Sur le polynôme principal d'une Algèbre

Sans résumé     

La C*-algèbre d'une quasi-représentation

We show that the C^* -algebra of the regular representation of a discrete group G onto a subset Σ of G is the reduced C^* -algebra of an r-discrete groupoid whose space of units is totally disconnected and contains Σ as a dense subset. The C^*-algebra of quasicrystals, some Cuntz-Krieger and crossed product algebras, and Wiener-Hopf algebras are particular cases of this construction     

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     

Sur l'algèbre enveloppante d'une algèbre pré-Lie

We construct an associative product on the symmetric module $S(L)$ of any pre-Lie algebra L. It turns $S(L)$ into a Hopf algebra which is isomorphic to the envelopping algebra of $L_{\mathrm{Lie}}$. Then we prove that in the case of rooted trees our construction is dual to that of Connes and Kreimer. We also show that symmetric brace algebras and pre-Lie algebras are the same. Finally, we give a similar interpretation of the Hopf algebra of planar rooted trees. To cite this article: J.-M. Oudom, D. Guin, C. R. Acad. Sci. Paris, Ser. I 340 (2005).     

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.     

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.     

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).     

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.

     

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).     

Aur la vacuité du spectre d'un élément d'une algèbre\mathcal{L}\mathcal{F}

We give an example of a complete commutative unitary and semi-simple topological algebra, which is a locally convex inductive limit of an increasing sequence of Fréchet algebras ( algebra), and which contains the field (X) of rational functions; so it contains elements which have empty spectrum and therefore does not contain any character, neither continuous nor non-continuous. This unitary algebra is not a division algebra, so it contains at least one non-trivial maximal ideal; but none of its maximal ideals is closed and they all have infinite codimension. The Gelfand-Mazur Theorem remains therefore unknown for algebras.

     

Relèvement d'une mesure ergodique par un codage

Summary Let be a coding: BA, to which is associated a mapping from the set of ergodic measures on B to the set of ergodic measures on A . We derive a relation between the number of ergodic solutions of the equation where P is a given ergodic measure on A , and the ambiguousness of decoding for each of the solutions Q.

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Laboratoire de Probabilités, tour 46–56, 3 ème étage, 4, Place Jussieu, F-75230 Paris Cedex 05. Laboratoire associé au CNRS n°224 «Processus Stochastiques et Applications»     

Suite spectrale d'une extension d'algèbres de Leibniz

Leibniz homology is a non-commutative homology theory for Lie algebras which can be extended to a larger class of algebras: the Leibniz algebras. We construct a “Leibniz version” of the Hochschild-Serre spectral sequence.     

Lambda-Opérations sur l''Homologie d''une Algèbre de Lie de Matrices

One extends lambda operations from the cyclic homology of A to the homology of the Lie algebra gl(A) using exterior powers of matrices. One shows a formula of giving their behavior with respect to the direct sum of matrices. It uses the coproduct and the structure of ring object induced by the tensor product of matrices.     

Une Neutralisation Explicite de L'algèbre de Weyl Quantique Complétée

Soit p un nombre premier. Nous établissons l'existence de neutralisations de divers complétés de l'algèbre de Weyl quantique spécialisée en une racine de l'unité primitive d'ordre p (qui est “génériquement” une algèbre d'Azumaya) et donnons en particulier un énoncé de neutralisation explicite relevant celui construit en caractéristique p dans [3 Gros , M. , Le Stum , B. , Quiros , A. ( 2010 ). A Simpson correspondence in positive characteristic . Publ. RIMS Kyoto Univ. 46 : 1 – 35 .[Crossref], [Web of Science ®] , [Google Scholar]].

Let p be a prime number. We establish the existence of neutralizations of various completions of the quantum Weyl algebra specialized at a primitive root of unity of prime order p (which is “generically” an Azumaya algebra) and, in particular, we give a statement of explicit neutralization similar to the one built in characteristic p in [3 Gros , M. , Le Stum , B. , Quiros , A. ( 2010 ). A Simpson correspondence in positive characteristic . Publ. RIMS Kyoto Univ. 46 : 1 – 35 .[Crossref], [Web of Science ®] , [Google Scholar]].     

Analogues non-commutatifs et quasi-symétriques des fonctions de Hall-Littlewood,et modules de Demazure d'une algèbre enveloppante quantique dégénérée

We establish Weyl and Demazure character formulae for certain degenerate quantum groups which admit quasi-symmetric functions as characters. We obtain in this way a new action of the Hecke Algebra on polynomials. This allows us to define quasi-symmetric and non-commutative analogues of Hall-Littlewood functions.