Algèbres pré-Lie et algèbres de Hopf liées à la renormalisation

We describe the rôle of the notion of pre-Lie algebra in the combinatorics of renormalization, as formalized by Connes and Kreimer, and in the study of flows of vector fields.     

Algèbres de Poisson et Algèbres de Lie Résolubles

Let 𝔤 be a solvable Lie algebra and Q an (ad 𝔤)-stable prime ideal of the symmetric algebra S(𝔤) of 𝔤. If E denotes the set of nonzero elements of S(𝔤)/Q which are eigenvectors for the adjoint action of 𝔤 on S(𝔤)/Q, then the localization (S(𝔤)/Q) _E has a natural structure of Poisson algebra. We study this algebra here.

Soient 𝔤 une algèbre de Lie résoluble et Q un idéal premier (ad 𝔤)-stable de l'algèbre symétrique S(𝔤) de 𝔤. Si E est l'ensemble des éléments non nuls de S(𝔤)/Q qui sont vecteurs propres pour l'action adjointe de 𝔤 dans S(𝔤)/Q, l'algèbre localisée (S(𝔤)/Q) _E a une structure naturelle d'algèbre de Poisson. On étudie ici cette algèbre.     

Train Algèbres Alternatives: Relation Entre Nilindice et Degré

We determine train polynomials for power associative algebras and for alternative train algebras. We show bonds between polynomials and nilindices of some factors of the Peirce decomposition.     

Algèbres de Lie quasi-réductives

We study the class of algebraic Lie algebras for which the generic stabilizer of the coadjoint action is reductive modulo the center.     

Algèbres de Sobolev Généralisées

We examine the structure of a generalized Sobolev spaces. We also exhibit a class of locally uniformlyA-convex algebras, the unitization of which are not the same type.

     

Représentations des algèbres de clifford des formes binaires et ternaires cubiques

Clifford algebras of polynomial forms of degreed>2 defined by N. Roby are infinite dimensional when the module has rank bigger than 1. So the study of their representations differs from that of Clifford algebras of quadratic forms. Some authors have established several results in this domain, especially in the case of binary cubic forms, which can be found in the bibliography. In the first section of this paper, we consider the case of binary cubic forms and the base fieldK is algebraically closed. We define an explicit family of irreducible representations of their Clifford algebras, indexed by only one parameter, such that every 3-dimensional representation is equivalent to an element of this family. When the form is ternary cubic, Revoy and Tesser on one hand, Van den Bergh on the other, proved in two different ways, in [7] and [9] respectively, the existence of 3-dimensional linearizations. In the last section, we show directly that the number of classes of 3-dimensional representations is finite and non zero.     

Dimension de Hochschild des algèbres graduées

It is a basic fact that the global dimension of a connected $\mathbb{N}$-graded algebra coincides with the projective dimension of the trivial module. This result is recovered by proving that the Hochschild dimension is equal to the projective dimension of the trivial module. Thus the result becomes more natural with bimodules entering into the picture. To cite this article: R. Berger, C. R. Acad. Sci. Paris, Ser. I 341 (2005).     

Invariance spectrale des algèbres d'opérateurs pseudodifférentiels

We construct and study several algebras of pseudodifferential operators that are closed under holomorphic functional calculus. This leads to a better understanding of the structure of inverses of elliptic pseudodifferential operators on certain non-compact manifolds. It also leads to decay properties for the solutions of these operators. To cite this article: R. Lauter et al., C. R. Acad. Sci. Paris, Ser. I 334 (2002) 1095–1099.     

Distances entre exponentielles et puissances d’éléments de certaines algèbres de Banach

Let A be a Banach algebra which does not contain any nonzero idempotent element, let γ > 0, and let . We show that if then . We also show, assuming a suitable spectral condition on x, that if , then Received: 12 July 2006 Revised: 31 January 2007     

Représentations semi-stables et modules fortement divisibles

Sans résumé Oblatum 1-IV-1998 &9-VI-1998 / Published online: 14 January 1999     

Equations des algèbres Lie triple qui sont des algèbres train

In this paper, we consider equations of Lie triple algebras that are train algebras. We obtain two different types of equations depending on assuming the existence of an idempotent or a pseudo-idempotent.In general Lie triple algebras are not power-associative. However we show that their train equation with an idempotent is similar to train equations of power-associative algebras that are train algebras and we prove that Lie triple algebras that are train algebras of rank $\le 4$ with an idempotent are Jordan algebras.Moreover, the set of non-trivial idempotents has the same expression in Peirce decomposition as that of $e$-stable power-associative algebras.We also prove that the algebra obtained by $2$-gametization process of a Lie triple algebra is a Lie triple one.     

Algèbres localement uniformémentA-pseudo-convexes advertiblement complètes

We examine the structure and the spectral properties of locally uniformlyA-pseudo-convex algebras.      

Erratum to: Algèbres de lie quasi-réductives

In Corollary 12(ii) and Theorem 13(v) of [1] we omitted the hypothesis dim $ \mathfrak{z}\leq 1 $ . Moreover, in some places the symbol $ \mathbb{K} $ must be replaced by the symbol $ {{\mathbb{K}}^{\times }} $ .     

Dérivations des Algèbres de Weyl Quantiques Multiparamétrées et de Certaines de leurs Localisations Simples

ABSTRACT

On calcule une base explicite de l'espace des dérivations modulo les dérivations intérieures des algèbres de Weyl quantiques et de certaines de leurs localisations simples. On applique les résultats obtenus à des questions de séparation à équivalence de Morita près de ces algèbres.

We compute an explicit basis of the space of derivations modulo inner derivations for quantum Weyl algebras and some of their simple localizations. We apply these results to separate these algebras up to Morita equivalence.