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<p>Quantum sl_3 projectors are morphisms in Kuperberg's sl_3 spider, a diagrammatically defined category equivalent to the full pivotal subcategory of the category of (type 1) finite-dimensional representations of the quantum group U_q (sl_3 ) generated by the defining representation, which correspond to projection onto (and then inclusion from) the highest weight irreducible summand. These morphisms are interesting from a topological viewpoint as they allow the combinatorial formulation of the sl_3 tangle invariant (in which tangle components are labelled by the defining representation) to be extended to a combinatorial formulation of the invariant in which components are labelled by arbitrary finite-dimensional irreducible representations. They also allow for a combinatorial description of the SU(3) Witten-Reshetikhin-Turaev 3-manifold invariant. </p><p>There exists a categorification of the sl_3 spider, due to Morrison and Nieh, which is the natural setting for Khovanov's sl_3 link homology theory and its extension to tangles. An obvious question is whether there exist objects in this categorification which categorify the sl_3 projectors. </p><p>In this dissertation, we show that there indeed exist such "categorified projectors," constructing them as the stable limit of the complexes assigned to k-twist torus braids (suitably shifted). These complexes satisfy categorified versions of the defining relations of the (decategorified) sl_3 projectors and map to them upon taking the Grothendieck group. We use these categorified projectors to extend sl_3 Khovanov homology to a homology theory for framed links with components labeled by arbitrary finite-dimensional irreducible representations of sl_3 .</p> / Dissertation

Made available in DSpace on 2015-03-26T13:45:36Z (GMT). No. of bitstreams: 1 texto completo.pdf: 610299 bytes, checksum: 974c87fb133c18b010f2adf40631a6b4 (MD5) Previous issue date: 2013-07-17 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / These algebras are defined by the tensor product of a Lie algebra and of a truncated polynornial ring. The rnain goal is to establish a criterion for the reducibility of the universal objects in the theory of highest weight representations, the so called Verma modules. ln his doctoral thesis, Benjamin J. Wilson proved that the reducibility of the Verrna rnodules of the truncated current Lie algebras depends only on one of their hornogeneous cornponents. This work consists in studying the criterion established by Wilson. / Neste trabalho estudamos representações de peso máxirno de álgebras de Lie correntes trancados. Estas álgebras são definidas corno o produto tensorial de urna álgebra de Lie por um anel de polinômios truncado. O objetivo principal é estabelecer um critério para a redutibilidade dos objetos universais da teoria de representações de peso máxirno, os chamados módulos Verme. Em sua tese de doutorado, Benjamin J. Wilson provou que a redutibilidade dos módulos Verma das álgebras de Lie correntes truncadas depende apenas de uma de suas componentes homogêneas. Nosso trabalho consiste em estudar o critério estabelecido por Wilson.

In this thesis we classify singular vectors in scalar parabolic Verma modules for those pairs (sl(n, C), p) of complex Lie algebras where the homogeneous space SL(n, C)/P is the Grassmannian of k-planes in Cn . We calculate cohomology of nilpotent radicals with values in certain unitarizable highest weight modules. According to [BH09] these modules have BGG resolutions with weights determined by this cohomology. Such resolutions induce complexes of invariant differential operators on sections of associated bundles over Hermitian symmetric spaces. We describe formal completions of unitarizable highest weight modules that one can use to modify method from [CD01] that constructs sequences of differential operators over any 1-graded (aka almost Hermitian) geometry. We suggest uniform description of octonionic planes that could serve as a basis for better understanding of the exceptional Hermitian symmetric space for group E6.

Les algèbres cellulaires furent introduite par J.J. Graham et G.I. Lehrer en 1996. Elles forment une famille d’algèbres associatives de dimension finie définies en termes de « données cellulaires » satisfaisant certains axiomes. Ces données cellulaires, lorsqu’elles sont identifiées pour une certaine algèbre, permettent une construction explicite de tous ses modules simples, à isomorphisme près, et de leurs couvertures projectives. Dans ce mémoire, nous définissons ces algèbres cellulaires en introduisant progressivement chacun des éléments constitutifs d’une façon axiomatique. Deux autres familles d’algèbres associatives sont discutées, à savoir les algèbres quasihéréditaires et celles dont les modules forment une catégorie de plus haut poids. Ces familles furent introduites durant la même période de temps, au tournant des années quatre-vingtdix. La relation entre ces deux familles ainsi que celle entre elles et les algèbres cellulaires sont prouvées. / Cellular algebras were introduced by J.J. Graham and G.I. Lehrer in 1996. They are a class of finite-dimensional associative algebras defined in terms of a “cellular datum” satisfying some axioms. This cellular datum, when made explicit for a given associative algebra, allows for the explicit construction of all its simple modules, up to isomorphism, and of their projective covers. In this work, we define these cellular algebras by introducing each building block of the cellular datum in a fairly axiomatic fashion. Two other families of associative algebras are discussed, namely the quasi-hereditary algebras and those whose modules form a highest weight category. These families were introduced at about the same period. The relationships between these two, and between them and the cellular ones, are made explicit.