Esta dissertação oferece uma introdução às já conhecidas álgebras de Krichever-Novikov se restringindo aos exemplos abordados previamente em Bremner (1995), Cox (2013), Cox e Jurisich (2013), Cox, Futorny e Martins (2014), Bueno, Cox e Furtony (2009), e as definições de estruturas que podem auxiliar a estudar estes espaços, incluindo álgebras de Lie afins, álgebras de loop e módulos de Verma. Considerando K uma álgebra de Krichever-Novikov do tipo 4-ponto, 3-ponto, elíptica ou DJKM e suas respectivas subálgebras de Heisenberg K\' = K hK , onde hK é a subálgebra de Cartan de K , nos Teoremas 3.2.3, 3.4.3, 3.6.3 e 3.8.3 são apresentados critérios explícitos de irredutibilidade para K\'-módulos do tipo -Verma. / This work gives an introduction to the already known Krichever-Novikov algebras limited only to the examples approached before in Bremner (1995), Cox (2013), Cox e Jurisich (2013), Cox, Futorny and Martins (2014), Bueno, Cox and Furtony (2009), and the structures definitions that could help us to study these spaces, including affine Lie algebras, loop algebras and Verma modules. Let K be a 4-point, 3-point, elliptic or DJKM Krichever-Novikov algebra and its respective Heisenberg subalgebras K\' = K hK , where hK is the K Cartan subalgebra. In the Theorems 3.2.3, 3.4.3, 3.6.3 and 3.8.3 we will give a explicit irreducibility criteria for -Verma K\'-modules.

Let $\overline{R}$ be a finitely bordered Riemann surface, and let $\mathfrak{E}_\rho(\overline{R})$ be a flat matrix $PU_n(\mathbb{C})$-bundle over $\overline{R}$. Let $\Gamma_c(\overline{R}, \mathfrak{E}(\overline{R}))$ denote the $C^*$-algebra of continuous cross-sections of $\mathfrak{E}(\overline{R})$, and let $\Gamma_h(\overline{R},\mathfrak{E}(\overline{R}))$ denote the subalgebra consisting of the continuous holomorphic sections, i.e.~the continuous cross-sections that are holomorphic on the interior of $\overline{R}$. The algebra $\Gamma_c(\overline{R}, \mathfrak{E}(\overline{R}))$ is an example of an $n$-homogeneous $C^*$-algebra, and the subalgebra $\Gamma_h(\overline{R},\mathfrak{E}(\overline{R}))$ is the principal object of study of this thesis. The algebras $\Gamma_h(\overline{R},\mathfrak{E}(\overline{R}))$ appeared in the earlier works \cite{Abrahamse1976} and \cite{Blecher2000}. Operators that can be viewed as elements in $\Gamma_h(\overline{R},\mathfrak{E}(\overline{R}))$ are the subject of \cite{Abrahamse1976}. The Morita theory of $\Gamma_h(\overline{R},\mathfrak{E}(\overline{R}))$, under the guise of a fixed-point algebra and in the special case of an annulus $R$, is studied in \cite[Ex.~8.3]{Blecher2000}. This thesis studies these algebras and their topological data $\mathfrak{E}_\rho(\overline{R})$ motivated by several problems in the theory of nonselfadjoint operator algebras. Boundary representations are an invariant of operator algebras that were introduced by Arveson in 1969. However, it took nearly 50 years to show that boundary representations existed in sufficient abundance in all cases. I show that every boundary representation of $\Gamma_c(\overline{R}, \mathfrak{E}(\overline{R}))$ for $\Gamma_h(\overline{R}, \mathfrak{E}(\overline{R}))$ is given by evaluation at some point $r \in \partial R$. As a corollary, the $C^*$-envelope of $\Gamma_h(\overline{R},\mathfrak{E}(\overline{R}))$ is $\Gamma_c(\partial R, \mathfrak{E}(\partial R))$. Using the $C^*$-envelope, I show that for certain choices of fibre and base space, $\Gamma_h(\overline{R}, \mathfrak{E}_\rho(\overline{R}))$ is not completely isometrically isomorphic to $A(\overline{R})\otimes M_n(\mathbb{C})$ unless the representation $\rho$ is the trivial representation. I also show that $\Gamma_h(\overline{R},\mathfrak{E}(\overline{R}))$ is an Azumaya over its center. Azumaya algebras are the ``pure-algebra'' analogues to $n$-homogeneous $C^*$-algebras \cite{Artin1969}. Thus the structure of the nonselfadjoint subalgebra $\Gamma_h(\overline{R},\mathfrak{E}(\overline{R}))$ reflects some of the structure of its $C^*$-envelope (which is $n$-homogeneous). Finally, I answer a question raised in \cite[Ex.~8.3]{Blecher2000} on the $cb$ and strong Morita theory of $\Gamma_h(\overline{R},\mathfrak{E}_\rho(\overline{R}))$, showing in particular that $\Gamma_h(\overline{R},\mathfrak{E}_\rho(\overline{R}))$ is $cb$ Morita equivalent to its center $A(\overline{R})$. As suggested in \cite[Ex.~8.3]{Blecher2000}, I provide additional evidence that $\Gamma_h(\overline{R},\mathfrak{E}_\rho(\overline{R}))$ may not be strongly Morita equivalent to its center. This evidence, in turn, suggests that there may be a Brauer group -like analysis for these algebras.

On étudie dans cette thèse les coefficients de structure et particulièrement leurs dépendancesen n dans le cadre d’une suite des algèbres de doubles-classes. Le premier chapitre est dédié à l’étude des coefficients de structure dans le cas général des centres d’algèbres de groupes finis et des algèbres de doubles-classes. On rappelle dans ce chapitre la théorie des représentationsdes groupes finiset son lien avec les coefficients de structure. On montre que l’étude des coefficients de structure des algèbres de doubles-classes est reliéeà la théorie des paires de Gelfand et auxfonctions sphériques zonales en donnant un théorème similaireà celui de Frobenius. Ce théorème exprime les coefficients de structure d’une algèbre de doubles-classes associée à une paire de Gelfand en fonction des fonctions sphériques zonales. Dans le deuxième chapitre, on rappellele théorème de Farahat et Higmann autour de la propriété de polynomialité des coefficients de structure du centre de l’algèbre du groupe symétriqueainsi que la preuve d’Ivanov et Kerov. On donne une preuvecombinatoire pour lapropriété de polynomialité des coefficients de structure de l’algèbre de Hecke de la paire (S2n, Bn) dans le troisième chapitre. On utilise dans notre preuve une algèbre universelle qui se projette sur l’algèbre de Hecke de la paire (S2n, Bn) pour tout n. On montre aussi que cette algèbre universelle est isomorphe à l’algèbre fonctions symétriques décalées d’ordre 2. Dans le dernier chapitre on présente un cadre général pour la forme des coefficients de structure dans le cas d’une suite des algèbres de doubles-classes.Ce cadre regroupe les propriétés de polynomialité des coefficients de structure du centre de l’algèbre du groupe symétrique et de l’algèbre de Hecke de la paire (S2n, Bn).De plus, on donne des propriétés de polynomialité pour les coefficients de structure du centre de l’algèbre du groupe hypéroctaédral et de l’algèbre de doubles-classes de diag (Sn-1) dans Sn x Sopp n-1. / In this thesis we studied the structure coefficients and especially their dependence on n in the case of a sequence of double-class algebras. The first chapter is dedicated to the study of the structure coefficients in the general cases of centers of group algebras and double-class algebras. We recall in it the representation theory of finite groups and its link with structure coefficients. We show also that the study of the structure coefficients of double-class algebras is related to the theory of Gelfand pairs and zonal spherical functions by giving, in the case of Gelfand pairs, a theorem similar to that of Frobenius which writes the structure coefficients of the double-class algebra associated to a Gelfand pair in terms of zonal spherical functions. In the second chapter, we recall the Farahat and Higman's theorem about the polynomiality of the structure coefficients of the center of the symmetric group algebra as well as the Ivanov and Kerov's approach to prove this theorem. We give a combinatorial proof to the polynomiality property of the structure coefficients of the Hecke algebra of thepair (S2n, Bn) in the third chapter. Our proof uses a universal algebra which projects on the Hecke algebra of (S2n, Bn) for each n. We show that this universal algebra is isomorphic to the algebra of 2-shifted symmetric functions. In the fourth and last chapter we build a general framework which gives us the form of the structure coefficients in the case of a sequence of double-class algebras. This framework implies the polynomiality property of the structure coefficients of both the center of the symmetric group algebra and the Hecke algebra of (S2n, Bn). In addition, we give a polynomiality property for the structure coefficients of both the center of the hyperoctahedral group algebra and the double-class algebra of diag (Sn-1) in Sn x Sopp n-1.

S.Skryabin demonstrou que qualquer álgebra de Lie simples de dimensão finita sobre um corpo de característica 2 possui posto toroidal 2. Duas 2- álgebras de Lie de dimensão 31 foram estudadas. Neste trabalho, mostramos que a primeira delas contem uma base toroidal absoluta de dimensão três, assim como a segunda, que foi estudada por Grishkov e Guerreiro anteriormente. Utilizando uma decomposicão de Cartan, exibimos um isomorfismo entre as duas 2- álgebras de Lie de dimensão 31. Este resultado foi sugerido depois de encontrar uma sub álgebra de dimensão 12 n ao solúvel e 7 isomorfas 2-sub álgebras de Lie de dimensão 7 nas duas álgebras. Finalmente, exploramos uma 2- álgebra de Lie de dimensão 34 como o fim de encontrar base toroidal absoluta de dimensão 4. Apoiamos os cálculos com algumas códigos no linguajem de MATLAB que permitiram optimizar e acelerar a pesquisa. / S.Skryabin showed that any finite dimensional simple Lie algebra over a field of characteristic 2 has absolute toral rank 2. Two 31-dimensional 2-algebras were known. In this work, we show that the first of these algebras, contains a 3-dimensional maximal toral subalgebra, as the second one, which was studied by Grishkov e Guerreiro previously. Using a Cartan decomposition we establish an isomorphism between the two 31-dimensional 2-algebras. This result was suggested after finding a 12-dimensional not soluble subalgebra and seven 7-dimensional isomorphic 2-subalgebras in both algebras. Finally, a 34-dimensional 2-Lie algebra was studied in order to find 4-dimensional maximal toral subalgebras. Some computations in this work were performed with help of MATLAB.

To the pair (E,c), where E is a countable Borel equivalence relation on a standard Borel space (X,A) and c a normalized Borel T-valued 2-cocycle on E, we associate a sequentially weakly closed Borel *-algebra Br*(E,c), contained in the bounded linear operators on L^2(E). Associated to Br*(E,c) is a natural (Borel) Cartan subalgebra (Definition 6.4.10) L(Bo(X)) isomorphic to the bounded Borel functions on X. Then L(Bo(X)) and its normalizer (the set of the unitaries u in Br*(E,c) such that u*fu in L(Bo(X)), f in L(Bo(X))) countably generates the Borel *-algebra Br*(E,c). In this thesis, we study Br*(E,c) and in particular prove that: i) If E is smooth, then Br*(E,c) is a type I Borel *-algebra (Definition 6.3.10). ii) If E is a hyperfinite, then Br*(E,c) is a Borel AF-algebra (Definition 7.5.1). iii) Generalizing Kumjian's definition, we define a Borel twist G over E and its associated sequentially closed Borel *-algebra Br*(G). iv) Let a Borel Cartan pair (B, Bo) denote a sequentially closed Borel *-algebra B with a Borel Cartan subalgebra Bo, where B is countably Bo-generated. Generalizing Feldman-Moore's result, we prove that any pair (B, Bo) can be realized uniquely as a pair (Br*(E,c), L(Bo(X))). Moreover, we show that the pair (Br*(E,c), L(Bo(X))) is a complete invariant of the countable Borel equivalence relation E. v) We prove a Krieger type theorem, by showing that two aperiodic hyperfinite countable equivalence relations are isomorphic if and only if their associated Borel *-algebras Br*(E1) and Br*(E2) are isomorphic.

The Banach-Stone Theorem (respectly, Kadison Theorem) says that two abelian (respectively, general) C*-algebras are isomorphic as C*-algebras (respectively, JB*-algebras) if and only if they are isomorphic as Banach spaces. We are interested in using different structures to determine C*-algebras. Here, we would like to study the disjointness structures of C*-algebras and investigate if it suffices to determine C*-algebras. There are at least four versions of disjointness structures: zero product, range orthogonality, domain orthogonality and doubly orthogonality. In this thesis, we first study these disjointness structures in the case of standard operator algebras. Then we extend these results to general C*-algebras, namely, C*-algebras with continuous trace.

This paper presents the theory of Noetherian filtrations, an important concept in commutative algebra. The paper describes many aspects of the theory of these objects, presenting basic results, examples and applications. In the study of Noetherian filtrations, a few other important concepts are introduced such as Rees algebras, essential powers filtrations, and filtrations on modules. Basic results on these are presented as well. This thesis discusses at length how Noetherian filtrations relate to important constructions in commutative algebra, such as graded rings and modules, dimension theory and associated primes. In addition, the paper presents an original proof of the finiteness of the intersection algebra of principal ideals in a UFD. It concludes by discussing possible applications of this result to other areas of commutative algebra.

No description available.

In Chapter One the theoretical basis for infinitesimal transformations is presented with particular emphasis on the central theme of this thesis which is the invariance of ordinary differential equations, and their first integrals, under infinitesimal transformations. The differential operators associated with these infinitesimal transformations constitute an algebra under the operation of taking the Lie Bracket. Some of the major results of Lie's work are recalled. The way to use the generators of symmetries to reduce the order of a differential equation and/or to find its first integrals is explained. The chapter concludes with a summary of the state of the art in the mid-seventies just before the work described here was initiated. Chapter Two describes the growing awareness of the algebraic properties of the paradigms of differential equations. This essentially ad hoc period demonstrated that there was value in studying the Lie method of extended groups for finding first integrals and so solutions of equations and systems of equations. This value was emphasised by the application of the method to a class of nonautonomous anharmonic equations which did not belong to the then pantheon of paradigms. The generalised Emden-Fowler equation provided a route to major development in the area of the theory of the conditions for the linearisation of second order equations. This was in addition to its own interest. The stage was now set to establish broad theoretical results and retreat from the particularism of the seventies. Chapters Three and Four deal with the linearisation theorems for second order equations and the classification of intrinsically nonlinear equations according to their algebras. The rather meagre results for systems of second order equations are recorded. In the fifth chapter the investigation is extended to higher order equations for which there are some major departures away from the pattern established at the second order level and reinforced by the central role played by these equations in a world still dominated by Newton. The classification of third order equations by their algebras is presented, but it must be admitted that the story of higher order equations is still very much incomplete. In the sixth chapter the relationships between first integrals and their algebras is explored for both first order integrals and those of higher orders. Again the peculiar position of second order equations is revealed. In the seventh chapter the generalised Emden-Fowler equation is given a more modern and complete treatment. The final chapter looks at one of the fundamental algebras associated with ordinary differential equations, the three element 8£(2, R), which is found in all higher order equations of maximal symmetry, is a fundamental feature of the Pinney equation which has played so prominent a role in the study of nonautonomous Hamiltonian systems in Physics and is the signature of Ermakov systems and their generalisations. / Thesis (Ph.D.)-University of Natal, 1995.

This paper presents the theory of Noetherian filtrations, an important concept in commutative algebra. The paper describes many aspects of the theory of these objects, presenting basic results, examples and applications. In the study of Noetherian filtrations, a few other important concepts are introduced such as Rees algebras, essential powers filtrations, and filtrations on modules. Basic results on these are presented as well. This thesis discusses at length how Noetherian filtrations relate to important constructions in commutative algebra, such as graded rings and modules, dimension theory and associated primes. In addition, the paper presents an original proof of the finiteness of the intersection algebra of principal ideals in a UFD. It concludes by discussing possible applications of this result to other areas of commutative algebra.