Thesis (Ph.D.)--University of Nebraska-Lincoln, 2007. / Title from title screen (site viewed Apr. 29, 2008). PDF text: 103 p. : ill. ; 493 K. UMI publication number: AAT 3283908. Includes bibliographical references. Also available in microfilm and microfiche formats.

Thesis (Ph.D.)--University of Nebraska-Lincoln, 2008. / Title from title screen (site viewed Jan. 13, 2009). PDF text: 40 p. ; 747 K. UMI publication number: AAT 3315330. Includes bibliographical references. Also available in microfilm and microfiche formats.

Thesis (Ph. D.)--Auburn University. / Abstract. Vita. Includes bibliographical references (p. 49-51).

The Lie algebra Vec(ℝ) of polynomial vector fields on the line acts naturally on ℂ[]. This action has a one-parameter family of deformations called the tensor density modules F_λ. The bounded indecomposable modules of Vec(ℝ) of length 2 composed of tensor density modules have been classified by Feigin and Fuchs. We present progress towards describing the annihilators of the unique indecomposable extension of F_λ by F_(λ+2) in the non-resonant case λ ≠ -½. We give the intersection of the annihilator and the subalgebra of lowest weight vectors of the universal enveloping algebra (Vec(ℝ)) of Vec(ℝ). This result is found by applying structural descriptions of the lowest weight vectors of (Vec(ℝ)).

Thesis (Ph.D.)--University of Nebraska-Lincoln, 2006. / Title from title screen (site viewed on Feb. 5, 2007). PDF text: 66 p. : ill. (some col.) UMI publication number: AAT 3216432. Includes bibliographical references. Also available in microfilm and microfiche format.

This thesis was concerned with classifying the real indecomposable solvable Lie algebras with codimension one nilradicals of dimensions two through seven. This thesis was organized into three chapters. In the first, we described the necessary concepts and definitions about Lie algebras as well as a few helpful theorems that are necessary to understand the project. We also reviewed many concepts from linear algebra that are essential to the research. The second chapter was occupied with a description of how we went about classifying the Lie algebras. In particular, it outlined the basic premise of the classification: that we can use the automorphisms of the nilradical of the Lie algebra to find a basis with the simplest structure equations possible. In addition, it outlined a few other methods that also helped find this basis. Finally, this chapter included a discussion of the canonical forms of certain types of matrices that arose in the project. The third chapter presented a sample of the classification of the seven-dimensional Lie algebras. In it, we proceeded step-by-step through the classification of the Lie algebras whose nilradical was one of four specifically chosen because they were representative of the different types that arose during the project. In the appendices, we presented our results in a list of the multiplication tables of the isomorphism classes found.

Neste trabalho aplicamos técnicas de combinatória infinitária e forcing na teoria dos espaços de Banach, investigando propriedades dos espaços de Banach da forma C(K), formado pelas funções reais contínuas sobre K com a norma do supremo, com poucos operadores, no sentido de que todo operador em C(K) é da forma gI+S, onde I é o operador identidade, g pertence a C(K) e S é fracamente compacto. Enfatizamos as construções onde K é conexo, o que implica que C(K) é indecomponível. Assumindo Axioma Diamante, um axioma combinatório mais forte que a Hipótese do Contínuo, construímos um espaço de Banach C(K) tal que C(L) tem poucos operadores, para todo L subespaço fechado de K. Sob a Hipótese do Contínuo construímos um espaço C(K) indecomponível com poucos operadores tal que K contém $\\beta N$ homeomorficamente. Em ZFC construímos um espaço C(K) com poucos operadores em um sentido estritamente mais fraco. Também mostramos a existência de pelo menos contínuo espaços de Banach C(K) indecomponíveis dois a dois essencialmente incomparáveis. Usando forcing provamos que existe consistentemente um espaço de Banach C(K) de densidade menor que contínuo com poucos operadores e um C(K) indecomponível de densidade menor que contínuo. / In this work we apply techniques of infinitary combinatorics and forcing in Banach spaces theory, investigating the compact topological spaces K such that the Banach space C(K), consisting of the continuous real-valued functions on K with the supremum norm, has few operators, in the sense that all operators on C(K) have the form gI+S, where I is the identity operator, g\\ belongs to C(K) and S is weakly compact. We emphasize the constructions where K is connected, which implies that C(K) is indecomposable. Assuming Diamond Axiom, a combinatoric axiom stronger than the continuum hypothesis, we construct a Banach space C(K) where C(L) has few operators, for every L closed subspace of K. Under continuum hypothesis we construct an indecomposable C(K) with few operators such that K contains $\\beta \\mathbb$ homeomorphically. In ZFC we construct a space C(K) with few operators in a strictly weaker sense. We also show the existence of at least continuum pairwise essentially incomparable indecomposable Banach spaces C(K). Using forcing, we prove that there exists consistently a Banach space C(K) of density smaller than continuum having few operators and an indecomposable C(K) of density smaller than continuum.

Neste trabalho, introduzimos as álgebras de Weyl clássicas A = A_n e as generalizadas A = D(sigma, a). Apresentamos algumas propriedades importantes dessas álgebras, dentre outras, que a n-ésima álgebra de Weyl A_n é um domínio simples Noetheriano à esquerda. Introduzimos os módulos de peso sobre A e estudamos os A-módulos de peso projetivos. Iniciamos a classificação dos A-módulos de peso simples (isto é, irredutíveis) através de uma categoria linear C_O e do seu esqueleto S_O cf. A classificação total dos A_infty-módulos de peso simples é dada utilizando a ação de certas localizações no anel de polinômios cf. Classificamos os blocos do tipo mansa na categoria dos A-módulos de peso localmente finitos e determinamos os A-módulos indecomponíveis nos blocos do tipo mansa. Seguindo, descrevemos os A-módulos de peso injetivos e projetivos indecomponíveis e deduzimos uma descrição dos blocos na categoria dos A-módulos de peso por quivers e relações. / In this dissertation, we introduce the classical Weyl algebras A = A_n and the generalized A = D(sigma, a). There are some important properties of these algebras, among others, that the n-th Weyl algebra A_n is a left Noetherian simple domain. We introduced the weight modules over A and study the projective weight A-modules. Started the classification of simple weight A-modules (this is, irreducible) by linear category C_O and its skeleton S_O in accordance with. The complete classification of simple weight A-modules is given using the action of certain localizations in the polynomial ring in accordance with. We classify the tame blocks in the category of locally-finite weight A-modules and determine the indecomposable A-modules in the tame blocks. Following, we describe indecomposable projective and injective weight A-modules and deduce the description of the blocks in the category of weight A-modules by quivers and relations.

In the usual definition of a Euclidean domain, a ring has a norm function whose codomain is the positive integers. It was noticed by Motzkin in 1949 that the codomain could be replaced by any well-ordered set. This motivated the study of transfinite Euclidean domains in which the codomain of the norm function is replaced by the class of ordinals. We prove that there exists a (transfinitely valued) Euclidean Domain with Euclidean order type for every indecomposable ordinal. Modifying the construction, we prove that there exists a Euclidean Domain with no multiplicative norm. Following a definition of Clark and Murty, we define a set of admissible primes. We develop an algorithm that can be used to find sets of admissible primes in the ring of integers of quadratic extensions of the rationals and provide some examples.

Neste trabalho, introduzimos as álgebras de Weyl clássicas A = A_n e as generalizadas A = D(sigma, a). Apresentamos algumas propriedades importantes dessas álgebras, dentre outras, que a n-ésima álgebra de Weyl A_n é um domínio simples Noetheriano à esquerda. Introduzimos os módulos de peso sobre A e estudamos os A-módulos de peso projetivos. Iniciamos a classificação dos A-módulos de peso simples (isto é, irredutíveis) através de uma categoria linear C_O e do seu esqueleto S_O cf. A classificação total dos A_infty-módulos de peso simples é dada utilizando a ação de certas localizações no anel de polinômios cf. Classificamos os blocos do tipo mansa na categoria dos A-módulos de peso localmente finitos e determinamos os A-módulos indecomponíveis nos blocos do tipo mansa. Seguindo, descrevemos os A-módulos de peso injetivos e projetivos indecomponíveis e deduzimos uma descrição dos blocos na categoria dos A-módulos de peso por quivers e relações. / In this dissertation, we introduce the classical Weyl algebras A = A_n and the generalized A = D(sigma, a). There are some important properties of these algebras, among others, that the n-th Weyl algebra A_n is a left Noetherian simple domain. We introduced the weight modules over A and study the projective weight A-modules. Started the classification of simple weight A-modules (this is, irreducible) by linear category C_O and its skeleton S_O in accordance with. The complete classification of simple weight A-modules is given using the action of certain localizations in the polynomial ring in accordance with. We classify the tame blocks in the category of locally-finite weight A-modules and determine the indecomposable A-modules in the tame blocks. Following, we describe indecomposable projective and injective weight A-modules and deduce the description of the blocks in the category of weight A-modules by quivers and relations.