Data

• Title: Categorification and applications in topology and representation theory
• Author: Daniel Tubbenhauer
• Status: published online by Universität Göttingen, Univ., Diss., 2013, DISS 2013 B 9481. Last update: Wed, 3 Jul 2013 13:43:42 GMT
• ArXiv link: http://arxiv.org/abs/1307.1011
• ArXiv version = 0.99 published version
• LaTex Beamer presentation: Slides1

Abstract

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 $\mathbb{Z}/2$-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.
In the second part of this thesis (which is based on joint work with Mackaay and Pan) 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 $H_n$.
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 $K^{\oplus}_0(K_S)$, 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.

A few extra words

Forced to reduce the thesis to one sentence, the author would choose:

The basic idea can be seen as follows. Take a “set-based” structure $S$ and try to find a “category-based” structure $\mathcal C$ such that $S$ is just a shadow of the category $\mathcal C$. If the category $\mathcal C$ is chosen in a “good” way, then one has an explanation of facts about the structure $S$ in a categorical language, that is certain facts in $S$ can be explained as special instances of natural constructions.
As an example, consider the following categorification of the integers $S=\mathbb{Z}$. We take $\mathcal C=$FinVec${}_K$ for a fixed field $K$. We observe that $K_0(\mathcal C)=\mathbb{Z}$. But a lot of information is lost from $\mathcal C$ to $K_0(\mathcal C)=\mathbb{Z}$, e.g. the first allows us to say how two vector spaces are isomorphic (think about non-trivial isomorphisms).
In our context this “richer structure” can e.g. be occur in the following example. A web, e.g. the bottom and top boundary of the figure below, can be seen as an intertwiner between tensors of certain representation of $U_q(\mathfrak{sl}_3)$, denote by $V_+$ and $V_-$. In the figure below we show one web between $V_+$ and $V_-$ (both as short hand notation just denoted $+$ and $-$). In the picture there is only one web, but a foam (a type of singular cobordism) can tell “how” they are equal.