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In this paper we show how the colored Khovanov–Rozansky sl N -matrix factorizations, due to Wu [45] and Y.Y. [46] , [47] , can be used to categorify the type A quantum skew Howe duality defined by Cautis, Kamnitzer and Morrison in [14] . In particular, we define sl N -web categories and 2-representations of Khovanov and Lauda's categorical quantum sl m on them. We also show that this implies that each such web category is equivalent to the category of finite-dimensional graded projective modules over a certain type A cyclotomic KLR-algebra.

In this paper, we discuss the generalization of finitary $2$-representation theory of finitary $2$-categories to finitary birepresentation theory of finitary bicategories. In previous papers on the subject, the classification of simple transitive $2$-representations of a given $2$-category was reduced to that for certain subquotients. These reduction results were all formulated as bijections between equivalence classes of $2$-representations. In this paper, we generalize them to biequivalences between certain $2$-categories of birepresentations. Furthermore, we prove an analog of the double centralizer theorem in finitary birepresentation theory., Significant revision of the original version

We classify simple transitive $2$-representations of certain $2$-sub\-ca\-te\-go\-ri\-es of the $2$-category of Soergel bimodules over the coinvariant algebra in Coxeter types $B_2$ and $I_2(5)$. In the $I_2(5)$ case it turns out that simple transitive $2$-representations are exhausted by cell $2$-representations. In the $B_2$ case we show that, apart from cell $2$-representations, there is a unique, up to equivalence, additional simple transitive $2$-representation and we give an explicit construction of this $2$-representation., Comment: revised version, 24 p., to appear in JPAA

We categorify the extended affine Hecke algebra and the affine quantum Schur algebra S(n, r) for 3

The main result of this paper establishes a bijection between the set of equivalence classes of simple transitive 2-representations with a fixed apex ${\mathcal{J}}$ of a fiat 2-category $\mathscr{C}$ and the set of equivalence classes of faithful simple transitive 2-representations of the fiat 2-subquotient of $\mathscr{C}$ associated with a diagonal ${\mathcal{H}}$-cell in ${\mathcal{J}}$. As an application, we classify simple transitive 2-representations of various categories of Soergel bimodules, in particular, completing the classification in types $B_{3}$ and $B_{4}$.

This paper is a follow-up to (MT13). In that paper we categorified the affine q-schur algebra b(n,r) for 2 < r < n, using a quotient of Khovanov and Lauda's categorification o f Uq(bln) (KL09, KL11, KL10). In this paper we categorify b(n,n) for n � 3, using an extension of the aforementioned quotient.

In this paper, we show an isomorphism of homological knot invariants categorifying the Reshetikhin-Turaev invariants for $\mathfrak{sl}_n$. Over the past decade, such invariants have been constructed in a variety of different ways, using matrix factorizations, category $\mathcal{O}$, 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 $\mathfrak{gl}_\ell$ 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 $\mathfrak{gl}_\ell\times \mathfrak{gl}_n$ on $\bigwedge\nolimits^{\!p}(\mathbb{C}^\ell\otimes \mathbb{C}^n)$ using diagrammatic bimodules. In this action, the functors corresponding to $\mathfrak{gl}_\ell$ and $\mathfrak{gl}_n$ are quite different in nature, but they will switch roles under Koszul duality., 62 pages. preliminary version, comments welcome

The quantum Satake correspondence relates dihedral Soergel bimodules to the semisimple quotient of the quantum $\mathfrak{sl}_2$ representation category. It also establishes a precise relation between the simple transitive $2$-representations of both monoidal categories, which are indexed by bicolored $\mathsf{ADE}$ Dynkin diagrams. Using the quantum Satake correspondence between affine $\mathsf{A}_{2}$ Soergel bimodules and the semisimple quotient of the quantum $\mathfrak{sl}_3$ representation category, we introduce trihedral Hecke algebras and Soergel bimodules, generalizing dihedral Hecke algebras and Soergel bimodules. These have their own Kazhdan-Lusztig combinatorics, simple transitive $2$-representations corresponding to tricolored generalized $\mathsf{ADE}$ Dynkin diagrams., Comment: 61 pages, many colored figures, revised version, comments welcome, to appear in Fund. Math

In this paper, which is a follow-up to [38] , I define and study sl N -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 sl N -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 sl N -webs and the split Grothendieck group of the corresponding sl N -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).

We associate a monoidal category $\mathcal{H}^\lambda$ to each dominant integral weight $\lambda$ of $\widehat{\mathfrak{sl}}_p$ or $\mathfrak{sl}_\infty$. These categories, defined in terms of planar diagrams, act naturally on categories of modules for the degenerate cyclotomic Hecke algebras associated to $\lambda$. We show that, in the $\mathfrak{sl}_\infty$ case, the level $d$ Heisenberg algebra embeds into the Grothendieck ring of $\mathcal{H}^\lambda$, where $d$ is the level of $\lambda$. The categories $\mathcal{H}^\lambda$ can be viewed as a graphical calculus describing induction and restriction functors between categories of modules for degenerate cyclotomic Hecke algebras, together with their natural transformations. As an application of this tool, we prove a new result concerning centralizers for degenerate cyclotomic Hecke algebras., Comment: 35 pages; v2: published version

In all finite Coxeter types but $I_2(12)$, $I_2(18)$ and $I_2(30)$, we classify simple transitive $2$-rep\-re\-sen\-ta\-ti\-ons for the quotient of the $2$-category of Soergel bimodules over the coinvariant algebra which is associated to the two-sided cell that is the closest one to the two-sided cell containing the identity element. It turns out that, in most of the cases, simple transitive $2$-representations are exhausted by cell $2$-representations. However, in Coxeter types $I_2(2k)$, where $k\geq 3$, there exist simple transitive $2$-representations which are not equivalent to cell $2$-representations., Revised version to appear in Trans. AMS

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 pages

We show that Rasmussen's invariant of knots, which is derived from Lee's variant of Khovanov homology, is equal to an analogous invariant derived from certain other filtered link homologies., Comment: There are two errors in the proof of Proposition 3.2. in this paper. These are indicated in an erratum added at the beginning. As a consequence the proof of Proposition 3.2 no longer holds and the proof of Theorem 4.2, which relies on it, is no longer valid

Using Bar-Natan's Khovanov homology we define a homology theory for coloured, oriented, framed links. We then compute this explicitly., 15 pages, 4 figures; minor revision

In this paper we define a class of state-sum invariants of compact closed oriented piece-wise linear 4-manifolds using finite groups. The definition of these state-sums follows from the general abstract construction of 4-manifold invariants using spherical 2-categories, as we defined in a previous paper, although it requires a slight generalization of that construction. We show that the state-sum invariants of Birmingham and Rakowski, who studied Dijkgraaf-Witten type invariants in dimension 4, are special examples of the general construction that we present in this paper. They showed that their invariants are non-trivial by some explicit computations, so our construction includes interesting examples already. Finally, we indicate how our construction is related to homotopy 3-types. This connection suggests that there are many more interesting examples of our construction to be found in the work on homotopy 3-types, such as by Brown, for example., Comment: LaTeX 44 pages, 10 eps.files, accepted for publication in Adv Math. Much better introduction explaining the fundamental ideas, equivalence relation on weak mon. 2-structures defined and theorem added

A categorification of the Beilinson-Lusztig-MacPherson form of the quantum sl(2) was constructed in the paper arXiv:0803.3652 by the second author. Here we enhance the graphical calculus introduced and developed in that paper to include two-morphisms between divided powers one-morphisms and their compositions. We obtain explicit diagrammatical formulas for the decomposition of products of divided powers one-morphisms as direct sums of indecomposable one-morphisms; the latter are in a bijection with the Lusztig canonical basis elements. These formulas have integral coefficients and imply that one of the main results of Lauda's paper---identification of the Grothendieck ring of his 2-category with the idempotented quantum sl(2)---also holds when the 2-category is defined over the ring of integers rather than over a field., 72 pages, LaTeX2e with xypic and pstricks macros

In this paper we define the 1,2-coloured HOMFLY-PT link homology and prove that it is a link invariant. We conjecture that this homology categorifies the coloured HOMFLY-PT polynomial for links whose components are labelled 1 or 2., 31 pages

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., 60 pages, lots of figures. v3: Substantial changes. To appear in Quantum Topology

In [14], Murakami, Ohtsuki and Yamada (MOY) developed a graphical calculus for the sl.N / link polynomial. In [7], Khovanov categorified the sl.3/ polynomial using singular cobordisms between webs called foams. Mackaay and Vaz [12] generalized Khovanov’s results to obtain the universal sl.3/ integral link homology, following an approach similar to the one adopted by Bar-Natan [1] for the original sl.2/ integral Khovanov homology. In [11] Khovanov and Rozansky defined a rational link homology which categorifies the sl.N / link polynomial using the theory of matrix factorizations.

We define the universal sl3-link homology, which depends on 3 parameters, following Khovanov's approach with foams. We show that this 3-parameter link homology, when taken with complex coefficients, can be divided into 3 isomorphism classes. The first class is the one to which Khovanov's original sl3-link homology belongs, the second is the one studied by Gornik in the context of matrix factorizations and the last one is new. Following an approach similar to Gornik's we show that this new link homology can be described in terms of Khovanov's original sl2-link homology., v2, 29pp, different proof of Lemma 3.10, extra comments below Def. 3.16 and minor corrections and typos

In this paper we establish a one-to-one correspondence between $S^1$-gerbes with connections, on the one hand, and their holonomies, for simply connected manifolds, or their parallel transports, in the general case, on the other hand. This result is a higher-order analogue of the familiar equivalence between bundles with connections and their holonomies for connected manifolds. The holonomy of a gerbe with group $S^1$ on a simply connected manifold $M$ is a group morphism from the thin second homotopy group to $S^1$, satisfying a smoothness condition, where a homotopy between maps from $[0,1]^2$ to $M$ is thin when its derivative is of rank $\leq 2$. For the non-simply connected case, holonomy is replaced by a parallel transport functor between two monoidal Lie groupoids. The reconstruction of the gerbe and connection from its holonomy is carried out in detail for the simply connected case. Our approach to abelian gerbes with connections holds out prospects for generalizing to the non-abelian case via the theory of double Lie groupoids., Final version. Improved readibility (hopefully). LaTeX, 60 pages, 14 figures. To appear in Advances in Mathematics

In this paper I define the notion of a non-degenerate finitely semi-simple semi-strict spherical 2-category of non-zero dimension. Given such a 2-category I define a state-sum for any triangulated compact closed oriented 4-manifold and show that this state-sum does not depend on the chosen triangulation by proving invariance under the 4D Pachner moves. This invariant of piece-wise linear closed compact oriented 4-manifolds generalizes the Crane-Yetter invariant and probably it is also a generalization of the Crane-Frenkel invariant. As an example we show how to obtain a 2-category of the right kind from a finite group and a 4-cocycle on this group. The invariant we obtain from this example looks like a four dimensional version of the Dijkgraaf-Witten invariant., 54 pages, Plain Tex, 14 eps.files