Categorification and applications in topology and representation theory
- 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
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 (), the variant of Lee () and other classical
link homologies. We show that our construction allows, over rings of characteristic , extensions
with no classical analogon, e.g. Bar-Natan's -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 webs and Khovanov's
foams to define a new algebra , which we call the
web algebra. It is the analogue of
Khovanov's arc algebra .
We prove that is a graded symmetric Frobenius algebra.
Furthermore, we categorify an instance of $q$-skew Howe duality,
which allows us to prove that is Morita equivalent to
a certain cyclotomic KLR-algebra. This allows us to determine
the split Grothendieck group , 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:
Interesting integers are shadows of richer structures in categories.
The basic idea can be seen as follows. Take a “set-based” structure and try to find a
“category-based” structure such that is just a shadow of the category
. If the category is chosen in a “good” way, then one has an explanation of
facts about the structure in a categorical language, that is certain facts in can be explained
as special instances of natural constructions.
As an example, consider the following categorification of the integers . We take FinVec
for a fixed field . We observe that . But a lot of information
is lost from to , 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 , denote by and .
In the figure below we show one web between and (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.
I am still a fool.
The arXiv version of this paper
The arXiv version of this paper
"There are two ways to do mathematics.
The first is to be smarter than everybody else.
The second way is to be stupider than everybody else - but persistent." -
based on a quotation from Raoul Bott.
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