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Categorification of quantum sl_3 projectors and the sl_3 Reshetikhin-Turaev invariant of framed tangles

dc.contributor.advisor Ng, Lenhard L Rose, David Emile Vatcher 2012-05-25T20:21:18Z 2013-05-20T04:30:06Z 2012
dc.description.abstract <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>
dc.subject Mathematics
dc.subject Categorification
dc.subject Highest weight projector
dc.subject Khovanov homology
dc.subject Quantum Group
dc.subject Spider
dc.title Categorification of quantum sl_3 projectors and the sl_3 Reshetikhin-Turaev invariant of framed tangles
dc.type Dissertation
dc.department Mathematics
duke.embargo.months 12

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