Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: A Categorification of Hall Algebras   In recent years, there has been great interest in the study of
categorification, specifically as it applies to the theory of quantum groups.
In this thesis, we would like to provide a new approach to this problem by
looking at Hall algebras. It is know, due to Ringel, that a Hall algebra is
isomorphic to a certain quantum group. It is our goal to describe a
categorification of Hall algebras as a way of doing so for their related
quantum groups. To do this, we will take the following steps. First, we
describe a new perspective on the structure theory of Hall algebras. This view
solves, in a unique way, the classic problem of the multiplication and
comultiplication not being compatible. Our solution is to switch to a different
underlying category $\Vect^K$ of vector spaces graded by a group $K$ called the
Grothendieck group. We equip this category with a nontrivial braiding which
depends on the $K$-grading. With this braiding and a given antipode, we find
that the Hall algebra does become a Hopf algebra object in $\Vect^K$. Second,
we will describe a categorification process, call `groupoidification', which
replaces vector spaces with groupoids and linear operators with `spans' of
groupoids. We will use this process to construct a braided monoidal bicategory
which categorifies $\Vect^K$ via the groupoidification program. Specifically,
graded vector spaces will be replaced with groupoids `over' a fixed groupoid
related to the Grothendieck group $K$. The braiding structure will come from an
interesting groupoid $\EXT(M,N)$ which will behave like the Euler
characteristic for the Grothendieck group $K$. We will finish with a
description of our plan to, in future work, apply the same concept to the
structure maps of the Hall algebra, which will eventually give us a Hopf
2-algebra object in our braided monoidal bicategory.

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The categories of this paper are: math-ph
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The categories of this paper are: q-fin
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The categories of this paper are: q-alg
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The categories of this paper are: gr-qc
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The categories of this paper are: chao-dyn
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The categories of this paper are: plasm-ph
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The categories of this paper are: eess
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The categories of this paper are: patt-sol
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The categories of this paper are: solv-int
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The categories of this paper are: acc-phys
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The categories of this paper are: cs
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The categories of this paper are: funct-an
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The categories of this paper are: mtrl-th
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The categories of this paper are: nlin
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The categories of this paper are: hep-ex
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The categories of this paper are: adap-org