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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.

O objetivo deste trabalho é estudar a estrutura dos módulos sobre uma álgebra trivial de dimensão dois na variedade M das álgebras comutativas de potências associativas. Em particular classificamos os módulos irredutíveis. Estes resultados nos permitem compreender melhor a estrutura das nilálgebras comutativas de dimensão finita e nilíndice 4. Finalmente classificamos, sob isomorfismos, as nilálgebras comutativas de potências associativas de dimensão n e nilíndice n. / The aim of this work is to study the structure of the modules over a trivial algebra of dimension two in the variety M of commutative and power-associative algebras. In particular we classify the irreducible modules. These results enables us to understand better the structure of finite-dimensional power-associative nilalgebras of nilindex 4. Finally, we classify, up to isomorphism, commutative power associative nilalgebras of nilindex n and dimension n.

O objetivo deste trabalho é estudar a estrutura dos módulos sobre uma álgebra trivial de dimensão dois na variedade M das álgebras comutativas de potências associativas. Em particular classificamos os módulos irredutíveis. Estes resultados nos permitem compreender melhor a estrutura das nilálgebras comutativas de dimensão finita e nilíndice 4. Finalmente classificamos, sob isomorfismos, as nilálgebras comutativas de potências associativas de dimensão n e nilíndice n. / The aim of this work is to study the structure of the modules over a trivial algebra of dimension two in the variety M of commutative and power-associative algebras. In particular we classify the irreducible modules. These results enables us to understand better the structure of finite-dimensional power-associative nilalgebras of nilindex 4. Finally, we classify, up to isomorphism, commutative power associative nilalgebras of nilindex n and dimension n.

La thèse porte sur la catégorification de généralisations de groupes de tresses. Nous étendons une représentation des groupes de tresses par complexes de bimodules de Soergel due à Rouquier. Nous généralisons d'abord ce résultat en type A aux monoïdes de tresses singulières, puis aux groupes de tresses virtuelles. Enfin nous définissons, puis catégorifions des groupes de tresses virtuelles de type B en nous fondant sur une description des groupes de tresses de type B donnée par tom Dieck utilisant des tresses symétriques.

abstract: In this thesis, I investigate the C*-algebras and related constructions that arise from combinatorial structures such as directed graphs and their generalizations. I give a complete characterization of the C*-correspondences associated to directed graphs as well as results about obstructions to a similar characterization of these objects for generalizations of directed graphs. Viewing the higher-dimensional analogues of directed graphs through the lens of product systems, I give a rigorous proof that topological k-graphs are essentially product systems over N^k of topological graphs. I introduce a "compactly aligned" condition for such product systems of graphs and show that this coincides with the similarly-named conditions for topological k-graphs and for the associated product systems over N^k of C*-correspondences. Finally I consider the constructions arising from topological dynamical systems consisting of a locally compact Hausdorff space and k commuting local homeomorphisms. I show that in this case, the associated topological k-graph correspondence is isomorphic to the product system over N^k of C*-correspondences arising from a related Exel-Larsen system. Moreover, I show that the topological k-graph C*-algebra has a crossed product structure in the sense of Larsen. / Dissertation/Thesis / Ph.D. Mathematics 2011

Dans cette thèse on montre que tout groupe compact peut être réalisé comme le groupe d'automorphismes extérieurs d'un facteur de type II_1. On montre également que la catégorie des représentations de tout groupe compact est équivalente à la catégorie des bimodules sur un facteur de type II_1. Plusieurs chapitres de cette thèse sont également consacrés à des rappels détaillés concernant la catégorie des bimodules sur un facteur de type II_1 ainsi que sur les actions minimales de groupes compacts sur des facteurs de type II_1.

On donne une preuve complète de la classification des représentations d'énergie positive unitaires de l'algèbre Neveu-Schwarz, de telle manière qu'on obtient directement les caractères de la séries discrètes. Ensuite, on explicite leur loi de fusion de Connes et on prouve que les sous-facteurs de Jones-Wassermann sont irréductibles d'indice fini, on donne leur formule.