A Categorification of Hall Algebras

TL;DR

This paper proposes a novel categorification of Hall algebras by using a braided monoidal bicategory and groupoidification, providing a new perspective on their structure and relation to quantum groups.

Contribution

It introduces a new approach to categorify Hall algebras through a braided monoidal bicategory and groupoidification, addressing compatibility issues in their algebraic structure.

Findings

Hall algebra becomes a Hopf algebra in a new categorical setting

Construction of a braided monoidal bicategory via groupoidification

Future plans to realize Hall algebra structure maps as a Hopf 2-algebra

Abstract

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.