determine which of the obtained classes appear as the nilradical of a rigid algebra.

On certain families of naturally graded Lie algebras

In this work large families of naturally graded nilpotent Lie algebras in arbitrary dimension and characteristic sequence (n; q; 1) with n ≡ 1(mod 2) satisfying the centralizer property are given. This centralizer property constitutes a generalization, for any nilpotent algebra, of the structural properties characterizing the Lie algebra Qn. By considering certain cohomological classes of the space H2(g;C), it is shown that, with few exceptions, the isomorphism classes of these algebras are given by central extensions of Qn by Cp which preserve the nilindex and the natural graduation.

Completable filiform Lie algebras

We determine the solvable complete Lie algebras whose nilradical is isomorphic to a filiform Lie algebra. Moreover we show that for any positive integer n there exists a solvable complete Lie algebra whose second cohomology group with values in the adjoint module has dimension at least n.

SYMPLECTIC FORMS AND PRODUCTS BY GENERATORS

ABSTRACT 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 Lie algebras of certain products by generators.

Classification of (n−5)-filiform Lie algebras

Abstract In this paper we consider the problem of classifying the (n−5) -filiform Lie algebras. This is the first index for which infinite parametrized families appear, as can be seen in dimension 7. Moreover we obtain large families of characteristic nilpotent Lie algebras with nilpotence index 5 and show that at least for dimension 10 there is a characteristic nilpotent Lie algebra with nilpotence index 4 which is the algebra of derivations of a nilpotent Lie algebra.

ON 2-ABELIAN (n – 5)-FILIFORM LIE ALGEBRAS

We classify the (n − 5)-filiform Lie algebras which have the additional property of a non-abelian derived subalgebra. Moreover we show that if a (n − 5)-filiform Lie algebra is characteristically nilpotent, then it must be 2-abelian.

An irreducible component of the variety of Leibniz algebras having trivial intersection with the variety of Lie algebras

We show the rigidity of a parameterized family of solvable Leibniz non-Lie algebras in arbitrary dimension, obtaining an irreducible component in the variety that does not intersect the variety of Lie algebras non-trivially. Moreover it is shown that for any the Abelian Lie algebra appears as the algebra of derivations of a solvable Leibniz algebra.We show the rigidity of a parameterized family of solvable Leibniz non-Lie algebras in arbitrary dimension, obtaining an irreducible component in the variety L epsilon(n) that does not intersect the variety of Lie algebras non-trivially. Moreover it is shown that for any n >= 3 the Abelian Lie algebra a(n) appears as the algebra of derivations of a solvable Leibniz algebra.

Completeness of quasi-filiform Lie algebras

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.