Knot invariants and higher representation theory

Abstract

We construct knot invariants categorifying the quantum knot variants for all representations of quantum groups. We show that these invariants coincide with previous invariants defined by Khovanov for sl_2 and sl_3 and by Mazorchuk-Stroppel and Sussan for sl_n. Our technique is to study 2-representations of 2-quantum groups (in the sense of Rouquier and Khovanov-Lauda) categorifying tensor products of irreducible representations. These are the representation categories of certain finite dimensional algebras with an explicit diagrammatic presentation, generalizing the cyclotomic quotient of the KLR algebra. When the Lie algebra under consideration is ${\mathfrak{sl}}_n$, we show that these categories agree with certain subcategories of parabolic category O for gl_k. We also investigate the finer structure of these 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 test objects for functors, as Vermas do in more classical representation theory. 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. In work of Reshetikhin and Turaev, the braiding and (co)evaluation maps between representations of quantum groups are used to define polynomial knot invariants. We show that the categorifications of tensor products are related by functors categorifying these maps, which allow the construction of bigraded knot homologies whose graded Euler characteristics are the original polynomial knot invariants.

99 pages. This is a significantly rewritten version of arXiv:1001.2020 and arXiv:1005.4559; both the exposition and proofs have been significantly improved. These earlier papers have been left up mainly in the interest of preserving references. v3: final version, to appear in Memoirs of the AMS. Proof of nondegeneracy moved to separate erratum

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• Representation theory of the cyclotomic Cherednik algebra via the Dunkl-Opdam subalgebra
• An Exceptional Collection For Khovanov Homology
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• Quantum gl(1|1) and tangle Floer homology
• Invariants of 4-manifolds from Khovanov-Rozansky link homology
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• The differential graded odd nilHecke algebra
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• Categorification of tensor powers of the vector representation of $U_q(\mathfrak{gl}(1|1))$
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• DAHA-Jones polynomials of torus knots
• Centers of KLR algebras and cohomology rings of quiver varieties
• Homological Knot Invariants from Mirror Symmetry
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• Cellularity and subdivision of KLR and weighted KLRW algebras
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• Knot Categorification from Mirror Symmetry, Part I: Coherent Sheaves
• Dualities and derived equivalences for category O
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• On some $p$-differential graded link homologies
• A diamond lemma for Hecke-type algebras
• Categorification via blocks of modular representations for sl(n)
• Categorical lifting of the Jones polynomial: a survey
• A hitchhiker's guide to Khovanov homology
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• Braid group actions from categorical symmetric Howe duality on deformed Webster algebras
• $\hat{Z}$- invariant for $SO(3)$ and $OSp(1|2)$ Groups
• Content systems and deformations of cyclotomic KLR algebras of type $A$ and $C$
• On category $\mathcal{O}$ for affine Grassmannian slices and categorified tensor products
• p-DG cyclotomic nilHecke algebras II
• Knot Categorification from Mirror Symmetry, Part II: Lagrangians
• Higher Representation Theory and Quantum Affine Schur-Weyl Duality
• Operator lift of Reshetikhin-Turaev formalism to Khovanov-Rozansky TQFTs
• A cobordism category attached to Khovanov-Rozansky link homologies based on operads
• DG-Enhanced Hecke and KLR Algebras
• Not even Khovanov homology
• String Theory and Math: Why This Marriage May Last
• Enhanced nilHecke algebras and baby Verma modules
• A topological theory of unoriented SL(4) foams
• A categorification of the skew Howe action on a representation category of $U_q(\mathfrak{gl}(m|n))$
• Trace and categorical sl(n) representations
• A remark on convolution products for quiver Hecke algebras
• Cellularity of KLR and weighted KLRW algebras via crystals
• Quantum supergroups IV. The modified form
• Empty runner removal theorem for Ariki-Koike algebras
• Asymptotic Grothendieck groups and c.b.l.f. positive dg-algebras