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Let $m$ be a positive integer and $D_m(\mathcal {A})$ be the $m$-periodicderived category of a finitary hereditary abelian category $\mathcal {A}$.Applying the derived Hall numbers of the bounded derived category $D^b(\mathcal{A})$, we define an $m$-periodic extended derived Hall algebra for$D_m(\mathcal {A})$, and use it to give a global, unified and explicitcharacterization for the algebra structure of Bridgeland's Hall algebra ofperiodic complexes. Moreover, we also provide an explicit characterization forthe odd periodic derived Hall algebra of $\mathcal {A}$ defined by Xu-Chen[22].

Periodic derived Hall algebras of hereditary abelian categories