APR tilting modules and graded quivers with potential

TL;DR

This paper explores the relationship between APR tilting modules and graded quivers with potential, showing how mutations of QPs induce tilting mutations and generate derived equivalences among algebras.

Contribution

It provides an explicit description of quivers with relations via graded QPs and mutations, linking algebraic cuts to tilting mutations and derived equivalences.

Findings

Mutations of QPs with algebraic cuts induce tilting mutations.

Mutations of QPs generate derived equivalence classes of algebras.

A sufficient condition for positive answers to Derksen-Weyman-Zelevinsky's question.

Abstract

We study the quiver with relations of the endomorphism algebra of an APR tilting module. We give an explicit description of the quiver with relations by graded quivers with potential (QPs) and mutations. The result also implies that mutations of QPs with algebraic cuts induce tilting mutations between the associated truncated Jacobian algebras. Consequently, mutations of QPs provide a rich source of derived equivalence classes of algebras. As an application, we give a sufficient condition of QPs such that Derksen-Weyman-Zelevinsky's question has a positive answer.