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Daniel Tubbenhauer

Publications and source records attributed to Daniel Tubbenhauer.

66 records · Page 4Linked to original sources
arXiv2018

Symmetric webs, Jones-Wenzl recursions and $q$-Howe duality

We define and study the category of symmetric $\mathfrak{sl}_2$-webs. This category is a combinatorial description of the category of all finite dimensional quantum $\mathfrak{sl}_2$-modules. Explicitly, we show that (the additive closure of) the symmetric $\mathfrak{sl}_2$-spider is (braided monoidally) equivalent to the latter. Our main tool is a quantum version of symmetric Howe duality. As a corollary of our construction, we provide new insight into Jones-Wenzl projectors and the colored Jones polynomials.

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arXiv2018

Webs and $q$-Howe dualities in types $\mathbf{B}\mathbf{C}\mathbf{D}$

We define web categories describing intertwiners for the orthogonal and symplectic Lie algebras, and, in the quantized setup, for certain orthogonal and symplectic coideal subalgebras. They generalize the Brauer category, and allow us to prove quantum versions of some classical type $\mathbf{B}\mathbf{C}\mathbf{D}$ Howe dualities.

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arXiv2018

The sl_3 web algebra

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 categorify an instance of $q$-skew Howe duality, which allows us to prove that $K^S$ is Morita equivalent to a certain cyclotomic KLR-algebra of level 3. This allows us to determine the split Grothendieck group $K^{\oplus}_0(\mathcal{W}^S)_{\mathbb{Q}(q)}$, to show that its center is isomorphic to the cohomology ring of a certain Spaltenstein variety, and to prove that $K^S$ is a graded cellular algebra.

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arXiv2017

Super $q$-Howe duality and web categories

We use super $q$-Howe duality to provide diagrammatic presentations of an idempotented form of the Hecke algebra and of categories of $\mathfrak{gl}_N$-modules (and, more generally, $\mathfrak{gl}_{N|M}$-modules) whose objects are tensor generated by exterior and symmetric powers of the vector representations. As an application, we give a representation theoretic explanation and a diagrammatic version of a known symmetry of colored HOMFLY--PT polynomials.

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arXiv2017

Cellular structures using $\textbf{U}_q$-tilting modules

We use the theory of $\textbf{U}_q$-tilting modules to construct cellular bases for centralizer algebras. Our methods are quite general and work for any quantum group $\textbf{U}_q$ attached to a Cartan matrix and include the non-semisimple cases for $q$ being a root of unity and ground fields of positive characteristic. Our approach also generalizes to certain categories containing infinite-dimensional modules. As applications, we give a new semisimplicty criterion for centralizer algebras, and recover the cellularity of several known algebras (with partially new cellular bases) which all fit into our general setup.

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arXiv2016

Semisimplicity of Hecke and (walled) Brauer algebras

We show how to use Jantzen's sum formula for Weyl modules to prove semisimplicity criteria for endomorphism algebras of $\textbf{U}_q$-tilting modules (for any field $\mathbb{K}$ and any parameter $q\in\mathbb{K}-\{0,-1\}$). As an application, we recover the semisimplicity criteria for the Hecke algebras of types $\textbf{A}$ and $\textbf{B}$, the walled Brauer algebras and the Brauer algebras from our more general approach.

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arXiv2015

Diagram categories for $\textbf{U}_q$-tilting modules at roots of unity

We give a diagrammatic presentation of the category of $\textbf{U}_q(\mathfrak{sl}_2)$-tilting modules $\mathfrak{T}$ for $q$ being a root of unity and introduce a grading on $\mathfrak{T}$. This grading is a "root of unity phenomenon" and might lead to new insights about link and $3$-manifold invariants deduced from $\mathfrak{T}$. We also give a diagrammatic category for the (graded) projective endofunctors on $\mathfrak{T}$, indicate how our results could generalize and collect some "well-known" facts to give a reasonably self-contained exposition.

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arXiv2014

Virtual Khovanov homology using cobordisms

We extend Bar-Natan's cobordism based categorification of the Jones polynomial to virtual links. Our topological complex allows a direct extension of the classical Khovanov complex ($h=t=0$), the variant of Lee ($h=0,t=1$) and other classical link homologies. We show that our construction allows, over rings of characteristic two, extensions with no classical analogon, e.g. Bar-Natan's $\mathbb{Z}/2$-link homology can be extended in two non-equivalent ways. Our construction is computable in the sense that one can write a computer program to perform calculations, e.g. we have written a Mathematica based program. Moreover, we give a classification of all unoriented TQFTs which can be used to define virtual link homologies from our topological construction. Furthermore, we prove that our extension is combinatorial and has semi-local properties. We use the semi-local properties to prove an application, i.e. we give a discussion of Lee's degeneration of virtual homology.

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arXiv2014

$\mathfrak{sl}_3$-web bases, intermediate crystal bases and categorification

We give an explicit graded cellular basis of the $\mathfrak{sl}_3$-web algebra $K_S$. In order to do this, we identify Kuperberg's basis for the $\mathfrak{sl}_3$-web space $W_S$ with a version of Leclerc-Toffin's intermediate crystal basis and we identify Brundan, Kleshchev and Wang's degree of tableaux with the weight of flows on webs and the $q$-degree of foams. We use these observations to give a "foamy" version of Hu and Mathas graded cellular basis of the cyclotomic Hecke algebra which turns out to be a graded cellular basis of the $\mathfrak{sl}_3$-web algebra. We restrict ourselves to the $\mathfrak{sl}_3$ case over $\mathbb{C}$ here, but our approach should, up to the combinatorics of $\mathfrak{sl}_N$-webs, work for all $N>1$ or over $\mathbb{Z}$.

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arXiv2013

Categorification and applications in topology and representation theory

This thesis splits into two major parts. The connection between the two parts is the notion of "categorification" which we shortly explain/recall in the introduction. In the first part of this thesis we extend Bar-Natan's cobordism based categorification of the Jones polynomial to virtual links. Our topological complex allows a direct extension of the classical Khovanov complex (h=t=0), the variant of Lee (h=0,t=1) and other classical link homologies. We show that our construction allows, over rings of characteristic 2, extensions with no classical analogon, e.g. Bar-Natan's Z/2Z-link homology can be extended in two non-equivalent ways. Our construction is computable in the sense that one can write a computer program to perform calculations, e.g. we have written a MATHEMATICA based program. Moreover, we give a classification of all unoriented TQFTs which can be used to define virtual link homologies from our topological construction. Furthermore, we prove that our extension is combinatorial and has semi-local properties. We use the semi-local properties to prove an application, i.e. we give a discussion of Lee's degeneration of virtual homology. In the second part of this thesis (which is based on joint work with Mackaay and Pan) we use Kuperberg's sl3 webs and Khovanov's sl3 foams to define a new algebra K_S, which we call the sl3 web algebra. It is the sl3 analogue of Khovanov's arc algebra H_n. We prove that K_S is a graded symmetric Frobenius algebra. Furthermore, we categorify an instance of q-skew Howe duality, which allows us to prove that K_S is Morita equivalent to a certain cyclotomic KLR-algebra. This allows us to determine the split Grothendieck group K_0(K_S), to show that its center is isomorphic to the cohomology ring of a certain Spaltenstein variety, and to prove that K_S is a graded cellular algebra.

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