La forme trace d'une algèbre simple centrale de degré 4

Let k be a field of characteristic different from 2 containing a primitive 4-th root of unity. We show that the trace quadratic form of any central simple k-algebra A of degree 4 decomposes in the Witt group of k as the sum of a 2-fold Pfister form $q_2$ and a 4-fold Pfister form $q_4$ which are uniquely determined by A. The form $q_2$ is the norm form of the quaternion algebra Brauer-equivalent to $A{?}_{k}A$, and $q_4$ is hyperbolic if and only if A is a symbol algebra. To cite this article: M. Rost et al., C. R. Acad. Sci. Paris, Ser. I 342 (2006).     

Minima, pentes et algèbre tensorielle

Slopes of an adelic vector bundle exhibit a behaviour akin to successive minima. Comparisons between the two amount to a Siegel lemma. Here we use Zhang’s version for absolute minima over the algebraic numbers. We prove a Minkowski-Hlawka theorem in this context. We also study the tensor product of two hermitian bundles bounding both its absolute minimum and maximal slope, thus improving an estimate of Chen. We further include similar inequalities for exterior and symmetric powers, in terms of some lcm of multinomial coefficients.     

Une caractérisation de l'algèbre de Fourier pour certains groupes localement compacts

Let G be a separable locally compact group with type-I left regular representation, $\stackrel{?}{G}$ its dual and $A(G)$ its Fourier algebra. We prove an analogue of Parseval's theorem and that the mapping

$T?u(x):=\underset{\stackrel{?}{G}}{\int }Tr[T(\pi )\pi {(x)}^{?1}]{\mathrm{d}}_{\mu }(\pi )$

is an isometric isomorphism of Banach spaces from $L^1(\stackrel{?}{G})$ onto $A(G)$.     

Invariants relatifs : une algèbre extérieure

Let G be a complex reflection group acting on V, M be a finite dimensional G-module and S be the coordinate ring of V. Generalizing results of Orlik and Solomon, and of Shepler, we build an exterior algebra structure on the set of relative invariants (associated to a linear character of G) of the algebra $S?\Lambda (M^1)$. To cite this article: V. Beck, C. R. Acad. Sci. Paris, Ser. I 342 (2006).     

La forme seconde trace d'une algèbre simple centrale de degré 4 de caractéristique 2

Let A be a central simple algebra of degree 4 over a field k of characteristic 2 and let $q_A$ be the quadratic form on A given by the second coefficient of the reduced characteristic polynomial. We show that A uniquely determines a 2-fold Pfister form $q_2$ and a 4-fold Pfister form $q_4$ such that $q_A=[1,1]+q_2+q_4$ in the Witt group of k, where $[1,1]$ is the form $x^2+xy+y^2$. The form $q_2$ is the norm form of the quaternion algebra Brauer-equivalent to $A{?}_{k}A$, and $q_4$ is hyperbolic if and only if A is cyclic. To cite this article: J.-P. Tignol, C. R. Acad. Sci. Paris, Ser. I 342 (2006).     

Le degré de similarité d'une algèbre d'opérateurs

Let H be a Hilbert space and let A ⊂B(H) be a unital closed subalgebra (not necessarily self adjoint). We introduce the “degree of similarity” of A, denoted by d(A), which is a non-negative integer or possibly ∞, and we study its significance in connection with the notions of amenability for groups and operator algebras.     

Sur l'algébre enveloppante d'une algébre de Lie nilpotente

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.     

Sur la Topologie m -Convexe d’une algèbre localement A -convexe

In a locallyA-convex algebra (E, τ) we consider the associatedm-convex topologym(τ). We show that the completion ofE with respect tom(τ) is always a locallyA-convex algebra contained in the complete locally convex space obtained from (E, τ). The topologym(τ) is also used to characterize locally boundedly multiplicatively convex algebras among locallyA-convex ones.

     

Algèbre Pré-Gerstenhaber à homotopie près

This paper is concerned by the concept of algebra up to homotopy for a structure defined by two operations . and $[,]$. An important example of such a structure is the Gerstenhaber algebra (i.e. commutatitve structure with degree 0 and Lie structure with degree ?1). The notion of Gerstenhaber algebra up to homotopy ($G_{\infty }$ algebra) is known: it is a codifferential bicogebra.Here, we give a definition of pre-Gerstenhaber algebra (pre-commutative and pre-Lie) allowing a similar construction for a $\text{pre}{G}_{\infty }$ algebra.Given a structure of pre-commutative (Zinbiel) and pre-Lie algebra and working over the corresponding Koszul dual operads, we will give an explicit construction of the associated pre-Gerstenhaber algebra up to homotopy: it is a bicogebra (Leibniz and permutative) equipped with a codifferential which is a coderivation for the two coproducts.     

Représentation de Weil associée à une représentation d'une algèbre de Jordan

In the present Note we construct the Weil representation of the Kantor-Koecher-Tits Lie algebra g associated to a simple real Jordan algebra V. Afterwards we introduce a family of integral operators intertwining the Weil representation with the infinitesimal representations of the degenerate principal series of the conformai group of a Jordan algebra V. We apply this result to the case conformal group SL(2r, ℝ).     

Systèmes récursifs et algèbre de hadamard de suites récurrentes linéaires sur des anneaux commutatifs.

Abstract

Let 𝒢 be a simple finite dimensional Lie algebra over the complex numbers and let 𝒢¯ = 𝒢₁ ⊕…⊕ 𝒢 _k be a regular semisimple subalgebra of 𝒢 with each 𝒢 _i being a simple algebra of type A or C. It is shown that the lattice of submodules of a generalized Verma 𝒢-module constructed by parabolic induction starting from a simple torsion free 𝒢¯-module is almost always isomorphic to the lattice of submodules of an associated module formed as a quotient of a classical Verma module by a sum of Verma submodules. In particular, it is shown that the Mathieu admissible Verma modules involved have maximal submodules which are the sum of Verma modules.     

Advertible complétude et structure deQ-algèbre

We give different characterizations of algebra types (Q-property, advertible completeness, MackeyQ-property) in terms of the relation ρ<-β and sequentiality notions in locally pseudo-convex algebras.

     

Les espaces Lp d'une algèbre de von Neumann définies par la derivée spatiale

This work answers a question raised by A. Connes (on the spatial theory of von Neumann algebras, preprint, Inst. Hautes Études Sci., France) and generalizes for a general von Neumann algebra the theory of non-commutative integration of J. Dixmier (Bull. Soc. Math. France81 (1953)) and I. Segal (Ann. of Math.57 (1973)).     

Sur le polynôme principal d'une Algèbre

Sans résumé     

Interprétation motivique de la formule d'excès d'intersection

Following Voevodsky, one can associate a Gysin triangle in the category of triangulated motives to every couple (X,Z) such that X is a smooth scheme and Z is a smooth closed subscheme of X. We study the functoriality of this triangle: if one considers a morphism $\text{f}\text{:Y→X}$ which is transversal to Z, this functoriality is simple. We are rather interested in the case where f is not quite transversal to Z. In that case, one obtains a motivic excess intersection formula and a ramification formula in the sense of equicharacteristic discrete valuation rings. To cite this article: F. Déglise, C. R. Acad. Sci. Paris, Ser. I 338 (2004).