Hall Algebras as Hopf Objects

TL;DR

This paper demonstrates that by changing the underlying category to a K-graded vector space with a specific braiding, Hall algebras can be realized as Hopf algebra objects, resolving previous compatibility issues.

Contribution

The authors introduce a new categorical framework with a K-grading and braiding, enabling Hall algebras to satisfy bialgebra and antipode conditions exactly.

Findings

Hall algebras satisfy bialgebra conditions in Vect^K

Existence of an antipode makes it a Hopf algebra object

Resolution of compatibility issues without using twisted structures

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.