Semi-derived Ringel-Hall algebras and Hall algebras of odd-periodic relative derived categories

TL;DR

This paper introduces the semi-derived Ringel-Hall algebra for hereditary abelian categories, describes its structure, and establishes an embedding of the derived Hall algebra for odd-periodic categories into it.

Contribution

It defines the semi-derived Ringel-Hall algebra from graded complexes, provides a basis and presentation, and links it to the derived Hall algebra for odd-periodic categories.

Findings

Established a natural basis for the semi-derived Ringel-Hall algebra.

Presented the algebra via generators and relations.

Proved an embedding of the derived Hall algebra into the semi-derived algebra for odd t.

Abstract

Let $t$ be a positive integer and $\mathcal{A}$ a hereditary abelian category satisfying some finiteness conditions. We define the semi-derived Ringel-Hall algebra of $\mathcal{A}$ from the category $\mathcal{C}_{\mathbb{Z}/t}(\mathcal{A})$ of $\mathbb{Z}/t$-graded complexes and obtain a natural basis of the semi-derived Ringel-Hall algebra. Moreover, we describe the semi-derived Ringel-Hall algebra by the generators and defining relations. In particular, if $t$ is an odd integer, we show that there is an embedding of derived Hall algebra of the odd-periodic relative derived category in the extended semi-derived Ringel-Hall algebra.