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Affine wreath product algebras with trace maps of generic parity
Get PDFO objetivo desse projeto é estudar teoria estrutural e representações de álgebras afins de produto entrelaçado An(F). Tais álgebras aparecem naturalmente em categorificações Heisenberg e generaliza outras importantes álgebras (álgebras de Hecke affim degenerada, álgebras de Sergeev afim e álgebras entrelaçadas de Hecke).A classe de algebras foi introduziada por D. Rosso e A. Savage em [RS17]. Em [Sav20], o segundo autor estudou teoria estrutural e de representações sobre a condição de que o traço de F fosse par. Nesse projeto estendemos a definição para o caso de traço ímpar, obtendo resultados análogos ímpares. Como nossa abordagem análoga a de Savage, nós consideramos os traços com paridade arbitrária e unificamos enunciados e demonstrações. Estudando a teoria estrutural, nós apresentamos uma base para An(F)e calculamos seu centro. Também introduzimos elementos de Jucys-Murphy e elementos entrelaçados. Considerando uma equivalência de categorias, descrevemos os An(F)-módulos simples em função de representações simples das álgebras de Hecke afim degenerada, álgebras de Sergeev afime álgebras de produto entrelaçantes. Definimos os quocientes ciclôtmicos ACn(F) de An(F) e mostramos que essas álgebras são álgebras de Frobenius com uma apropriada escolha de traço.Enunciamos um ciclotômico Mackey teorema e mostramos que ACn(F)é uma extensão de Frobenius de ACn1(F)The goal of this project is to study the structure and representation theory of affine wreath product algebras An(F). These algebras appear naturally in Heisenberg categorification and generalize many important algebras (degenerate affine Hecke algebras, affine Sergeev algebras and wreath Hecke algebras). The whole class was introduced by D. Rosso and A. Savage in [RS17]. In [Sav20] the second author studied both structure and representations under the condition that the trace map of F is even. In this project we extend the definition for the case where the trace is odd, obtaining ``odd-analogous\'\' results. Since our approach is analogous to Savage\'s, we consider the trace map being of arbitrary parity and unify statements and proofs. Studing the strucure theory, we present a basis for An(F) and compute its center. We also introduce Jucys-Murphy and intertwining elements. Considering a equivalence of categories, we describe the simple An(F)-modules in terms of simple representations of degenerate affine Hecke algebras, affine Sergeev algebras and wreath product algebras. We define the cyclotomic quotients AnC(F) for An(F) and we show that these are Frobenius algebras with an appropriately chosen trace map. We also state a cyclotomic Mackey Theorem and show that AnC(F) is a Frobenius extension of An-1C(F)$
g-Módulos U(h)-finitamente gerados e famílias coerentes
No full textThis thesis investigates the relationship between two categories of representations over a finite-dimensional simple Lie algebra g: the category admW of admissible weight modules, consisting of g-modules on which a fixed Cartan subalgebra h acts semi-simply with weight spaces of uniformly bounded dimension; and the category frkA of U(h)-finite g-modules, consisting of modules finitely generated over U(h), the universal enveloping algebra of h. We establish that both categories admW and frkA have objects with Gelfand-Kirillov dimension bounded by the rank of g. As a consequence of this result, we prove that frkA shares several structural properties with admW, such as the finite length of its objects, a block decomposition corresponding to generalized central characters and the existence of infinite dimensional objects only when g is of type A or C. The weighting functor W provides another link between these categories, mapping a U(h)-finite modules to admissible weight modules. By studying W and its left derived functors, we transfer properties of simple admissible weight modules to frkA. In particular, we prove that infinite dimensional simple U(h)-modules are U(h)-torsion free and locally free at all but finitely many maximal ideals of U(h). We introduce the concept of almost equivalence of weight modules and show that, if M in frkA is simple of infinite dimension, then W(M) is almost equivalent to a finite number of copies, called coherent-multiplicity, of a unique irreducible semi-simple coherent family - a class of module introduced and classified by Mathieu in \\cite. Using this framework, we classify simple U(h)-finite modules with coherent-multiplicity one, except for those with regular central character when g=sl(n+1) for n>=3. Finally, for each positive integer m, we construct a family of simple sl(n+1)-modules with coherent-multiplicity m. Moreover, we show that any U(h)-finite module with coherent-multiplicity greater than one cannot be a weight module with respect to any Cartan subalgebra h\'.Esta tese investiga a relação entre duas categorias de representações de uma álgebra de Lie simples de dimensão finita g: a categoria \\admW dos módulos de pesos admissíveis, cujos objetos são g-módulos tais quais uma subálgebra de Cartan fixa h age semi-simplesmente e que seus espaços de pesos tem dimensão uniformemente limitada; e a categoria frkA dos g-módulos U(h)-finitos, consistido de módulos finitamente gerados sobre U(h) (a álgebra envolvente universal de h). Estabelecemos que ambas as categorias admW e frkA possuem objetos com dimensão de Gelfand-Kirillov limitada pelo posto de g. Como consequência desse resultado, provamos que frkA compartilha várias propriedades estruturais com admW, tais como objetos com comprimento finito, admite decomposição em blocos correspondentes a caracteres centrais generalizados e a existência de objetos de dimensão infinita ocorre apenas quando g é do tipo A ou C. O functor de ponderação W fornece outra ligação entre essas categorias, mapeando um módulo U(h)-finito para um módulo de pesos admissíveis. Ao estudar W e seus functores derivados à esquerda, transferimos propriedades dos módulos de pesos admissíveis simples para objetos de frkA. Em particular, provamos que módulos U(h)-finitos simples de dimensão infinita são livres de torção sobre U(h) e localmente livres em todos os ideais maximais de U(h), exceto em um número finito deles. Introduzimos o conceito de quase equivalência de módulos de pesos e mostramos que se M em frkA é simples de dimensão infinita, então W(M) é quase equivalente a um número finito de cópias, chamado de multiplicidade coerente, de uma única família coerente semi-simples irredutível uma classe de módulos introduzida e classificada por Mathieu em \\cite. Usando essas ferramentas, classificamos módulos U(h)-finitos simples com multiplicidade coerente um, exceto para aqueles com caractere central regular quando g = sl(n+1) para n>=3. Finalmente, para cada inteiro positivo m, construímos uma família de sl(n+1)-módulos simples com multiplicidade coerente m. Além disso, mostramos que qualquer módulo U(h)-finito com multiplicidade coerente maior que um não pode ser um módulo de pesos com respeito a qualquer subálgebra de Cartan h\'
Representation of Kac-Moody algebras
No full textNeste trabalho, apresentamos uma técnica geral para a construção de novos módulos de peso irredutíveis para álgebras de Kac-Moody afins, utilizando a técnica da indução parabólica. Nosso objetivo principal foi superar as restrições, vistas em \\cite, que limitavam os módulos induzidos, oferecendo uma visão unificada e generalizada dessas estruturas para certas categorias. Exploramos a aplicação da indução parabólica em situações onde o fator Levi de uma subálgebra parabólica é infinito-dimensional e a carga central não é nula. Os resultados principais deste trabalho incluem um critério de irredutibilidade para o \\widehat{\\mathfrak g}-módulo M_(V), em que V é um (G+H)-módulo de peso irredutível com carga central não nula, Teorema ef. Além disso, estabelecemos que a indução parabólica define um funtor \\mathbb^{\\lambda} que preserva a irredutibilidade para a categoria dos \\widehat{\\mathfrak}-módulos de peso. Outro resultado significativo é a demonstração da irredutibilidade dos \\widehat{\\mathfrak}-módulos de Verma imaginário generalizado M_{a, \\widehat{\\mathfrak}}(V) com carga central não nula, em que V é um \\widehat{\\mathfrak}-módulo de peso, tensor e irredutível com carga central não nula, Teorema ef, fornecendo uma ferramenta valiosa para a construção de novos módulos irredutíveis em álgebras de Kac-Moody afins. Esses resultados representam avanços importantes na teoria dos módulos de álgebras de Kac-Moody. A abordagem desenvolvida neste trabalho tem o potencial de abrir novas perspectivas de pesquisa e promover o entendimento mais profundo de outras estruturas.In this work, we develop a general technique for constructing new irreducible weight modules for affine Kac-Moody algebras using the parabolic induction. Our main goal was to overcome the restrictions, as seen in \\cite, that limited induced modules, offering a unified and generalized view of these structures for certain categories. We explore the application of parabolic induction in situations where the Levi factor of a parabolic subalgebra is infinite-dimensional and the central charge is nonzero. The main results of this work include a criterion of irreducibility for the \\widehat{\\mathfrak g}-module M_(V), where V is an irreducible weight module with nonzero central charge, Theorem ef. Furthermore, we establish that parabolic induction defines a functor \\mathbb^{\\lambda} that preserves irreducibility for the category of weight \\widehat{\\mathfrak}-modules. Another significant result is the demonstration of the irreducibility of generalized imaginary Verma modules M_{a, \\widehat{\\mathfrak}}(V) with nonzero central charge, where V is a weight, tensor, and irreducible \\widehat{\\mathfrak}-module with nonzero central charge, Theorem ef, providing a valuable tool for constructing new irreducible modules in affine Kac-Moody algebras. These results represent important advances in the theory of modules of affine Kac-Moody algebras. The approach developed in this work has the potential to open new research perspectives and promote a deeper understanding of other structures
Going Beyond Counting First Authors in Author Co-citation Analysis
Get PDFThe present study examines one of the fundamental aspects of author co-citation analysis (ACA) - the way co-citation
counts are defined. Co-citation counting provides the data on which all subsequent statistical analyses and mappings
are based, and we compare ACA results based on two different types of co-citation counting - the traditional type that
only counts the first one among a cited work's authors on the one hand and a non-traditional type that takes into
account the first 5 authors of a cited work on the other hand. Results indicate that the picture produced through this non-traditional author co-citation counting contains more coherent author groups and is therefore considerably clearer. However, this picture represents fewer specialties in the research field being studied than that produced through the traditional first-author co-citation counting when the same number of top-ranked authors is selected and analyzed. Reasons for these effects are discussed
The three graces in the Tits--Kantor--Koecher category
Get PDFA metaphor of Jean-Louis Loday describes Lie, associative, and commutative
associative algebras as ``the three graces'' of the operad theory. In this
article, we study the three graces in the category of -modules
that are sums of copies of the trivial and the adjoint representation. That
category is not symmetric monoidal, and so one cannot apply the wealth of
results available for algebras over operads. Motivated by a recent conjecture
of the second author and Mathieu, we embark on the exploration of the extent to
which that category ``pretends'' to be symmetric monoidal. To that end, we
examine various homological properties of free associative algebras and free
associative commutative algebras, and study the Lie subalgebra generated by the
generators of the free associative algebra.Comment: 17 pages, comments are welcom
Variations on the Author
Get PDF“Variations on the Author” discusses two of Eduardo Coutinho’s recent films (Um Dia na Vida, from 2010, and Últimas Conversas, posthumously released in 2015) and their contribution to the general question of documentary authorship. The director’s filmography is characterized by a consistent yet self-effacing form of authorial self-inscription: Coutinho often features as an interviewer that rather than express opinions propels discourses; an interviewer that is good at listening. This mode of self-inscription characterizes him as an author who is not expressive but who is nonetheless markedly present on the screen. In Um Dia na Vida, however, Coutinho is completely absent form the image, while Últimas Conversas, on the contrary, includes a confessional prologue that moves the director from the margins to the center of his films. This article examines the ways in which these works stand out in the filmography of a director who offers new insights into the notion of cinematic authorship
Appropriate Similarity Measures for Author Cocitation Analysis
Get PDFWe provide a number of new insights into the methodological discussion about author cocitation analysis. We first argue that the use of the Pearson correlation for measuring the similarity between authors’ cocitation profiles is not very satisfactory. We then discuss what kind of similarity measures may be used as an alternative to the Pearson correlation. We consider three similarity measures in particular. One is the well-known cosine. The other two similarity measures have not been used before in the bibliometric literature. Finally, we show by means of an example that our findings have a high practical relevance.information science;Pearson correlation;cosine;similarity measure;author cocitation analysis
Dispelling the Myths Behind First-author Citation Counts
Get PDFWe conducted a full-scale evaluative citation analysis study of scholars in the XML research field to explore just how different from each other author rankings resulting from different citation counting methods actually are, and to demonstrate the capability of emerging data and tools on the Web in supporting more realistic citation counting methods. Our results contest some common arguments for the continued
use of first-author citation counts in the evaluation of scholars, such as high correlations between author rankings by first-author citation counts and other citation
counting methods, and high costs of using more realistic citation counting methods that are not well-supported by the ISI databases. It is argued that increasingly available digital full text research papers make it possible for citation analysis studies to go beyond what the ISI databases have directly supported and to employ more
sophisticated methods
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