Periodic derived Hall algebras of hereditary abelian categories

TL;DR

This paper introduces an $m$-periodic extended derived Hall algebra for hereditary abelian categories, providing a unified algebraic framework for periodic complexes and connecting to Bridgeland's Hall algebra.

Contribution

It defines a new $m$-periodic derived Hall algebra and offers explicit characterizations for it and for Xu-Chen's odd periodic derived Hall algebra, unifying various algebraic structures.

Findings

Explicit algebraic structures for $m$-periodic derived Hall algebras.

Unified characterization of Bridgeland's Hall algebra of periodic complexes.

Connection to Xu-Chen's odd periodic derived Hall algebra.

Abstract

Let $m$ be a positive integer and $D_m(\mathcal {A})$ be the $m$-periodic derived 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 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 $\mathcal {A}$ defined by Xu-Chen [22].