Hall algebras associated to root categories

TL;DR

This paper constructs Hall algebras for root categories of finitary hereditary abelian categories using derived Hall numbers, establishing their isomorphism to Drinfeld double Hall algebras and introducing a 1-periodic variant.

Contribution

It introduces a new Hall algebra for root categories and proves its isomorphism to the Drinfeld double Hall algebra, also defining a 1-periodic derived Hall algebra.

Findings

Hall algebra for root category is isomorphic to Drinfeld double Hall algebra

Defined 1-periodic derived Hall algebra using derived Hall numbers

Established foundational structures linking derived categories and Hall algebras

Abstract

Let $\mathcal {A}$ be a finitary hereditary abelian category. We define a Hall algebra for the root category of $\mathcal {A}$ by applying the derived Hall numbers of the bounded derived category $D^b(\mathcal {A})$, which is proved to be isomorphic to the Drinfeld double Hall algebra of $\mathcal {A}$. In the appendix, we also define the 1-periodic derived Hall algebra via the derived Hall numbers of $D^b(\mathcal {A})$.