Periodic derived Hall algebras of hereditary abelian categories

Let m be a positive integer and Dm(A) be the m-periodic derived category of a finitary hereditary abelian category A. Applying the derived Hall numbers of the bounded derived category Db(A), we define an m-periodic extended derived Hall algebra for Dm(A), and use it to give a global, unified and explicit characterization for the algebra structure of Bridgeland's Hall algebra of periodic complexes. Moreover, we also provide an explicit characterization for the odd periodic derived Hall algebra of A defined by Xu-Chen [24].