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High-input agriculture has been associated with a drastic reduction of within-field crop genetic diversity, while plant (mostly functional) diversity in natural ecosystems has been shown to promote ecosystem functioning. Increasing intra-specific diversity in agroecosystems is a promising strategy to stabilize crop productivity and promote the associated diversity of fauna and microbiota. We investigated the effect of the within-field diversity of bread wheat varieties on arbuscular mycorrhizal fungi (AMF) and two enzymatic activities involved in organic nitrogen and phosphorus mineralization. A field experiment was designed to test whether the number of mixed wheat varieties in a plot, or their functional diversity (previously assessed), influence the abundance and diversity of AMF and the activity of leucine aminopeptidases and phosphatases in the root zone. The AMF abundance was measured by quantitative polymerase chain reaction, community composition was analyzed by Illumina metabarcoding on two AMF specific markers, and potential microbial activities were quantified by biochemical assays. Wheat traits related to root morphology and susceptibility to fungal diseases previously quantified for each variety were also used. Number of varieties significantly increased AMF abundance in roots, whereas functional cluster number did not, with no impact of root morphology. Functional cluster number influenced AMF diversity, though weakly and not linearly, responding most to binary mixtures. Both wheat variety and functional group number increased the potential leucine amino-peptidase activities in the root zone, while no effect was observed for phosphatase activities. Our results highlight that increasing crop intra-specific diversity triggered changes in key processes involved in nutrient acquisition.

International audience; Alley cropping agroforestry systems are complex agroecosystems highlighted for their positive effects on soil quality. However, the potential spatial heterogeneity of soil quality created by tree rows at the plot scale has seldom been studied. The aim of this study was to evaluate soil quality at the plot scale, under tree rows and along transects perpendicular to the tree row and to compare alley cropping systems with monocropping systems. This study was performed on an alley cropping system that combined hybrid walnut trees (21 years old) and peas. Topsoil was sampled at tree rows between 1 and 2, 2 and 4 and 4 and 6.5 m from the tree row in the alley cropping system, as well as in a neighbouring monocropping plot. Physical, chemical and microbiological indicators of soil quality were measured. Tree row implantation induced spatial heterogeneity in the chemical indicators, microbial biomass, activities and community structure at the alley cropping plot scale. Alley cropping not only improved microbiological soil quality indicators within the tree rows but also in the interrows when compared to a monocropping system. These indicators were then integrated into one soil quality index (SQI) built through a statistical approach. The soil quality index was calculated for the monocropping plot and for each position within the alley cropping plot. After 21 years of agroforestry practice, tree rows and permanent grass cover improved the SQI until 2 m in the interrow. Weighted SQIs were calculated relative to the surface area of each location for the entire alley cropping plot (i.e., tree row + interrow positions) and for the entire alley cropping interrow (i.e., removing the tree row surface area). The weighted SQI of the entire alley cropping plot significantly increased compared with that of the monocropping plot.

International audience; The Kontsevich formality can be viewed as a non-linear map ℱ from the L∞ algebra of poly-vector fields on ℝd to the space of poly-differential operators. The space of the half-homogenous poly-vector fields is a sub-L∞ algebra. We prove here that the restriction of ℱto this subspace is weakly analytic.

Background & Aims: Better understanding of belowground interactions in agroforestry systems is crucial for the success of plant co-existence. Beyond root competition, associated arbuscular mycorrhizal (AM) fungi can also be involved in plant to plant interactions. Thus far, the contribution of each agroforestry component (trees, herbaceous vegetation beneath trees -HVbT- and crops) in the establishment and maintenance of AM communities is poorly documented, particularly in temperate areas. This study investigates the spatio-temporal dynamics of both roots and AM fungi in two alley-cropping sites located in southwestern France. Methods: Over a one-year period, (i) root length density, production and distribution, (ii) AM activity (root mycorrhization rate and extra-radical hyphal production) and (iii) AM diversity (metabarcoding) were assessed at different distances from tree rows in two agroforestry systems. Results: The mycorrhization rate and hyphal production increased at the interface between tree rows and cultivated alleys, showing a positive effect of the presence of a perennial system (tree and HVbT) and of plant diversity. Compared to HVbT, tree roots colonized farther into superficial layers of the cultivated alleys. However, due to higher root densities and well-established AM fungi observed throughout all the year, HVbT appeared to be more relevant in maintaining an active source of AM inoculum for newly developing crop roots in winter. Conclusion: The spatial proximity of roots and common AM fungi provides new perspectives in deciphering the significance of arbuscular mycorrhizal communities in crop nutrition and yield in agroforestry systems.

Resume. On etudie le concept d'algebre a homotopie pres pour une structure definie par deux operations . et [ , ]. Un exemple important d'une telle structure est celui d'algebre de Gerstenhaber (avec une structure commutative de degre 0 et une structure de Lie de degre −1). La notion d'algebre de Gerstenhaber a homotopie pres (G∞ algebre) est connue : c'est une bicogebre codifferentielle. Ici nous proposons une definition d'algebre pre-Gerstenhaber (pre-commutative et pre-Lie) permettant la construction similaire d'une preG∞ algebre. Partant d'une structure pre-commutative (Zinbiel) et pre-Lie, on utilise les operades duales correspondantes, qui sont de Koszul. Nous donnons la construction explicite de l'algebre a homotopie pres associee. Celle-ci est une bicogebre (Leibniz et permutative), munie d'une codifferentielle qui est une coderivation des deux coproduits. Abstract. 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∞ 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 preG∞ 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.

Arbuscular mycorrhizal (AM) fungi are crucial for plant nutrition and the sustainability of agroforestery systems. However, the contribution of each agroforestry component (i.e. trees, under-tree herbaceous vegetation -UtHV- and crops) in the establishment and maintenance of AM communities is poorly documented particularly in temperate areas. This study investigates the spatio-temporal dynamics of AM fungi colonizing roots of the three agroforestry components in southwest France. Standing fine root length density and production, AM activity (colonization and extraradical hyphal growth) and diversity of walnut trees, UtHV and soft wheat were assessed over one year in two agroforestry systems at different distances from the perennial tree-UtHV alley. Compared to UtHV, trees showed a higher ability to colonize superficial layers far into the cultivated alleys due to their wider root system in summer. However, due to higher root densities and well established AM fungi observed throughout all the year, UtHV appeared to be more ecologically relevant to maintain an active source of AM inoculum for newly developing crop roots in winter. High degree of root proximity and similarity of AM fungal communities among the three agroforestry components provide new perspectives in deciphering the significance of common mycorrhizal networks in plant to plant interactions.

We give a concrete realization of the Plancherel measure for a semi-direct product $$N \rtimes H$$ where N and H are vector groups for which the linear action of H on N is almost everywhere regular. A procedure using matrix reductions produces explicit (orbital) parameters by which a continuous field of unitary irreducible representations is realized and the almost all of the dual space of $$N \rtimes H$$ naturally has the structure of a smooth manifold. Using the simplest possible field of positive semi-invariant operators, the Plancherel measure is obtained via an explicit volume form on a smooth cross-section $$\Sigma $$ for almost all H-orbits. The associated trace characters are also shown to be tempered distributions.

International audience; In this paper, we study the restriction of the Kontsevich formality to the subalgebra of the linear polyvectors in the algebra of polyvector fields on Rd.We prove that this formality is an analytic map.; Dans ce papier nous étudions la restriction de la formalité de Kontsevich à la sous-algèbre des tenseurs linéaires, dans l’algèbre de Lie des tenseurs surRd. On établit que c’est une formalité analytique.

The simple $GL(n,\mathbb {C})$ -modules are described by using semistandard Young tableaux. Any semistandard skew tableau can be transformed into a well defined semistandard tableau by a combinatorial operation, the Schutzenberger jeu de taquin. Associated to the classical Lie groups $SP(2n,\mathbb {C})$ , $SO(2n+1,\mathbb {C})$ , there are other notions of semistandard Young tableaux and jeux de taquin. In this paper, we study these various jeux de taquin, proving that each of them has a simple and explicit formulation as a step-by-step sliding. Any of these jeux de taquin is the restriction of the orthogonal one, associated to $SO(2n+1,\mathbb {C})$ .

International audience; In this work, we describe the H-invariant, so(n)-relative cohomology of a natural class of osp(n|2)-modules M, for n ≠ 2. The Lie superalgebra osp(n|2) can be realized as a superalgebra of vector fields on the superline R1|n. This yields canonical actions on spaces of densities and differential operators on the superline. The above result gives the zero, first, and second cohomology spaces for these modules of densities and differential operators.

In this paper, we recall combinatorial basis for shape and reduced shape algebras of the Lie algebras gl ( n ) , sp ( 2 n ) and so ( 2 n + 1 ) . They are given by semistandard and quasistandard tableaux. Then we generalize these constructions to the case of the Lie superalgebra spo ( 2 n , 2 m + 1 ) . The main tool is an extension of Schutzenberger's jeu de taquin to these algebras.

The online version of this article (doi:10.1007/s11104-016-3127-3) contains supplementary material, which is available to authorized users.; AimsPhosphorus (P) is frequently limiting crop production in agroecosystems. Large progress was achieved in understanding root traits associated with P acquisition efficiency (PAE, i.e. P uptake achieved under low P conditions). Most former studies were performed in controlled environments, and avoided the complexity of soil-root interactions. This may lead to an oversimplification of the root-soil relations. The aim of the present study was, therefore, to identify the dominant root and rhizosphere-related traits determining PAE, in contrasting soil conditions in the field. Methods Twenty-three maize hybrids were grown at two contrasting P levels of a long-term P-fertilizer trial in two adjacent soil types: alkaline and neutral. Bulk soil, rhizosphere and root parameters were studied in relation to plant P acquisition. ResultsSoil type had robust effect on PAE. Hybrids’ performance in one soil type was not related to that in the other soil type. In the neutral soil, roots exhibited higher specific root length, higher root/shoot ratio but lower PAE. Best performing hybrids in the neutral soil were characterized by top soil exploration, i.e., greater root surface and topsoil foraging. In contrast, in the alkaline soil, PAE and foraging traits were not correlated, P availability in the rhizosphere was greater than the bulk soil and phosphatase activity was higher, suggesting a ‘mining strategy’ in that case (i.e. traits that facilitate elevated P availability). Conclusions These results indicate the key role of environmental factors for roots traits determining high PAE. The study highlights the need to consider soil properties when breeding for high PAE, as various soil types are likely to require different crop ideotypes.

Presenting the structure equation of a hom-Lie algebra 𝔤, as the vanishing of the self commutator of a coderivation of some associative comultiplication, we define up to homotopy hom-Lie algebras, which yields the general hom-Lie algebra cohomology with value in a module. If the hom-Lie algebra is quadratic, using the Pinczon bracket on skew symmetric multilinear forms on 𝔤, we express this theory in the space of forms. If the hom-Lie algebra is symmetric, it is possible to associate to each module a quadratic hom-Lie algebra and describe the cohomology with value in the module.

International audience; Let V be a finite dimensional real vector space, let g be the real span of a finite set of commuting endomorphisms of V, and G = exp g. We study the orbit structure in elements of a finite partition of V into explicit G-invariant connected sets. In particular, we prove that either there is an open conull G-invariant subset Omega of V in which every G-orbit is regular, or there is a G-invariant, conull, G(delta) subset of V in which every orbit is not regular. We present an explicit computable necessary and sufficient condition for almost everywhere regularity. Finally in the case of regularity we construct an explicit topological cross-section for the orbits in Omega.

Let \(G\) be a connected and simply connected Lie group with Lie algebra \(\mathfrak g \). We say that a subset \(X\) in the set \(\mathfrak g ^\star / G\) of coadjoint orbits is convex hull separable when the convex hulls differ for any pair of distinct coadjoint orbits in \(X\). In this paper, we define a class of solvable Lie groups, and we give an explicit construction of an overgroup \(G^+\) and a quadratic map \(\varphi \) sending each generic orbit in \(\mathfrak g ^\star \) to a \(G^+\)-orbit in \(\mathfrak{g ^+}^\star \), in such a manner that the set \(\varphi (\mathfrak g ^\star _{gen}){/ G^+}\) is convex hull separable. We then call \(G^+\) a weak quadratic overgroup for \(G\). Thanks to this construction, we prove that any nilpotent Lie group, with dimension at most 7 admits such a weak quadratic overgroup. Finally, we produce different examples of solvable Lie groups, having weak quadratic overgroups, but which are not in our class of Lie groups and for which usual constructions fail to hold.

For class R, type I solvable groups of the form NH, N nilpotent, H abelian, we construct an explicit layering with cross-sections for coadjoint orbits. We show that any ultrafine layer Ω has a natural structure of fiber bundle. The description of this structure allows us to build explicit local canonical coordinates on Ω.

Any unitary irreducible representation π of a Lie group G defines a moment set Iπ, subset of the dual g⁎ of the Lie algebra of G. Unfortunately, Iπ does not characterize π. If G is exponential, there exists an overgroup G+ of G, built using real-analytic functions on g⁎, and extensions π+ of any generic representation π to G+ such that Iπ+ characterizes π.In this paper, we prove that, for many different classes of group G, G admits a quadratic overgroup: such an overgroup is built with the only use of linear and quadratic functions.

Given an exponential Lie group G, we show that the constructions of B. Currey, 1992, go through for a less restrictive choice of the Jordan-Holder basis. Thus we obtain a stratification of g * into G-invariant algebraic subsets, and for each such subset Ω, an explicit cross-section Σ C Ω for coadjoint orbits in Ω, so that each pair (Ω, Σ) behaves predictably under the associated restriction maps on g * . The cross-section mapping σ: Ω → Σ is explicitly shown to be real analytic. The associated Vergne polarizations are not necessarily real even in the nilpotent case, and vary rationally with ∈ Ω. For each Ω, algebras e 0 (Ω) and e 1 (Ω) of polarized and quantizable functions, respectively, are defined in a natural and intrinsic way. Now let 2d > 0 be the dimension of coadjoint orbits in Ω. An explicit algorithm is given for the construction of complex-valued real analytic functions {q 1 , q 2 , ..., q d } and {p 1 ,p 2 ,...,p d } such that on each coadjoint orbit O in Ω, the canonical 2-form is given by Σdp k ^ dq k . The functions {q 1 ,q 2 , ..., q d } belong to e 0 (Ω), and the functions {p 1 ,p 2 , ... ,p d } belong to e 1 (Ω). The associated geometric polarization on each orbit O coincides with the complex Vergne polarization, and a global Darboux chart on O is obtained in a simple way from the coordinate functions (p 1 , ... , p d , q 1 , ... , q d ) (restricted to O). Finally, the linear evaluation functions l ↦ l(X) are shown to be quantizable as well.

Using a formality on a Poisson manifold, we construct a star product and for each Poisson vector field a derivation of this star product. Starting with a Poisson action of a Lie group, we are able under a natural cohomological assumption to define a representation of its Lie algebra in the space of derivations of the star product. Finally, we use these results to define some generically tangential star products on duals of Lie algebra as in [1] but in a more realistic context.

We introduce the Chevalley cohomology for the graded Lie algebra of polyvector ?elds on Rd. This cohomology occurs naturally in the problem of construction and classi?cation of formalities on the space Rd. Considering only graph formalities, that is, formalities de?ned with the help of graphs as in the original construction of Kontsevich, we de?ne (as the ?rst and third authors did earlier for the Hochschild cohomology) the Chevalley cohomology directly on spaces of graphs. More precisely, observing ?rst a noteworthy property for Kontsevich?s explicit formality on Rd, we restrict ourselves to graph formalities with that property. With this restriction, we obtain some simple expressions for the Chevalley coboundary operator; in particular, we can write this cohomology directly on the space of purely aerial, nonoriented graphs. We also give examples and applications.

A formality on a manifold M is a quasi isomorphism between the space of polyvector fields (Tpoly(M)) and the space of multidifferential operators (Dpoly(M)). In the case M=R d , such a mapping was explicitly built by Kontsevich, using graphs drawn in configuration spaces. Looking for such a construction step by step, we have to consider several cohomologies (Hochschild, Chevalley, and Harrison and Chevalley) for mappings defined on Tpoly. Restricting ourselves to the case of mappings defined with graphs, we determine the corresponding coboundary operators directly on the spaces of graphs. The last cohomology vanishes.

In the paper entitled “Separation of representations with quadratic overgroups”, we defined the notion of quadratic overgroups, and announced that the 6-dimensional nilpotent Lie algebra g 6 , 20 admits such a quadratic overgroup. There is a mistake in the proof. The present Erratum explains that the proposed overgroup is only weakly quadratic, and g 6 , 20 does not admit any natural quadratic overgroup.

We consider Kontsevich star products on the duals of Lie algebras. Such a star product is relative if, for any Lie algebra, its restriction to invariant polynomial functions is the usual pointwise product. Let \(\mathfrak{g}\) be a fixed Lie algebra. We shall say that a Kontsevich star product is \(\mathfrak{g}\)-relative if, on \(\mathfrak{g}\)*, its restriction to invariant polynomial functions is the usual pointwise product. We prove that, if \(\mathfrak{g}\) is a semi-simple Lie algebra, the only strict Kontsevich \(\mathfrak{g}\)-relative star products are the relative (for every Lie algebras) Kontsevich star products.

On considere un groupe de Lie exponentiel G, d'algebre de Lie g, et une orbite coadjointe O d'un point ξ de g * , dual de g, munie d'une carte de Darboux globale (p i ,q i ). On designe par A l'ensemble des restrictions a O des fonctions polynomiales complexes sur g * qu'on identifie a un sous espace d'une completion A* de C[p i , q i ] et on montre que la completion de l'ensemble des fonctions quantifiables contient les generateurs p i et q i . Enfin on montre que deux cartes globales (p i , q i ) et (p' i ,q' i ) de l'orbite O, associees a une meme polarisation h definissent la meme completion A de A et deux produits star * et *' equivalents sur A.

Resume Dans [3], Maxim Kontsevich construit explicitement, pour chaque structure de Poisson a sur l'espace ℝd, un produit star pilote par α. Lorsque a est lineaire, donc lorsque ℝd est le dual d'une algebre de Lie g, Kontsevich compare ce produit avec celui defini dans [2] par S. Gutt. La presence de ≪ roues ≫ dans les graphes de Kontsevich fait que ces produits different. Si g est nilpotente, les roues n'apparaissent pas et les deux produits coincident. Le but de cette Note est d'etablir directement ce resultat en calculant les coefficients du produit de Kontsevich dans ce cas particulierement simple.

Soit g une algebre de Lie simple complexe et O min son orbite coadjointe (non triviale) de dimension minimale. En utilisant un star-produit sur un revetement de O min (ou d'un ouvert dense de O min ), on donne une construction explicite des realisations minimales de g.

We give a new proof of a weak Paley-Wiener theorem for nilpotent Lie groups due to Lipsman and Rosenberg and we introduce a general notion of Q.U.P for any unimodular locally compact group.

Let $T_{\operatorname{poly}}(\mathbb{R}^d)$ denote the space of skew-symmetric polyvector fields on $\mathbb{R}^d$, turned into a graded Lie algebra by means of the Schouten bracket. Our aim is to explore the cohomology of this Lie algebra, with coefficients in the adjoint representation, arising from cochains defined by linear combination of aerial Kontsevich graphs. We prove that this cohomology is localized at the space of graphs without any isolated vertex, any "hand" or any "foot". As an application, we explicitly compute the cohomology of the "ascending graphs" quotient complex.

Soit G un groupe de Lie exponentiel, d'algebre de Lie g. Soit f un point du dual g* de g. Si h1 et h2 sont des polarisations en f qui satisfont la condition de Pukanszky, alors les representations π i = ind G H i χ f ( i = 1, 2, H i = exp h i , χ f ( exp X ) = e if ( X ) pour X dans g) sont irreductibles et equivalentes. Le principal obstacle a la construction d'un operateur d'entrelacement pour ces representations est de prouver la convergence d'une certaine integrale. Dans cet article, nous presentons un operateur d'entrelacement explicite pour π1 et π2 qui evite cette difficulte.

Soient $G$ un groupe de Lie connexe et $ \pi$ une représentation unitaire de $G$ dans un espace de Hilbert ${\mathcal H}_{\pi}$ non séparable. Pour chaque ensemble infini $I$, on définit la représentation unitaire $\hat{\pi}_I=(\# I) \pi$ de $G$ dans l'espace ${\mathcal H}_{\hat{\pi}_I}=(\# I){\mathcal H}_{\pi}$. Alors, on montre que l'ensemble moment généralisé de $\hat{\pi}_I$ caractérise $\pi$ à quasi-équivalence près.

Resume Si n + est le facteur nilpotent d'une algebre semi-simple g , le cone diamant de g est la description combinatoire d'une base d'un n + module indecomposable naturel. Cette notion a ete introduite par N.J. Wildberger pour sl ( 3 ) , le cone diamant de sl ( n ) est decrit dans Arnal (2006) [2] , celui des algebres semi-simples de rang 2 dans Agrebaoui (2008) [1] . Dans cet article, nous generalisons ces constructions au cas des algebres de Lie sp ( 2 n ) . Les tableaux de Young semi-standards symplectiques ont ete definis par C. De Concini (1979) [4] , ils forment une base de l'algebre de forme de sp ( 2 n ) . Nous introduisons ici la notion de tableaux de Young quasi standards symplectiques, ces derniers decrivent le cone diamant de sp ( 2 n ) .

The space of linear polyvector fields on $$\mathbb{R}^d$$ is a Lie subalgebra of the (graded) Lie algebra $$T_{\rm poly}(\mathbb{R}^d)$$ , equipped with the Schouten bracket. In this paper, we compute the cohomology of this subalgebra for the adjoint representation in $$T_{\rm poly}(\mathbb{R}^d)$$ , restricting ourselves to the case of cochains defined with purely aerial Kontsevich’s graphs, as in Pac. J. Math. 218(2):201–239, 2005. We find a result which is very similar to the cohomology for the vector case Pac. J. Math. 229(2):257–292, 2007.

In \cite{W}, there is a graphic description of any irreducible, finite dimensional $\mathfrak{sl}(3)$ module. This construction, called diamond representation is very simple and can be easily extended to the space of irreducible finite dimensional ${\mathcal U}\_q(\mathfrak{sl}(3))$-modules. In the present work, we generalize this construction to $\mathfrak{sl}(n)$. We show this is in fact a description of the reduced shape algebra, a quotient of the shape algebra of $\mathfrak{sl}(n)$. The basis used in \cite{W} is thus naturally parametrized with the so called quasi standard Young tableaux. To compute the matrix coefficients of the representation in this basis, it is possible to use Groebner basis for the ideal of reduced Pl\"{u}cker relations defining the reduced shape algebra.

The space of smotth functions and vector fields on $\R^d$ is a Lie subalgebra of the (graded) Lie algebra $T\_{poly}(\R^d)$, equipped with the Scouten bracket. In this paper, we compute the cohomology of this subalgebra for the adjoint representation in $T\_{poly}(\R^d)$, restricting ourselves to the case of cochains defined with purely aerial Kontsevich's graphs, as in [AGM]. We find results which are very similar to the classical Gelfand-Fuchs and de Wilde-Lecomte one.

In a recent article, Cattaneo, Felder and Tomassini explained how the notion of formality can be used to construct flat Fedosov connections on formal vector bundles on a Poisson manifold $M$ and thus a star product on $M$ through the original Fedosov method for symplectic manifolds. In this paper, we suppose that $M$ is a fibre bundle manifold equipped with a Poisson tensor tangential to the fibers. We show that in this case the construction of Cattaneo-Felder- Tomassini gives tangential (to the fibers) star products.

We first generalize a result by Bavula on the sl(2) cohomology to the osp(1∣2) cohomology and then we entirely compute the cohomology for a natural class of osp(1∣2) modules M. We study the restriction to the sl(2) cohomology of M and apply our results to the module M=Dλ,μ of differential operators on the superline acting on densities.

The contractions of Lie groups and Lie algebras and their representations are studied geometrically. We prove they can be defined by deformations in Poisson algebras of symplectic manifolds on which the groups act. These deformations are given by Dirac constraints which induce on C∞ functions on the deformed manifold an associative twisted product, characterizing the contracted group or its representations. We treat the contractions of SO(n) to E(n) and apply this theory to thermodynamical limits in spin systems.

An algebraic study of the irreducible representations of the complex Lie algebra sl(2) is presented in this article. This study generalizes a former series of works of W. Miller. Though the list is not complete, it gives hints as to the construction of a very vast family of representations.

Resume Un prototype des algebres de Gerstenhaber est l'espace T poly ( R d ) des champs de tenseurs sur R d muni du produit exterieur et du crochet de Schouten. Dans cet article, on decrit explicitement la structure de la G ∞ algebre enveloppante d'une algebre de Gerstenhaber. Cette structure permet de definir une cohomologie de Chevalley–Harrison sur cette algebre. On montre que cette cohomologie a valeur dans R n'est pas triviale dans le cas de la sous algebre de Gerstenhaber des tenseurs homogenes T poly hom ( R d ) .