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Ribnere, Evija. « Engelbedingungen für nilpotente und auflösbare Gruppen ». [S.l.] : [s.n.], 2007. http://deposit.ddb.de/cgi-bin/dokserv?idn=983427631.
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Peters, Christoph. « Blätterungen von Nilmannigfaltigkeiten ». [S.l. : s.n.], 2003. http://deposit.ddb.de/cgi-bin/dokserv?idn=967209927.
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Gomez, John Hermes Castillo. « Propriedades de Lie de elementos simétricos sob involuções orientadas em álgebras de grupo ». Universidade de São Paulo, 2012. http://www.teses.usp.br/teses/disponiveis/45/45131/tde-04012013-170011/.
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Sejam $F$ um corpo de característica diferente de $2$ e $G$ um grupo. A partir da involução clássica, que envia cada elemento em seu inverso, e uma orientação do grupo $G$ é possível definir uma involução clássica orientada na álgebra de grupo $FG$. O objetivo desta tese é estudar propriedades de Lie do conjunto dos elementos simétricos $(FG)^+$ e, em alguns casos, do conjunto dos elementos anti-simétricos $(FG)^-$. Primeiro, abordamos o caso quando $G$ não tem elementos de ordem $2$. Aqui, mostramos que se $(FG)^+$ (ou $(FG)^-$) é Lie nilpotente ou Lie $n$-Engel, então $FG$ também é Lie nilpotente ou Lie $m$-Engel, respectivamente. Depois, consideramos o caso quando $G$ contém uma cópia do grupo quatérnio de ordem $8$. Neste caso, caracterizamos completamente as álgebras de grupo tais que $(FG)^+$ é fortemente Lie nilpotente, Lie nilpotente e Lie $n$-Engel. Como consequência, provamos que o conjunto das unidades simétricas deste tipo de grupos é nilpotente. Estudamos também o caso em que quando $G$ não contém uma cópia do grupo quatérnio de ordem $8$. Em particular, apresentamos um exemplo que mostra que os resultados obtidos em pesquisas anteriores, com a involução clássica, não devem ser esperados ao trabalhar com involuções clássicas orientadas. Não entanto, damos alguns casos especiais de grupos nos quais esses resultados são obtidos. Finalmente, estudamos o índice de Lie nilpotência de $(FG)^+$. Estabelecemos uma condição necessária e suficiente, para que o índice de Lie nilpotência de $(FG)^+$ e a classe de nilpotência das unidades simétricas de uma álgebra de grupo Lie nilpotente sejam o maior possível. Além disso, consideramos a situação em que o grupo $G$ contém uma cópia de $Q_8$.Let $F$ be a field of characteristic different from $2$ and $G$ a group. From the classical involution, which sends each element in its inverse and an orientation of $G$, it is possible to define an oriented classical involution on the group algebra $FG$. The goal of this thesis is to study Lie properties of the set of symmetric elements $(FG)^+$ and, in some cases, of the set of skew-symmetric elements $(FG)^-$. We first deal with the case when $G$ does not have elements of order $2$. In this situation, we show that if $(FG)^+$ (or $(FG)^-$) is Lie nilpotent or Lie $n$-Engel, then the whole group algebra $FG$ satisfies the same property. Later we consider the case when $G$ contains a copy of the quaternion group of order $8$. In this instance, we give a complete description of the group algebras such that $(FG)^+$ is strongly Lie nilpotent, Lie nilpotent and Lie $n$-Engel. As a consequence, we get that the set of symmetric units of this kind of groups is nilpotent. Furthermore, we study the case when $G$ does not contain a copy of the quaternion group of order $8$. Here, we present an example that shows that the previews results obtained in former works, with the classical involution, may not hold with an oriented classical involution. However, we give some kinds of groups for which those results are achieved. Finally, we study the Lie nilpotency index of $(FG)^+$. It is given a necessary and sufficient condition to the Lie nilpotency index of $(FG)^+$ and the nilpotency class of the symmetric units to be maximal, in a Lie nilpotent group algebra. In addition, we consider the situation when $G$ contains a copy of the quaternion group of order $8$.
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Jöllenbeck, Michael. « Algebraic discrete Morse theory and applications to commutative algebra (Algebraische diskrete Morse-Theorie und Anwendungen in der kommutativen Algebra) / ». [S.l. : s.n.], 2005. http://archiv.ub.uni-marburg.de/diss/z2005/0108/.
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Silva, Andre Ricardo Belotto da. « Análise das bifurcações de um sistema de dinâmica de populações ». Universidade de São Paulo, 2010. http://www.teses.usp.br/teses/disponiveis/45/45132/tde-18082010-122313/.
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Nesta dissertação, tratamos do estudo das bifurcações de um modelo bi-dimensional de presa-predador, que estende e aperfeiçoa o sistema de Lotka-Volterra. Tal modelo apresenta cinco parâmetros e uma função resposta não monotônica do tipo Holling IV: $$ \\left\\{\\begin \\dot=x(1-\\lambda x-\\frac{\\alpha x^2+\\beta x +1})\\\\ \\dot=y(-\\delta-\\mu y+\\frac{\\alpha x^2+\\beta x +1}) \\end ight. $$ Estudamos as bifurcações do tipo sela-nó, Hopf, transcrítica, Bogdanov-Takens e Bogdanov-Takens degenerada. O método dos centros organizadores é usado para estudar o comportamento qualitativo do diagrama de bifurcação.In this work are studied the bifurcations of a bi-dimensional predator-prey model, which extends and improves the Volterra-Lotka system. This model has five parameters and a non-monotonic response function of Holling IV type: $$ \\left\\{\\begin \\dot=x(1-\\lambda x-\\frac{\\alpha x^2+\\beta x +1})\\\\ \\dot=y(-\\delta-\\mu y+\\frac{\\alpha x^2+\\beta x +1}) \\end ight. $$ They studied the sadle-node, Hopf, transcritic, Bogdanov-Takens and degenerate Bogdanov-Takens bifurcations. The method of organising centers is used to study the qualitative behavior of the bifurcation diagram.
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ZAHID, ABOUBEKRE. « Les endomorphismes k - finis des modules de whittaker. Orbite nilpotente minimale en type g2 et operateurs differentiels ». Paris 6, 1990. http://www.theses.fr/1990PA066722.
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Soit g une algebre de lie semi-simple complexe, wh un module de whittaker simple. On donne une decomposition de l'algebre des endomorphismes k-finis de wh, en somme directe de modules de harish-chandra simples. On obtient alors une famille de suralgebres de certains quotients primitifs de l'algebre enveloppante
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Terpereau, Ronan. « Schémas de Hilbert invariants et théorie classique des invariants ». Phd thesis, Université de Grenoble, 2012. http://tel.archives-ouvertes.fr/tel-00748952.
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Pour toute variété affine W munie d'une opération d'un groupe réductif G, le schéma de Hilbert invariant est un espace de modules qui classifie les sous-schémas fermés de W, stables par l'opération de G, et dont l'algèbre affine est somme directe de G-modules simples avec des multiplicités finies préalablement fixées. Dans cette thèse , on étudie d'abord le schéma de Hilbert invariant, noté H, qui paramètre les sous-schémas fermés GL(V)-stables Z de W=n1 V oplus n2 V^* tels que k[Z] est isomorphe à la représentation régulière de GL(V) comme GL(V)-module. Si dim(V)<3,on montre que H est une variété lisse, et donc que le morphisme de Hilbert-Chow gamma: H -> W//G est une résolution des singularités du quotient W//G. En revanche, si dim(V)=3, on montre que H est singulier. Lorsque dim(V)<3, on décrit H par des équations et aussi comme l'espace total d'un fibré vectoriel homogène au dessus d'un produit de deux grassmanniennes. On se place ensuite dans le cadre symplectique en prenant n1=n2 et en remplaçant W par la fibre en 0 de l'application moment mu: W -> End(V). On considère alors le schéma de Hilbert invariant H' qui paramètre les sous-schémas contenus dans mu^{-1}(0). On montre que H' est toujours réductible, mais que sa composante principale Hp' est lisse lorsque dim(V)<3. Dans ce cas, le morphisme de Hilbert-Chow est une résolution (parfois symplectique) des singularités du quotient mu^{-1}(0)//G. Lorsque dim(V)<3, on décrit Hp' comme l'espace total d'un fibré vectoriel homogène au dessus d'une variété de drapeaux. Enfin, on obtient des résultats similaires lorsque l'on remplace GL(V) par un autre groupe classique (SL(V), SO(V), O(V), Sp(V)) que l'on fait opérer d'abord dans W=nV, puis dans la fibre en 0 de l'application moment.
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Santos, Edson Carlos Licurgo. « Estruturas complexas comauto-espaços nilpotentes e soluveis ». [s.n.], 2007. http://repositorio.unicamp.br/jspui/handle/REPOSIP/305823.
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Orientador: Luiz Antonio Barrera San MartinTese (doutorado) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação Cientifica
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Resumo: Seja (g; [·,·]) uma álgebra de Lie com uma estrutura complexa integrável J. Os ± i-auto-espaços de J são subálgebras complexas de gC isomorfas a álgebra (g; [*]J ) com colchete [X * Y ]J = ½ ([X, Y ] - [JX, JY ]). Consideramos, no capítulo 2, o caso onde estas subálgebras são nilpotentes e mostramos que a álgebra de Lie original (g, [·,·]) é solúvel. Consideramos também o caso 6-dimensional e determinamos explicitamente a única álgebra de Lie possível (g; [*]J ). Finalizamos esse capítulo pruduzindo vários exemplos ilustrando diferentes situações, em particular mostramos que para cada s existe g com estrutura complexa J tal que (g; [*]J ) é s-passos nilpotente. Exemplos similares para estruturas hipercomplexas são também construidos. No capítulo 3 consideramos o caso onde os ±i-auto-espaços de J são subálgebras complexas solúveis e a álgebra complexa é uma álgebra de Lie semi-simples. Mostramos que, se a álgebra real é compacta, uma tal estrutura complexa depende unicamente de um subespaço da subálgebra de Cartan. Finalizamos esse capítulo considerando o caso em que as subálgebras solúveis complexas estão contidas em subálgebras de Borel de uma órbita aberta da ação dos automorfismos internos da álgebra real. Mostramos que, assim como no caso compacto, as estruturas complexas são determinandas, de modo único, por subespaços da subálgebra de Cartan. Ao final da tese apresentamos um procedimento, elaborado em MAPLE, que possibilita testar a identidade de Jacobi quando os colchetes de Lie são dados pelas constantes de estrutura
Abstract: Let (g; [·,·]) be a Lie algebra with an integrable complex structure J. The ±i eigenspaces of J are complex subalgebras of gC isomorphic to the algebra (g; [*]J )with bracket [X * Y ]J = ½ ([X, Y ] - [JX, JY ]). We consider, in chapter three, thecase where these subalgebras are nilpotent and prove that the original Lie algebra(g, [·,·]) must be solvable. We consider also the 6-dimensional case and determineexplicitly the possible nilpotent Lie algebras (g; [*]J ). We finish this chapter byproducing several examples illustrating different situations, in particular we showthat for each given s there exists g with complex structure J such that (g; [*]J ) iss-step nilpotent. Similar examples of hypercomplex structures are also built.In Chapter 3 we consider the case where the ± i eigenspaces of J are solvablecomplex subalgebras and gC is a semisimple Lie algebra. We prove that, if g is compact, such a complex structure comes from a subspace of the Cartan subalgebra.We finish this chapter by considering the case where the solvable complex subalgebras are contained in Borel subalgebras of an open orbit of the action of inner automorphisms of the real algebra.At the end of the thesis we present an algorithm, made in MAPLE, that allowus to verify the Jacobi identity when the Lie brackets are defined by the structureconstants
Doutorado
Mestre em Matemática
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MELO, Emerson Ferreira de. « Sobre Anéis de Lie Admitindo Automorfismos de Ordens Finitas e Álgebras de Lie Quase Nilpotentes ». Universidade Federal de Goiás, 2011. http://repositorio.bc.ufg.br/tede/handle/tde/1938.
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Previous issue date: 2011-02-28In this work we present a study on Lie rings and algebras admitting an automorphism of finite order. We emphasize questions on nilpotency. We prove important results of this theory, for example the Higman, Kreknin and Kostrikin s Theorem. Furthermore, let L be a finite dimensional Lie algebra over an algebraically closed field of characteristic 0. Suppose that L admits a nilpotent Lie algebra D with n weights in L, and let m be the dimension of the Fitting null component with respect to D. Then L is almost nilpotent, namely, L contains a nilpotent subalgebra N of {m,n}-bounded codimension and of nbounded nilpotency class. If m = 0, then L is nilpotent of bounded class by a function of n. This theorem was published by E. I. Khukhro and P. Shumyatsky in the paper entitled Lie Algebras with Almost Constant-Free Derivations .
Nesta dissertação apresentamos um estudo sobre anéis e álgebras de Lie admitindo um automorfismo de ordem finita, com ênfase em questões sobre nilpotência. Demonstramos resultados importantes desta teoria, como por exemplo o Teorema de Higman, Kreknin e Kostrikin. Além disso, considere L uma álgebra de Lie de dimensão finita sobre um corpo algebricamente fechado de característica 0. Suponha que L admita uma álgebra de derivações nilpotente D com n pesos em L, e seja m a dimensão da componente nula de Fitting com respeito a D. Então L é quase nilpotente, ou seja, L contém uma subálgebra N de codimensão {m,n}-limitada e classe de nilpotência n-limitada. Se m = 0, então L é nilpotente de classe limitada por uma função de n. Este teorema foi publicado por E. I. Khukhro e P. Shumyatsky num artigo intitulado Lie Algebras with almost constant-free derivations .
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Rodrigues, Claudenir Freire. « Grupos abelianos-por-nilpotentes do tipo homologico 'FP IND.3' ». [s.n.], 2006. http://repositorio.unicamp.br/jspui/handle/REPOSIP/306915.
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Orientador: Dessislava H. KochloukovaTese (doutorado) - Universidade Estadual de Campinas. Instituto de Matematica, Estatistica e Computação Cientifica
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Resumo: Neste trabalho estudamos grupos abstratos finitamente gerados G que são extensões cindidas de um grupo abeliano A por um grupo Q nilpotente de classe 2. Mostramos que se G tem tipo homológico F P3, então o quociente G/N também tem tipo homológico F P3 onde N é o fecho normal do centro de Q em G. Observamos que não existe classificação quando G pode ter tipo FP3, nem classificação para tipo F P2 ou ser finitamente apresentável. Por causa disso nós trabalhamos com um quociente especifico de G. Ainda fica em aberto se cada quociente de G tem tipo FP3 quando G tem tipo FP3. Observamos que isso vale quando G é grupo metabeliano, nesse caso a teoria de Bieri-Strebel pode ser aplicada
Abstract: We study abstract finitely generated groups G that are split extensions from A abelian group by Q nilpotent group of class two. We show that if G has homological type FP3 then the quotient group GjN has homological type FP3 too, where N is the normal closure of the center of Q in G. Since there is no classification when G is of type FP3, nor when G is of type FP2 or finitely presented we work with one specific quotient. It is an open problem whether every quotient of G has type F P3. This holds if G is a metabelian group and in this case the Bieri-Strebel theory applies
Doutorado
Doutor em Matemática
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Barucchieri, Bianca. « Affine Hermite-Lorentz manifolds ». Thesis, Bordeaux, 2019. http://www.theses.fr/2019BORD0153/document.
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Dans ce travail nous nous intéressons aux groupes cristallographiques, i.e. aux sous-groupes du groupe des transformations affines qui agissent proprement discontinûment et de façon cocompacte sur l’espace affine. Ce sont les groupes fondamentaux des variétés affines compactes et complètes. Nous classifions les groupes cristallographiques dont la partie linéaire préserve une forme hermitienne de signature (n,1). Grunewald et Margulis ont prouvé que ces groupes cristallographiques sont virtuellement résolubles (la conjecture d’Auslander affirme que c’est toujours le cas). Notre classification est effectuée pour n ≤ 3. Elle correspond à la classification, à revêtement fini près, des variétés Hermite-Lorentz plates, compactes et complètes en dimension complexe inférieure ou égale à4. Ce travail est inspiré par ceux menés par Bieberbach, puis Fried, et enfin Grunewald et Margulis sur les groupes cristallographiques dont la partie linéaire préserve une forme quadratique définie positive ou lorentzienne. En effectuant cette classification, nous avons été amené à étudier certains familles d’algèbres de Lie nilpotentes de dimension 8. Nous avons ensuite étendu cette classification à celle de toutes les algèbres de Lie 3-nilpotentes de dimension 8 ayant l’algèbre de Lie libre 3-nilpotente à 3générateurs pour quotient. Ce résultat peut être vu comme un pas dans la direction d’une classification des algèbres de Lie nilpotentes de dimension 8. Ensuite nous nous sommes demandé lesquelles de ces algèbres admettent une métrique pseudo-riemannienne plate et nous avons donné une réponse partielleIn this work we deal with crystallographic groups, i.e. the subgroups of the group of affine transformations that act properly discontinuously and cocompactly on affine space. In otherwords they are the fundamental groups of compact and complete affine manifolds. In this thesis we classify such groups with the additional hypothesis that the linear part preserves a Hermitian form of signature (n,1). Grunewald and Margulis proved that such crystallographic groups are virtually solvable (the Auslander conjecture states that this is always true). Our classification is for n ≤ 3. It corresponds to a classification, up to finite covering, and for complex dimension at most 4, of flat compact complete Hermite-Lorentz manifolds. This is inspired by the works done by Bieberbach,then Fried, and finally Grunewald and Margulis who classified crystallographic groups whose line arpart preserves a positive definite or Lorentzian quadratic form. Making this classification we had to classify a family of 8-dimensional nilpotent Lie algebras. We then extended this classification toall the 8-dimensional 3-step nilpotent Lie algebras having the free 2-step nilpotent Lie algebra on 3generators as quotient. This result can be seen as a step in the direction of a general classification of nilpotent Lie algebras of dimension 8. We then wondered which of these Lie algebras admit flat pseudo-Riemannian metrics and gave a partial answer to this question
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Williams, Michael Peretzian. « Nilpotent N-Lie Algebras ». NCSU, 2004. http://www.lib.ncsu.edu/theses/available/etd-02162004-083708/.
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In 1986, Kasymov introduced the concept of nilpotent $n$-Lie algebras, proved an analogue of Engel's Theorem and later proved an analog of Jacobson's refinement of Engel's Theorem. Despite these achievements, the subject of nilpotency in $n$-Lie algebras has not been examined in great detail in the literature since. We shall explore the concept of nilpotent $n$-Lie algebras by examining, and proving where possible, other classical nilpotent group theory and nilpotent Lie algebra results, in the $n$-Lie algebra setting.
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Bolgar, J. R. « Nilpotent left-symmetric algebras ». Thesis, University of Oxford, 1994. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.259773.
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Sorkatti, Layla Hamad Elnil Mugbil. « Nilpotent symplectic alternating algebras ». Thesis, University of Bath, 2015. https://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.669034.
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Abramov, Gueorgui. « Nilpotent Class Field Theory ». Doctoral thesis, Humboldt-Universität zu Berlin, Mathematisch-Naturwissenschaftliche Fakultät II, 1999. http://dx.doi.org/10.18452/14361.
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Itikawa, Jackson. « O problema do centro-foco para singularidades nilpotentes no plano ». Universidade de São Paulo, 2012. http://www.teses.usp.br/teses/disponiveis/55/55135/tde-05122012-144434/.
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O estudo dos pontos singulares em campos vetoriais analíticos é um problema quase completamente resolvido. O único caso que ainda permanece insolúvel é o caso monodrômico, em que as órbitas circundam a singularidade. Em sistemas diferenciais analíticos, se p é singularidade monodrômica, então p ou é um centro, ou é um foco. O problema do centro-foco consiste em determinar condições que diferenciem os casos em que p é um foco, daqueles em que p é um centro. O tema central desta dissertação é a investigação do problema do centro-foco em sistemas diferenciais analíticos com singularidade nilpotente. Este problema é bastante estudado, uma vez que ainda não existe um algoritmo eficiente para este caso, tal como ocorre em sistemas com singularidades não degeneradas. Estudamos duas técnicas bastante distintas. A primeira faz uso da teoria das formas normais e aborda o problema da maneira clássica, dividindo-o na investigação da monodromia e no estudo da estabilidade. O outro método investiga os sistemas diferenciais com singularidades nilpotentes como limite de sistemas com singularidades não degeneradas. A fim de avaliarmos sua eficiência e compreendermos as possíveis obstruções envolvidas, aplicamos os métodos a famílias concretas de sistemas diferenciaisThe study of singular points in planar analytic vector fields is a problem almost completely solved. The only case that remains open is the monodromic one, in which the orbits turn around the singularity. In analytic differential systems, if p is a monodromic singular point, then p is either a center or a focus. The center-focus problem consists in determining conditions for distinguishing between a center and a focus. The main purpose of this work is the investigation of the center-focus problem in analytic differential systems with nilpotent singular points. This problem is still widely studied, since there is no algorithm for such case, comparable to the Lyapunov method for the case of non-degenerate singularities. We studied two different methods. The first makes use of the normal form theory and deals with the problem in the classic way, splitting it up in two parts: the investigation of the monodromy and the study of the stability. The latter investigates the differential analytic systems with nilpotent singular points as limit of differential systems with nondegenerate singularities. In order to evaluate the efficiency and understand possible obstructions, we applied the two techniques to concrete families of differential systems
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Wagner, Heily. « Extensões cindidas por ideais nilpotentes ». Universidade de São Paulo, 2008. http://www.teses.usp.br/teses/disponiveis/45/45131/tde-21102010-205202/.
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Consideremos A e B duas álgebras de Artin tais que é uma extensão cindida de A pelo ideal Q, onde é um ideal nilpotente de B. Estudamos algumas propriedades homológicas das categorias modA e modB, tais como dimensão projetiva e injetiva. A partir disso mostramos que se B pertence a uma das seguintes classes: hereditária, laura, fracamente shod, shod, quase inclinada, colada à esquerda, colada à direita ou disfarçada; então A pertence a mesma classe. Além disso, restringindo nosso estudo para álgebras de dimensão finita sobre um corpo algebricamente fechado, comparamos as respectivas aljavas ordinárias, bem como suas apresentações. Finalmente, após caracterizarmos o ideal Q, exibimos alguns exemplos de extensões no contexto de álgebras de caminhos com relações, que mostram que A pode ser de uma das classes citadas sem que B o sejaLet A and B be two Artin algebras such that B is a split-by-nilpotent extension of A by Q, were Q is a nilpotent ideal of B. We study some homological properties of the categories mod A and mod B such that the projetive and the injetive dimensions of their objects. Using this we show that if B belongs to one of this classes: hereditary, laura, weakly shod, shod, quasi-tilted, left glued, right glued or concealed; then A belongs to same class. Moreover restricting our study to finite dimensional algebras over algebraically closed fields, we compare the ordinary quivers and presentations of the corresponding algebras. Finally, after giving a characterization of ideal Q as above, we exhibit some exemples of split extensions in the context of path algebras bounded by relations, which shows that A can be one of the above cited algebras without B so
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Gilg, Marc. « Super-algèbres de Lie nilpotentes ». Mulhouse, 2000. http://www.theses.fr/2000MULH0604.
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Dans ce travail, on s'intéresse aux propriétés et à la classification des super-algèbres de Lie nilpotentes. On les caractérise à l'aide des suites centrales puis en utilisant l'invariant de Goze, élargi aux super-algèbres de Lie nilpotentes. On y donne aussi la définition des super-algèbres de Lie filiformes et des propriétés générales concernant les super-algèbres de Lie nilpotentes. Dans la suite, les super-algèbres filiformes s'obtiennent par déformation linéaire d'une super-algèbre de Lie filiforme modèle, notée Ln,m. Ces déformations sont construites à partir des 2-cocycles paires de Ln,m, ce qui nous conduit à l'étude de ces cocycles. Du point de vue géométrique, on en déduit la dimension de l'orbite de Ln,m et une estimation de la dimension d'une composante irréductible contenant Ln,m dans la variété des super-algèbres de Lie nilpotentes. On établit, dans le dernier chapitre, la classification à isomorphisme près des super-algèbres de Lie filiformes dans les cas suivants : G = G0 ○+ G1 avec dim G0 = n + 1 et dim G1 = m où (n,m) ∈ {(1,m) ; (2,2) ; (2,3) ; (3,2) ; (4,2) ; (5,2)}.
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Goddard, Russell. « Commuting varieties and nilpotent orbits ». Thesis, University of Birmingham, 2017. http://etheses.bham.ac.uk//id/eprint/7476/.
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Let \(G\) be a reductive algebraic group over an algebraically closed field \(k\) of good characteristic, let \(g\)=Lie(\(G\)) be the Lie algebra of \(G\), and let \(P\) be a parabolic subgroup of \(G\) with \(p\)=Lie(\(P\)). We consider the commuting variety \(C\)(\(p\)) of \(p\) and obtain two criteria for \(C\)(\(p\)) to be irreducible. In particular we classify all cases when the commuting variety \(C\)(\(b\)) is irreducible, for \(b\) a Borel subalgebra of \(g\). We then let \(G\) be a classical group and let \(O\)\(_1\) and \(O\)\(_2\) be nilpotent orbits of \(G\) in \(g\). We say that \(O\)\(_1\) and \(O\)\(_2\) commute if there exists a pair (\(X\), \(Y\)) ∈ \(O\)\(_1\)×\(O\)\(_2\) such that [\(X\),\(Y\)]=0. For \(g\)=\(s\)\(p\)\(_2\)\(_m\)(\(k\)) or \(g\)=\(s\)\(o\)\(_n\)(\(k\)), we describe the orbits that commute with the regular orbit, and classify (with one exception) the orbits that commute with all other orbits in \(g\). This extends previously-known results for \(g\)=\(g\)\(l\)\(_n\)(\(k\)). Finally let φ be a Springer isomorphism, that is, a \(G\)-equivariant isomorphism from the unipotent variety \(U\) of \(G\) to the nilpotent variety \(N\) of \(g\). We show that polynomial Springer isomorphisms exist when \(G\) is of type G\(_2\), but do not exist for types E\(_6\) and E\(_7\) for \(k\) of small characteristic.
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Morris, Thomas Bembridge Slater. « Nilpotent injectors in finite groups ». Thesis, University of Birmingham, 2011. http://etheses.bham.ac.uk//id/eprint/3066/.
Texte intégralRésumé :
We prove that the odd nilpotent injectors (a certain type of maximal nilpotent subgroup) f a minimal simple group are all conjugate, extending the result from soluble groups. We also prove conjugacy in GU(\_3\)(q) and SU(\_3\)(q). In a minimal counterexample to the onjecture that the odd nilpotent injectors of an arbitrary ¯nite group are all conjugate we show that there must be a component, which cannot be of type A\(_n\) except possibly 3 ¢ A(\_6\) or 3 ¢ A(\_7\). Finally, we produce a partial result on minimal simple groups for a more general type of nilpotent injector.
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Mihov, Diko. « Quantization of nilpotent coadjoint orbits ». Thesis, Massachusetts Institute of Technology, 1996. http://hdl.handle.net/1721.1/38410.
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Wilson, Aaron Thomas. « Co-growth in nilpotent groups ». Thesis, University of Bath, 2001. https://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.341677.
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Lachand, Armand. « Entiers friables et formes binaires ». Thesis, Université de Lorraine, 2014. http://www.theses.fr/2014LORR0189/document.
Texte intégralRésumé :
Un entier est dit y-friable si tous ses facteurs premiers n'excèdent pas y. Les valeurs friables de formes binaires interviennent de manière essentielle dans l'algorithme de factorisation du crible algébrique (NFS). Dans cette thèse, nous obtenons des formules asymptotiques pour le nombre de représentations des entiers friables par différentes familles de polynômes. Nous considérons dans la première partie les formes binaires qui se décomposent comme produit d'une forme linéaire et d'une forme quadratique. Nous combinons pour cela le principe d'inclusion-exclusion à des idées issues de travaux sur la distribution multiplicative de certaines suites d'entiers représentés par des formes quadratiques développés par Fouvry et Iwaniec, puis Balog, Blomer, Dartyge et Tenenbaum. Dans un second temps, nous nous concentrons sur les valeurs friables de formes cubiques irréductibles. En adaptant les travaux de Heath-Brown et Moroz sur les nombres premiers représentés par de tels polynômes, nous obtenons des formules asymptotiques valides dans un vaste domaine de friabilité. Notre méthode permet également d'évaluer des moyennes sur les valeurs d'une forme cubique pour d'autres fonctions arithmétiques comprenant en particulier les fonctions de Möbius et de Liouville. Dans le dernier chapitre, nous étudions les corrélations de l'indicatrice des friables avec les nilsuites. En employant la méthode nilpotente de Green et Tao, nous en déduisons une formule pour le nombre de valeurs friables d'un produit de formes affines deux à deux affinement indépendantesAn integer is called y-friable if its largest prime factor does not exceed y. Friable values of binary forms play a central role in the integer factoring algorithm NFS (Number Field Sieve). In this thesis, we obtain some asymptotic formulas for the number of representations of friable integers by various classes of polynomials. In the first part, we focus on binary forms which split as a product of a linear form and a quadratic form. To achieve this, we combine the inclusion-exclusion principle with ideas based on works of Fouvry and Iwaniec and Balog, Blomer, Dartyge and Tenenbaum related to the distribution of some sequences of integers represented by quadratic forms. We then take a closer look at friable values of irreducible cubic forms. Extending some previous works of Heath-Brown and Moroz concerning primes represented by such polynomials, we provide some asymptotic formulas which hold in a large range of friability. With this method, we also evaluate some means over the values of an irreducible cubic form for other multiplicative functions including the Möbius function and the Liouville function. In the last chapter, we investigate the correlations between nilsequences and the characteristic function of friable integers. By using the nilpotent method of Green and Tao, our work provides a formula for the number of friable integers represented by a product of affine forms such that any two forms are affinely independent
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Sanselme, Luc. « Algorithmes quantiques dans les groupes nilpotents ». Paris 11, 2008. http://www.theses.fr/2008PA112297.
Texte intégralRésumé :
Dans cette thèse, nous commençons par donner avec précision une définition formelle des groupes boîtes noires, et nous rappelons les principaux algorithmes existant dans ce cadre. Dans un deuxième temps, nous proposons une définition nouvelle d’un groupe boîte noire quantique. Nous formalisons, par ailleurs, précisément cette définition et donnons les principaux algorithmes quantiques connus dans ce cadre. Ensuite, nous donnons un certain nombre d’algorithmes de calcul de théorie algorithmique quantique des groupes dans les groupes résolubles, et dans certaines sous-classes particulières de ces groupes. Enfin, nous présentons un résultat original, démontré au cours de l’élaboration de cette thèse. Nous expliquons comment résoudre efficacement le problème du sous-groupe caché dans les groupes extraspéciaux et nilpotents de classe deux, en calcul quantique. Au passage, nous donnons un certain nombre de réductions du problème du sous-groupe caché, valable dans un groupe nilpotent de classe supérieure. Le dernier chapitre, un peu à part dans cette thèse, explique comment résoudre efficacement un système d’équations quadratiques dans un corps fini, résultat nécessaire pour résoudre le problème du sous-groupe caché dans les groupes nilpotents de classe 2We start off this Ph. D. Thesis with giving the definition of a black-box group and reminding some algorithm associated with this group representation. Then, we put forward a new definition of a quantum black-box group. We explain precisely this new approach and we enumerate the main algorithms associated to this notion. After that, we give some algorithm of quantum computational group theory in solvable groups and in some subclasses of these solvable groups such as nilpotent groups, p-groups or extraspecial groups. Finally, we present a new result that was proved during this thesis. We show that we can solve efficiently, with a quantum computer, the hidden subgroup problem in extraspecial and nilpotent group of class 2. In addition, we give some reduction of the Hidden subgroup problem in nilpotent groups of higher classes. The last chapter of this thesis shows how to solve some system of quadratic equations over a finite field. This result is needed to solve the Hidden subgroup problem in nilpotent groups of class 2
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Wang, Zhiqing. « Locally nilpotent derivations of polynomial rings ». Thesis, National Library of Canada = Bibliothèque nationale du Canada, 1999. http://www.collectionscanada.ca/obj/s4/f2/dsk1/tape9/PQDD_0018/NQ48119.pdf.
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Grenham, Dermot. « Some topics in nilpotent group theory ». Thesis, University of Oxford, 1988. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.329954.
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Fowler, Russell Adam. « Spherical nilpotent orbits in positive characteristic ». Thesis, University of Birmingham, 2007. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.446564.
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Smith, Jeremy Francis. « Topics in products of nilpotent groups ». Thesis, University of Warwick, 1998. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.340502.
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Tay, Kian Boon. « Nilpotent orbits and multiplicty-free representations ». Thesis, Massachusetts Institute of Technology, 1994. http://hdl.handle.net/1721.1/28092.
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Lampetti, Enrico. « Nilpotent orbits in semisimple Lie algebras ». Bachelor's thesis, Alma Mater Studiorum - Università di Bologna, 2021. http://amslaurea.unibo.it/23595/.
Texte intégralRésumé :
This thesis is dedicated to the introductory study of the so-called nilpotent orbits in a semisimple complex Lie algebra g, i.e., the orbits of nilpotent elements under the adjoint action of the adjoint group Gad with Lie algebra g. These orbits have an extremely rich structure and lie at the interface of Lie theory, algebraic geometry, symplectic geometry, and geometric representation theory.
The Jacobson and Morozov Theorem relates the orbit of a nilpotent element X in a semisimple complex Lie algebra g with a triple {H,X,Y} that generates a subalgebra of g isomorphic to sl(2,C). There is a parabolic subalgebra associated to this triple that permits to attach a weight to each node of the Dynkin diagram of g. The resulting diagram is called a weighted Dynkin diagram associated with the nilpotent orbit of X. This is a complete invariant of the orbit that one can use in order to
show that there are only _nitely many nilpotent orbits in g.
The thesis is organized as follows: the first three chapters contain some preliminary material on Lie algebras (Chapter 1), on Lie groups (Chapter 3) and on the representation theory of sl(2,C) (Chapter 2). Chapter 4 and 5 are the heart of the thesis. Namely,
Jacobson-Morozov, Kostant and Mal'cev Theorems are proved in Chapter 4 and Chapter 5 is dedicated to the construction of weighted Dynkin diagrams. As an example the
conjugacy classes of nilpotent elements in sl(n,C) are described in detail and a formula for their dimension is given. In this case, as well as in the case of all classical Lie algebras, the description of the orbits can be done in terms of partitions and
tableaux.
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Milian, Dagmara. « Locally nilpotent 5-Engel p-groups ». Thesis, University of Oxford, 2010. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.561122.
Texte intégralRésumé :
In this thesis we investigate the structure of locally nilpotent 5-Engel p-groups. We show that for p > 7, locally nilpotent 5-Engel p-groups have class at most 10. This is a global theorem, where the result is not dependent on the number of generators of the group. The proof uses new and established Lie methods and a custom C++ implementation of an algorithm that constructs minimal generating sets and structure constants of multi- graded Lie algebras in a variety defined by three multilinear relations, which hold in the Lie rings associated with 5-Engel p-groups. We obtain our results by calculating in the set Q(p) = {~ I x E Z, yE Z+, Y # 0 modulo any p f/. p} (where p is a set of excluded primes and x, y are arbitrarily large integers), as well as the fields Zp, p prime. We introduce several reduction theorems, making the result possible. We also present results about the normal closure of elements in these groups. We use a Higman reduction theorem and the same custom C++ program to show that locally nilpotent 5-Engel p-groups, p 2: 5, are Fitting, with Fitting degree at most 4 if p > 7, at most 5 if p = 7 and at most 6 if p = 5. These results are best possible.
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Aydin, Hueseyin. « Recurrence relations in finite nilpotent groups ». Thesis, University of Bath, 1991. https://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.292809.
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Oudghiri, Mourad. « Sur le théorème de Weyl ». Lille 1, 2004. http://www.theses.fr/2004LIL10079.
Texte intégralRésumé :
Situé dans le cadre de la Théorie des opérateurs bornés sur les espaces de Banach, ce travail s'articule sur deux grands axes. Le premier consiste à aborder le théorème de Weylpour une classe d'opérateurs vérifiant la SVEP. Cependant, le deuxième axe concerne le problème de stabilité du théorème de Weyl sous perturbations. Dans une prémière partie on montre que le théorème de Browder, version affaiblie du théorème de Weyl), est satisfait par tout opérateur jouissant de la SVEP; et on fournie plusieurs conditions nécessaires et suffisantes pour que ces opérateurs vérifient le théorème de Weyl. On introduit aussi une classe d'opérateurs P englobant la plupart des opérateurs étudiés dans la littérature en connexion avec le théorème de Weyl et. On prouve que si T est un opérateur borné et h est une fonction analytique sur un voisinage du spectre de T, non identiquement constante sur chaque composante connexe de son domaine de définition et téls que h(T) [appartient à] P, alors le théorème de Weyl est satisfait par f(T) et f(T*) pour toute fonction f analytique sur un voisinage du spectre de T. Compte tenu du lien fort existant entre le théorème de Weyl et la notion de la déscente, on consacre la seconde partie à l'étude de cette notion. On établit qu'un espace de Banach est de dimension infinie si, et seulement si, le commutant de tout opérateur contient un opérateur non algébrique. En se basant sur ce résultat, on donne une réponse positive à une question conjecturée par M. Kaashoek et D. Lay: Soit F un opérateur borné tel que pour tout opérateur T commutant avec F, la descente de T est finie si, et seulement si, la descente de T + F est finie, alors une puissance de F est de rang finie. Dans la troisième partie de ce travail, on montre la stabilité du théorème de Weyl sous perturbations de Riesz commutatives pour les opérateurs isoloides finis. Cependant, pour la classe P, on généralise ce résultat aux perturbations commutatives par les opérateurs polynomialement de Riesz.
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Rojas, Yerko Contreras. « Sobre a influência dos centralizadores dos automorfismos de ordem dois em grupos de ordem ímpar ». Universidade Federal de Goiás, 2013. http://repositorio.bc.ufg.br/tede/handle/tede/3090.
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Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES
This document presents an approach and development of some of the results of Shumyatsky in [14, 15, 16, 17, 18], where he worked with automorphisms of order two in finite groups of odd order, mainly showing the influence that the structure of the centralizer has on that of Group. Let G be a group with odd order, and ϕ an automorphism on G, of order two, where G = [G,ϕ], and given a limitation in the order of the centralizer of ϕ regard to G, CG(ϕ), which induces a limitation in the order of derived group G′ of group G, and we also verified that G has a normal subgroup H that is ϕ-invariant, such that H′ ≤ Gϕ and its index [G : H] is bounded with the initial limitation. With the same hypothesis of the group G and with the same limitation of the order of the centralizer of the automorphism, let V a abelian p-group such that G⟨ϕ⟩ act faithful and irreductible on V, then there is a bounded constant k, limitated by a function depending only on the parameter m, where m is tha limitation in the order of CG(ϕ), and elements x1, ...xk ∈ G−ϕ such that V = ρϕx 1,...,xk(V−ϕ).
O trabalho baseia-se na apresentação e desenvolvimento de alguns resultados expostos por Shumyatsky em [14, 15, 16, 17, 18], onde trabalha com automorfismos de ordem dois em grupos de ordem ímpar, mostrando fundamentalmente a influência da estrutura do centralizador do automorfismo na estrutura do grupo. Seja G um grupo de ordem ímpar e ϕ um automorfismo de G, de ordem dois, tal que G = [G,ϕ], dada uma limitação na ordem do centralizador de ϕ em G, CG(ϕ), a mesma induz uma limitação na ordem do grupo derivado G′ do grupo G, além disso verificamos que G tem um subgrupo H normal ϕ-invariante, tal que H′ ≤ Gϕ e o índice [G : H] é limitado dependendo da limitação inicial de CG(ϕ). Nas mesmas hipóteses do grupo G e com a mesma limitação da ordem do centralizador do automorfismo, seja V um p-grupo abeliano, tal que G⟨ϕ⟩ age fiel e irredutivelmente sobre V, então existe uma constante k, limitada por uma função que depende só da limitação de CG(ϕ), e elementos x1, ...xk ∈ G−ϕ, tal que V = ρϕx 1,...,xk(V−ϕ).
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Fernandes, Tharso Dominisini. « Problema do subgrupo oculto em grupos nilpotentes ». Laboratório Nacional de Computação Científica, 2008. http://www.lncc.br/tdmc/tde_busca/arquivo.php?codArquivo=153.
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Computadores quânticos prometem resolver certos problemas assintoticamente mais rápido do que os computadores clássicos. Algoritmos quânticos, como o algoritmo de Shor, podem ser considerados casos particulares do chamado Problema do Subgrupo Oculto(PSO). O PSO consiste em encontrar um subgrupo H de um grupo G por meio de avaliações de uma função f que é constante em classes laterais de H e distinta em classes laterais diferentes. O PSO em grupos abelianos é resolvido eficientemente em um computador quântico, mas será que os computadores quânticos podem resolver o PSO em grupos não abelianos? Esta questão tem sido discutida regularmente pela comunidade científica devido a importantes aplicações, como é o caso do problema de isomorfismo de grafos e do problema do menor vetor em um reticulado. Nesta dissertação é feita uma revisão do trabalho de Ivanyos et al. (2007a), o qual apresenta uma solução para o PSO em grupos nilpotentes de classe 2. Com esta finalidade, é elaborada uma breve revisão sobre a Computação Quântica; são mostradas algumas características dos grupos nilpotentes e dos grupos solúveis, dando uma atenção especial aos grupos nilpotentes de classe 2; é exposto o método padrão de solução do PSO em grupos abelianos; também são exibidas as principais características de sequencias policıclicas e reduçõesde grupos nilpotentes usando as propriedades de sequencias policıclicasQuantum computers may solve certain problems asymptotically faster than the classical computers. Quantum algorithms, such as Shors algorithm, may be considered as a particular case of the Hidden Subgroup Problem (HSP). The HSP consists in finding a subgroup H of a group G by evaluating a function f, which is constant in cosets of H and distinct for each coset. The HSP for Abelian groups is efficiently solved in a quantum computer, but is quantum computers can solve the HSP in non-Abelian groups efficiently? This question has been regularly discussed by the scientific community due to the importance of some applications, such as the graph isomorphism problem and the short vector in a lattice. In this dissertation we review the Ivanyos et al. (2007a) that address HSP in nilpotent groups of class 2. We make a brief review on Quantum Computing; we address some characteristics of nilpotent groups and solvable groups, with special attention to nilpotent groups of class 2; we discuss the standard method of solution of the HSP in Abelian groups; we present the main characteristics of the polycyclic sequences and important reductions of the HSP in classes of nilpotent groups using the properties of polycyclic sequences. Finally, we present an efficient algorithm to solve the HSP in nilpotent groups of class 2.
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Silva, Leonardo de Amorin e. 1980. « Grupos abelianos-por-(nilpotentes de classe 2) ». [s.n.], 2014. http://repositorio.unicamp.br/jspui/handle/REPOSIP/306919.
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Orientador: Dessislava Hristova KochloukovaTese (doutorado) - Universidade Estadual de Campinas, Instituto de Matemática Estatística e Computação Científica
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Resumo: Nesta tese consideramos uma extensão cindida G de um grupo abeliano A por um grupo nilpotente (de classe 2) Q e provamos dois resultados. Primeiro, se Q age nilpotentemente sobre A e G tem tipo FP2, calculamos o sigma invariante de G em dimensão 2. Segundo, se G tem tipo FP4, mostramos que cada quociente de G tem tipo FP4
Abstract: In this thesis we consider a split extension G of an abelian group A by a nilpotent group (class 2) Q and prove two results. First, if Q acts nilpotently on A and G has type FP2, compute the sigma invariant of G in dimension 2. Second, if G has type FP4, we show that every quotient G has type FP4
Doutorado
Matematica
Doutora em Matemática
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Costa, Eudes Antonio da. « Álgebras associativas Lie nilpotentes de classe 4 ». reponame:Repositório Institucional da UnB, 2013. http://repositorio.unb.br/handle/10482/14973.
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Tese (doutorado)—Universidade de Brasília, Instituto de Ciências Exatas, Departamento de Matemática, 2013.Submitted by Jaqueline Ferreira de Souza (jaquefs.braz@gmail.com) on 2014-01-14T09:50:38Z No. of bitstreams: 1 2013_EudesAntoniodaCosta_Parcial.pdf: 44872 bytes, checksum: 10ca621a1fee4cef0211600d2862047a (MD5)
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Sejam K um anel associativo, comutativo e unitário e (K) X a K-álgebra associativa livre num conjunto não-vazio X de geradores livres. Defina um comutador normado à esquerda [x1;x2; : : : ;xn] por [a;b] = ab−ba e [a;b;c] = [ [a;b];c ] . Para n ≥ 2, seja T(n) o ideal bilateral em K(X) gerado pelos comutadores [a1;a2; : : : ;an] (ai ∈ K(X)). A álgebra quociente K(X)=T(n+1) pode ser vista como a K-álgebra universal associativa Lie nilpotente de classe n gerada por X. É fácil ver que o ideal T(2) é gerado, como um ideal bilateral em K(X), pelos comutadores [x1;x2] (xi ∈ X). É bem conhecido que o ideal T(3) é gerado pelos polinômios [x1;x2;x3] e [x1;x2][x3;x4]+[x1;x3][x2;x4] (xi ∈ X). Um conjunto similar de geradores para T(4) é também conhecido. O resultado principal do presente trabalho é exibir um conjunto semelhante de geradores para T(5). Nós provaremos que o ideal T(5) é gerado, como um ideal bilateral em K(X), pelos seguintes polinômios: [x1;x2;x3;x4;x5]; [x1;x2;x3][x4;x5;x6]; [x1;x2;x3][x4;x5;x6;x7]; [x1;x2][x3;x4;x5;x6]+[x6;x2][x3;x4;x5;x1]; ( [x1;x2][x3;x4]+[x1;x3][x2;x4] ) [x5;x6;x7]; [ [x1;x2][x3;x4]+[x1;x3][x2;x4];x5;x6 ] ; [ [x1;x2][x3;x4]+[x1;x3][x2;x4];x5 ] [x6;x7]+ [ [x1;x2][x3;x4]+[x1;x3][x2;x4];x6 ] [x5;x7]; ( [x1;x2][x3;x4]+[x1;x3][x2;x4] )( [x5;x6][x7;x8]+[x5;x7][x6;x8] ) ; com xi ∈ X para todo i. Nós também descreveremos algumas componentes multilineares de Z(X)=L3 e Z(X)=L4, sendo Ln o n-ésimo termo da série central inferior de Z(X) visto como um anel de Lie . ______________________________________________________________________________ ABSTRACT
Let K be a unital associative and commutative ring and let K(X) be the free associative K-algebra on a non-empty set X of free generators. Define a left-normed commutator [x1;x2; : : : ;xn] by [a;b] = ab−ba and [a;b;c] = [ [a;b];c ] . For n ≥ 2, let T(n) be the two-sided ideal in K(X) generated by all commutators [a1;a2; : : : ;an] (ai ∈ K(X)). The quotient algebra K(X)=T(n+1) can be viewed as the universal Lie nilpotent associative K-algebra of class n generated by X. It can be easily seen that the ideal T(2) is generated, as a two-sided ideal in K(X), by the commutators [x1;x2] (xi ∈ X). It is well-known that T(3) is generated by the polynomials [x1;x2;x3] and [x1;x2][x3;x4]+[x1;x3][x2;x4] (xi ∈ X). A similar generating set for T(4) is also known. The aim of the present work is to exhibit a similar generating set for T(5). We prove that the ideal T(5) is generated, as a two-sided ideal in K(X), by the following polynomials: [x1;x2;x3;x4;x5]; [x1;x2;x3][x4;x5;x6]; [x1;x2;x3][x4;x5;x6;x7]; [x1;x2][x3;x4;x5;x6]+[x6;x2][x3;x4;x5;x1]; ( [x1;x2][x3;x4]+[x1;x3][x2;x4] ) [x5;x6;x7]; [ [x1;x2][x3;x4]+[x1;x3][x2;x4];x5;x6 ] ; [ [x1;x2][x3;x4]+[x1;x3][x2;x4];x5 ] [x6;x7]+ [ [x1;x2][x3;x4]+[x1;x3][x2;x4];x6 ] [x5;x7]; ( [x1;x2][x3;x4]+[x1;x3][x2;x4] )( [x5;x6][x7;x8]+[x5;x7][x6;x8] ) ; where xi ∈ X for all i. We also describe some multilinear components of Z(X)=L3 and Z(X)=L4 where Ln is the n-th term of the lower central series of Z(X) viewed as a Lie ring.
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Delboni, Bruno de Assis. « Unificação assimétrica módulo operadores nilpotentes com homomorfismo ». reponame:Repositório Institucional da UnB, 2017. http://repositorio.unb.br/handle/10482/24161.
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Dissertação (mestrado)—Universidade de Brasília, Instituto de Ciências Exatas, Departamento de Matemática, 2017.Submitted by Raquel Almeida (raquel.df13@gmail.com) on 2017-06-20T21:15:09Z No. of bitstreams: 1 2017_BrunodeAssisDelboni.pdf: 1000608 bytes, checksum: 0c7c86de2221eca903edd2ffda11ee2c (MD5)
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Esta dissertação tem como foco o estudo do problema de unificação módulo uma teoria equacional cuja assinatura contém um operador binário que satisfaz as identidades Associatividade, Comutatividade, Unidade e Nilpotência (ACUN), e que pode ou não conter um operador unário que satisfaz a identidade de homomorfismo (ACUNh), que é a teoria equacional do operador , amplamente utilizado em diversas ferramentas criptográficas, como MAUDE-NPA[10] que utiliza uma encriptação de grupos abelianos, incluindo ( ou exclusivo ), exponenciação e encriptação homomórfica. Primeiro apresentaremos alguns critérios para existência de soluções para problemas de ACUN(h)-unificação elementar com constantes que consiste em associar o problema de unificação à um sistema de equações lineares cujos coeficientes são elementos de ou , dependendo se o homomorfismo é ou não considerado. Segundo, apresentaremos um algoritmo para resolver problemas de ACUN(h)- unificação geral que retorna sempre um conjunto completo de unificadores. Finalmente, apresentaremos o estudo de um novo paradigma de unificação, a dizer, \emph{unificação assimétrica}, que consiste de obter unificadores de um problema de unificação com a propriedade de preservar formas normais do lado direito de cada equação de com relação a um sistema de reescrita convergente e coerente módulo uma teoria equacional . No caso particular da teoria equacional ACUN construiremos um algoritmo de conversão de ACUN-unificadores para ACUN-unificadores assimétricos.
This dissertation focuses on the study of unification problems modulo an equational theory whose signature contains a binary operator , which satisfies the identities of Associativity, Commutativity, Unity and Nilpotence (ACUN), and which may or not contain a unary operator satisfying the homomorphism identity (ACUNh), which is the equational theory for the operator XOR, Widely used on many cryptographic tools, like MAUDE[10], which uses group encryption, including XOR ( exclusive or ), exponentiation and homomorphic encryption. First we will present some criteria to the existence of solutions for elementary with constants ACUN(h)-unification problems which consist of associating a unification problem to a linear equation system whose coefficients are elements of or , depending one we are considering homomorphism or not. Second, we will present an algorithm to solve general ACUNh-unification problems which always returns a complete set of most general unifiers. Finally, we will present the study of a new unification paradigm, to say so, asymmetric unification, which consist of obtaining unifiers from the unification problem , with the property of preserving the normal form from of the right hand side of each equation in , considering a convergent and coherent rewriting system. In the particular case of the equational theory ACUN, we will also present an algorithm which takes as input ACUN-unifiers and outputs ACUN-asymmetric unifiers.
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Eberlin, Valerien. « Centroïdes et algèbres de Lie dimensionnellement nilpotentes ». Montpellier 2, 1997. http://www.theses.fr/1997MON20084.
Texte intégralRésumé :
Il est connu que les algebres de lie dimensionnellement nilpotentes, non simples de dimension 3, sont resolubles et que les algebres de lie dimensionnellement nilpotentes, non nilpotentes sont des algebres locales d'ideal maximal nilpotent. Ces resultats fondamentaux obtenus par leger et manley nous ont permis de trouver quelques proprietes elementaires des algebres de lie dimensionnellement nilpotentes et d'etudier les types associes a l'ideal maximal nilpotent de ces algebres. Ainsi, nous montrons que si cet ideal maximal nilpotent n'est pas abelien ou de heisenberg, il est filiforme gradue engendre par ses elements de degre 1 ou produit semi-direct d'une filiforme graduee engendree par ses elements de degre 1 par une droite c'est a dire de type de la forme (n,2). Ce resultat est interessant parce'que michele vergne dans l'etude des algebres de lie filiformes engendrees par ses elements de degre 1, a montre qu'il n'y a que deux classes d'isomorphismes possibles pour ces algebres : les structures des algebres de lie dimensionnellement nilpotentes non nilpotentes sont donc presque toutes connues. Cela permet de calculer le centroide d'une algebre de lie dimensionnellement nilpotente quelconque, non nilpotente et de montrer qu'il est petit. Une etude des algebres de lie 2-nilpotentes dimensionnellement nilpotentes est aussi abordee ou nous etablissons une condition necessaire et suffisante pour qu'une algebre de lie 2-nilpotente soit dimensionnellement nilpotente. Un calcul explicite, decrit de facon precise les constantes de structures, sur une base dite adaptee, des algebres de lie 2-nilpotentes dimensionnellement nilpotentes dont le centre et l'ideal derive coincident ; cette description nous a permis de fournir la classification de ces algebres de lie en dimension 5,6 et 8.
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Vechetová, Jana. « Geometrické postupy v řízení robotických hadů ». Master's thesis, Vysoké učení technické v Brně. Fakulta strojního inženýrství, 2018. http://www.nusl.cz/ntk/nusl-392853.
Texte intégralRésumé :
Tato diplomová práce se zabývá popisem řiditelnosti specifického robotického hada, který se nazývá trident snake robot. Tento robot je řazen mezi neholonomní systémy. Model je převeden do jazyka diferenciální geometrie a řízen pomocí vektorových polí a operace na nich zavedené (Lieova závorka). Je také uvažována aproximace řídicí distribuce. Dále jsou formulovány pohyby hada ve směru vektorových polí a jejich kombinace, které zajišťují základní pohyby v prostoru (rotace a translace). Tyto pohyby jsou na závěr simulovány v prostředí V-REP.
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Speh, Peter (Peter Daniel). « A classification of real and complex nilpotent orbits of reductive groups in terms of complex even nilpotent orbits ». Thesis, Massachusetts Institute of Technology, 2012. http://hdl.handle.net/1721.1/73443.
Texte intégralRésumé :
Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2012.Cataloged from PDF version of thesis.
Includes bibliographical references (p. 83).
Let g be a complex, reductive Lie algebra. We prove a theorem parametrizing the set of nilpotent orbits in g in terms of even nilpotent orbits of subalgebras of g and show how to determine these subalgebras and how to explicitly compute this correspondence. We prove a theorem parametrizing nilpotent orbits for strong involutions of G in terms of even nilpotent orbits of complex subalgebras of g and show how to explicitly compute this correspondence.
by Peter Speh.
Ph.D.
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Popov, Vladimir L., et popov@ppc msk ru. « Self-Dual Algebraic Varieties and Nilpotent Orbits ». ESI preprints, 2001. ftp://ftp.esi.ac.at/pub/Preprints/esi978.ps.
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Chitayat, Michael. « Locally Nilpotent Derivations and Their Quasi-Extensions ». Thesis, Université d'Ottawa / University of Ottawa, 2016. http://hdl.handle.net/10393/35072.
Texte intégralRésumé :
In this thesis, we introduce the theory of locally nilpotent derivations and use it to compute certain ring invariants.
We prove some results about quasi-extensions of derivations and use them to show that certain rings are non-rigid.
Our main result states that if k is a field of characteristic zero, C is an affine k-domain and
B = C[T,Y] / < T^nY - f(T) >, where n >= 2 and f(T) \in C[T] is such that
delta^2(f(0)) != 0 for all nonzero locally nilpotent derivations delta of C,
then ML(B) != k.
This shows in particular that the ring B is not a polynomial ring over k.
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Lin, Wan. « Automorphism groups of free metabelian nilpotent groups ». Thesis, National Library of Canada = Bibliothèque nationale du Canada, 1999. http://www.collectionscanada.ca/obj/s4/f2/dsk1/tape9/PQDD_0012/NQ42998.pdf.
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Iraghi, Moghaddam Gholamhossein. « Minimal presentations of free metabelian nilpotent groups ». Thesis, National Library of Canada = Bibliothèque nationale du Canada, 2000. http://www.collectionscanada.ca/obj/s4/f2/dsk2/ftp02/NQ52738.pdf.
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Sommers, Eric Nathan 1971. « Nilpotent orbits and the affine flag manifold ». Thesis, Massachusetts Institute of Technology, 1997. http://hdl.handle.net/1721.1/42772.
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Matthiesen, Lilian. « Applications of the nilpotent Hardy-Littlewood method ». Thesis, University of Cambridge, 2012. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.610152.
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Amantini, Andrea. « Fraïssé-Hrushovski predimensions on nilpotent Lie algebras ». Doctoral thesis, Humboldt-Universität zu Berlin, Mathematisch-Naturwissenschaftliche Fakultät II, 2011. http://dx.doi.org/10.18452/16345.
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In dieser Arbeit wird das Fraïssé-Hrushowskis Amalgamationsverfahren in Zusammenhang mit nilpotenten graduierten Lie Algebren über einem endlichen Körper untersucht. Die Prädimensionen die in der Konstruktion auftauchen sind mit dem gruppentheoretischen Begriff der Defizienz zu vergleichen, welche auf homologische Methoden zurückgeführt werden kann. Darüber hinaus wird die Magnus-Lazardsche Korrespondenz zwischen den oben genannten Lie Algebren und nilpotenten Gruppen von Primzahl-Exponenten beschrieben. Dabei werden solche Gruppen durch die Baker-Haussdorfsche Formel in den entsprechenden Algebren definierbar interpretiert. Es wird eine omega-stabile Lie Algebra von Nilpotenzklasse 2 und Morleyrang omega + omega erhalten, indem man eine unkollabierte Version der von Baudisch konstruierten "new uncountably categorical group" betrachtet. Diese wird genau analysiert. Unter anderem wird die Unabhängigkeitsrelation des Nicht-Gabelns durch die Konfiguration des freien Amalgams charakterisiert. Mittels eines induktiven Ansatzes werden die Grundlagen entwickelt, um neue Prädimensionen für Lie Algebren der Nilpotenzklassen größer als zwei zu schaffen. Dies erweist sich als wesentlich schwieriger als im Fall 2. Wir konzentrieren uns daher auf die Nilpotenzklasse 3, als Induktionsbasis des oben genannten Prozesses. In diesem Fall wird die Invariante der Defizienz auf endlich erzeugte Lie Algebren adaptiert. Erstes Hauptergebnis der Arbeit ist der Nachweis dass diese Definition zu einem vernüftigen Begriff selbst-genügender Erweiterungen von Lie Algebren führt und sehr nah einer gewünschten Prädimension im Hrushovskischen Sinn ist. Wir zeigen – als zweites Hauptergebnis – ein erstes Amalgamationslemma bezüglich selbst-genügender Einbettungen.In this work, the so called Fraïssé-Hrushowski amalgamation is applied to nilpotent graded Lie algebras over the p-elements field with p a prime. We are mainly concerned with the uncollapsed version of the original process. The predimension used in the construction is compared with the group theoretical notion of deficiency, arising from group Homology. We also describe in detail the Magnus-Lazard correspondence, to switch between the aforementioned Lie algebras and nilpotent groups of prime exponent. In this context, the Baker-Hausdorff formula allows such groups to be definably interpreted in the corresponding algebras. Starting from the structures which led to Baudisch’ new uncountably categorical group, we obtain an omega-stable Lie algebra of nilpotency class 2, as the countable rich Fraïssé limit of a suitable class of finite Lie algebras. We study the theory of this structure in detail: we show its Morley rank is omega+omega and a complete description of non-forking independence is given, in terms of free amalgams. In a second part, we develop a new framework for the construction of deficiency-predimensions among graded Lie algebras of nilpotency class higher than 2. This turns out to be considerably harder than the previous case. The nil-3 case in particular has been extensively treated, as the starting point of an inductive procedure. In this nilpotency class, our main results concern a suitable deficiency function, which behaves for many aspects like a Hrushovski predimension. A related notion of self-sufficient extension is given. We also prove a first amalgamation lemma with respect to self-sufficient embeddings.
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Stoll, Michael. « Asymptotics of some number theoretic functions and an application to the growth of nilpotent groups ». Bonn : [s.n.], 1994. http://catalog.hathitrust.org/api/volumes/oclc/31760796.html.
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Zipperer, Jörg. « Kohomologie von Kurven und geometrische Realisierung nilpotenter Gruppen ». [S.l. : s.n.], 2002. http://deposit.ddb.de/cgi-bin/dokserv?idn=965506657.
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