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An n-parameter Fibonacci AF-algebra is determined by a constant incidence matrix K of a special form. The form of the matrix K is defined by a given n-parameter Fibonacci sequence. We compute the K-theory of certain Fibonacci AF-algebra, and relate their K-theory to the K-theory of an AF-algebra defined by incidence matrices that are the transpose of K.

An n-tuple of operators V=(V_1,…,V_n) acting on a Hilbert space H is said to be isometric if the corresponding row operator is an isometry. A free semigroup algebra is the weakly closed algebra generated by an isometric n-tuple V. The structure of a free semigroup algebra contains a great deal of information about V. Thus it is natural to study this algebra in order to study V. A free semigroup algebra is said to be analytic if it is isomorphic to the noncommutative analytic Toeplitz algebra, which is a higher-dimensional generalization of the classical algebra of bounded analytic functions on the complex unit disk. This notion of analyticity is of central importance in the general theory of free semigroup algebras. A vector x in H is said to be wandering for an isometric n-tuple V if the set of words in the entries of V map x to an orthonormal set. As in the classical case, the analytic structure of the noncommutative analytic Toeplitz algebra is determined by the existence of wandering vectors for the generators of the algebra. In the first part of this thesis, we prove the following dichotomy: either an isometric n-tuple V has a wandering vector, or the free semigroup algebra it generates is a von Neumann algebra. This implies the existence of wandering vectors for every analytic free semigroup algebra. As a consequence, it follows that every free semigroup algebra is reflexive, in the sense that it is completely determined by its invariant subspace lattice. In the second part of this thesis we prove a decomposition for an isometric tuple of operators which generalizes the classical Lebesgue-von Neumann-Wold decomposition of an isometry into the direct sum of a unilateral shift, an absolutely continuous unitary and a singular unitary. The key result is an operator-algebraic characterization of an absolutely continuous isometric tuple in terms of analyticity. We show that, as in the classical case, this decomposition determines the weakly closed algebra and the von Neumann algebra generated by the tuple.

An n-tuple of operators V=(V_1,…,V_n) acting on a Hilbert space H is said to be isometric if the corresponding row operator is an isometry. A free semigroup algebra is the weakly closed algebra generated by an isometric n-tuple V. The structure of a free semigroup algebra contains a great deal of information about V. Thus it is natural to study this algebra in order to study V. A free semigroup algebra is said to be analytic if it is isomorphic to the noncommutative analytic Toeplitz algebra, which is a higher-dimensional generalization of the classical algebra of bounded analytic functions on the complex unit disk. This notion of analyticity is of central importance in the general theory of free semigroup algebras. A vector x in H is said to be wandering for an isometric n-tuple V if the set of words in the entries of V map x to an orthonormal set. As in the classical case, the analytic structure of the noncommutative analytic Toeplitz algebra is determined by the existence of wandering vectors for the generators of the algebra. In the first part of this thesis, we prove the following dichotomy: either an isometric n-tuple V has a wandering vector, or the free semigroup algebra it generates is a von Neumann algebra. This implies the existence of wandering vectors for every analytic free semigroup algebra. As a consequence, it follows that every free semigroup algebra is reflexive, in the sense that it is completely determined by its invariant subspace lattice. In the second part of this thesis we prove a decomposition for an isometric tuple of operators which generalizes the classical Lebesgue-von Neumann-Wold decomposition of an isometry into the direct sum of a unilateral shift, an absolutely continuous unitary and a singular unitary. The key result is an operator-algebraic characterization of an absolutely continuous isometric tuple in terms of analyticity. We show that, as in the classical case, this decomposition determines the weakly closed algebra and the von Neumann algebra generated by the tuple.

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.

O principal objetivo deste trabalho é o estudo de certas álgebras de funções complexas infinitamente deriváveis. Denomidas Álgebras de Dales-Davie, apresentamos a construção dessas álgebras que foram definidas por Dales e Davie em 1973, como também apresentamos os principais resultados envolvendo sua completude e naturalidade, tópicos que foram estudados por diversos autores, e.g. Abtahi, Honary, et al. Por fim, apresentamos os resultados obtidos por Lourenço e Vieira a respeito do estudo da diferença das álgebras de Dales-Davie com a Álgebra de Disco. / The main purpose of this work is to study the Dales-Davie algebras . We shall present the construction of this algebras defined by Dales and Davie in 1973. Following, the main results about its completeness and naturality as an normed algebra, this topics were studied by some authors, e.g. Abtahi, Honary, et al. Furthermore, we study the results found by Lourenço e Vieira about the difference of Dales-Davie algebras and Disk algebra.

<p>This thesis, consisting of three papers and a summary, studies properties of stratified algebras and representations of Lie superalgebras.</p><p>In Paper I we give a characterization when the Ringel dual of an SSS-algebra is properly stratified.</p><p>We show that for an SSS-algebra, whose Ringel dual is properly stratified, there is a (generalized) tilting module which allows one to compute the finitistic dimension of the SSS-algebra, and moreover, it gives rise to a new covariant Ringel-type duality.</p><p>In Paper II we give a characterization of standardly stratified algebras in terms of certain filtrations of (left or right) projective modules, generalizing the corresponding theorem of V. Dlab. We extend the notion of Ringel duality to standardly stratified algebras and estimate their finitistic dimension in terms of endomorphism algebras of standard modules.</p><p>Paper III deals with the queer Lie superalgebra and the corresponding BGG-category O. We show that the typical blocks correspond to standardly stratified algebras, and we generalize Kostant's Theorem to the queer Lie superalgebra.</p>

This thesis, consisting of three papers and a summary, studies properties of stratified algebras and representations of Lie superalgebras. In Paper I we give a characterization when the Ringel dual of an SSS-algebra is properly stratified. We show that for an SSS-algebra, whose Ringel dual is properly stratified, there is a (generalized) tilting module which allows one to compute the finitistic dimension of the SSS-algebra, and moreover, it gives rise to a new covariant Ringel-type duality. In Paper II we give a characterization of standardly stratified algebras in terms of certain filtrations of (left or right) projective modules, generalizing the corresponding theorem of V. Dlab. We extend the notion of Ringel duality to standardly stratified algebras and estimate their finitistic dimension in terms of endomorphism algebras of standard modules. Paper III deals with the queer Lie superalgebra and the corresponding BGG-category O. We show that the typical blocks correspond to standardly stratified algebras, and we generalize Kostant's Theorem to the queer Lie superalgebra.

Orientador: Plamen Emilov Kochloukov / Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matemática Estatística e Computação Científica / Made available in DSpace on 2018-08-25T04:30:00Z (GMT). No. of bitstreams: 1 Machado_GustavoGrings_D.pdf: 808427 bytes, checksum: 4482c43f5d1998040317e1873220ce8c (MD5) Previous issue date: 2014 / Resumo: Neste trabalho estudamos o invariante denominado dimensão de Gelfand-Kirillov para álgebras com identidades polinomiais, sobretudo para álgebras não-associativas, com o objetivo de melhor compreender a estrutura das identidades polinomiais. Ultimamente este invariante tem ganhado importância, uma vez que ele é relativamente fácil de calcular e, de certa forma, é capaz de diferenciar o crescimento de duas álgebras. Para álgebras associativas a GK-dimensão mostrou-se muito útil ao detectar que álgebras que por um lado são PI-equivalentes sobre corpos de característica zero pelo Teorema do Produto Tensorial de Kemer, por outro lado não são PI-equivalentes quando a característica do corpo infinito é positiva. Isto aponta para o surgimento de novos ????-ideais, conjuntos de identidades satisfeitas por uma álgebra, que são ???? -primos para corpos infinitos de característica positiva. Ainda é um problema em aberto a classificação e a compreensão destes ????-ideais em característica positiva, embora seja bem compreendida para PI-Álgebras associativas em característica zero, segundo a teoria de Kemer. Entretanto a situação é ainda menos clara para variedades de álgebras não-associativas como Álgebras de Jordan ou Álgebras de Lie. Sabe-se muito pouco sobre resultados que apontem para uma classificação de ????-ideais fora do caso associativo, até mesmo sobre corpos de característica zero. Inclusive se conhece pouco sobre o comportamento dos ????-ideais, mesmo de álgebras simples. Aqui damos um passo, calculando algumas GK-dimensões para álgebras relativamente livres de posto finito a partir da expressão da série de Hilbert. Destacamos em especial que calculamos a dimensão de Gelfand-Kirillov da álgebra relativamente livre de qualquer posto finito da álgebra de Lie das matrizes 2 × 2 de traço zero sobre um corpo infinito de característica diferente de 2. Acreditamos que estes resultados permitirão ajudar a compreender melhor o comportamento dos ????-ideais em álgebras não-associativas / Abstract: In this thesis we study the invariant called Gelfand-Kirillov Dimension for algebras with polynomial identities, mainly for non-associative algebras, aiming at better understanding the structure of the polynomial identities. This invariant has gained importance lately since in many cases it is relatively easy to calculate and, surprisingly, it is capable of distinguishing the growth of two algebras. For associative algebras GK-dimension was found to be very useful to detect that algebras which on one hand are PI-equivalent over fields of characteristic zero, according to Tensor Product Theorem of Kemer, on the other hand are not PI-equivalent when the characteristic of the infinite base field is positive. This points towards the rise of new ????-ideals, sets of identities satisfied by an algebra, which are ????-prime for infinite fields of positive characteristic. The classification and the understanding of such ????-ideals in positive characteristic are still open problems, although it is well understood for associative PI-Algebras in characteristic zero, using Kemer¿s theory. The situation is much less clear for varieties of non-associative algebras like Jordan Algebras or Lie Algebras. Very little is known about results towards a classification of ????-ideals outside the associative case, even over fields of characteristic zero. Accordingly little is known concerning the behavior of ????-ideals, even for simple algebras. Here we make a step towards this goal by computing some GK-dimensions of some relatively free algebras of finite rank by using the expression of the Hilbert series. In particular we compute the Gelfand-Kirillov dimension of the relatively free algebra of any finite rank generated by the Lie Algebra of the 2 × 2 traceless matrices over an infinite field of characteristic different from 2. We hope that results in this direction will contribute to a better understanding of the behavior of ????-ideals in non-associative algebras / Doutorado / Matematica / Doutor em Matemática

Orientadores: Daniela Mariz Silva Vieira, Jorge Tulio Mujica Ascui / Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matemática, Estatística e Computação Científica / Made available in DSpace on 2018-08-19T00:22:09Z (GMT). No. of bitstreams: 1 Nachtigall_Cicero_D.pdf: 625263 bytes, checksum: 7dc27b9956ffbb6e717aa95776b7b8c8 (MD5) Previous issue date: 2011 / Resumo: O resumo, na íntegra, poderá ser visualizado no texto completo da tese digital. / Abstract: The complete abstract is available with the full electronic digital thesis or dissertations. / Doutorado / Matematica / Doutor em Matemática