No full-text available
To read the full-text of this research,
you can request a copy directly from the authors.
A topological property of the set of characteristically nilpotent Lie algebras
Abstract
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 Elsevier SAS. Tous droits réservés.
ResearchGate has not been able to resolve any citations for this publication.
We describe the class of complex filiform nilpotent Lie algebras provided with a not trivial external torus of derivations.
We prove also that, for dimensions greater than 8, any algebraic irreducible component of the variety of complex nilpotent
filiform laws of Lie algebra contains an open set whose elements are characteristically nilpotent laws.
We develop the notion of deformations using a valuation ring as ring of coefficients. This permits to consider in particular the classical Gerstenhaber deformations of associative or Lie algebras as infinitesimal deformations and to solve the equation of deformations in a polynomial frame. We consider also the deformations of the enveloping algebra of a rigid Lie algebra and we define valued deformations for some classes of non associative algebras.
Before giving a axiomatic description of the non classical or non standard Analysis (N.S.A.), we present some examples which demonstrate its efficiency and elegance-. They are particularly revealing as they can be treated with Infinitesimal methods in a way which is economic and precise while the classical approach leads to results whose statements and proofs are sometimes heavy and clumsy. These examples are purely algebraic; they concern the roots of polynomials. Non Standard Analysis provides well adapted tools (for example, the concept of perturbation, the decomposition of a point …), thus avoiding the use of the classical arsenal; although this is universal and hence very powerful, it involves deep results in analysis which are peripheral to our algebraic interest (e.g. the Implicit functions theorem, inverse function theorem).
In this paper we develop a method of construction of complex rigid solvable Lie algebras which is independent of coho-mological techniques or classification of Lie algebras. As an application, we classify all rigid solvable Lie algebras in dimension less or equal than eight, and we obtain partial results in dimension nine. Moreover, we give several examples of families of rigid Lie algebras in arbitrary dimension, some of them having its second cohomology group, in the Chevalley cohomology, non trivial.
The aim of this paper is to present remarkable classes of Lie-admissible algebras containing in particular the associative algebras, the Vinberg algebras and pre-Lie algebras. We determine the associated quadratic operads and their dual operads.
Sur les algèbres caractéristiquement nilpotentes
- R Carles
- Carles, R.
R. Carles, Sur les algèbres caractéristiquement nilpotentes, Publ. Univ. Poitiers 5 (1984).
On the rigidity of solvable Lie algebras, Deformation Theory of Algebras and Structures and Applications
- J M Ancochea
- Ancochea, J.M.
Perturbations of Lie algebra structures, Deformation Theory of Algebras and Structures and Applications
- M Goze
- Goze, M.