Marco Mackaay

Marco Mackaay
Universidade do Algarve | UALG · Departamento de Matemática

18.02
 · 
PhD

About

41
Publications
1,414
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770
Citations
Research Experience
January 2000 - present
Universidade do Algarve
Position
  • Assistant Professor

Publications

Publications (41)
Preprint
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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 reducti...
Preprint
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In this paper we study the graded 2-representation theory of Soergel bimodules for a finite Coxeter group. We establish a precise connection between the graded 2-representation theory of this non-semisimple 2-category and the 2-representation theory of the associated semisimple asymptotic bicategory. This allows us to formulate a conjectural classi...
Article
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The quantum Satake correspondence relates dihedral Soergel bimodules to the semisimple quotient of the quantum sl2 representation category. It also establishes a precise relation between the simple transitive 2-representations of both monoidal categories, which are indexed by bicolored ADE Dynkin diagrams. Using the quantum Satake correspondence...
Article
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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 $\cC$ and the set of equivalence classes of faithful simple transitive $2$-representations of the fiat $2$-subquotient of $\cC$ associated with a diagonal $\math...
Article
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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 $\mathfra...
Article
We categorify the extended affine Hecke algebra and the affine quantum Schur algebra Ŝ(n, r) for 3 ≤ r < n, using results on diagrammatic categorification in affine type A by Elias-Williamson, that extend the work of Elias-Khovanov for finite type A, and Khovanov-Lauda respectively. We also define 2-representations of these categorifications on an...
Article
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For any fiat 2-category C, we show how its simple transitive 2-representations can be constructed using co-algebra 1-morphisms in the injective abelianization of C. Dually, we show that these can also be constructed using algebra 1-morphisms in the projective abelianization of C. We also extend Morita-Takeuchi theory to our setup and work out sever...
Article
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In this paper we complete the ADE-like classification of simple transitive 2-representations of Soergel bimodules in finite dihedral type, under the assumption of gradeability. In particular, we use bipartite graphs and zigzag algebras of ADE type to give an explicit construction of a graded (non-strict) version of all these 2-representations. More...
Article
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 t...
Article
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, ap...
Article
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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 tenso...
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This is a follow-up to the paper in which we categorified the affine quantum Schur algebra S(n,r) for 2 < r < n, using a quotient of Khovanov and Lauda's categorification of the affine quantum sl_n. In this paper we categorify S(n,n) for n > 2, using an extension of the aforementioned quotient.
Article
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In this paper, which is a follow-up to my paper with Yonezawa "The sl(N)-web categories and categorified skew Howe duality", I define and study sl(N)-web algebras for any N greater than one. For N=2 these algebras are isomorphic to Khovanov's arc algebras and for N=3 they are Morita equivalent to the sl(3)-web algebras which I defined and studied w...
Article
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In this paper we show how the colored Khovanov–Rozansky slN-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 slN-web categories and 2-representations of Khovanov and Lauda's categorical quantum slm on the...
Article
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We categorify the extended affine Hecke algebra and the affine quantum Schur algebra S(n,r) for 2 < r < n, using Elias-Khovanov and Khovanov-Lauda type diagrams. We also define the affine analogue of the Elias-Khovanov and the Khovanov-Lauda 2-representations of these categorifications into an extension of the 2-category of affine (singular) Soerge...
Article
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In this paper we use Kuperberg’s $\mathfrak {sl}_3$ -webs and Khovanov’s $\mathfrak {sl}_3$ -foams to define a new algebra $K^S$ , which we call the $\mathfrak {sl}_3$ -web algebra. It is the $\mathfrak {sl}_3$ analogue of Khovanov’s arc algebra. We prove that $K^S$ is a graded symmetric Frobenius algebra. Furthermore, we categori...
Article
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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...
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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 form...
Article
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For each N≥4, we define a monoidal functor from Elias and Khovanov's diagrammatic version of Soergel's category of bimodules to the category of sl(N) foams defined by Mackaay, Stošić, and Vaz. We show that through these functors Soergel's category can be obtained from the sl(N) foams.
Article
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For each N > 3, we define a monoidal functor from Elias and Khovanov's diagrammatic version of Soergel's category of bimodules to the category of sl(N) foams defined by Mackaay, Stosic and Vaz. We show that through these functors Soergel's category can be obtained from the sl(N) foams. Comment: v2, minor corrections, 17 pages, lots of figures
Article
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In this paper I define certain interesting 2-functors from the Khovanov-Lauda 2-category which categorifies quantum sl(k), for any k>1, to a 2-category of universal sl(3) foams with corners. For want of a better name I use the term "foamation" to indicate those 2-functors. I conjecture the existence of similar 2-functors to the 2-category of sl(n)...
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We use foams to give a topological construction of a rational link homology categorifying the sl(N) link invariant, for N ≥ 4. To evaluate closed foams we use the Kapustin-Li formula adapted to foams by Khovanov and Rozansky [7]. We show that for any link our homology is isomorphic to the Khovanov-Rozansky [6] homology. 1.
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In this paper we compute the reduced HOMFLY-PT homologies of the Conway and the Kinoshita-Terasaka knots and show that they are isomorphic.
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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.
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We prove that the universal rational sl3 link homologies which were constructed by Khovanov in [sl(3) link homology, Algebr Geom Topol 4 (2004) 1045-1081] and by the authors in [The universal sl3 link homology, Algebr Geom Topol 7 (2007) 1135 -1169], using foams, and by Khovanov and Rozansky in [Virtual crossings, convolutions and a categorificatio...
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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 stu...
Article
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.
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Using Bar-Natan's Khovanov homology we define a homology theory for coloured, oriented, framed links. We then compute this explicitly.
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The representation theory for categorical groups is constructed. Each categorical group determines a monoidal bicategory of representations. Typically, these categories contain representations which are indecomposable but not irreducible. A simple example is computed in explicit detail.
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In this paper, we establish a one-to-one correspondence between U(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 holon...
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Recently twisted K-theory has received much attention due to its applications in string theory and the announced result by Freed, Hopkins and Telemann relating the twisted equivariant K-theory of a compact Lie group to its Verlinde algebra. Rather than considering gerbes as separate objects, in twisted K-theory one considers a gerbe as being part o...
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In this paper we give a short introduction to our results on the holonomy of gerbe-connections and explain our motivation coming from state-sum models.
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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...
Article
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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 mo...
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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...
Article
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 mo...
Article
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In order to construct a representation of the tangle category one needs an enhanced R-matrix. In this paper we define a sufficient and necessary condition for enhancement that can be checked easily for any R-matrix. If the R-matrix can be enhanced, we also show how to construct the additional data that define the enhancement. As a direct consequenc...

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