Tesis: "Semisimple algebraic groups"

1

Mohrdieck, Stephan. "Conjugacy classes of non-connected semisimple algebraic groups". [S.l. : s.n.], 2000. http://www.sub.uni-hamburg.de/disse/172/diss.pdf.

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2

Hazi, Amit. "Semisimple filtrations of tilting modules for algebraic groups". Thesis, University of Cambridge, 2018. https://www.repository.cam.ac.uk/handle/1810/271774.

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Let $G$ be a reductive algebraic group over an algebraically closed field $k$ of characteristic $p > 0$. The indecomposable tilting modules $\{T(\lambda)\}$ for $G$, which are labeled by highest weight, form an important class of self-dual representations over $k$. In this thesis we investigate semisimple filtrations of minimal length (Loewy series) of tilting modules. We first demonstrate a criterion for determining when tilting modules for arbitrary quasi-hereditary algebras are rigid, i.e. have a unique Loewy series. Our criterion involves checking that $T(\lambda)$ does not have certain subquotients whose composition factors extend more than one layer in the radical or socle series. We apply this criterion to show that the restricted tilting modules for $SL_4$ are rigid when $p \geq 5$, something beyond the scope of previous work on this topic by Andersen and Kaneda. Even when $T(\lambda)$ is not rigid, in many cases it has a particularly structured Loewy series which we call a balanced semisimple filtration, whose semisimple subquotients or "layers" are symmetric about some middle layer. Balanced semisimple filtrations also suggest a remarkably straightforward algorithm for calculating tilting characters from the irreducible characters. Applying Lusztig's character formula for the simple modules, we show that the algorithm agrees with Soergel's character formula for the regular indecomposable tilting modules for quantum groups at roots of unity. We then show that these filtrations really do exist for these tilting modules. In the modular case, high weight tilting modules exhibit self-similarity in their characters at $p$-power scales. This is due to what we call higher-order linkage, an old character-theoretic result relating modular tilting characters and quantum tilting characters at $p$-power roots of unity. To better understand this behavior we describe an explicit categorification of higher-order linkage using the language of Soergel bimodules. Along the way we also develop the algebra and combinatorics of higher-order linkage at the de-categorified level. We hope that this will provide a foundation for a tilting character formula valid for all weights in the modular case when $p$ is sufficiently large.

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3

Kenneally, Darren John. "On eigenvectors for semisimple elements in actions of algebraic groups". Thesis, University of Cambridge, 2010. https://www.repository.cam.ac.uk/handle/1810/224782.

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Let G be a simple simply connected algebraic group defined over an algebraically closed field K and V an irreducible module defined over K on which G acts. Let E denote the set of vectors in V which are eigenvectors for some non-central semisimple element of G and some eigenvalue in K*. We prove, with a short list of possible exceptions, that the dimension of Ē is strictly less than the dimension of V provided dim V > dim G + 2 and that there is equality otherwise. In particular, by considering only the eigenvalue 1, it follows that the closure of the union of fixed point spaces of non-central semisimple elements has dimension strictly less than the dimension of V provided dim V > dim G + 2, with a short list of possible exceptions. In the majority of cases we consider modules for which dim V > dim G + 2 where we perform an analysis of weights. In many of these cases we prove that, for any non-central semisimple element and any eigenvalue, the codimension of the eigenspace exceeds dim G. In more difficult cases, when dim V is only slightly larger than dim G + 2, we subdivide the analysis according to the type of the centraliser of the semisimple element. Here we prove for each type a slightly weaker inequality which still suffices to establish the main result. Finally, for the relatively few modules satisfying dim V ≤ dim G + 2, an immediate observation yields the result for dim V < dim B where B is a Borel subgroup of G, while in other cases we argue directly.

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4

Gandhi, Raj. "Oriented Cohomology Rings of the Semisimple Linear Algebraic Groups of Ranks 1 and 2". Thesis, Université d'Ottawa / University of Ottawa, 2021. http://hdl.handle.net/10393/42566.

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In this thesis, we compute minimal presentations in terms of generators and relations for the oriented cohomology rings of several semisimple linear algebraic groups of ranks 1 and 2 over algebraically closed fields of characteristic 0. The main tools we use in this thesis are the combinatorics of Coxeter groups and formal group laws, and recent results of Calm\`es, Gille, Petrov, Zainoulline, and Zhong, which relate the oriented cohomology rings of flag varieties and semisimple linear algebraic groups to the dual of the formal affine Demazure algebra.

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5

Maccan, Matilde. "Sous-schémas en groupes paraboliques et variétés homogènes en petites caractéristiques". Electronic Thesis or Diss., Université de Rennes (2023-....), 2024. https://ged.univ-rennes1.fr/nuxeo/site/esupversions/2e27fe72-c9e0-4d56-8e49-14fc84686d6c.

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Cette thèse achève la classification des sous-schémas en groupes paraboliques des groupes algébriques semi-simples sur un corps algébriquement clos, en particulier de caractéristique deux et trois. Dans un premier temps, nous présentons la classification en supposant que la partie réduite de ces sous-groupes soit maximale, avant de passer au cas général. Nous parvenons à une description quasiment uniforme : à l'exception d'un groupe de type G₂ en caractéristique deux, chaque sous-schémas en groupes parabolique est obtenu en multipliant des paraboliques réduits par des noyaux d'isogénies purement inséparables, puis en prenant l'intersection. En conclusion, nous discutons quelques implications géométriques de cette classification
This thesis brings to an end the classification of parabolic subgroup schemes of semisimple groups over an algebraically closed field, focusing on characteristic two and three. First, we present the classification under the assumption that the reduced part of these subgroups is maximal; then we proceed to the general case. We arrive at an almost uniform description: with the exception of a group of type G₂ in characteristic two, any parabolic subgroup scheme is obtained by multiplying reduced parabolic subgroups by kernels of purely inseparable isogenies, then taking the intersection. In conclusion, we discuss some geometric implications of this classification

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6

Oriente, Francesco. "Classifying semisimple orbits of theta-groups". Doctoral thesis, Università degli studi di Trento, 2012. https://hdl.handle.net/11572/368303.

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I consider the problem of classifying the semisimple orbits of a theta-group. For this purpose, once a preliminary presentation of the theoretical subjects where my problem arises from, I first give an algorithm to compute a Cartan subspace; subsequently I describe how to compute the little Weyl group.

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7

Oriente, Francesco. "Classifying semisimple orbits of theta-groups". Doctoral thesis, University of Trento, 2012. http://eprints-phd.biblio.unitn.it/731/1/tesi.pdf.

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I consider the problem of classifying the semisimple orbits of a theta-group. For this purpose, once a preliminary presentation of the theoretical subjects where my problem arises from, I first give an algorithm to compute a Cartan subspace; subsequently I describe how to compute the little Weyl group.

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8

Lampetti, Enrico. "Nilpotent orbits in semisimple Lie algebras". Bachelor's thesis, Alma Mater Studiorum - Università di Bologna, 2021. http://amslaurea.unibo.it/23595/.

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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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9

Nishiyama, Kyo. "Representations of Weyl groups and their Hecke algebras on virtual character modules of a semisimple Lie group". 京都大学 (Kyoto University), 1986. http://hdl.handle.net/2433/86366.

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10

Athapattu, Mudiyanselage Chathurika Umayangani Manike Athapattu. "Chevalley Groups". OpenSIUC, 2016. https://opensiuc.lib.siu.edu/theses/1986.

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In this thesis, we construct Chevalley groups over arbitrary fields. The construction is based on the properties of semi-simple complex Lie algebras, the existence of Chevalley bases and notion of universal enveloping algebras. Using integral lattices in universal enveloping algebras and integral properties of Chevalley bases, we present a method which produces, for any complex simple Lie group, an analogous group over an arbitrary field.

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11

Popov, Vladimir L. y vladimir@popov msk su. "Generators and Relations of the Affine Coordinate Rings of Connected". ESI preprints, 2000. ftp://ftp.esi.ac.at/pub/Preprints/esi972.ps.

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12

Caprace, Pierre-Emmanuel. ""Abstract" homomorphisms of split Kac-Moody groups". Doctoral thesis, Universite Libre de Bruxelles, 2005. http://hdl.handle.net/2013/ULB-DIPOT:oai:dipot.ulb.ac.be:2013/210962.

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Cette thèse est consacrée à une classe de groupes, appelés groupes de Kac-Moody, qui généralise de façon naturelle les groupes de Lie semi-simples, ou plus précisément, les groupes algébriques réductifs, dans un contexte infini-dimensionnel. On s'intéresse plus particulièrement au problème d'isomorphismes pour ces groupes, en vue d'obtenir un analogue infini-dimensionnel de la célèbre théorie des homomorphismes 'abstraits' de groupes algébriques simples, due à Armand Borel et Jacques Tits.

Le problème d'isomorphismes qu'on étudie s'avère être un cas particulier d'un problème plus général, qui consiste à caractériser les homomorphismes de groupes algébriques vers les groupes de Kac-Moody, dont l'image est bornée. Ce problème peut à son tour s'énoncer comme un problème de rigidité pour les actions de groupes algébriques sur les immeubles, via l'action naturelle d'un groupe de Kac-Moody sur une paire d'immeubles jumelés. Les résultats partiels, relatifs à ce problème de rigidité, que nous obtenons, nous permettent d'apporter une solution complète au problème d'isomorphismes pour les groupes de Kac-Moody déployés.

En particulier, on obtient un résultat de dévissage pour les automorphismes de ces objets. Celui-ci fournit à son tour une description complète de la structure du groupe d'automorphismes d'un groupe de Kac-Moody déployé sur un corps de caractéristique~$0$.

Nos arguments permettent également de traiter de façon analogue certaines formes anisotropes de groupes de Kac-Moody complexes, appelées formes unitaires. On montre en particulier que la topologie Hausdorff naturelle que portent ces formes est un invariant de leur structure de groupe abstrait. Ceci généralise un résultat bien connu de H. Freudenthal pour les groupes de Lie compacts.

Enfin, l'on s'intéresse aux homomorphismes de groupes de Kac-Moody à image fini-dimensionnelle, et l'on démontre la non-existence de tels homomorphismes à noyau central, lorsque le domaine est un groupe de Kac-Moody de type indéfini sur un corps infini. Ceci réduit un problème ouvert, dit problème de linéarité pour les groupes de Kac-Moody, au cas de corps de base finis.
Doctorat en sciences, Spécialisation mathématiques
info:eu-repo/semantics/nonPublished

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13

Gruson, Caroline. "Sur les super groupes de Lie". Paris 7, 1993. http://www.theses.fr/1993PA077056.

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La première partie est une adaptation au cadre des super groupes de Lie du théorème du à Cartier qui assure que les groupes formels sont lisses en caractérisque zéro. La seconde partie donne une description des super groupes de Lie dits vraiment classiques comme groupes d'automorphismes des super algèbres semi-simples à involution, selon une méthode de Weil. La troisième partie est consacrée à l'étude de l'idéal définissant l'orbite d'un vecteur de plus haut poids d'une représentation simple de dimension finie d'une super algèbre de Lie basique classique complexe.

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14

Junior, Fernando Martins Antoneli. "Subalgebras maximais das álgebras de Lie semisimples, quebra de simetria e o código genético". Universidade de São Paulo, 1998. http://www.teses.usp.br/teses/disponiveis/45/45132/tde-01092009-171526/.

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O propósito deste trabalho é dar uma contribuição ao projeto iniciado por Hornos & Hornos que visa explicar as degenerescências do código genético como resultado de sucessivas quebras de simetria ocorridas durante sua evolução. O modelo matemático usado requer a construção de todas as representações irredutíveis de dimensão 64 das álgebras de Lie simples (chamadas representações de códons) e a análise de suas regras de ramicação sob redução a subalgebras. A classicação de todas as possibilidades é baseada na classicação das subalgebras maximais das álgebras de Lie semisimples obtida por Dynkin. No presente trabalho, os resultados de Dynkin são apresentados em linguagem e notação moderna e são aplicados ao problema de construir todas as possíveis cadeias de subalgebras maximais das álgebras de Lie simples B_6 = so(13) e D_7 = so(14) e de identicar aquelas que reproduzem as degenerescências do código genético.
The purpose of this work is to make a contribution to the project initiated by Hornos & Hornos which aims at explaining the degeneracy of the genetic code as the result of a sequence of symmetry breaking that occurred during its evolution. The mathematical model employed requires the construction of all 64-dimensional irreducible representations of simple Lie algebras (called codon representations) and the analysis of their branching rules under reduction to sub-algebras. The classification of all possibilities is based on Dynkins classification of the maximal sub-algebras of semi-simple Lie algebras. In the present work, Dynkins results are presented in modern language and notation and are applied to the problem of constructing all possible chains of maximal sub-algebras of the simple Lie algebras B_6 = so(13) and D_7 = so(14) and of identifying all those that reproduce the degeneracies of the genetic code.

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15

Santos, Ricardo Leite dos. "Teoremas de Maschke". Universidade Federal de Santa Maria, 2013. http://repositorio.ufsm.br/handle/1/9981.

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Coordenação de Aperfeiçoamento de Pessoal de Nível Superior
In representation theory, having a representation of a group G is equivalent to having a kG-module. Since |G-modules which are sums of irreducible kG-modules form a very important class in the theory of modules, to know conditions for a kG-module be irreducible or completely reducible from the particularities of the field k and the group G become a very important issue, whose solution was originally presented by the German mathematician Heinrich Maschke which proved that if the order of G is not a multiple of the characteristic of the field k, then kG is completely reducible (or semisimple). From there, issues unrelated to representation theory, but that concern the semisimplicity of cross products in general are treated as Maschke-type theorem. Our goal in this dissertation is to present some versions of this theorem, starting with classic versions involving cross products for actions of groups on algebras and then versions for Hopf algebras and smash products.
Na teoria de representações de grupos, ter uma representação de um grupo G é equivalente a ter um kG-módulo. Desde que kG-módulos que são somas de kG-módulos irredutíveis formam uma classe bastante importante na teoria de módulos, conhecer condições para que um kG-módulo seja irredutível ou completamente redutível a partir das particularidades do corpo k e do grupo G passou a ser um problema bastante importante. Problema este cuja solução foi originalmente apresentada pelo matemático alemão Heinrich Maschke que provou que se a ordem do grupo G não for múltiplo da característica do corpo k, então kG é completamente redutível (ou semissimples). A partir daí, questões independentes a teoria de representações, mas que dizem respeito a semissimplicidade de produtos cruzados em geral são tratados como Teorema tipo-Maschke. Nosso objetivo neste trabalho é apresentar algumas versões deste teorema. Iniciamos com versões mais clássicas envolvendo produtos cruzados globais e parciais para em seguida estudarmos versões em álgebras de Hopf e produtos smash.

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16

Petrov, Viktor [Verfasser]. "J-invariant of semisimple algebraic groups / vorgelegt von Viktor Petrov". 2006. http://d-nb.info/983036160/34.

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17

Mohrdieck, Stephan [Verfasser]. "Conjugacy classes of non-connected semisimple algebraic groups / vorgelegt von Stephan Mohrdieck". 2000. http://d-nb.info/959425802/34.

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18

Egea, Leandro Ginés. "Teorema del peso máximo". Bachelor's thesis, 2011. http://hdl.handle.net/11086/38.

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Tesis (Lic. en Matemática)--Universidad Nacional de Córdoba. Facultad de Matemática, Astronomía y Física, 2010.
Este trabajo es sobre la clasificación, módulo equivalencia, de las representaciones irreducibles de dimensión finita de álgebras de Lie complejas semisimples de dimensión finita. El Teorema del peso máximo describe a las clases de equivalencia como un octante (pesos dominantes) de un reticulado (pesos enteros), en el dual de una subálgebra de Cartan del álgebra de Lie. De este teorema, también se deduce la clasificación de todas las representaciones irreducibles de grupos de Lie compactos.
Leandro Ginés Egea.

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Tesis sobre el tema "Semisimple algebraic groups"

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