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    Knot invariants and higher representation theory I: diagrammatic and geometric categorification of tensor products

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    In this paper, we study 2-representations of 2-quantum groups (in the sense of Rouquier and Khovanov-Lauda) categorifying tensor products of irreducible representations. Our aim is to construct knot homologies categorifying Reshetikhin-Turaev invariants of knots for arbitrary representations, which will be done in a follow-up paper. We consider an algebraic construction of these categories, via an explicit diagrammatic presentation, generalizing the cyclotomic quotient of the quiver Hecke algebra and a geometric construction given by Zheng. One of our primary results is that these categories coincide when both are defined. We also investigate finer structure of these categories. Like many similar representation-theoretic categories, they are standardly stratified and satisfy a double centralizer property with respect to their self-dual modules. The standard modules of the stratification play an important role, as Vermas do in more classical representation theory, as test objects for functors. The existence of these representations has consequences for the structure of previously studied categorifications; it allows us to prove the non-degeneracy of Khovanov and Lauda's 2-category (that its Hom spaces have the expected dimension) in all symmetrizable types, and that the cyclotomic quiver Hecke algebras are symmetric Frobenius.

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    Description

    Title : Knot invariants and higher representation theory I: diagrammatic and geometric categorification of tensor products
    Author(s) : Ben Webster
    Abstract : In this paper, we study 2-representations of 2-quantum groups (in the sense of Rouquier and Khovanov-Lauda) categorifying tensor products of irreducible representations. Our aim is to construct knot homologies categorifying Reshetikhin-Turaev invariants of knots for arbitrary representations, which will be done in a follow-up paper. We consider an algebraic construction of these categories, via an explicit diagrammatic presentation, generalizing the cyclotomic quotient of the quiver Hecke algebra and a geometric construction given by Zheng. One of our primary results is that these categories coincide when both are defined. We also investigate finer structure of these categories. Like many similar representation-theoretic categories, they are standardly stratified and satisfy a double centralizer property with respect to their self-dual modules. The standard modules of the stratification play an important role, as Vermas do in more classical representation theory, as test objects for functors. The existence of these representations has consequences for the structure of previously studied categorifications; it allows us to prove the non-degeneracy of Khovanov and Lauda's 2-category (that its Hom spaces have the expected dimension) in all symmetrizable types, and that the cyclotomic quiver Hecke algebras are symmetric Frobenius.
    Keywords : geometric topology, quantum algebra, representation theory

    Subject : unspecified
    Area : Other
    Language : English
    Year : 2010

    Affiliations Mathematics, Northeastern University, Boston
    Journal : Main
    Pages : 69
    Url : http://arxiv.org/abs/1001.2020

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