Submitted by Johnny Rodrigues (johnnyrodrigues@ufcg.edu.br) on 2018-08-09T18:04:07Z No. of bitstreams: 1 ANTONIO MARCOS DUARTE DE FRANÇA - DISSERTAÇÃO 2014..pdf: 1066992 bytes, checksum: 6e270db1611e61d65507f5f99e9bd161 (MD5) / Made available in DSpace on 2018-08-09T18:04:07Z (GMT). No. of bitstreams: 1 ANTONIO MARCOS DUARTE DE FRANÇA - DISSERTAÇÃO 2014..pdf: 1066992 bytes, checksum: 6e270db1611e61d65507f5f99e9bd161 (MD5) Previous issue date: 2014-10 / Capes / Para ler o resumo deste trabalho recomendamos o download do arquivo, uma vez que o mesmo possui fórmulas e caracteres matemáticos que não foram possíveis trascreve-los aqui. / To read the summary of this work we recommend downloading the file, since it has formulas and mathematical characters that were not possible to transcribe them here.

Submitted by Johnny Rodrigues (johnnyrodrigues@ufcg.edu.br) on 2018-08-10T16:27:27Z No. of bitstreams: 1 ALAN DE ARAÚJO GUIMARÃES - DISSERTAÇÃO PPGMAT 2014..pdf: 389630 bytes, checksum: 8fee4901dc2c6f4008991c541e1728b0 (MD5) / Made available in DSpace on 2018-08-10T16:27:27Z (GMT). No. of bitstreams: 1 ALAN DE ARAÚJO GUIMARÃES - DISSERTAÇÃO PPGMAT 2014..pdf: 389630 bytes, checksum: 8fee4901dc2c6f4008991c541e1728b0 (MD5) Previous issue date: 2014-12 / Capes / O tema central da presente dissertação é o estudo das graduações de um grupo G nas álgebras UTn(F) eUT(d1,...,dm).Inicialmente, no Capítulo 2, supondo o grupo G abeliano e infnito e o corpo F algebricamente fechado e de característica zero, provamos que qualquer graduação em UTn(F) é elementar (a menos de automorfismo G-graduado). Ainda no Capítulo 2,sem fazer qualquer suposição sobre o grupo G e ocorpo F, chegamos à mesma conclusão. Para tanto, foi necessário utilizar técnicas mais sutis na demonstração. No Capítulo 3, novamente supondo o grupo G abeliano e infinito e o corpo F algebricamente fechado e de característica zero,classificamos as G-graduações da F-álgebra UT(d1,...,dm). Veremos que,neste caso, existe uma decomposição d1 = tp1,...,dm = tpm talqueUT(d1,...,dm) é isomorfa, como álgebra G-graduada ,ao produto tensorial Mt(F)⊗UT(p1,...,pm), onde Mt(F) tem uma G-graduação na e UT(p1,...,pm) tem uma G-graduação elementar. / The central theme of this dissertation is the study the of the gradings of a group G in the algebras UTn(F) and UT(d1, . . . , dm). Initially, in Chapter 2, assuming G a nite abelian group and F an algebraically closed eld and of characteristic zero, we prove that any grading in UTn(F) is elementary (up to graded isomorphism). Still in Chapter 2, without making any assumption about the group G and the eld F, we obtain the same conclusion. To prove this was necessary to use more subtle techniques in demonstration. In Chapter 3, again assuming G a nite abelian group and F an algebraically closed eld of characteristic zero, we classify the gradings of the algebra UT(d1, . . . , dm). We will see that there is a decomposition d1 = tp1, . . . , dm = tpm such that UT(d1, ..., dm) is isomorphic, as graded algebra, to the tensor product Mt(F) ⊗ UT(p1, . . . , pm), where Mt(F) has a ne grading and UT(p1, . . . , pm) has a elementary grading.

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.

This thesis has two distinguish parts. The first part concerns the study of n-ary algebras. A n-ary algebra is a vector space with a multiplication on n arguments. Classically the multiplications are binary, but the use of ternary multiplication in theoretical physic like for Nambu brackets led mathematicians to investigate these type of algebras. Two classes of n-ary algebras are fundamental: the associative n-ary algebras and the Lie n-ary algebras. We are interested by both classes. Concerning the associative n-ary algebras we are mostly interested in 3-ary partially associative 3-ary algebras, that is, algebras whose multiplication satisfies ((xyz)tu)+(x(yzt)u)+(xy(ztu))=0. This type is interesting because the previous woks on this subject was not distinguish the even and odd cases. We show in this thesis that the case n=3 can not be treated as the even cases. We investigate in detail the free partially associative 3-ary algebra on k generators. This algebra is graded and we compute the dimensions of the 7 first components. In the general case, we give a spanning set such as the sub family of non zero vector is a basis. The main consequences are the free partially associative 3-ary algebra is solvable. In the free commutative partially associative 3-ary algebra any product on 9 elements is trivial. The operad for partially associative 3-ary algebra do not satisfy the Koszul property. Then we study n-ary products on the tensors. The simplest example is given by a internal product of non square matrices. We can define a 3-ary product by taking A . ^tB . C. We show that we have to generalize a bit the definition of partial associativity for n-ary algebras. We then introduce the products -partially associative where  is a permutation of the symmetric group of degree n. Concerning the n-ary algebras, two classes have been defined: Filipov algebras (also called recently Lie-Nambu algebras) and some more general class, the n-Lie algebras. Filipov algebras are very important in the study of the mechanic of Nambu-Poisson, and is a particular case of the other. So to define an approach of Maurer-Cartan type, that is, define a scalar cohomology, we consider in this work Fillipov as n-Lie algebras and develop such a calculus in the n-Lie algebras frame work. We also give some classifications of n-ary nilpotent algebras. The last chapter of this part concerns my work in Master on the Poisson algebras on polynomials. We present link with the Lie algebras is clear. Thus we extend our study to Poisson algebras which associated Lie algebra is rigid and we apply these results to the enveloping algebras of rigid Lie algebras. The second part concerns intervals arithmetic. The interval arithmetic is used in a lot of problems concerning robotic, localization of parameters, and sensibility of inputs. The classical operations of intervals are based of the rule : the result of an operation of interval is the minimal interval containing all the result of this operation on the real elements of the concerned intervals. But these operations imply many problems because the product is not distributive with respect the addition. In particular it is very difficult to translate in the set of intervals an algebraic functions of a real variable. We propose here an original model based on an embedding of the set of intervals on an associative algebra. Working in this algebra, it is easy to see that the problem of non distributivity disappears, and the problem of transferring real function in the set of intervals becomes natural. As application, we study matrices of intervals and we solve the problem of reduction of intervals matrices (diagonalization, eigenvalues, and eigenvectors).

O objetivo deste trabalho é estudar a estrutura dos módulos sobre uma álgebra trivial de dimensão dois na variedade M das álgebras comutativas de potências associativas. Em particular classificamos os módulos irredutíveis. Estes resultados nos permitem compreender melhor a estrutura das nilálgebras comutativas de dimensão finita e nilíndice 4. Finalmente classificamos, sob isomorfismos, as nilálgebras comutativas de potências associativas de dimensão n e nilíndice n. / The aim of this work is to study the structure of the modules over a trivial algebra of dimension two in the variety M of commutative and power-associative algebras. In particular we classify the irreducible modules. These results enables us to understand better the structure of finite-dimensional power-associative nilalgebras of nilindex 4. Finally, we classify, up to isomorphism, commutative power associative nilalgebras of nilindex n and dimension n.

Submitted by Johnny Rodrigues (johnnyrodrigues@ufcg.edu.br) on 2018-08-13T14:12:26Z No. of bitstreams: 1 LAISE DIAS ALVES ARAÚJO - DISSERTAÇÃO PPGMAT 2017..pdf: 818445 bytes, checksum: 666322e4502e880db6af0ea641df08f7 (MD5) / Made available in DSpace on 2018-08-13T14:12:26Z (GMT). No. of bitstreams: 1 LAISE DIAS ALVES ARAÚJO - DISSERTAÇÃO PPGMAT 2017..pdf: 818445 bytes, checksum: 666322e4502e880db6af0ea641df08f7 (MD5) Previous issue date: 2017-06 / Capes / Nesta dissertação estudamos as graduações elementares (ou boas graduações) e as identidades polinomiais graduadas correspondentes em álgebras de matrizes triangulares superiores em blocos. Uma graduação elementar por um grupo G na álgebra A = UT(α1, α2, ..., αr) de matrizes triangulares superiores em blocos é determinada por uma n-upla em Gn, onde n = α1+· · ·+αr. Mostraremos que as graduações elementares em A determinadas por duas n-uplas em Gnsão isomorfas se, e somente se, as n-uplas estão na mesma órbita da bi-ação canônica em Gn com o grupo Sα1 × · · · × Sαr agindo à esquerda e G à direita. Em seguida utilizamos estes resultados para mostrar que, sob certas hipóteses (por exemplo, se o grupo G tem ordem prima), duas álgebras de matrizes triangulares superiores em blocos, graduadas pelo grupo G, satisfazem as mesmas identidades graduadas se, e somente se, são isomorfas (como álgebras graduadas). / In this dissertation we study elementary (or good) gradings in upper block triangular matrix algebras and the corresponding graded polynomial identities. An elementary grading by a group G on the algebra A = UT(α1, α2, ..., αr) of upper block triangular matrices is determined by an n-tuple in Gn, where n = α1 + · · · + αr. It will be proved that the elementary gradings on A determined by two n-tuples in Gn are isomorphic if and only if the n-tuples are in the same orbit in the canonical bi-action on Gn with the group Sα1 × · · · × Sαr acting on the left and the group G acting on the right. These results will be used to prove that under suitable hypothesis (for example if the group G has prime order) two upper block triangular matrix algebras, graded by the group G, satisfy the same graded identities if and only if they are isomorphic (as graded algebras).

Submitted by Erika Demachki (erikademachki@gmail.com) on 2015-03-11T19:37:56Z No. of bitstreams: 2 Dissertação - Victor Hugo López Solís - 2012.pdf: 755677 bytes, checksum: ab64efbb1cbb6b6d5b9683cad6f75d6e (MD5) license_rdf: 23148 bytes, checksum: 9da0b6dfac957114c6a7714714b86306 (MD5) / Approved for entry into archive by Erika Demachki (erikademachki@gmail.com) on 2015-03-13T18:58:33Z (GMT) No. of bitstreams: 2 Dissertação - Victor Hugo López Solís - 2012.pdf: 755677 bytes, checksum: ab64efbb1cbb6b6d5b9683cad6f75d6e (MD5) license_rdf: 23148 bytes, checksum: 9da0b6dfac957114c6a7714714b86306 (MD5) / Made available in DSpace on 2015-03-13T18:58:59Z (GMT). No. of bitstreams: 2 Dissertação - Victor Hugo López Solís - 2012.pdf: 755677 bytes, checksum: ab64efbb1cbb6b6d5b9683cad6f75d6e (MD5) license_rdf: 23148 bytes, checksum: 9da0b6dfac957114c6a7714714b86306 (MD5) Previous issue date: 2012-03-19 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / Our work has as main objective, to establish conditions for a canonical form for elements of a ring, semigroup or algebraic structure similar. This result is obtained through the main Theorem 3.10 (The Lemma of Bergman’s Diamond) with applications. / O nosso trabalho tem como objetivo principal, estabelecer condições para obter uma forma canônica para os elementos de um anel, semigrupo ou estrutura algébrica similar. Isto é obtido através do resultado principal, o Teorema 3.10 (O Lema do Diamante de Bergman), com aplicações.

O objetivo deste trabalho é estudar a estrutura dos módulos sobre uma álgebra trivial de dimensão dois na variedade M das álgebras comutativas de potências associativas. Em particular classificamos os módulos irredutíveis. Estes resultados nos permitem compreender melhor a estrutura das nilálgebras comutativas de dimensão finita e nilíndice 4. Finalmente classificamos, sob isomorfismos, as nilálgebras comutativas de potências associativas de dimensão n e nilíndice n. / The aim of this work is to study the structure of the modules over a trivial algebra of dimension two in the variety M of commutative and power-associative algebras. In particular we classify the irreducible modules. These results enables us to understand better the structure of finite-dimensional power-associative nilalgebras of nilindex 4. Finally, we classify, up to isomorphism, commutative power associative nilalgebras of nilindex n and dimension n.

In this thesis we study gradings on blocks of group algebras. The motivation to study gradings on blocks of group algebras and their transfer via derived and stable equivalences originates from some of the most important open conjectures in representation theory, such as Broue’s abelian defect group conjecture. This conjecture predicts the existence of derived equivalences between categories of modules. Some attempts to prove Broue’s conjecture by lifting stable equivalences to derived equivalences highlight the importance of understanding the connection between transferring gradings via stable equivalences and transferring gradings via derived equivalences. The main idea that we use is the following. We start with an algebra which can be easily graded, and transfer this grading via derived or stable equivalence to another algebra which is not easily graded. We investigate the properties of the resulting grading. In the first chapter we list the background results that will be used in this thesis. In the second chapter we study gradings on Brauer tree algebras, a class of algebras that contains blocks of group algebras with cyclic defect groups. We show that there is a unique grading up to graded Morita equivalence and rescaling on an arbitrary basic Brauer tree algebra. The third chapter is devoted to the study of gradings on tame blocks of group algebras. We study extensively the class of blocks with dihedral defect groups. We investigate the existence, positivity and tightness of gradings, and we classify all gradings on these blocks up to graded Morita equivalence. The last chapter deals with the problem of transferring gradings via stable equivalences between blocks of group algebras. We demonstrate on three examples how such a transfer via stable equivalences is achieved between Brauer correspondents, where the group in question is a TI group.

Ce mémoire comporte deux parties distinctes. La première partie concerne une étude d'algèbres n-aires. Une algèbre n-aire est un espace vectoriel sur lequel est définie une multiplication sur n arguments. Classiquement les multiplications sont binaires, mais depuis l'utilisation en physique théorique de multiplications ternaires, comme les produits de Nambu, de nombreux travaux mathématiques se sont focalisés sur ce type d'algèbres. Deux classes d'algèbres n-aires sont essentielles: les algèbres n-aires associatives et les algèbres n-aires de Lie. Nous nous intéressons aux deux classes. Concernant les algèbres n-aires associatives, on s'intéresse surtout aux algèbres 3-aires partiellement associatives, c'est-à-dire dont la multiplication vérifie l'identité ((xyz)tu)+(x(yzt)u)+(xy(ztu))=0 Ce cas est intéressant car les travaux connus concernant ce type d'algèbres ne distinguent pas les cas n pair et n-impair. On montre dans cette thèse que le cas n=3 ne peut pas être traité comme si n était pair. On étudie en détail l'algèbre libre 3-aire partiellement associative sur un espace vectoriel de dimension finie. Cette algèbre est graduée et on calcule précisément les dimensions des 7 premières composantes. On donne dans le cas général un système de générateurs ayant la propriété qu'une base est donnée par la sous famille des éléments non nuls. Les principales conséquences sont L'algèbre libre 3-aire partiellement associative est résoluble. L'algèbre libre commutative 3-aire partiellement associative est telle que tout produit concernant 9 éléments est nul. L'opérade quadratique correspondant aux algèbres 3-aires partiellement associatives ne vérifient pas la propriété de Koszul. On s'intéresse ensuite à l'étude des produits n-aires sur les tenseurs. L'exemple le plus simple est celui d'un produit interne sur des matrices non carrées. Nous pouvons définir le produit 3aire donné par A . ^tB . C. On montre qu'il est nécessaire de généraliser un peu la définition de partielle associativité. Nous introduisons donc les produits -partiellement associatifs où  est une permutation de degré p. Concernant les algèbres de Lie n-aires, deux classes d'algèbres ont été définies: les algèbres de Fillipov (aussi appelées depuis peu les algèbres de Lie-Nambu) et les algèbres n-Lie. Cette dernière notion est très générale. Cette dernière notion, très important dans l'étude de la mécanique de Nambu-Poisson, est un cas particulier de la première. Mais pour définir une approche du type Maurer-Cartan, c'est-à-dire définir une cohomologie scalaire, nous considérons dans ce travail les algèbres de Fillipov comme des algèbres n-Lie et développons un tel calcul dans le cadre des algèbres n-Lie. On s'intéresse également à la classification des algèbres n-aires nilpotentes. Le dernier chapitre de cette partie est un peu à part et reflète un travail poursuivant mon mémoire de Master. Il concerne les algèbres de Poisson sur l'algèbre des polynômes. On commence à présenter le crochet de Poisson sous forme duale en utilisant des équations de Pfaff. On utilise cette approche pour classer les structures de Poisson non homogènes sur l’algèbre des polynômes à trois variables . Le lien avec les algèbres de Lie est clair. Du coup on étend notre étude aux algèbres de Poisson dont l'algèbre de Lie sous jacent est rigide et on applique les résultats aux algèbres enveloppantes des algèbres de Lie rigides. La partie 2 concerne l'arithmétique des intervalles. Cette étude a été faite suite à une rencontre avec une société d'ingénierie travaillant sur des problèmes de contrôle de paramètres, de problème inverse (dans quels domaines doivent évoluer les paramètres d'un robot pour que le robot ait un comportement défini). [...] / This thesis has two distinguish parts. The first part concerns the study of n-ary algebras. A n-ary algebra is a vector space with a multiplication on n arguments. Classically the multiplications are binary, but the use of ternary multiplication in theoretical physic like for Nambu brackets led mathematicians to investigate these type of algebras. Two classes of n-ary algebras are fundamental: the associative n-ary algebras and the Lie n-ary algebras. We are interested by both classes. Concerning the associative n-ary algebras we are mostly interested in 3-ary partially associative 3-ary algebras, that is, algebras whose multiplication satisfies ((xyz)tu)+(x(yzt)u)+(xy(ztu))=0. This type is interesting because the previous woks on this subject was not distinguish the even and odd cases. We show in this thesis that the case n=3 can not be treated as the even cases. We investigate in detail the free partially associative 3-ary algebra on k generators. This algebra is graded and we compute the dimensions of the 7 first components. In the general case, we give a spanning set such as the sub family of non zero vector is a basis. The main consequences are the free partially associative 3-ary algebra is solvable. In the free commutative partially associative 3-ary algebra any product on 9 elements is trivial. The operad for partially associative 3-ary algebra do not satisfy the Koszul property. Then we study n-ary products on the tensors. The simplest example is given by a internal product of non square matrices. We can define a 3-ary product by taking A . ^tB . C. We show that we have to generalize a bit the definition of partial associativity for n-ary algebras. We then introduce the products -partially associative where  is a permutation of the symmetric group of degree n. Concerning the n-ary algebras, two classes have been defined: Filipov algebras (also called recently Lie-Nambu algebras) and some more general class, the n-Lie algebras. Filipov algebras are very important in the study of the mechanic of Nambu-Poisson, and is a particular case of the other. So to define an approach of Maurer-Cartan type, that is, define a scalar cohomology, we consider in this work Fillipov as n-Lie algebras and develop such a calculus in the n-Lie algebras frame work. We also give some classifications of n-ary nilpotent algebras. The last chapter of this part concerns my work in Master on the Poisson algebras on polynomials. We present link with the Lie algebras is clear. Thus we extend our study to Poisson algebras which associated Lie algebra is rigid and we apply these results to the enveloping algebras of rigid Lie algebras. The second part concerns intervals arithmetic. The interval arithmetic is used in a lot of problems concerning robotic, localization of parameters, and sensibility of inputs. The classical operations of intervals are based of the rule : the result of an operation of interval is the minimal interval containing all the result of this operation on the real elements of the concerned intervals. But these operations imply many problems because the product is not distributive with respect the addition. In particular it is very difficult to translate in the set of intervals an algebraic functions of a real variable. We propose here an original model based on an embedding of the set of intervals on an associative algebra. Working in this algebra, it is easy to see that the problem of non distributivity disappears, and the problem of transferring real function in the set of intervals becomes natural. As application, we study matrices of intervals and we solve the problem of reduction of intervals matrices (diagonalization, eigenvalues, and eigenvectors).