A Note on Odd Periodic derived Hall algebras

TL;DR

This paper establishes an algebra embedding from the derived Hall algebra of an odd-periodic derived category to its extended version, providing a basis-level homomorphism for the case of odd positive integers.

Contribution

It introduces a basis-level algebra embedding between derived Hall algebras for odd periodic derived categories, extending prior algebraic structures.

Findings

Proves an algebra embedding for odd m in derived Hall algebras.

Provides explicit basis element mapping for the algebra homomorphism.

Extends the understanding of algebraic structures in periodic derived categories.

Abstract

Let $m$ be an odd positive integer and $D_m(\mathcal {A})$ be the $m$-periodic derived category of a finitary hereditary abelian category $\mathcal {A}$. In this note, we prove that there is an embedding of algebras from the derived Hall algebra of $D_m(\mathcal {A})$ defined by Xu-Chen [13] to the extended derived Hall algebra of $D_m(\mathcal {A})$ defined in [16]. This homomorphism is given on basis elements, rather than just on generating elements.