A Categorification of Hall Algebras (1304.0219v2)

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 withspans' 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.