The Bridgeland's Ringel-Hall algebra associated to an algebra with global dimension at most two

TL;DR

This paper establishes an embedding from the twisted Ringel-Hall algebra to Bridgeland's Ringel-Hall algebra for algebras with global dimension at most two, including tilted and canonical algebras.

Contribution

It introduces a new embedding result connecting twisted and Bridgeland's Ringel-Hall algebras for specific classes of algebras.

Findings

Embedding exists for algebras with global dimension ≤ 2

Applicable to tilted and canonical algebras

Provides a structural link between two types of Ringel-Hall algebras

Abstract

For any finitely dimensional associative algebra with global dimension $\leq 2$, we show that there is an embedding from the twisted Ringel-Hall algebra to the Brigeland's Ringel-Hall algebra. In particular, this result is true for tilted algebras and canonical algebras.