Hall Algebras as Hopf Objects

Abstract

One problematic feature of Hall algebras is the fact that the standard multiplication and comultiplication maps do not satisfy the bialgebra compatibility condition in the underlying symmetric monoidal category Vect. In the past this problem has been resolved by working with a weaker structure called a `twisted' bialgebra. In this paper we solve the problem differently by first switching 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, we find that the Hall algebra does satisfy the bialgebra condition exactly for the standard multiplication and comultiplication, and can also be equipped with an antipode, making it a Hopf algebra object in Vect^K.