A note on Bridgeland's Hall algebra of two-periodic complexes (Q269896)

Let \(\mathcal{A}\) be an abelian category satisfying the conditions in the main theorem in this paper. Denote by \(\mathcal{DH}(\mathcal{A})\) the Bridgeland's Hall algebra associated to \(\mathcal{A}\), which is the localization of the twisted Hall algebra of two-periodic complexes of projective objects in \(\mathcal{A}\). Denote by \(\tilde{\mathcal{H}}(\mathcal{A})\) the twisted extended Ringel-Hall algebra associated to \(\mathcal{A}\), which is a bialgebra with a Hopf pairing. In this paper, the author proves that the Bridgeland's Hall algebra \(\mathcal{DH}(\mathcal{A})\) is isomorphic to the Drinfeld double of \(\tilde{\mathcal{H}}(\mathcal{A})\). At last, the author also discusses about the invariance of Bridgeland's Hall algebras under derived equivalences.