Modified Ringel-Hall Algebras, Green's formula and Derived Hall Algebras

TL;DR

This paper introduces a modified Ringel-Hall algebra for hereditary categories, providing new proofs of Green's formula and embedding derived Hall algebras, showing invariance under derived equivalences.

Contribution

It defines the modified Ringel-Hall algebra from bounded complexes, offers a new proof of Green's formula, and demonstrates an embedding of derived Hall algebras in certain cases.

Findings

New proof of Green's formula using the algebra's associative multiplication

Embedding of derived Hall algebra into the modified Ringel-Hall algebra in certain cases

Modified Ringel-Hall algebra is isomorphic to tensor algebra of derived Hall algebra and torus, invariant under derived equivalences

Abstract

In this paper we define the modified Ringel-Hall algebra $\cm\ch(\ca)$ of a hereditary abelian category $\ca$ from the category $C^b(\mathcal{A})$ of bounded $\mathbb{Z}$-graded complexes. Two main results have been obtained. One is to give a new proof of Green's formula on Ringel-Hall numbers by using the associative multiplication of the modified Ringel-Hall algebra. The other is to show that in certain twisted cases the derived Hall algebra can be embedded in the modified Ringel-Hall algebra. As a consequence of the second result, we get that in certain twisted cases the modified Ringel-Hall algebra is isomorphic to the tensor algebra of the derived Hall algebra and the torus of acyclic complexes and so the modified Ringel-Hall algebra is invariant under derived equivalences.