Thesis (Ph. D.)--Hong Kong University of Science and Technology, 2003. / Includes bibliographical references (leaves 148-150). Also available in electronic version. Access restricted to campus users.

Thesis (Ph. D.)--University of Oregon, 2007. / Typescript. Includes vita and abstract. Includes bibliographical references (leaves 130-131). Also available for download via the World Wide Web; free to University of Oregon users.

In this thesis, we consider the question of describing the eigenvalues of a distinguished family of invariant differential operators associated to a Lie superalgebra g and a g-module W, called the "Capelli basis", via evaluation of certain classes of supersymmetric functions, called the interpolation super Jack polynomials. Finding the eigenvalues of the Capelli basis is referred to the Capelli Eigenvalue Problem. The eigenvalue formula depends on the chosen parametrization of the highest weight vectors in the decomposition of the superpolynomial algebra P(W), and consequently on the choice of a Borel subalgebra. In this thesis, we give a solution for each conjugacy class of Borel subalgebras, which we call a refined solution to the Capelli Eigenvalue Problem. Given the pair (g, W), we investigate the formulae for the eigenvalues of the Capelli operators associated to the completely reducible and multiplicity-free modules for two cases: diagonal and symmetric cases. In the former case, we show that we can express the eigenvalue of the Capelli operator on the irreducible component of the multiplicity-free decomposition of P(W) as a polynomial function of the b-highest weight of the irreducible component for any Borel subalgebra b. In the latter case, we show with a concrete counterexample that we cannot expect the results to be as strong as in the first case for all Borel subalgebras. We then express the eigenvalue of the Capelli operator on the irreducible component of the multiplicity-free decomposition of P(W) as a polynomial function of a piecewise affine map on the span of b-highest weights of the irreducible submodules of P(W), with respect to different decreasing Borel subalgebras b.

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Thesis (M.Sc.)--Memorial University of Newfoundland, 2002. / Bibliography: leaves 42-43.

Thesis (Ph. D.)--University of Oregon, 2004. / Typescript. Includes vita and abstract. Includes bibliographical references (leaves 123-125). Also available for download via the World Wide Web; free to University of Oregon users.

Orientador: Adriano Adrega de Moura / Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matemática Estatística e Computação Científica / Made available in DSpace on 2018-08-22T08:28:52Z (GMT). No. of bitstreams: 1 Calixto_LucasHenrique_M.pdf: 1707951 bytes, checksum: a7576ec9f19a4faf6e8bd959192baeb8 (MD5) Previous issue date: 2013 / Resumo: O principal objetivo dessa dissertação é explicar a classificação dos módulos irredutíveis de dimensão finita para qualquer super álgebra de funções sobre uma super álgebra de Lie básica. Os principais resultados dizem que um módulo irredutível de dimensão finita ou é uma representação de avaliação ou é um módulo de Kac para certo módulo de avaliação generalizado. Para chegar a tal objetivo, também fazemos uma revisão detalhada da classificação das super álgebras de Lie básicas / Abstract: The goal of this dissertation is to explain the classification of the irreducible finite-dimensional representations of a map superalgebra whose underlying simple Lie superalgebra is basic. The main result says that an irreducible finite-dimensional module is either an evaluation module or a Kac module associated to a certain generalized evaluation module. We also give a detailed review of the classification of the basic Lie superalgebras / Mestrado / Matematica / Mestre em Matemática

<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.

Ce texte étudie le groupe quantique Uξ sl(2|1) associé à la superalgèbre de Lie sl(2|1) et une catégorie de ses représentations de dimension finie. L'objectif est de construire des invariants topologiques de 3-variétés en utilisant la notion de trace modifiée. D'abord nous prouvons que la H catégorie CH des modules de poids nilpotents sur Uξ sl(2|1) est enrubannée et qu'il existe une trace modifiée sur son idéal des modules projectifs. De plus CH possède une structure relativement G-prémodulaire ce qui est une condition suffisante pour construire un invariant de 3-variétés à la Costantino-Geer-Patureau. Cet invariant est le cœur d'une 1+1+1-TQFT (Topological Quantum Field Theory). D'autre part Hennings a proposé à partir d'une algèbre de Hopf de dimension finie une construction d’invariants qui dispense de considérer la catégorie de H l l ses représentations. Nous montrons que le groupe quantique déroulé Uξ sl(2|1)/(e1 , f1 ) possède une complétion qui est une algèbre de Hopf enrubannée topologique. Nous construisons un invariant de 3-variétés à la Hennings en utilisant cette structure algébrique, une transformation de Fourier discrète et la notion de G-intégrales. L'intégrale dans une algèbre de Hopf est centrale dans la construction de Hennings. La notion de trace modifiée dans une catégorie s'est récemment révélée être une généralisation des intégrales dans les algèbres de Hopf de dimension finie. Dans un contexte plus général d'algèbre de Hopf de dimension infinie nous prouvons la relation formulée entre la trace modifiée et la G -intégrale. / This text studies the quantum group Uξ sl(2|1) associated with the Lie superalgebra sl(2|1) and a category of finite dimensional representations. The aim is to construct the topological invariants of 3-manifolds using the notion of modified trace. We first prove that the category CH of the nilpotent weight modules over Uξ sl(2|1) is ribbon and that there exists a modified trace on its ideal of projective modules. Furthermore, CH possesses a relative G-premodular structure which is a sufficient condition to construct an invariant of 3-manifolds of Costantino-Geer-Patureau type. This invariant is the heart of a 1+1+1-TQFT (Topological Quantum Field Theory). Next Hennings proposed from a finite dimensional Hopf algebra, a construction of invariants which does not require to consider the category of its representations. We show that the unrolled H l l quantum group Uξ sl(2|1)/(e1 , f1 ) has a completion which is a topological ribbon Hopf algebra. We construct an invariant of 3-manifolds of Hennings type using this algebraic structure, a discrete Fourier transform, and the notion of G-integrals. The integral in a Hopf algebra is central in the construction of Hennings. The notion of modified trace in a category has recently been revealed to be a generalization of the integrals in a finite dimensional Hopf algebra. In a more general context of infinite dimensional Hopf algebras we prove the relation formulated between the modified trace and the G-integral.