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With every irreducible finite root system, one can associate the corresponding Drinfel'd-Jimbo quantum group. This is a Hopf algebra, which can be thought of as a deformation of the universal enveloping algebra of the Lie algebra of the same Cartan type. It naturally comes equipped with a universal R-matrix, thus providing solutions of the Yang-Baxter equation which plays a definitional role in the theoryof quantum integrable systems and underlies the algebraic Bethe ansatz.In case the initial root system is affine instead of finite, the resulting Drinfel'd-Jimbo quantum groups are known as quantum affine algebras. Drinfel'd proposedan alternative presentation of these algebras though, closer in spirit to their classiccurrent or loop presentation as Lie algebras. It is now widely referred to as the Drinfel'd presentation and was rigorously established by Damiani and Beck, making crucial use of Lusztig's affine braid group symmetries ; a quantum analogue of the classical Weyl group symmetries of simple Lie algebras. As one expects in view of the classical current Lie algebra case, Drinfel'd's presentation only depends on the underlying finite root system, i.e. the one with the extra affine simple root removed. Now it turns out that this inherently affine presentation still makes sense if, insteadof a finite root system, one takes an affine root system. In that case, the doubly affine algebra one obtains is known as a quantum toroidal algebra. Although the latter are believed to be relevant in various areas of theoretical physics,ranging from quantum integrable systems to CFT, not much is presently known about their representation theory. From a more mathematical perspective, the interest in these algebras essentially stems from the fact that, in type an_1, they are known to be Frobenius-Schur duals of the widely studied doubly affine Hecke algebras or DAHA originally introduced by Cherednik in order to prove MacDonald's conjectures. In this thesis, we study quantum toroidal algebras and their representation theory. In the first section, we construct a new presentation of the algebra using the braid group action on the generators and show the existence of an isomorphism between both presentations. This allows us to define a new triangular decomposition. Using these results, we define and classify highest t-weight representations. Finally, we generalize the action of the braid group to any root system., A toute algèbre de Lie sur le corps des complexes, nous pouvons lui associer le groupe quantique considéré comme généralisation de l’algèbre. C’est la déformation de l’algèbre enveloppante universelle. En prenant la limite q tend vers 1, nous retrouvons l’algèbre enveloppante universelle. L’algèbre de Lie possède une généralisation naturelle en dimension infinie qui est l’algèbre de Lie affine. La déformation de l’algèbre enveloppante d’une algèbre de Lie affine non-tordue nous permet de définir les algebres affines quantiques. Due à V.G. Drinfel’d les algèbres affines quantiques possèdent une deuxième réalisation en terme de générateurs de Drinfel’d. Cet isomorphisme est prouvé Par I. Damiani et J. Beck. Ceci nous permet de dire qu’on peut effectuer l’affinisation avant ou bien après la quantification. On a un diagramme commutative. En plus, on peut definir la quantification affine qui nous permet d’associer à toute algèbre de Lie de type finie une algèbre quantique affine dans la réalisation de Drinfel’d. Le procédé de quantification affine peut être effectué sur une algèbre affine non tordue. Ceci est la definition des algèbres toroidales quantiques. Le résultat est une algèbre qui est doublement affine. Dans cette thèse nous étudions les algèbres toroidales quantiques et leurs représentations. La première partie est consacrée à l’étude de l’algèbre toroidale quantique de type A1. Par action du groupe des tresses, nous construisons une nouvelle presentation de l’algèbre qui nous donne une nouvelle décomposition triangulaire. Dans la seconde partie, nous utilisons ce résultat pour définir et classifier les représentations simples de plus hauts t-poids. Finalement, nous généralisons les résultats de la première partie pour obtenir une action du groupe des tresses sur tout autres systèmes de racines.

Davydov--Yetter (DY) cohomology classifies infinitesimal deformations of the monoidal structure of tensor functors and tensor categories. In this paper we provide new tools for the computation of the DY cohomology for finite tensor categories and exact functors between them. The key point is to realize DY cohomology as relative Ext groups. In particular, we prove that the infinitesimal deformations of a tensor category $\mathcal{C}$ are classified by the 3-rd self-extension group of the tensor unit of the Drinfeld center $\mathcal{Z}(\mathcal{C})$ relative to $\mathcal{C}$. From classical results on relative homological algebra we get a long exact sequence for DY cohomology and a Yoneda product for which we provide an explicit formula. Using the long exact sequence and duality, we obtain a dimension formula for the cohomology groups based solely on relatively projective covers which reduces a problem in homological algebra to a problem in representation theory, e.g. calculating the space of invariants in a certain object of $\mathcal{Z}(\mathcal{C})$. Thanks to the Yoneda product, we also develop a method for computing DY cocycles explicitly which are needed for applications in the deformation theory. We apply these tools to the category of finite-dimensional modules over a finite-dimensional Hopf algebra. We study in detail the examples of the bosonization of exterior algebras $\Lambda\mathbb{C}^k \rtimes \mathbb{C}[\mathbb{Z}_2]$, the Taft algebras and the small quantum group of $\mathfrak{sl}_2$ at a root of unity., Comment: v2: 54 pages, fixed a gap in the proof of Prop 3.10; v3: 63 pages, new results for factorisable Hopf algebras in Sec 5.5 and 6.4, Intro and Abstract improved, Prop 3.2 extended and references updated

For G an algebraic (or more generally, a bornological) quantum group and B a closed quantum subgroup of G, we build in this paper an induction module by explicitly defining an inner product which takes its value in the convolution algebra of B, as in the original approach of Rieffel [Rie74]. In this context, we study the link with the induction functor defined by Vaes. In the last part we illustrate our result with parabolic induction of complex semi-simple quantum groups with the approach suggested by Clare [Cla13][CCH16].

Inspired by the work of Resnikoff, which is described if full details in the first part of this two-part paper, we give a quantum description of the space P of perceived colors. We show that P is the effect space of a rebit, a real quantum qubit, whose state space S is isometric to the hyperbolic Klein disk K. This chromatic state space of perceived colors can be represented as a Bloch disk of real dimension 2 that coincides with the Hering disk given by the color opponency mechanism. Attributes of perceived colors, hue and saturation, are defined in terms of Von Neumann entropy.

We study unitary representations of semidirect products of a compact quantum group with a finite group. We give a classification of all irreducible unitary representations, a description of the conjugate representation of irreducible unitary representations in terms of this classification, and the fusion rules for the semidirect product., Comment: version submitted for publication (compiled with pdflatex to fix overful lines with references)

We apply the canonical forms (barcodes) of Morse complexes to explore topology of loss surfaces. We present a novel algorithm for calculations of the objective function's barcodes of minima. We have conducted experiments for calculating barcodes of local minima forbenchmark functions and for loss surfaces of neural networks. Our experiments confirm two principal observations: (1) the barcodes of minima are located in a small lower part of the range of values of loss function of neural networks and (2) increase of the neural network's depth brings down the minima's barcodes. This has natural implications for the neural network learning and the ability to generalize.

This article is an extended version of the minicourse given by the second author at the summer school of the conference "Interactions of quantum affine algebras with cluster algebras, current algebras and categorification", held in June 2018 in Washington. The aim of the minicourse, consisting of three lectures, was to present a number of results and conjectures on certain monoidal categories of finite-dimensional representations of quantum affine algebras, obtained by exploiting the fact that their Grothendieck rings have the natural structure of a cluster algebra., v2: minor corrections. Will appear in the proceedings of the conference "Interactions of quantum affine algebras with cluster algebras, current algebras and categorification"

We calculate the Plancherel formula for complex semisimple quantum groups, that is, Drinfeld doubles of $ q $-deformations of compact semisimple Lie groups. As a consequence we obtain a concrete description of their associated reduced group $ C^* $-algebras. The main ingredients in our proof are the Bernstein-Gelfand-Gelfand complex and the Hopf trace formula., Comment: 21 pages. Minor revision. Accepted for publication in Ann. Sci. Ec. Norm. Sup

The localization of the EA-matrix integrals from [B4], is calculated via determinants and tau functions of KP-type hierarchies in the case of associative algebras with even scalar product.

Let $\Sigma$ be a compact connected oriented surface with one boundary component and let $\mathcal{M}$ denote the mapping class group of $\Sigma$. By considering the action of $\mathcal{M}$ on the fundamental group of $\Sigma$ it is possible to define different filtrations of $\mathcal{M}$ together with some homomorphisms on each term of the filtration. The aim of this paper is twofold. Firstly we study a filtration of $\mathcal{M}$ introduced recently by Habiro and Massuyeau, whose definition involves a handlebody bounded by $\Sigma$. We shall call it the "alternative Johnson filtration", and the corresponding homomorphisms are referred to as "alternative Johnson homomorphisms". We provide a comparison between the alternative Johnson filtration and two previously known filtrations: the original Johnson filtration and the Johnson-Levine filtration. Secondly, we study the relationship between the alternative Johnson homomorphisms and the functorial extension of the Le-Murakami-Ohtsuki invariant of $3$-manifolds. We prove that these homomorphisms can be read in the tree reduction of the LMO functor. In particular, this provides a new reading grid for the tree reduction of the LMO functor., Comment: 62 pages, several figures. v_2 minor changes

We consider the trigonometric classical $r$-matrix for $\mathfrak{gl}_N$ and the associated quantum Gaudin model. We produce higher Hamiltonians in an explicit form by applying the limit $q\to 1$ to elements of the Bethe subalgebra for the $XXZ$ model., 14 pages

International audience; The algorithm for calculation of "canonical form" = "persistence barcodes/diagrams" invariants from S.Barannikov "The Framed Morse complex and its invariants" Adv. in Sov. Math., vol 21, AMS transl, (1994), is described.

The Hamiltonian of a quantum mechanical system has an affiliated spectrum, and in order for this spectrum to be locally observable, the Hamiltonian should be Hermitian. Non-Hermitian Hamiltonians can be observed non-locally via parity, i.e. by taking the expectation value of the Wigner distribution evaluated at the orgin in phase space. Studies such as these quantum nonlocality analogies have led to the Bender-Brody-M\"uller (BBM) conjecture, which involves a non-Hermitian Hamiltonian eigenequation whose eigenvalues are the nontrivial zeros of the Riemann zeta function. Herein it is shown from symmetrization of the BBM Hamiltonian that the eigenvalues are not locally observable, i.e. \textit{the analytic continuation of the Riemann zeta function is not an analytically computable function at $\sigma=1/2$}. In the present case, the Riemann zeta function is analogous to chaotic quantum systems, as the harmonic oscillator is for integrable quantum systems. As such, herein we perform a symmetrization procedure of the BBM Hamiltonian to obtain a Hermitian Hamiltonian using a similarity transformation, and provide a trivial analytical expression for the eigenvalues of the results using Green's functions. A nontrivial expression for the eigensolution of the eigenequation is also obtained. A Gelfand triplet is then used to ensure that the eigensolution is well defined. The holomorphicity of the resulting eigenvalue spectrum is demonstrated, and it is shown that that the expectation value of the Hamiltonian operator is periodically zero such that the nontrivial zeros of the Riemann zeta function are not observable, i.e., the Riemann Hypothesis is not decidable. Moreover, a second quantization of the resulting Schr\"odinger equation is performed, and a convergent solution for the nontrivial zeros of the analytic continuation of the Riemann zeta function is obtained. Finally, from the holomorphicity of the eigensolution it is shown that the real part of every nontrivial zero of the Riemann zeta function exists at $\sigma=1/2$, and a general solution is obtained by performing an invariant similarity transformation.

Motivated by Maulik-Okounkov stable maps associated to quiver varieties, we define and construct algebraic stable maps on tensor products of representations in the category O of the Borel subalgebra of an arbitrary untwisted quantum affine algebra. Our representation-theoretical construction is based on the study of the action of Cartan-Drinfeld subalgebras. We prove the algebraic stable maps are invertible and depend rationally on the spectral parameter. As an application, we obtain new R-matrices in the category O and we establish that a large family of simple modules, including the prefundamental representations associated to Q-operators, generically commute as representations of the Cartan-Drinfeld subalgebra. We also establish categorified QQ*-systems in terms of the R-matrices we construct., Comment: 33 pages; v3 : final version accepted in Representation Theory

The Polynomial Chaos theory, being a less expensive and more efficient alternative of the Monte Carlo Simulation, remains limited to the polynomials of Gaussian variables. We present a Hilbertian method that generalizes this theory and we establish the conditions of existence and convergence of an expansion in Generalized Fourier Series. Then, we present the Statistics of Things that allows studying the statistical characteristics of a set of random infinite-dimensional objects. By computing the distances between the hypervolumes, namely the distance of Hausdorff, this method allows determining the median object, the quantile objects and a confidence interval at a given level for a finite set of random objects. In the third section, we address a method for simulating a large size sample of a random object at a much reduced computational cost, and calculating its mean without using the distance between the hypervolumes., La théorie des polynômes de chaos, étant une alternative moins onéreuse et plus efficace de la simulation de Monte Carlo, reste limitée aux polynômes de variables gaussiennes. On présente une méthode de type hilbertien qui généralise cette théorie et on établit les conditions d’existence et de convergence d’une expansion en Série de Fourier Généralisée. Ensuite, on présente la Statistique des Objets qui permet d’étudier les caractéristiques statistiques d’un ensemble d’objets aléatoires en dimension infinie. En calculant les distances entre les hypervolumes, notamment la distance de Hausdorff, cette méthode permet de déterminer l’objet médian, les objets quantiles et un intervalle de confiance à un seuil donné pour un ensemble fini d’objets aléatoires. Une méthode pour simuler un échantillon de grande taille d’un objet aléatoire à coût computationnel très réduit, et calculer sa moyenne sans faire appel à la distance entre les hypervolumes, fait l’objet de la troisième partie.

This text studies the quantum group $\mathcal{U}_{\xi}^{H}\mathfrak{sl}(2|1)$ associated with the Lie superalgebra $\mathfrak{sl}(2|1)$ and a category of finite dimensional representations. The aim is to construct the topological invariants of $3$-manifolds using the notion of {\em modified trace}. We first prove that the category $\mathscr{C}^H$ of the nilpotent weight modules over $\mathcal{U}_{\xi}^{H}\mathfrak{sl}(2|1)$ is ribbon and that there exists a modified trace on its ideal of projective modules. Furthermore, $\mathscr{C}^H$ possesses a relative $G$-premodular structure which is a sufficient condition to construct an invariant of $3$-manifolds of Costantino-Geer-Patureau type. This invariant is the heart of a $1 + 1 + 1$-TQFT (Topological Quantum Field Theory). Next Hennings proposed from a finite dimensional Hopf algebra, a construction of invariants which does not require to consider the category of its representations. We show that the unrolled quantum group $\mathcal{U}_{\xi}^{H}\mathfrak{sl}(2|1)/(e_1^{\ell}, f_1^{\ell})$ has a completion which is a topological ribbon Hopf algebra. We construct an invariant of $3$-manifolds of Hennings type using this algebraic structure, a discrete Fourier transform, and the notion of $G$-integrals. The integral in a Hopf algebra is central in the construction of Hennings. The notion of modified trace in a category has recently been revealed to be a generalization of the integrals in a finite dimensional Hopf algebra. In a more general context of infinite dimensional Hopf algebras we prove the relation formulated between the modified trace and the $G$-integral., Ce texte étudie le groupe quantique $\mathcal{U}_{\xi}^{H}\mathfrak{sl}(2|1)$ associé à la superalgèbre de Lie $\mathfrak{sl}(2|1)$ et une catégorie de ses représentations de dimension finie. L'objectif est de construire des invariants topologiques de $3$-variétés en utilisant la notion de {\em trace modifiée}. D'abord nous prouvons que la catégorie $\mathscr{C}^H$ des modules de poids nilpotents sur $\mathcal{U}_{\xi}^{H}\mathfrak{sl}(2|1)$ est enrubannée et qu'il existe une trace modifiée sur son idéal des modules projectifs. De plus $\mathscr{C}^H$ possède une structure relativement $G$-prémodulaire ce qui est une condition suffisante pour construire un invariant de $3$-variétés à la Costantino-Geer-Patureau. Cet invariant est le coeur d'une $1+1+1$-TQFT (Topological Quantum Field Theory). D'autre part Hennings a proposé à partir d'une algèbre de Hopf de dimension finie une construction d'invariants qui dispense de considérer la catégorie de ses représentations. Nous montrons que le groupe quantique déroulé $\mathcal{U}_{\xi}^{H}\mathfrak{sl}(2|1)/(e_1^{\ell}, f_1^{\ell})$ possède une complétion qui est une algèbre de Hopf enrubannée topologique. Nous construisons un invariant de $3$-variétés à la Hennings en utilisant cette structure algébrique, une transformation de Fourier discrète et la notion de $G$-intégrales. L'intégrale dans une algèbre de Hopf est centrale dans la construction de Hennings. La notion de trace modifiée dans une catégorie s'est récemment révélée être une généralisation des intégrales dans les algèbres de Hopf de dimension finie. Dans un contexte plus général d'algèbre de Hopf de dimension infinie nous prouvons la relation formulée entre la trace modifiée et la $G$-intégrale.

For a non Archimedean local field which is not of characteristic $2$, nor an extension of $\mathbb Q_2$, we construct a pseudo-differential calculus covariant under a unimodular subgroup of the affine group of the field. Our phase space is a quotient group of the covariance group. Our main result is a generalisation on that context of the Calder\'on-Vaillancourt estimate. Our construction can be thought as the non Archimedean version of Unterberger's Fuchs calculus and our methods are mainly based on Wigner functions and on coherent states transform., Comment: to appear in JFG

In this thesis, following the work initiated by V. Drinfeld and pursued by B. Enriquez, then by the latter together with D. Calaque and P. Etingof, we study the universal twisted elliptic (ellipsitomic in short) KZB connection, associated to the moduli space of elliptic curves with n marked points and a (M,N)-level structure. The flatness of this connection allows us to study monodromy relations satisfied by this connection, opening the way to a general theory of ellipsitomic associators and Grothendieck-Teichmüller groups corresponding to them, which is released via the use of the formalism of operads (and some of their variants) basing ourselves on the work of B. Fresse. On the one hand, this formalism allows us to study the structure of associators in higher genus. On the other hand, the ellipsitomic KZB associator allows us to derive a theory of elliptic multiple zeta values at torsion points, from which some of their first associator-like properties are distinguished. We will begin by setting up the operadic machinery necessary to define the ellipsitomic associators starting successively with the genus 0 situation, which is well-known, then the genus 1 situation and their cyclotomic variants. Then, in light of this formalism, we will release a definition of genus gassociators. Next, we will go into the details of the construction of the universal ellipsitomic KZB connection, first over the (M,N)-twisted configuration space of an elliptic curve and then over the moduli space of elliptic curves with a level structure. We will associate this connection to its realized version by means of the use of double affine Hecke algebras and of classical dynamical r-matrices. Finally we will present the applications of this construction, namely : the formality of certain subgroups of the braid group on the torus, the ellipsitomic KZB associator, elliptic multiple zeta values at points of torsion as well as an application in representations of cyclotomic Cherednik algebras., Dans cette thèse, en suivant les travaux initiés par V. Drinfeld, poursuivis par B. Enriquez, puis par ce dernier, D. Calaque et P. Etingof, nous étudions la connexion KZB elliptique cyclotomique (ellipsitomique en plus court) universelle, associée à l’espace de modules des courbes elliptiques avec n points marqués et une structure de (M,N)-niveau. La platitude de cette connexion nous permet d’étudier des relations de monodromie, ouvrant la voie à une théorie générale des associateurs ellipsitomiques et des groupes de Grothendieck-Teichmüller qui lui correspondent, que l’on dégage via l’utilisation du formalisme des opérades (et certaines de leurs variantes) en nous basant sur les travaux de B. Fresse à ce sujet. D’une part, ce formalisme nous permet par ailleurs d’étudier la structure des associateurs en genre supérieur. D’autre part, l’associateur KZB ellipsitomique nous permet de dégager une théorie des valeurs multizêta elliptiques en des points de torsion, dont on démarque quelques unes de leurs premières propriétés du type associateurs. On commencera par mettre en place la machinerie opéradique nécessaire pour définir les associateurs ellipsitomiques en partant tour à tour de la situation déjà connue en genre 0, puis de celle en genre 1 et ensuite de leurs variantes cyclotomiques. Enfin, grâce à ce formalisme, nous dégagerons une définition des associateurs en tout genre. Ensuite, nous entrerons dans le détail de la construction de la connexion KZB ellipsitomique universelle, en premier temps sur l’espace de configurations (M,N)-décorées d’une courbe elliptique puis sur les espaces de modules des courbes à niveau, nous la lieront à sa version réalisée via l’utilisation des algèbres de Hecke doublement affines et des r-matrices classiques dynamiques. Pour finir nous présenterons les applications de cette construction, à savoir : formalité de certains sous-groupes de tresses sur le tore, l’associateur KZB ellipsitomique, valeurs multizêta elliptiques en des points de torsion ainsi qu’une application en représentations d’algèbres de Cherednik cyclotomiques.

This memoir is a summary of recent work, including collaborations with Erik van Erp, Christian Voigt and Marco Matassa, compiled for the "Habilitation \`a diriger des recherches". We present various different approaches to constructing algebras of pseudodifferential operators adapted to filtered and multifiltered manifolds and some quantum analogues. A general goal is the study of index problems in situations where standard elliptic theory is insufficient. We also present some applications of these constructions. We begin by presenting a characterization of pseudodifferential operators on filtered manifolds in terms of distributions on the tangent groupoid which are essentially homogeneous with respect to the natural $\mathbb{R}^\times_+$-action. Next, we describe a rudimentary multifiltered pseudodifferential theory on the full flag manifold $\mathcal{X}$ of a complex semisimple Lie group $G$ which allows us to simultaneously treat longitudinal pseudodifferential operators along every one of the canonical fibrations of $\mathcal{X}$ over smaller flag manifolds. The motivating application is the construction of a $G$-equivariant $K$-homology class from the Bernstein-Gelfand-Gelfand complex of a semisimple group. Finally, we discuss pseudodifferential operators on two classes of quantum flag manifolds: quantum projective spaces and the full flag manifolds of $SU_q(n)$. In particular, on the full flag variety of $SU_q(3)$ we obtain an equivariant fundamental class from the Bernstein-Gelfand-Gelfand complex., Comment: Memoir for the "Habilitation \`a diriger des recherches"

The construction of cohomology classes of compactified moduli spaces of curves from asymptotic expansions of EA matrix integrals, described in the works of speaker, is presented. The construction defines a cohomological field theory, which conjecturally coincides with the all genus Gromov-Witten invariants of the mirror manifold. The construction is based on the speaker’s theorem identifying the cell complex of the compactified moduli space of curves with the Feynman transform of the operad of permutation group algebras. As a simple application a new formula for generating function for products of psi classes in the total cohomology of the compactified moduli spaces of curves is described. Relevant papers: arxiv:1803.11549 / HAL-00429963 (2009), arxiv:0912.5484 / hal-00102085(2006)

International audience

Concurrent separation logic (CSL) is a specification logic for concurrent imperative programs with shared memory and locks. In this paper, we develop a concurrent and interactive account of the logic inspired by asynchronous game semantics. To every program $C$, we associate a pair of asynchronous transition systems $[C]_S$ and $[C]_L$ which describe the operational behavior of the Code when confronted to its Environment or Frame --- both at the level of machine states ($S$) and of machine instructions and locks ($L$). We then establish that every derivation tree $\pi$ of a judgment $\Gamma\vdash\{P\}C\{Q\}$ defines a winning and asynchronous strategy $[\pi]_{Sep}$ with respect to both asynchronous semantics $[C]_S$ and $[C]_L$. From this, we deduce an asynchronous soundness theorem for CSL, which states that the canonical map $\mathcal{L}:[C]_S\to[C]_L$ from the stateful semantics $[C]_S$ to the stateless semantics $[C]_L$ satisfies a basic fibrational property. We advocate that this provides a clean and conceptual explanation for the usual soundness theorem of CSL, including the absence of data races., Comment: Full version of an extended abstract published at LICS 2018

In a recent paper the authors Beliakova, Blanchet and Gainutdinov have shown that the modified trace on the category $H$-pmod of the projective modules corresponds to the symmetrised integral on the finite dimensional pivotal Hopf algebra $H$. We generalize this fact to the context of $G$-graded categories and Hopf $G$-coalgebra studied by Turaev-Virelizier. We show that the symmetrised $G$-integral on a finite type pivotal Hopf $G$-coalgebra induces a modified trace in the associated $G$-graded category.

We establish ring isomorphisms between quantum Grothendieck rings of certain remarkable monoidal categories of finite-dimensional representations of quantum affine algebras of types $A_{2n-1}^{(1)}$ and $B_n^{(1)}$. Our proof relies in part on the corresponding quantum cluster algebra structures. Moreover, we prove that our isomorphisms specialize at $t = 1$ to the isomorphisms of (classical) Grothendieck rings obtained recently by Kashiwara, Kim and Oh by other methods. As a consequence, we prove a conjecture formulated by the first author in 2002 : the multiplicities of simple modules in standard modules in the categories above for type $B_n^{(1)}$ are given by the specialization of certain analogues of Kazhdan-Lusztig polynomials and the coefficients of these polynomials are positive., Comment: v1 : 63 pages. v2 : 64 pages. Minor modification. To appear in Adv. Math

After introducing some motivations for this survey, we describe a formalism to parametrize a wide class of algebraic structures occurring naturally in various problems of topology, geometry and mathematical physics. This allows us to define an "up to homotopy version" of algebraic structures which is coherent (in the sense of $\infty$-category theory) at a high level of generality. To understand the classification and deformation theory of these structures on a given object, a relevant idea inspired by geometry is to gather them in a moduli space with nice homotopical and geometric properties. Derived geometry provides the appropriate framework to describe moduli spaces classifying objects up to weak equivalences and encoding in a geometrically meaningful way their deformation and obstruction theory. As an instance of the power of such methods, I will describe several results of a joint work with Gregory Ginot related to longstanding conjectures in deformation theory of bialgebras, $E_n$-algebras and quantum group theory., References added, some corrections in Sections 3 and 4. This survey will eventually appear in the MATRIX book series http://www.matrix-inst.org.au/book-series/ . 43 pages, comments welcome

The notion of nondegenerate critical point in the BV formalism is studied. The analogs of the Morse and Darboux theorems in the BV formalism are proven. The theorem on the normal form of an arbitrary quadratic function on odd symplectic space is proven. This can be viewed as an analog of Jordan type decomposition for a pair of a symmetric pairing on vector space and an anti-symmetric pairing on the dual space. The last result was used for the construction of propagators in the BV formalism in the equivariant setting, see [2, 3, 4]

We present a general formula for obtaining the zeros of the analytic continuation formula of the Riemann zeta function from Euler's quadratic equation and show that these zeros are all real. Furthermore, substituting this formula into the analytic continuation formula, the root of the Analytic Continuation Formula (ACF) of the Riemann zeta function is determined.

This thesis is concerned with the construction of a family of quantum transformation groupoids in the algebraic framework in the form of measured multiplier Hopf *-algebroids in the sense of Timmermann and Van Daele and also in the context of operator algebras in the form of Hopf C*-bimodules on a C*-base in the sense of Timmermann. In the purely algebraic context, we first give a definition of a braided commutative Yetter-Drinfeld *-algebra over an algebraic quantum group in the sense of Van Daele and a Yetter-Drinfeld integral on it. Then, using these objects we construct a measured multiplier Hopf *-algebroid, we call to this new object an algebraic quantum transformation groupoid. In order to pass to the operator algebra framework, we give some conditions on the Yetter-Drinfeld integral inspired by the properties of KMS-weights on C*-algebras which will allow us to use the Gelfand–Naimark–Segal construction to extend all the purely algebraic objects to the C*-algebraic level. At this level, we construct in a similar way to that used in the work of Enock and Timmermann, a new mathematical object that we call a C*-algebraic quantum transformation groupoid, which is defined using the language of Hopf C*-bimodules on C*-bases., Cette thèse porte sur la construction d'une famille de groupoïdes quantiques de transformations qui dans le cadre algébrique sont des algébroïdes de Hopf de multiplicateurs mesurés au sens de Timmermann et Van Daele et qui dans le cadre des algèbres d'opérateurs sont des C*-bimodules de Hopf sur une C*-base au sens de Timmermann. Dans le contexte purement algébrique, nous définissons d'abord une algèbre involutive de Yetter-Drinfeld tressée commutative sur un groupe quantique algébrique au sens de Van Daele et une intégrale de Yetter-Drinfeld sur elle. En utilisant ces objets nous construisons après un algébroide de Hopf de multiplicateurs involutif mesuré, ce nouvel objet nous l'appellons groupoïde quantique algébrique de transformations.Pour être capables de passer au cadre des algèbres d'opérateurs, nous donnons des conditions sur l'intégral de Yetter-Drinfeld qui vont nous permettre d'utiliser la construction Gelfand–Naimark–Segal pour étendre tous nos objets purement algébriques en des objets C*-algébriques. Dans ce contexte, notre construction se fait d'une manière similaire à celle présentée dans le travail de Enock et Timmermann, nous obtenons un nouvel objet mathématique que nous appellons un groupoïde quantique C*-algébrique de transformations, qui est définit en utilisant le langage des C*-bimodules de Hopf sur une C*-base.

Given an algebraic structure on the homology of a chain complex, we define its realization space as a Kan complex whose vertices are the structures up to homotopy realizing this structure at the homology level. Our algebraic structures are parametrised by props and thus include various kinds of bialgebras. We give a general formula to compute subsets of equivalences classes of realizations as quotients of automorphism groups, and determine the higher homotopy groups via the cohomology of deformation complexes. As a motivating example, we compute subsets of equivalences classes of realizations of Poincar\'e duality for several examples of manifolds., Comment: Typo corrections, explanations improved for some technical parts. Correction of a mistake concerning the result about connected components of realization spaces, and another one concerning relative realization spaces. 40 pages, to appear in IMRN

This text studies the quantum group Uξ sl(2|1) associated with the Lie superalgebra sl(2|1) and a category of finite dimensional representations. The aim is to construct the topological invariants of 3-manifolds using the notion of modified trace. We first prove that the category CH of the nilpotent weight modules over Uξ sl(2|1) is ribbon and that there exists a modified trace on its ideal of projective modules. Furthermore, CH possesses a relative G-premodular structure which is a sufficient condition to construct an invariant of 3-manifolds of Costantino-Geer-Patureau type. This invariant is the heart of a 1+1+1-TQFT (Topological Quantum Field Theory). Next Hennings proposed from a finite dimensional Hopf algebra, a construction of invariants which does not require to consider the category of its representations. We show that the unrolled H l l quantum group Uξ sl(2|1)/(e1 , f1 ) has a completion which is a topological ribbon Hopf algebra. We construct an invariant of 3-manifolds of Hennings type using this algebraic structure, a discrete Fourier transform, and the notion of G-integrals. The integral in a Hopf algebra is central in the construction of Hennings. The notion of modified trace in a category has recently been revealed to be a generalization of the integrals in a finite dimensional Hopf algebra. In a more general context of infinite dimensional Hopf algebras we prove the relation formulated between the modified trace and the G-integral., Ce texte étudie le groupe quantique Uξ sl(2|1) associé à la superalgèbre de Lie sl(2|1) et une catégorie de ses représentations de dimension finie. L'objectif est de construire des invariants topologiques de 3-variétés en utilisant la notion de trace modifiée. D'abord nous prouvons que la H catégorie CH des modules de poids nilpotents sur Uξ sl(2|1) est enrubannée et qu'il existe une trace modifiée sur son idéal des modules projectifs. De plus CH possède une structure relativement G-prémodulaire ce qui est une condition suffisante pour construire un invariant de 3-variétés à la Costantino-Geer-Patureau. Cet invariant est le cœur d'une 1+1+1-TQFT (Topological Quantum Field Theory). D'autre part Hennings a proposé à partir d'une algèbre de Hopf de dimension finie une construction d’invariants qui dispense de considérer la catégorie de H l l ses représentations. Nous montrons que le groupe quantique déroulé Uξ sl(2|1)/(e1 , f1 ) possède une complétion qui est une algèbre de Hopf enrubannée topologique. Nous construisons un invariant de 3-variétés à la Hennings en utilisant cette structure algébrique, une transformation de Fourier discrète et la notion de G-intégrales. L'intégrale dans une algèbre de Hopf est centrale dans la construction de Hennings. La notion de trace modifiée dans une catégorie s'est récemment révélée être une généralisation des intégrales dans les algèbres de Hopf de dimension finie. Dans un contexte plus général d'algèbre de Hopf de dimension infinie nous prouvons la relation formulée entre la trace modifiée et la G -intégrale.

We review arXiv:1308.3444 and arXiv:1104.1891. The structure of the spectrum of a quantum integrable system is crucial to understand its properties. In his seminar 1971 paper, Baxter observed that the spectrum of the "ice model" has a very remarkable form involving polynomials. Then it was conjectured that analog polynomials can be used to describe the spectra of more general quantum integrable systems (arXiv:math/9810055). We discuss how these (generalized Baxter's) polynomials arise naturally in terms of representation theory. This result lead recently to the proof of the conjecture., Comment: Review paper, in french. 15 pages, 4 figures

In the past 20 years the optimal transport theory revealed to be an efficient tool to study the asymptotic behavior for diffusion equations, to prove functional inequalities, to extend geometrical properties in extremely general spaces like metric measure spaces, etc. The curvature-dimension of the Bakry-Émery theory appears as the cornerstone of those applications. Just think to the easier and most important case of the quadratic Wasserstein distance W2: contraction of the heat flow in W2 characterizes uniform lower bounds for the Ricci curvature; the transport Talagrand inequality, comparing W2 to the relative entropy is implied and implies via the HWI inequality the log-Sobolev inequality; McCann geodesics in the Wasserstein space (P2(Rn),W2) allow to prove important functional properties like convexity, and standard functional inequalities, such as isoperimetry, measure concentration properties, the Prékopa Leindler inequality and so on. However the lack of regularity of optimal maps, requires non-smooth analysis arguments. The Schrödinger problem is an entropy minimization problem with marginal constraints and a fixed reference process. From the Large deviation theory, when the reference process is driven by the Brownian motion, its minimal value A converges to W2 when the temperature goes to zero. The entropic interpolations, solutions of the Schrödinger problem, are characterized in terms of Markov semigroups, hence computation along them naturally involves Γ2 computations and the curvature-dimension condition. Dating back to the 1930s, and neglected for decades, the Schrödinger problem recently enjoys an increasing popularity in different fields, thanks to this relation to optimal transport, smoothness of solutions and other well performing properties in numerical computations. The aim of this work is twofold. First we study some analogy between the Schrödinger problem and optimal transport providing new proofs of the dual Kantorovich and the dynamic Benamou-Brenier formulations for the entropic cost A. Secondly, as an application of these connections we derive some functional properties and inequalities under curvature-dimensions conditions. In particular, we prove the concavity of the exponential entropy along entropic interpolations under the curvature-dimension condition CD(0, n) and regularity of the entropic cost along the heat flow. We also give different proofs the Evolutionary Variational Inequality for A and contraction of the heat flow in A, recovering as a limit case the classical results in W2, under CD(κ,∞) and also in the flat dimensional case. Finally we propose an easy proof of the Gaussian concentration property via the Schrödinger problem as an alternative to classical arguments as the Marton argument which is based on optimal transport, Au cours des 20 dernières années, la théorie du transport optimal s’est revelée être un outil efficace pour étudier le comportement asymptotique dans le cas des équations de diffusion, pour prouver des inégalités fonctionnelles et pour étendre des propriétés géométriques dans des espaces extrêmement généraux comme des espaces métriques mesurés, etc. La condition de courbure-dimension de la théorie Bakry-Emery apparaît comme la pierre angulaire de ces applications. Il suffit de penser au cas le plus simple et le plus important de la distance quadratique de Wasserstein W2 : la contraction du flux de chaleur en W2 caractérise les bornes inférieures uniformes pour la courbure de Ricci ; l’inégalité de Talagrand du transport, comparant W2 à l’entropie relative est impliquée et implique, par l’inégalité HWI, l’inégalité log-Sobolev ; les géodésiques de McCann dans l’espace de Wasserstein (P2(Rn),W2) permettent de prouver des propriétés fonctionnelles importantes comme la convexité, et des inégalités fonctionnelles standards telles que l’isopérymétrie, des propriétés de concentration de mesure, l’inégalité de Prékopa-Leindler et ainsi de suite. Néanmoins, le manque de régularité des plans minimisation nécessite des arguments d’analyse non lisse. Le problème de Schrödinger est un problème de minimisation de l’entropie avec des contraintes marginales et un processus de référence fixes. À partir de la théorie des grandes déviations, lorsque le processus de référence est le mouvement Brownien, sa valeur minimale A converge vers W2 lorsque la température est nulle. Les interpolations entropiques, solutions du problème de Schrödinger, sont caractérisées en termes de semigroupes de Markov, ce qui implique naturellement les calculs Γ2 et la condition de courbure-dimension. Datant des années 1930 et négligé pendant des décennies, le problème de Schrodinger connaît depuis ces dernières années une popularité croissante dans différents domaines, grâce à sa relation avec le transport optimal, à la regularité de ses solutions, et à d’autres propriétés performantes dans des calculs numériques. Le but de ce travail est double. D’abord, nous étudions certaines analogies entre le problème de Schrödinger et le transport optimal fournissant de nouvelles preuves de la formulation duale de Kantorovich et de celle, dynamique, de Benamou-Brenier pour le coût entropique A. Puis, en tant qu’application de ces connexions, nous dérivons certaines propriétés et inégalités fonctionnelles sous des conditions de courbure-dimension. En particulier, nous prouvons la concavité de l’entropie exponentielle le long des interpolations entropiques sous la condition de courbure-dimension CD(0, n) et la régularité du coût entropique le long du flot de la chaleur. Nous donnons également différentes preuves de l’inégalité variationnelle évolutionnaire pour A et de la contraction du flux de la chaleur en A, en retrouvant comme cas limite, les résultats classiques en W2, sous CD(κ,∞) et CD(0, n). Enfin, nous proposons une preuve simple de la propriété de concentration gaussienne via le problème de Schrödinger comme alternative aux arguments classiques tel que l’argument de Marton basé sur le transport optimal

30 pages, 12 figures, We study two different objects attached to an arbitrary quadrangulation of a regular polygon. The first one is a poset, closely related to the Stokes polytopes introduced by Baryshnikov. The second one is a set of some paths configurations inside the quadrangulation, satisfying some specific constraints. These objects provide a generalisation of the existing combinatorics of cluster algebras and nonnesting partitions of type A., On étudie deux objets attachés à une quadrangulation quelconque d'un polygone régulier. Le premier objet est un ensemble partiellement ordonné, fortement lié aux polytopes de Stokes introduits par Barysknikov. Le second est un ensemble de configurations de chemins dans la quadrangulation. Ces deux objets généralisent respectivement les aspects combinatoires des algèbres amassées et des partitions non-emboitées de type A.

We introduce a functor ${\sf As}$ from the category of posets to the category of nonsymmetric binary and quadratic operads, establishing a new connection between these two categories. Each operad obtained by the construction ${\sf As}$ provides a generalization of the associative operad because all of its generating operations are associative. This construction has a very singular property: the operads obtained from ${\sf As}$ are almost never basic. Besides, the properties of the obtained operads, such as Koszulity, basicity, associative elements, realization, and dimensions, depend on combinatorial properties of the starting posets. Among others, we show that the property of being a forest for the Hasse diagram of the starting poset implies that the obtained operad is Koszul. Moreover, we show that the construction ${\sf As}$ restricted to a certain family of posets with Hasse diagrams satisfying some combinatorial properties is closed under Koszul duality., 40 pages

We show that the Gerstenhaber-Schack cohomology of a Hopf algebra determines its Hochschild cohomology, and in particular its Gerstenhaber-Schack cohomological dimension bounds its Hochschild cohomological dimension, with equality of the dimensions when the Hopf algebra is cosemisimple of Kac type. Together with some general considerations on free Yetter-Drinfeld modules over adjoint Hopf subalgebras and the monoidal invariance of Gerstenhaber-Schack cohomology, this is used to show that both Gerstenhaber-Schack and Hochschild cohomological dimensions of the coordinate algebra of the quantum permutation group are 3., 21 pages.V3: title has changed, because the main theoretical results relating Hochschild and Gerstenhaber cohomologies are in fact valid for any Hopf algebra. However, because of a gap in part of Theorem 5.2 of the previous version, the computations in Section 6 now only cover the cosemisimple examples

Given a Hopf algebra $A$ graded by a discrete group together with an action of the same group preserving the grading, we define a new Hopf algebra, which we call the graded twisting of $A$. If the action is adjoint, this new Hopf algebra is a twist of $A$ by a pseudo-$2$-cocycle. Analogous construction can be carried out for monoidal categories. As examples we consider graded twistings of the Hopf algebras of nondegenerate bilinear forms, their free products, hyperoctahedral quantum groups and $q$-deformations of compact semisimple Lie groups. As applications, we show that the analogues of the Kazhdan-Wenzl categories in the general semisimple case cannot be always realized as representation categories of compact quantum groups, and for genuine compact groups, we analyze quantum subgroups of the new twisted compact quantum groups, providing a full description when the twisting group is cyclic of prime order., Comment: v3: minor corrections, to appear in Ann. Inst. Fourier; v2: compilation error fix and minor phrasing changes; v1: 25 pages

Ce mémoire d'habilitation porte sur les invariants de la topologie en petite dimension. Qu’il s’agisse de la théorie des nœuds, et plus généralement des objets noués (entrelacs, tresses, enlacements d’intervalles, enchevêtrements, graphes noués), ou de l’étude des variétés de dimension 3, ce sont en effet là des branches de la topologie qui sont activement étudiées depuis plus d’un siècle, et pour lesquelles ont été développés des outils toujours plus nombreux et sophistiqués. Cette diversité s’est significativement accentuée dans les 30 dernières années, avec plusieurs avancées spectaculaires. Il est donc devenu nécessaire d’étudier cette multitude d’invariants, de tenter de comprendre les liens qui existent entre eux.Le mémoire est constitué de deux chapitres principaux, le premier traitant des objets noués et le second des variétés de dimension 3. Chacun de ces deux chapitres commence par une section contenant les rappels nécessaires; ces sections de rappel sont organisées de façon strictement parallèle, mettant en évidence les fortes analogies existant entre les deux contextes. Un troisième et dernier chapitre regroupe une série de questions ou pistes de recherches, plus ou moins directement liées aux travaux présentés ici.

We obtain a presentation of the t-deformed Grothendieck ring of a quantum loop algebra of Dynkin type A, D, E. Specializing t at the the square root of the cardinality of a finite field F, we obtain an isomorphism with the derived Hall algebra of the derived category of a quiver Q of the same Dynkin type. Along the way, we study for each choice of orientation Q a tensor subcategory whose t-deformed Grothendieck ring is isomorphic to the positive part of a quantum enveloping algebra of the same Dynkin type, where the classes of simple objects correspond to Lusztig's dual canonical basis., Comment: version 2: minor corrections; some examples added; section 9 on quiver varieties has been improved. version 3: minor corrections; final version to appear in Crelle

For the class of quantum integrable models generated from the $q-$Onsager algebra, a basis of bispectral multivariable $q-$orthogonal polynomials is exhibited. In a first part, it is shown that the multivariable Askey-Wilson polynomials with $N$ variables and $N+3$ parameters introduced by Gasper and Rahman [1] generate a family of infinite dimensional modules for the $q-$Onsager algebra, whose fundamental generators are realized in terms of the multivariable $q-$difference and difference operators proposed by Iliev [2]. Raising and lowering operators extending those of Sahi [3] are also constructed. In a second part, finite dimensional modules are constructed and studied for a certain class of parameters and if the $N$ variables belong to a discrete support. In this case, the bispectral property finds a natural interpretation within the framework of tridiagonal pairs. In a third part, eigenfunctions of the $q-$Dolan-Grady hierarchy are considered in the polynomial basis. In particular, invariant subspaces are identified for certain conditions generalizing Nepomechie's relations. In a fourth part, the analysis is extended to the special case $q=1$. This framework provides a $q-$hypergeometric formulation of quantum integrable models such as the open XXZ spin chain with generic integrable boundary conditions ($q\neq 1$)., Comment: 33 pages. v3: Theorem 3.3 removed. References reorganized according to the journal standards

We compute the zeta function of the Hilbert scheme of $n$ points on a two-dimensional torus and show it satisfies a remarkable functional equation. To this end we establish an explicit formula for the number $C_n(q)$ of ideals of codimension $n$ of the algebra $\textbf{F}_q[x,y,x^{-1}, y^{-1}]$ of Laurent polynomials in two variables over a finite field of cardinality $q$. This number is a palindromic polynomial of degree $2n$ in $q$. Moreover, $C_n(q) = (q-1)^2 P_n(q)$, where $P_n(q)$ is another palindromic polynomial, whose integer coefficients turn out to be non-negative and which is a $q$-analogue of the sum of divisors of $n$. Each coefficient of $P_n(q)$ counts the divisors of $n$ in a certain interval. We also give arithmetical interpretations of the values of $C_n(q)$ and of $P_n(q)$ at $q = -1$ and at roots of unity of order $3$, $4$, $6$.

Quantum dots are nanostructures confined in the three space directions. Since many decades, numerous studies have been devoted to these structures for their interesting electronic and optical properties.In this thesis, we model the electronic behaviour of quantums dots thanks to a type Bloch model derived il the Heisenberg formalism. The closure of equations leads to a non linear model stemming from Coulomb and electron--phonon interactions. We study the qualitative properties of the obtained Bloch models (trace, hermicity, positivitiveness) and the Cauchy problem for the semi-classical model coupling Bloch and Maxwell equations to describe laser--quantum dot interaction. We derive also formally rate equations from the non-linear Bloch equations. The discretizations of one-dimensionnal Maxwell--Bloch equations involve splitting methods for the Bloch equations, which enable the preservation of the qualitative properties of the continuous model. The validation of the model and the study of the relevancy of some simplification is performed thanks to self-induced transparency and coherence-transfert test cases., Les boîtes quantiques sont les nanostructures confinées suivant les trois directions de l'espace. Depuis quelques décennies, de nombreuses études sont consacrées à des boîtes pour leurs propriétés électroniques et optiques intéressantes.Dans cette thèse, nous modélisons le comportement électronique de boîtes quantiques par un modèle de type Bloch dérivé dans le formalisme de Heisenberg. La fermeture des équations du modèle aboutit à un modèle non-linéaire issu des interactions coulombiennes et des interactions entre les électrons et les phonons. Nous étudions les propriétés qualitatives de la solution des modèles de Bloch obtenus (trace, hermicité, positivité) ainsi que le problème de Cauchy associé au couplage semi-couplage avec les équations de Maxwell. Nous dérivons également formellement des équations de taux à partir des modèles de Bloch non-linéaires. La discrétisation des modèles unidimensionnels de Maxwell--Bloch fait appel à une méthode de splitting (méthode par pas fractionnaires) pour les équations de Bloch préservant les propriétés qualitatives du modèle continu. La validation du modèle et l'étude de pertinence de certaines simplifications sont effectuées grâce à des cas tests de transparence auto-induite et de transfert de cohérence.

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Generalizing the notion of matched pair of groups, we define and study matched pairs of locally compact groupoids endowed with Haar systems, in order to give new examples of measured quantum groupoids., In this new version paragraph 4.3 has been modified and completed in order to correct a previous technical error which does not really affect our main result

34 pages; The existence of common structures in physics and computer sciences theories has become an active field of research. In the present paper the scope is enlarged, in particular in Economics, to any theory which is formalized in quantitative models meeting some precise but general conditions. The first section is dedicated to characterize the main components of any scientific theory, using the concepts of epistemology, and puts in prominent positions models, association of objects with properties which are represented by quantitative variables. In a second section we show that a large class of models can be represented in Hilbert spaces. In this framework we precise the concepts of observables, with its relation with measures, and develop several tools to characterize the evolution of systems, interacting systems and phases transitions.

9 pages; In this note, inspired by the proof of the Kirillov-Reshetikhin conjecture, we consider tensor products of Kirillov-Reshetikhin modules of a fixed node and various level. We fix a positive integer and attach to each of its partitions such a tensor product. We show that there exists an embedding of the tensor products, with respect to the classical structure, along with the reverse dominance relation on the set of partitions.

We establish, for all simply laced types, a q,t-character formula, first conjectured by Nakajima. It relates, on one hand, the structure of the $\ell$-weight spaces of standard modules regarded as modules over the Heisenberg subalgebra of some quantum affine algebra and, on the other hand, the t-dependence of their q,t-characters, as originally defined in terms of the Poincar\'e polynomials of certain Lagrangian subvarieties in quiver varieties of the corresponding simply laced type. Our proof is essentially geometric and generalizes to arbitrary simply laced types earlier representation theoretical results for standard $\mathrm{U}_q (\widehat{\mathfrak{sl}}_2)$-modules., Comment: 33 pages

An operad structure on certain bicoloured noncrossing configurations in regular polygons is studied. Motivated by this study, a general functorial construction of enveloping operad, with input a coloured operad and output an operad, is presented. The operad of noncrossing configurations is shown to be the enveloping operad of a coloured operad of bubbles. Several suboperads are also investigated, and described by generators and relations.

This thesis is devoted to the study of modular representations of Ariki-Koike algebras, and of the connections with Kashiwara's crystal and canonical bases theory via Ariki's categorification theorem. First, we study, using combinatorial tools, the decomposition matrices associated to these algebras, generalising the works of Geck and Jacon. We fully classify the cases of existence and non-existence of canonical basic sets, and we explicitly construct these sets when they exist. Next, we make explicit the crystal isomorphisms for Fock spaces representations of the quantum affine algebra of affine type A. We then construct of a particular isomorphism, so-called canonical, which gives, inter alia, a non-recursive description of any connected component of the crystal. We also stress the links with the combinatorics of words underlying the crystal structure of Fock spaces, by describing notably an analogue of the Robinson-Schensted-Knuth correspondence for affine type A., Cette thèse est consacrée à l'étude des représentations modulaires des algèbres d'Ariki-Koike, et des liens avec la théorie des cristaux et des bases canoniques de Kashiwara via le théorème de catégorification d'Ariki. Dans un premier temps, on étudie, grâce à des outils combinatoires, les matrices de décomposition de ces algèbres en généralisant les travaux de Geck et Jacon. On classifie entièrement les cas d'existence et de non-existence d'ensembles basiques, en construisant explicitement ces ensembles lorsqu'ils existent. On explicite ensuite les isomorphismes de cristaux pour les représentations de Fock de l'algèbre affine quantique de type A affine. On construit alors un isomorphisme particulier, dit canonique, qui permet entre autres une caractérisation non-récursive de n'importe quelle composante connexe du cristal. On souligne également les liens avec la combinatoire des mots sous-jacente à la structure cristalline des espaces de Fock, en décrivant notamment un analogue de la correspondance de Robinson-Schensted-Knuth pour le type A affine.

We consider subgroups of the braid groups which are generated by $k$-th powers of the standard generators and prove that any infinite intersection (with even $k$) is trivial. This is motivated by some conjectures of Squier concerning the kernels of Burau's representations of the braid groups at roots of unity. Furthermore, we show that the image of the braid group on 3 strands by these representations is either a finite group, for a few roots of unity, or a finite extension of a triangle group, by using geometric methods., 18p. Former arXiv:0907.0568 is now split as two separate papers: this one records the first half and the title is changed accordingly. The second half is contained in arxiv:1108.4904