Sur les a Algèbres Préhilbertiennes Vérifiant ∥a 2∥ ≤ ∥a∥2

Résumé.  Soit A une algèbre réelle sans diviseurs de zéro. On suppose que l’espace vectoriel A est muni d’une norme ∥.∥ préhilbertienne vérifiant ∥a ²∥ ≤ ∥a∥² pour tout . Alors A est de dimension finie dans chacun des quatre cas suivants :

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.     

Résultats combinatoires sur les modules de Demazure

In this paper, we introduce the notion of polytopes with integral structure, which generalizes the notion of usual polytopes whose vertices have integral coordinates. Then we consider a symmetrizable Kac-Moody algebra. For every dominant weight λ and every element of the Weyl group w, we construct a polytope with integral structure such that its Ehrhart polynomial describes the dimension of the Demazure module of highest weight nλ associated to w.     

Propriétés Analytiques du Spectre dans les Algèbres de Jordan-Banach

If f() is an analytic function from a domain D of the complex plane into a Jordan-Banach algebra we prove that Sp f() is an analytic multivalued function. From this derives the subharmonicity of Log (f()), where denotes the spectral radius. We apply these results to prove that a Jordan-Banach algebra A is associative if and only if the spectral radius is subadditive and submultiplicative on A and to prove that A/Rad A is isomorphic to the complex plane if and only if each element of A has only one point in its spectrum.

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Le travail du permier auteur a été subventionné par le Conseil de recherches en sciences naturelles et en génie du Canada (subvention A 7668)     

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.

     

Critère d’existence d’idempotent basé sur les algèbres de Rétrocroisement

We study the relationship of backcrossing algebras with mutation algebras and algebras satisfying ω-polynomial identities: we show that in a backcrossing algebra every element of weight 1 generates a mutation algebra and that for any polynomial identity f there is a backcrossing algebra satisfying f. We give a criterion for the existence of idempotent in the case of baric algebras satisfying a nonhomogeneous polynomial identity and containing a backcrossing subalgebra. We give numerous genetic interpretations of the algebraic results.     

Identités Multilinéaires de Degré 4 pour les Algèbres de Bernstein et de Mutation Non Commutatives

We give the identities generating the spaces of degree 4 multilinear identities satisfied by the non commutative 0th-order Bernstein algebras and the non commutative mutation algebras. For it, we use a method reducing the search for multilinear identities to the resolution of linear systems.     

Quelques résultats sur les déformations équivariantes des courbes stables

Let G be a finite group, let g≥2 and g ′ ≥ 0 be integers. We introduce the algebraic stack classifying the stable curves of genus g endowed with an action of G faithful in each geometric fiber and such that the quotient of each fiber is a semi-stable curve of genus g′. We study the completion of the local rings of this algebraic stack. They are closely related to universal equivariant deformation rings R_C,G of stable curves endowed with a faithful action of G. A useful tool for this purpose is a local-global principle generalizing the one obtained by Bertin and Mézard in [BM00]. We then use the results we already proved in [Mau03b] and [Mau03a] to describe some properties of the space (purity, dimension).     

Résolutions de type Bernstein-Gel'fand-Gel'fand pour les super-algèbres de Virasoro N = 1

In this note, we study the structure of Verma modules and construct Bernstein-Gel'fandGel' fand type resolutions for Neveu-Schwarz and Ramond algebras.     

Théorèmes sur les groupes de substitutions

Sans résumé     

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.

     

Sur les identités polynomiales vérifiées par les algèbres de rétrocroisement

We study the ideal of polynomial identities of a single indeterminate satisfied by all backcrossing algebras. For this we distinguish two categories according to whether or not these algebras satisfy an identity for the plenary powers. For each category, we give the generators for the vector space of identities, a condition for any object belonging to one of these two categories verify a given identity, a necessary and su?cient condition that a polynomial is an identity and we study the existence of an idempotent element. We give a method which brings the search of identities satified by the backcrossing algebras to the solution of linear systems and we illustrate this method by constructing generators of homogeneous and non homogeneous identities of degrees less than 8.     

Quelques remarques sur les actions analytiques des réseaux des groupes de Lie de rang supérieur

All the actions considered here are (real) analytic. Let г be a subgroup of finite index of SL(n,). We prove, in particular, the (global) homotopical rigidity, for both its standard affine action on the torus T^a (n ≥ 3), and its standard projective action on the sphere Sⁿ⁻¹ (n, ≥ 4).     

Sur les systèmes distingués de représentants et sur les compléments normaux des sous-groupes de Hall

Sans résumé Présenté par G. Hajós     

Problème de la courbure scalaire prescrite sur les variétés riemanniennes complètes

On a complete Riemannian manifold of dimension n⩾3, we study the prescribed scalar curvature problem, especially in the null case. Under some hypotheses, we show, when the scalar curvature is zero, the existence of ε>0 such that any f∈C^∞ satisfying |f|<εinf[1,r^−(2+λ)], is the scalar curvature of a complete conformal metric; here λ>0 and r denotes the distance to a fixed point.