La thèse comporte six chapitres. Dans le premier chapitre, on rappelle les bases de la théorie et on étudie la structure des algèbres Hom-associatives ainsi que les différentes constructions comme la composition avec des endomorphismes qui nous permet de construire de nouveaux objets et d’établir certaines nouvelles propriétés. Parmi les résultats originaux, on peut signaler l’étude des algèbres Hom-associatives simples ainsi que leurs constructions. On a montré que toutes les algèbres Hom-associatives multiplicatives simples s’obtiennent par composition d’algèbres simples et d’automorphismes. Dans le deuxième chapitre, on commence par étudier les propriétés des changements de base dans ces structures algébriques. On a calculé la base de Gröbner de l’idéal engendrant la variété algébrique des algèbres Hom-associatives de dimension 2 où la multiplication µ et l’application linéaire α sont identifiées à leurs constantes de structure relativement à une base donnée. La classification, à isomorphisme près, des algèbres Hom-associatives unitaires et non unitaires est établie en dimension 2 et 3. On a aussi décrit les algèbres de type associatif en se basant sur le théorème de twist de Yau. Dans le troisième chapitre, on étudie certaines propriétés et invariants comme les dérivations, αk-dérivations où k est un entier positif. Dans le quatrième chapitre, on établit la cohomologie de ces algèbres. On a pu lister les algèbres rigides grâce à leur classe de cohomologie puis on s'est 'intéressé aux déformations infinitésimales et dégénérations. D’une part, la cohomologie et déformation de ces algèbres nous a permis d’identifier les algèbres rigides dont le deuxième groupe de cohomologie est nulle, et d’autre part de caractérisation de composante irréductible. Dans le cinquième chapitre, on s’intéresse aux structures Rota-Baxter de poids λ ϵK de ces algèbres. Enfin, dans le dernier chapitre, on a travaillé sur les structures Hom-bialgèbres et leurs invariants. / The purpose of this thesis is to study the structure of Hom-associative algebras and provide classifications. Among the results obtained in this thesis, we provide 2-dimensional and 3-dimensional Hom-associative algebras and give a characterization of multiplicative simple Hom-associative algebras. Moreover we compute some invariants and discuss irreducible components of the corresponding algebraic varieties. The thesis is organized as follows. In the first chapter we give the basics about Hom-associative algebras and provide some new properties. Moreover, we discuss unital Hom-associative algebras. Chapter 2 deals with simple multiplicative Hom-associative algebras. We present one of the main results of this paper, that is a characterization of simple multiplicative Hom-associative algebras. Indeed, we show that they are all obtained by twistings of simple associative algebras. Chapter 3 is dedicated to describe algebraic varieties of Hom-associative algebras and provide classifications, up to isomorphism, of 2-dimensional and 3-dimensional Hom-associative algebras. In chapter 4, we compute their derivations and twisted derivations, whereas in chapter 5, we compute their Hom-Type Hochschild cohomology. In the last section of this chapter, we consider the geometric classification problem using one-parameter formel deformations, and describe the irreducible components. In chapter 6, we compute Rota-Baxter structures of weight k of Hom-associative algebras appearing in our classification. In chapter 7, We work out Hom-bialgebras structures as well as their invariants. Properties and classifications, as well as the calculation of certain invariants such as the first and second cohomology groups, were studied.

The international community is in agreement that a lepton collider in the TeV centre of mass energy range is required to leverage discoveries made at the Large Hadron Collider and expand the physics programme. Two future colliders are proposed. The International Linear Collider (ILC) will collide electron and positron bunches at a centre of mass energy of 500 GeV, upgradable to 1 TeV. The Compact Linear Collider (CLIC) is designed to reach 3 TeV. This thesis investigates the wakefields, which degrade the beam quality, and beam dynamics in the main linacs of the ILC, presenting the first direct comparison of beam dynamics for linacs made up of the alternative high gradient superconducting cavity designs - the Reentrant and Ichiro cavities. Higher order modes of the electromagnetic field in the cavities, which will be excited by the passage of the bunches, are calculated using finite difference and finite element techniques. A trapped dipole mode in the Ichiro cavity at 2.4498 GHz is identified. These modes are used as the basis for the beam dynamics studies. These simulations have demonstrated that ILC linacs made up of the new high gradient cavities, with targeted damping, would meet wakefield requirements for delivering high quality beams for particle physics studies. This result is important since any upgrade of the ILC from 500 GeV to 1 TeV centre of mass energy would make use of one of these high gradient cavity designs in the extension to the linacs. Beam dynamics in the CLIC beam delivery system (BDS), are also detailed. Simulations included deflecting mode Crab Cavities required to maximise collision luminosity when there is a crossing angle, and verify analytic results for the required deflecting voltage and tolerances to phase differences. The tolerance to crab cavity roll angle is found to be extremely tight, at 5.9 millidegrees. Additionally, results in this thesis uncover a problem with the BDS magnet layout which must be addressed.

In this thesis we look at hom-associative algebras (which turn out to be exactly the G1-hom-associative algebras), by, in two and three dimensions, trying to find the structure constants for which an algebra becomes hom-associative when the homomorphism 𝛼 is defined as different matrix units. These algebras are also hom-Lie admissible (or G6-hom-associative, which turn out to be the same thing) with a commutator, so we also find the commutator for each of these hom-Lie admissible algebras. We end up finding every hom-associative and hom-Lie algebra for 𝛼 defined as each 2×2 matrix unit in two dimensions, each 3×3 matrix unit in three dimensions when the problem is mapped to one dimension, for three 3×3 matrix units in three dimensions when the problem is mapped to two dimensions (but with the commutators not having been calculated), and only a few hom-associative algebras and hom-Lie algebras for one 3×3 matrix unit in the full three dimensions. We also compare the results for the different values of 𝛼, and find that in 𝑛 dimensions it is possible to find the values of the structure constants for all 𝑛2 different 𝛼:s simply by finding all of the solutions for 𝑛 different 𝛼:s (chosen in a specific way) and then permutating all of the indices.

Le but de cette thèse est d'étudier quelques aspects algébriques du problème de quantification par déformation. On considère d'une part la formalité dans le cas des algèbres libres et de l'algèbre de Lie so(3), et on s'intéresse d'autre part à la quantification par déformation pour des structures Hom-algébriques. Suivant le résultat de formalité de Kontsevich en 1997 pour les algèbres symétriques, on étudie dans la première partie de cette thèse les algèbres libres, qui sont un cas particulier d'algèbres enveloppantes, et on montre qu'il n'y a pas formalité en général, sauf dans les cas triviaux. On montre aussi qu'il n'y a pas formalité pour l'algèbre de Lie so(3). Les techniques utilisées sont de type homologiques. On calcule la cohomologie de ces algèbres et on procède à la construction du L-infini-quasi-isomorphisme entre l'algèbre de Lie différentielle graduée des cochaînes de Hochschild munie du crochet de Gerstenhaber et l'algèbre de la cohomologie munie du crochet de Schouten. Dans la seconde partie de ce travail, on utilise un principe de déformation par twist pour les structures Hom-algébriques, pour construire de nouvelles structures de même type, ou encore pour déformer une structure classique en une Hom-structure correspondante à l'aide d'un morphisme d'algèbres. En particulier, on applique ce procédé aux structures de Poisson et aux star-produits de Moyal-Weyl. Par ailleurs, on établit une correspondance entre les algèbres enveloppantes d'algèbres Hom-Lie possédant une structure Hom-coPoisson et les bigèbres Hom-Lie. / This thesis aims to study some algebraic aspects of the deformation quantization problem. On one hand, we consider formality for free algebras and the Lie algebra so(3), and on the other hand we study deformation quantization for Hom-algebraic structures. Following Kontsevich's formality result in 1997 for symmetric algebras, we study in the first part free algebras, which are a particular case of envelopping algebras, and show that there is no formality, except for the trivial cases. We also show that there is no formality for the Lie algebra so(3). The tools used are homological ones. We compute the cohomology of these algebras and proceed to the construction of the L-infinity-quasi-isomorphism between the differential graded Lie algebra of the Hochschild cochains endowed with the Gerstenhaber bracket and the cohomology algebra endowed with the Schouten bracket. In the second part of this work, we use a principle of deformation by twist for Hom-algebraic structures, to construct new structures of the same type, or to deform a classical structure in the corresponding Hom-structure using an algebra morphism. In particular, we apply this method to Poisson structures and Moyal-Weyl star-products. Besides, we establish a correspondance between enveloping algebras of Hom-Lie algebras endowed with a Hom-coPoisson structure and Hom-Lie bialgebras.

No description available.

Thesis (M. Phil.)--University of Hong Kong, 1985.

Thesis (M. Sc.(Urb. Plan.))--University of Hong Kong, 1995. / Includes bibliographical references (leaf 216-221).

Thesis--Brunswick. / In Periodical Room.

In this thesis, we consider hom-associative algebras, which is an algebra with multiplication that is not necessarily commutative nor associative, but obeys a twisted version of associativity by a linear homomorphism. We will give some conditions for associativity, which helps us determine commuting elements. Under other conditions, such as different types of unitality conditions, we can also state some results regarding commuting elements in the general, non-associative case.

No description available.