The main theme of this thesis is higher algebraic structures that come from operads and props. The first chapter is an introduction to the mathematical framework needed for the content of this thesis. The chapter does not contain any new results. The second chapter is concerned with the construction of a configuration space model for a particular 2-colored differential graded operad encoding the structure of two A∞ algebras with two A∞ morphisms and a homotopy between the morphisms. The cohomology of this operad is shown to be the well-known 2-colored operad encoding the structure of two associative algebras and of an associative algebra morphism between them. The third chapter is concerned with deformation quantization of (potentially) infinite dimensional (quasi-)Poisson manifolds. Our proof employs a variation on the transcendental methods pioneered by M. Kontsevich for the finite dimensional case. The first proof of the infinite dimensional case is due to B. Shoikhet. A key feature of the first proof is the construction of a universal L∞ structure on formal polyvector fields. Our contribution is a simplification of B. Shoikhet proof by considering a more natural configuration space and a simpler choice of propagator. The result is also put into a natural context of the dg Lie algebras coming from graph complexes; the L∞ structure is proved to come from a Maurer-Cartan element in the oriented graph complex. The fourth chapter also deals with deformation quantization of (quasi-)Poisson structures in the infinite dimensional setting. Unlike the previous chapter, the methods used here are purely algebraic. Our main theorem is the possibility to deformation quantize quasi-Poisson structures by only using perturbative methods; in contrast to the transcendental methods employed in the previous chapter. We give two proofs of the theorem via the theory of dg operads, dg properads and dg props. We show that there is a dg prop morphism from a prop governing star-products to a dg prop(erad) governing (quasi-)Poisson structures. This morphism gives a theorem about the existence of a deformation quantization of (quasi-)Poisson structure. The proof proceeds by giving an explicit deformation quantization of super-involutive Lie bialgebras and then lifting that to the dg properad governing quasi-Poisson structures. The prop governing star-products was first considered by S.A. Merkulov, but the properad governing quasi-Poisson structures is a new construction. The second proof of the theorem employs the Merkulov-Willwacher polydifferential functor to transfer the problem of finding a morphism of dg props to that of finding a morphism of dg operads.We construct an extension of the well known operad of A∞ algebras such that the representations of it in V are equivalent to an A∞ structure on V[[ħ]]. This new operad is also a minimal model of an operad that can be seen as the extension of the operad of associative algebras by a unary operation. We give an explicit map of operads from the extended associative operad to the operad we get when applying the Merkulov-Willwacher polydifferential functor to the properad of super-involutive Lie bialgebras. Lifting this map so as to go between their respective models gives a new proof of the main theorem.

Dans cette thèse, nous nous intéressons à la renormalisation de Connes et Kreimer dans le contexe des algèbres de Hopf de graphes de Feynman spécifiés. Nous construisons une structure d'algèbre de Hopf $\mathcal{H}_\mathcal{T}$ sur l'espace des graphes de Feynman spécifié d'une théorie quantique des champs $\mathcal{T}$. Nous définissons encore un dédoublement $\wt\mathcal{D}_\mathcal{T}$ de la bigèbre de graphes de Feynman spécifiés, un produit de convolution \divideontimes et un groupe de caractères de cette algèbre de Hopf à valeurs dans une algèbre commutative qui prend en compte la dépendance en les moments extérieurs. Nous mettons en place alors la renormalisation décrite par A. Connes et D. Kreimer et la décomposition de Birkhoff pour deux schémas de renormalisation : le schéma minimal de renormalisation et le schéma de développement de Taylor. Nous rappelons la définition des intégrales de Feynman associées à un graphe. Nous montrons que ces intégrales sont holomorphes en une variable complexe D dans le cas des fonctions de Schwartz, et qu'elles s'étendent en une fonction méromorphe dans le cas des fonctions de types Feynman. Nous pouvons alors déterminer les parties finies de ces intégrales en utilisant l'algorithme BPHZ après avoir appliqué la procédure de régularisation dimensionnelle. / In this thesis, we study the renormalization of Connes-Kreimer in the contex of specified Feynman graphs Hopf algebra. We construct a Hopf algebra structure $\mathcal{H}_\mathcal{T}$ on the space of specified Feynman graphs of a quantum field theory $\mathcal{T}$. We define also a doubling procedure for the bialgebra of specified Feynman graphs, a convolution product and a group of characters of this Hopf algebra with values in some suitable commutative algebra taking momenta into account. We then implement the renormalization described by A. Connes and D. Kreimer and the Birkhoff decomposition for two renormalization schemes: the minimal subtraction scheme and the Taylor expansion scheme.We recall the definition of Feynman integrals associated with a graph. We prove that these integrals are holomorphic in a complex variable D in the case oh Schwartz functions, and that they extend in a meromorphic functions in the case of a Feynman type functions. Finally, we determine the finite parts of Feynman integrals using the BPHZ algorithm after dimensional regularization procedure.

On the category of representations of a given quiver we define a tensor product point-wise and arrow-wise. The corresponding Clebsch-Gordan problem of how the tensor product of indecomposable representations decomposes into a direct sum of indecomposable representations is the topic of this thesis. The choice of tensor product is motivated by an investigation of possible ways to modify the classical tensor product from group representation theory to the case of quiver representations. It turns out that all of them yield tensor products which essentially are the same as the point-wise tensor product. We solve the Clebsch-Gordan problem for all Dynkin quivers of type A, D and E6, and provide explicit descriptions of their respective representation rings. Furthermore, we investigate how the tensor product interacts with Galois coverings. The results obtained are used to solve the Clebsch-Gordan problem for all extended Dynkin quivers of type Ãn and the double loop quiver with relations βα=αβ=αn=βn=0.

<p>On the category of representations of a given quiver we define a tensor product point-wise and arrow-wise. The corresponding Clebsch-Gordan problem of how the tensor product of indecomposable representations decomposes into a direct sum of indecomposable representations is the topic of this thesis.</p><p>The choice of tensor product is motivated by an investigation of possible ways to modify the classical tensor product from group representation theory to the case of quiver representations. It turns out that all of them yield tensor products which essentially are the same as the point-wise tensor product.</p><p>We solve the Clebsch-Gordan problem for all Dynkin quivers of type A, D and E₆, and provide explicit descriptions of their respective representation rings. Furthermore, we investigate how the tensor product interacts with Galois coverings. The results obtained are used to solve the Clebsch-Gordan problem for all extended Dynkin quivers of type Ã_n and the double loop quiver with relations βα=αβ=αⁿ=βⁿ=0.</p>

Peaks andThe main subject of this thesis is diagram rewriting.This is a generalisation to dimension~$2$ of word rewriting (in dimension~$1$). In a first time, we give the first convergent diagrammatic presentation of the PRO of linear maps in arbitrary field. Then we study the convergent diagrammatic presentation of matrix of isometries of $RR^n$. We focus especially on a rule similar to the Yang-Baxter equation, described by a certain map $h$. We use the confluence of critical the parametric diagrams, To study the algebraic properties of $h$, Finally, we present the $Sigma$-diagrams, an alternative approach for calculation in bialgebras. We illustrate this approach with examples. The last two chapters have been already published: Diagram rewriting for orthogonal matrices: a study of critical peaks, avec Yves Lafont, Lecture Notes in Computer Science 5117, p. 232-245, 2008 Properties of co-operations: diagrammatic proofs, Mathematical Structures in Computer Science 22(6), p. 970-986, 2012. / The main subject of this thesis is diagram rewriting.This is a generalisation to dimension~$2$ of word rewriting (in dimension~$1$). In a first time, we give the first convergent diagrammatic presentation of the PRO of linear maps in arbitrary field. Then we study the convergent diagrammatic presentation of matrix of isometries of $RR^n$. We focus especially on a rule similar to the Yang-Baxter equation, described by a certain map $h$. We use the confluence of criticalthe parametric diagrams, To study the algebraic properties of $h$, Finally, we present the $Sigma$-diagrams, an alternative approach for calculation in bialgebras. We illustrate this approach with examples. The last two chapters have been already published: Diagram rewriting for orthogonal matrices: a study of critical peaks, avec Yves Lafont, Lecture Notes in Computer Science 5117, p. 232-245, 2008 Properties of co-operations: diagrammatic proofs, Mathematical Structures in Computer Science 22(6), p. 970-986, 2012.

<p dir="ltr">In this thesis, we develop an algebraic framework for colored, colored connected, semi-grouplike-flavored, and pathlike co-/bi-/Hopf algebras, which are essential in combinatorics, topology, number theory, and physics. Moreover, we introduce and explore simply colored comonoid, which generalises the notion of colored conilpotent coalgebra. The simply colored structure captures the essence of being connected and give unified treatment of all connected co-/bi-algebras. </p><p dir="ltr">As a consequence, we establish precise conditions for the invertibility of characters essential for renormalization in the Connes-Kreimer formulation, supported by examples from these fields. In order to construct antipodes, we discuss formal localization constructions and quantum deformations. These allow to define and explain the appearance of Brown style coactions. We also investigate the relation between pointed coalgebras and color conilpotent coalgebras. </p><p dir="ltr">Using these results, we interpret all relevant coalgebras through categorical constructions, linking the bialgebra structures to Feynman categories and applying our developed theory in this context. This comprehensive framework provides a robust foundation for future research in mathematical physics and algebra.</p>