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.     

Représentations induites d'algèbres de Hecke affines et singularités de R-matrices

We give an explicit criterion for the irreducibility of some induction products of evaluation modules of affine Hecke algebras of type A. This allows to describe the form of the zeroes and poles of the trigonometric R-matrix associated to any evaluation module of U_ν(sl_N).     

Sur les représentations linéaires des groupes clos

Sans résumé     

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.     

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.     

Variété des lois d'algèbres de Lie nilpotentes

Let N _nbe the variety of n-dimensional complex Lie algebra laws. For n12, two irreducible components in the variety N _nare described. The characteristically nilpotent filiform Lie algebras are studied.     

Sur l'approximation uniforme des opérateurs linéaires compacts par des opérateurs non linéaires de rang fini

Sans résumé     

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.     

Homogénéisation de problèmes non linéaires dans des milieux quasi-stratifiés

This Note deals with the homogenization of some nonlinear problems. The aim is to characterize the weak limit of solutions of a particular nonlinear type of equations such that - div (Q^∈ G(x, N^∈ ∇u^∈)) = ∫^∈, where the matrices Q^ɛ and (N^ɛ)⁻¹ have specific dependence on coordinates. Under suitable assumptions, we prove that the weak limit also satisfies an equation of such type.     

Représentations extraordinaires

Cet article constitue une partie importante de ma thèse. Je l'ai écrit sous la direction avisée de Henri Carayol, à qui je dois un très grand merci car ce sont ses conseils et son optimisme qui m'ont permis d'en venir à bout. Je tiens également à remercier Laurent Clozel pour toutes ses remarques, utiles et pertinentes.Correspondance de Langlands entre représentations de GL(2) et représentations galoisiennes aux places extraordinaires Université Louis Pasteur     

Deux théorèmes de non-annulation de valeurs spéciales de fonctions L

In answer to questions recently raised by Merel [Mer], we prove two non-vanishing theorems for the central value of automorphic L-functions: let p be prime and let χ be a primitive character modulo p. Then for all p large enough 1. If χ is not quadratic and even, there exists a primitive weight 2 form f of level p with . 2. If χ is quadratic and even, then there exists a primitive weight 2 form f of level p with . Received: 12 March 2000 / Revised version: 26 September 2000     

Sur la cohomologie des super algèbres de Lie étranges

In this paper we compute the cohomology with trivial coefficients for the Lie superalgebraspsl(n, n), p (n) andq(2n); we show that the cohomology ring ofq(2n+1) is of Krull dimension 1 and we calculate the ring forq(3) andq(5). The last section is devoted to a result about the cohomology of a Lie superalgebra with reductive even part with coefficients in a finite dimensional moduleM.     

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).