A note on Bridgeland Hall algebras

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 proof of the main theorem in 17 Yanagida, S. (2016). A note on Bridgeland’s Hall algebra of two-periodic complexes. Math. Z. 282(3):973–991.Crossref], Web of Science ®] , Google Scholar]], which states that the Bridgeland Hall algebra of 2-cyclic complexes of projective objects in 𝒜 is isomorphic to the Drinfeld double Hall algebra of 𝒜.