Sur les idéaux induits dans les algèbres enveloppantes

Résumé Soient une algèbre de Lie de dimension finie et une sousalgèbre de Lie de . Je montre que l'induction, qui envoie les idéaux de l'algèbre enveloppante de dans les idéaux de l'algèbre enveloppante de , commute, à une torsion près, à l'anti-automorphisme principal. La résolution libre standard de la représentation triviale d'une algèbre de Lie joue un rôle important dans la démonstration.

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Summary Let be a finite dimensional Lie algebra, and a Lie subalgebra. I show that the induction map, between the sets of ideals in the enveloping algebras of and , commutes, up to a twisting, with the principal anti-automorphism. The standard free resolution of the trivial representation of a Lie algebra plays an important role in the proof.

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Convention. On fixe un corps commutatifk. Tous les espaces vectoriels, produits tensoriels et algèbres considérés dans cet article sont surk.

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Notations. Si g est une algèbre de Lie, on noteU(g) son algèbre enveloppante. On noteu l'anti-automorphisme principal deU(g): c'est l'antiautomorphisme tel que pourXg. Si est une forme linéaire sur g, nulle sur [g,g], on noteuu l'automorphisme deU(g) tel queX =X+(X) pourXg. Si g est de dimension finie, la fonction module de g est la forme linéaireXtradX.     

Autour de la condition d'engel dans les algèbres de bernstein

We know that, in general, an algebra satisfying an Engel's condition is a nilalgebra. But an Engel's condition don't implies necessarily the nilpotency of the algebra. In this paper we show that every Bernstein algebra satisfying the second or the third Engel's condition is genetic, that is, the kernel of his weight fonction is nilpotent. This is also the case for a Bernstein algebra satisfying the second weak Engel's condition.

On sait que, en général, toute algèbre vérifiant une condition d'Engel est une nilalgèbre. Cependant, une condition d'Engel n'entraîne pas nécessairement la nilpotence de l'algèbre. Nous montrons, dans ce papier, que taute algèbre de Bernstein vérifiant la deuxième ou la troisième condition d'Engel est génétique, c'est-à-dire, le noyau de sa pondération est nilpotent. C'est aussi le cas pour la deuxième condition faible d'Engel.     

Factorisation et existence de l'unité dans les algèbres de Banach commutatives

We give two theorems on the existence of the unit in commutative Banach algebras. As corollaries, we obtain results of V. Runde and P. G. Dixon.     

Homologie de Quillen pour les algèbres de Poisson

We describe an explicit complex which calculates the Quillen homology for Poisson algebras. As a consequence, we show that in the smooth case the Quillen homology coincides with the Poisson homology introduced by Koszul and Brylinski.     

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.     

Décomposition spectrale dans des algèbres munies d'un star-produit

Summary The purpose of this paper is to study, in intrinsic way, the Moyal's product, defined in the flat space R ²ⁿ. This product is defined here with the twisted convolution and the Fourier transform. The S(R ²ⁿ) and L²(R ²ⁿ) spaces are_*5-algebras. Because of this definition, the_*V-product of some tempered distributions is defined. Let O _M ^v be the set of multiplication operators in S(R ²ⁿ). By transposition, the S(R ²ⁿ) space is a right-module on O _M ^v . The support of f_*v g is different from the support of f·g; under large enough hypotheses, there is a Taylor's formula for the star-product function of the v variable. The ^v space of the multiplication operators in L²(R ²ⁿ) is defined here as the space of tempered distributions, the image of which is the set of bounded operators in L²(R ²ⁿ) by the Weyl map. After the study of ^v space, it is possible to show the spectral resolution of the real elements of ^v or of O _M ^v , which satisfies a, probably superfluous, hypothesis.     

Niveau hermitien de Certaines algèbres de quaternions

Let D [d] =(a,b/F) a quaternion divisior algebra over a field F of characteristic ? 2. Denote 1, i, j , k the basis of D, such that i²[d] n, j²[d] b, ij [d] -ji [d] k and A :D → D the involution given by i [d] -i, j [d] j (and k [d] k). In [LE] D. LEWIS asks the following question :Does there exist a quadratic Pfister form [S p. 721 [d] such that the hermitian form [d] [d] D is isotropic over (D, [d]) but not hyperbolic &; In this note, we show that the answer of this question is negative, so that the hermitien level [§I], when it is finite, of (D, A) is a power of two. This result holds for quaternion algebras with standard involution [LE].     

Sur les algèbres de Lie quasi-filiformes admettant un tore de dérivations

Résumé  Dans ce travail on décrit les algèbres de Lie quasi-filiformes de rang non nul. De plus, on rappelle et corrige la classification des algèbres de Lie filiformes admettant un tore de dérivations, ainsi que la liste des algèbres graduées naturellement et quasi-filiformes.      

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).     

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.     

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.     

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.     

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)