A note on Bridgeland's Hall algebras

TL;DR

This paper provides a simplified proof that Bridgeland's Hall algebra of 2-cyclic complexes in a hereditary category is isomorphic to the Drinfeld double Hall algebra, clarifying key steps in existing proofs.

Contribution

It offers a new, straightforward proof of an isomorphism between Bridgeland's Hall algebra and the Drinfeld double Hall algebra, simplifying previous complex arguments.

Findings

Simplified proof of the isomorphism between Bridgeland's Hall algebra and the Drinfeld double Hall algebra.

Streamlined the proof of a key theorem in the theory of Hall algebras.

Clarified the structure of Hall algebras in hereditary categories.

Abstract

In this note, let $\A$ be a finitary hereditary abelian category with enough projectives. By using the associativity formula of Hall algebras, we give a new and simple proof of the main theorem in \cite{Yan}, which states that the Bridgeland's Hall algebra of 2-cyclic complexes of projective objects in $\A$ is isomorphic to the Drinfeld double Hall algebra of $\A$. In a similar way, we give a simplification of the key step in the proof of Theorem 4.11 in \cite{LP}.