Didier Arnal

University of Burgundy | UB · Institut de Mathématiques de Bourgogne (IMB)

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 F to this subspace is weakly analytic.

Cambridge Core - Abstract Analysis - Representations of Solvable Lie Groups - by Didier Arnal

The diamond cone is a combinatorial description for a basis of a natural indecom-posable n-module, where n is the nilpotent factor of a complex semisimple Lie algebra g. After N. J. Wildberger who introduced this notion, this description was achieved for g = sl(n) , the rank 2 semisimple Lie algebras and g = sp(2n). In this work, we generalize thes...

The diamond cone is a combinatorial description for a basis of an indecomposable module for the nilpotent factor n of a complex semi-simple Lie algebra. After N. J. Wildberger who introduced this notion, this description was achieved for sl}(n), the rank 2 semi-simple Lie algebras and sp(2n). In this work, we generalize these constructions to the L...

Describing the Gelfand construction for the analytic states on an universal enveloping algebra, we characterize pure states and re-find the main result of a preceding work with L. Abdelmoula and J. Ludwig on the separation of unitary irreducible representations of a connected Lie group by their generalized moment sets. Mathematics Subject Classific...

We give a concrete realization of the Plancherel measure for a semi-direct product N⋊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 a...

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 z...

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 Schützenberger jeu de taquin. Associated to the classical Lie groups \(SP(2n,\mathbb {C})\), \(SO(2n+1,\mathbb {C})\), there are othe...

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 symmet...

Résumé On étudie le concept d'algèbre à homotopie près pour une structure définie par deux opérations . et [,]. Un exemple important d'une telle structure est celui d'algèbre de Gerstenhaber (avec une structure commutative de degré 0 et une structure de Lie de degré −1). La notion d'algèbre de Gerstenhaber à homotopie près (G∞ algèbre) est connue :...

Given a symmetric non degenerated bilinear form b on a vector space V , G. Pinczon and R. Ushirobira defined a bracket { , } on the space of multilinear skewsymmetric forms on V. With this bracket, the quadratic Lie algebra structure equation on (V, b) becomes simply {Ω, Ω} = 0. We characterize similarly quadratic associative, commutative or pr...

Let V be a finite dimensional real vector space, let (Formula presented.) be the real span of a finite set of commuting endomorphisms of V, and (Formula presented.). 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 su...

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

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 gi...

We study the regularity of orbits for the natural action of a Lie subgroup G of GL(V), where V is a finite dimensional real vector space. When G is connected, abelian, and satisfies a certain rationality condition, we show that there are two possibilities: either there is a G-invariant Zariski open set Ω in which every orbit is regular, or there is...

Let Tpoly(Rd) denote the space of skew-symmetric polyvector fields on Rd, 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 co...

In this paper, we first study the shape algebra and the reduced shape algebra for the Lie superalgebra $\mathfrak{sl}(m,n)$. We define the quasistandard tableaux, their collection is the diamond cone for $\mathfrak{sl}(m,n)$, which is a combinatorial basis for the reduced shape algebra. We realize a bijection between the set of semistandard tableau...

After recalling the construction of a graded Lie bracket on the space of cyclic multilinear forms on a vector space V, due to Georges Pinczon and Rosane Ushirobira, we prove this construction gives a structure of quadratic associative algebra, up to homotopy, on V. In the associative case, it is easy to refind the associated usual Hochshild cohomol...

The diamond cone is a combinatorial description for a basis of an indecomposable module for the nilpotent factor $\mathfrak n$ of a semi simple Lie algebra. After N. J. Wildberger who introduced this notion, this description was achevied for $\mathfrak{sl}(n)$, the rank 2 semi-simple Lie algebras and $\mathfrak{sp}(2n)$. In the present work, we gen...

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 (commutatitve and Lie). The notion of Gerstenhaber algebra up to homotopy ($G_\infty$ algebra) is known. Here, we give a definition of pre-Gerstenhaber algebra (...

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 g6,20g6,20 admits such a quadratic overgroup. There is a mistake in the proof. The present Erratum explains that the proposed overgroup is only weakly quadratic, a...

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 mom...

Let G be a connected Lie group and it a unitary representation of G on a non separable Hilbert space H(pi). For any infinite set I, we define the representation (pi) over cap (I) = (#I)pi of G on H((pi) over capI) = (#I)H(pi). Then, we show that the generalized moment set of (pi) over cap (I) characterize pi up to quasi-equivalence.

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. KeywordsSolvable Lie grou...

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+ characte...

A way to separate irreducible unitary representations pi for a Lie group G by moment sets is to use an infinite-dimensional overgroup (G) over tilde and extensions of each representation pi to a representation (pi) over tilde of (G) over tilde, in such a manner that the moment set of (pi) over tilde characterizes pi. In this paper we propose a univ...

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

Si est le facteur nilpotent d'une algèbre semi-simple , le cône diamant de est la description combinatoire d'une base d'un module indécomposable naturel. Cette notion a été introduite par N.J. Wildberger pour , le cône diamant de est décrit dans Arnal (2006) [2], celui des algèbres semi-simples de rang 2 dans Agrebaoui (2008) [1].

The space $T_{poly}(\mathbb R^d)$ of all tensor fields on $\mathbb R^d$, equipped with the Schouten bracket is a Lie algebra. The subspace of ascending tensors is a Lie subalgebra of $T_{poly}(\mathbb R^d)$. In this paper, we compute the cohomology of the adjoint representations of this algebra (in itself and $T_{poly}(\mathbb R^d)$), when we restr...

The diamond cone is a combinatorial description for a basis in a indecomposable module for the nilpotent factor n+ of a semi simple Lie algebra. After N.J. Wildberger who introduced this notion for sl(3), this description was achevied by N. Bel Baraka, N.J. Wildberger and D. A. for sl(n) and by B. Agrebaoui and ourselves for the rank 2 semi-simple...

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-Hölder basis. Thus we obtain a stratification of g* into G-invariant algebraic subsets, and for each such subset Ω, an explicit cross-section ∑ ⊂ Ω for coadjoint orbits in Ω, so that each pair (Ω ∑) behaves pre...

The diamond cone is a combinatorial description for a basis in a indecomposable module for the nilpotent factor n+ of a semi simple Lie algebra. After N.J. Wildberger who introduced this notion for sl(3), this description was achevied by N. Bel Baraka, N.J. Wildberger and D. A. for sl(n) and by B. Agrebaoui and ourselves for the rank 2 semi-simple...

We entirely compute the cohomology for a natural and large class of $\mathfrak{osp}(1|2)$ modules $M$. We study the restriction to the $\mathfrak{sl}(2)$ cohomology of $M$ and apply our results to the module $M={\mathfrak D}_{\lambda,\mu}$ of differential operators on the super circle, acting on densities.

Let $\pi$ be an unitary irreducible representation of a Lie group $G$. $\pi$ defines a moment set $I_\pi$, subset of the dual $\mathfrak g^*$ of the Lie algebra of $G$. Unfortunately, $I_\pi$ does not characterize $\pi$. However, we sometimes can find an overgroup $G^+$ for $G$, and associate, to $\pi$, a representation $\pi^+$ of $G^+$ in such a m...

The fundamental example of Gerstenhaber algebra is the space $T_{poly}({\mathbb R}^d)$ of polyvector fields on $\mathbb{R}^d$, equipped with the wedge product and the Schouten bracket. In this paper, we explicitely describe what is the enveloping $G_\infty$ algebra of a Gerstenhaber algebra $\mathcal{G}$. This structure gives us a definition of the...

The present work is a part of a larger program to construct explicit combinatorial models for the (indecomposable) regular representation of the nilpotent factor $N$ in the Iwasawa decomposition of a semi-simple Lie algebra $\mathfrak g$, using the restrictions to $N$ of the simple finite dimensional modules of $\mathfrak g$. Such a description is...

Looking to the separation of irreducible unitary representations of an exponential Lie group G through the image of their moment map, we propose here a new way: instead to extend the moment map to the universal enveloping algebra of G, we define a non linear mapping Φ from the dual of the Lie algebra gg of G to the dual g+∗g+∗ of a larger solvable...

Looking to the separation of irreducible unitary representations of an exponential Lie group G through the image of their moment map, we propose here a new way: instead to extend the moment map to the universal enveloping algebra of G, we define a non linear mapping Phi from the dual of the Lie algebra g of G to the dual g(+*) of a larger solvable...

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 t...

The space of linear polyvector fields on \mathbbRd\mathbb{R}^d is a Lie subalgebra of the (graded) Lie algebra Tpoly(\mathbbRd)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 Tpoly(\mathbbRd)T_{\rm poly}(\mathbb{R}^d), restricting our...

The space of smooth 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. Here 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 graphs, as in Arna...

We show that every unitary representation π of a connected Lie group G is characterized up to quasi-equivalence by its complete moment set.Moreover, irreducible unitary representations π of G are characterized by their moment sets.

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 $...

The space of smooth functions and vector fields on ℝd is a Lie subalgebra of the (graded) Lie algebra Tpoly(ℝd), equipped with the Scouten bracket. Here we compute the cohomology of this subalgebra for the adjoint representation in Tpoly(ℝd), restricting ourselves to the case of cochains defined with purely aerial Kontsevich graphs, as in Arnal, Ga...

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 equi...

We introduce the Chevalley cohomology for the graded Lie algebra Tpoly(Rd) of polyvector fields on Rd. This cohomology occurs naturally in the problem of construction and classification of formalities on the space Rd. Consi- dering only graph formalities, i.e. formalities defined with the help of graphs like in the original construction of Kontsevi...

We show that when the methods of (2) are combined with the explicit stratification and orbital parameters of (9) and (10), the result is a construction of explicit analytic canonical coordinates for any coadjoint orbit O of a completely solvable Lie group. For each layer in the stratification, the canonical coordinates and the orbital cross-section...

We consider a type I, solvable Lie group G, of the form R ◊ Rd. We show that every irreducible unitary representation of G is characterized by its generalized moment set.

A formality on a manifold M is a quasi isomorphism between the space of polyvector fields (T poly(M)) and the space of multidifferential operators (D poly(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...

In this Note, we consider a principal fibre bundle P → M with structural group G, endowed with a flat connection. Supposing there is a G invariant formality ℱ on P, we can define a quotient formality ℱV on the basis M of our fibre bundle. We give a few examples, especially for the spheres Sd. If d = 2, this defines a canonical differential star-pro...

We describe geometrically the classical and quantum inhomogeneous groups $G_0=(SL(2, \BbbC)\triangleright \BbbC^2)$ and $G_1=(SL(2, \BbbC)\triangleright \BbbC^2)\triangleright \BbbC$ by studying explicitly their shape algebras as a spaces of polynomial functions with a quadratic relations.

The existence of star products on any Poisson manifold M is a consequence of Kontsevich's formality theorem, the proof of which is based on an explicit formula giving a formality quasi-isomorphism in the at case M = R(d). We propose here a coherent choice of orientations and signs in order to carry on Kontsevich's proof in the R(d) case, i.e., prov...

We determine the Hochschild cohomology for the Kontsevich's graphs. As usual, that cohomology is localized on totally antisymmetric graphs with as many feet as legs. Using this cohomology, we reinterpret the formality equation for the space ℝd.

In this paper we apply star products to the invariant theory for mul- tiplicity free actions. The space of invariants for a compact linear multiplicity free action has two canonical bases which are orthogonal with respect to two dierent inner products. One of these arises in connection with the star product. We use this fact to determine the elemen...

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 \(\mathfrak{g}\) *,...

The explicit realization of M. Kontsevich's formality on $R^d$ is the main step of the proof of formality theorem on any manifold. We present here a coherent choice of orientations and signs in order to write completely M. Kontsevich's proof for $R^d$, including all the signs appearing in the formality equation.

In this Letter, we consider Kontsevich's wheel operators for linear Poisson structures, i.e. on the dual of Lie algebras \mathfrakg\mathfrak{g} . We prove that these operators vanish on each invariant polynomial function on \mathfrakg\mathfrak{g} *. This gives a characterization of the Kontsevich star products which are deformations relative to...

We show that every irreducible unitary representation π of an exponential Lie group G = exp g is characterized by its generalized moment set.

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 montr...

Following the ideas presented in q-alg/9709040, we give the definition of Kontsevich star products for linear Poisson structures on d. We prove that all these structures are equivalent and can be defined by integral formulae. Finally, we characterize, among these star products, the Gutt and Duflo star products.

In [3], Maxim Kontsevich defines explicitely, for each Poisson structure n on the space ℝd, a star product on ℝd. If α is linear, i.e. if (ℝd, α) is the dual of a Lie agebra g, Kontsevich compares his star product with the product defined by S. Gutt in [2]. Due to the existence of “wheels” in the Kontsevich's graphs, these two deformations are dist...

In [3], Maxim Kontsevich defines explicitely, for each Poisson structure α on the space ℝd, a star product on ℝd. If α is linear, i.e. if (ℝd, α) is the dual of a Lie agebra g, Kontsevich compares his star product with the product defined by S. Gutt in [2]. Due to the existence of "wheels" in the Kontsevich's graphs, these two deformations are dist...

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.

Soit G un groupe de Lie exponentiel, d'algèbre 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 représentations πi = ind GH i χf (i = 1, 2, Hi = exp hi, χf ( expX) = eif (X) pour X dans g) sont irréductibles et équivalentes. Le principal obstacle à la construction...

The deformation program (the use of star products in harmonic analysis) leads to the definition of an adapted Fourier transform, unitary transformation between spaces of square integrable functions of the group G and on the dual \mathfrakg* \mathfrak{g}^* of its Lie algebra, describing the unitary dual of G and its Plancherel transform. This Let...

Using a parametrization for the universal covering O-0 of any coadjoint orbit O of a solvable (connected and simply connected) Lie group G, we prove that the Moyal product on O-0 gives a realization of the unitary representations canonically associated to the orbit O in Pukanszky's theory by deformation of the algebra of C-infinity functions on O-0...

. We show that the sum of two adjoint orbits in the Lie algebra of an exponential Lie group coincides with the Campbell-Baker-Hausdorff product of these two orbits. Introduction N. Wildberger and others have recently investigated the structure of the hypergroup of the adjoint orbits in relation with the class hypergroup of compact Lie groups. A gen...

We give a complete classification of the class of connected, simply connected Lie groups whose coadjoint orbits are of dimension smaller or equal to two.

On démontre l'existence de produits-star covariants sur les orbites de la représentation coadjointe d'un groupe de Lie qui admettent des polarisations.

We give a detailed study of the enveloping algebra of the Lie superalgebra sl(2, 1), including classification of irreducible Harish-Chandra modules, completeness of finite dimensional irreducible, explicit computation of center, and classification of primitive ideals.

We construct an algebraic star product on the minimal nilpotent coadjoint orbit of a simple complex Lie group with a Lie algebra which is not of typeA n. According to the deformation program, we study the representations of the Lie algebra associated to this orbit.

Let π be a unitary representation of a Lie group G. The moment mapping Ψπ of π assigns to every C∞ vector ξ in the Hilbert space of π the linear functional Ψπ(ξ) of the Lie algebra g of G by the rule In this paper, we study the moment set Iπ of π, i.e., the closure of the image of Ψπ. It is shown that for solvable G, Iπ is always convex and that if...

We suggest a construction of a Moyal type * product on Hermitian symmetric spaces ; this construction is motivated by what happens for coadjoint orbits of the Heisenberg group.

We give a complete and explicit realization of the unitary irreducible representations of any exponential group G by deformation of the associative and Poisson algebra of functions on the dual ∗ of the Lie algebra of G. We define an adapted Fourier transform which is a deformation of the usual one and which gives a natural description of the harmon...

We give a complete and explicit realization of the unitary irreducible representations of the universal covering group G of E(2), the Euclidean group in two dimensions, by deformation of the algebra of functions on the dual g * of the Lie algebra of G. We define an adapted Fourier transform for G which gives a natural description of the harmonic an...

On introduit une notion de forme normale pour une representation non lineaire formelle T, dans C n , d'une algebre de Lie complexe de dimension finie. Cette notion est optimale dans le sens qu'elle est canonique si la partie lineaire de T est completement reductible, qu'elle generalise les notions introduites precedemment et qu'un theoreme de norma...

We realise the *-representation program for compact semi simple Lie groups, using a Kodaira's map and Berezin's symbols. We study the Fourier transform it induces. We show that injectivity of Kodaira's map selects the data of geometric quantization. In this case an inverse of Kodaira's map is given by a moment map. This relates our construction to...

This talk follows the S.Gutt’s lecture. We study here what we call the *-exponential function considered first by Bayen, Flato, Fronsdal, Lichnerowicz and Sternheimer in [1] and by many other authors (Bayen, Maillard [2] for instance). This function is defined on a symplectic or Poisson manifold W, endowed with a * product. Here we consider only (o...

Let be a finite dimensional real Lie algebra and ∗ its dual. ∗ is a Poisson manifold. Thus the space C∞(∗) of C∞ functions on ∗ has an associative and a Lie algebra structure. The problem of formal deformations of such a structure needs the determination of some cohomology groups of C∞(∗), considered as a module on itself for left multiplication or...

A deformation of the polynomial algebra S()on * when S() is a free I() module (I() = algebra of invariant polinomials). This deformation restricts nicely to a large class of orbits. We also give an example to show that deformations of S() restricting to orbits may not always be defined by bidifferential operators.

On each orbit W of the coadjoint representation of any nilpotent (connected, simply connected) Lie group G, we construct *-products and associated Von Neumann algebras G. G acts canonically on G by automorphisms. In the unique faithful, irreducible representation of G, this action is implemented by the unitary irreducible representation of G corres...

We study the nilpotent Fourier transform on spaces of distributions. We use it to prove the equivalence between *-products on g* for nilpotent g.

On each orbit W of the coadjoint representation of a nilpotent, connected and simply connected Lie group G, there exist * products which are relative quantizations for the Lie algebra g of G. Choosing one of these * products, we first define a *-exponential for each X in g. These *-exponentials are formal power series and, with the * product, they...

The aim of this Letter is to show that the Poincare-Dulac theorem for holomorphic finite-dimensional representation, is valid for any nilpotent Lie algebrag. We reduce the classification problem of representations with a semisimple linear part satisfying the Poincar condition to an algebraic problem. We develop a complete computation in a particula...

Several notions of invariance and covariance for ∗ products with respect to Lie algebras and Lie groups are investigated. Some examples, including the Poincaré group, are given. The passage from the Lie‐algebra invariance to the Lie‐group covariance is performed. The compact and nilpotent cases are treated.

We expose here some results which are obtained by a team at the University of Dijon. This team included Jean-Claude Cortet, Georges Pinczon and myself.

In this contribution we briefly review the deformation (phase-space) approach to quantization in both quantum mechanics and quantum field theory. This leads to a new framework for group representations by star-products rather than operators; some examples of these are studied more in details, in particular induced star-representations of the Poinca...