62 research outputs found

    The foam and the matrix factorization sl3 link homologies are equivalent

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    We prove that the foam and matrix factorization universal rational sl3 link homologies are naturally isomorphic as projective functors from the category of link and link cobordisms to the category of bigraded vector spaces.Comment: We have filled a gap in the proof of Lemma 5.2. 28 page

    Categorified skew Howe duality and comparison of knot homologies

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    In this paper, we show an isomorphism of homological knot invariants categorifying the Reshetikhin-Turaev invariants for sln\mathfrak{sl}_n. Over the past decade, such invariants have been constructed in a variety of different ways, using matrix factorizations, category O\mathcal{O}, affine Grassmannians, and diagrammatic categorifications of tensor products. While the definitions of these theories are quite different, there is a key commonality between them which makes it possible to prove that they are all isomorphic: they arise from a skew Howe dual action of gl\mathfrak{gl}_\ell for some \ell. In this paper, we show that the construction of knot homology based on categorifying tensor products (from earlier work of the second author) fits into this framework, and thus agrees with other such homologies, such as Khovanov-Rozansky homology. We accomplish this by categorifying the action of gl×gln\mathfrak{gl}_\ell\times \mathfrak{gl}_n on  ⁣p(CCn)\bigwedge\nolimits^{\!p}(\mathbb{C}^\ell\otimes \mathbb{C}^n) using diagrammatic bimodules. In this action, the functors corresponding to gl\mathfrak{gl}_\ell and gln\mathfrak{gl}_n are quite different in nature, but they will switch roles under Koszul duality.Comment: 62 pages. preliminary version, comments welcom

    Operadores de Yang-Baxter e a categoria dos emaranhados

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    Trabalho de síntese para efeitos de preparação de provas de aptidão pedagógica e capacidade científica, Matemática, Unidade de Ciências Exactas e Humanas, Universidade do Algarve, 1998Todos nós conhecemos o famoso truque do prestidigitador: mostra uma corda com um grande nó no meio e de repente puxa as duas pontas da corda cada uma para o seu lado e o nó desaparece misteriosamente

    TheslN-web algebras and dual canonical bases

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    In this paper, which is a follow-up to [38], I define and study SIN-web algebras, for any N >= 2. For N = 2 these algebras are isomorphic to Khovanov's [22] arc algebras and for N = 3 they are Morita equivalent to the sl(3)-web algebras which I defined and studied together with Pan and Tubbenhauer [34]. The main result of this paper is that the SIN-web algebras are Morita equivalent to blocks of certain level-N cyclotomic KLR algebras, for which I use the categorified quantum skew Howe duality defined in [38]. Using this Morita equivalence and Brundan and Kleshchev's [4] work on cyclotomic KLR-algebras, I show that there exists an isomorphism between a certain space of SIN-webs and the split Grothendieck group of the corresponding SIN-web algebra, which maps the dual canonical basis elements to the Grothendieck classes of the indecomposable projective modules (with a certain normalization of their grading).info:eu-repo/semantics/publishedVersio

    A diagrammatic categorification of the q-Schur algebra

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    In this paper we categorify the q-Schur algebra S(n,d) as a quotient of Khovanov and Lauda's diagrammatic 2-category U(sln). We also show that our 2-category contains Soergel's monoidal category of bimodules of type A, which categorifies the Hecke algebra H(d), as a full sub-2-category if d does not exceed n. For the latter result we use Elias and Khovanov's diagrammatic presentation of Soergel's monoidal category of type A.Comment: 60 pages, lots of figures. v3: Substantial changes. To appear in Quantum Topolog
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