Selfinjective quivers with potential and 2-representation-finite algebras

TL;DR

This paper explores selfinjective quivers with potential, establishing their connection to 2-representation-finite algebras, and investigates their mutation behavior and derived equivalences.

Contribution

It characterizes selfinjective QPs as truncated Jacobian algebras of 2-representation-finite algebras and introduces planar QPs as a new source of selfinjective QPs.

Findings

Selfinjective QPs are truncated Jacobian algebras of 2-representation-finite algebras.

Selfinjectivity of QPs is preserved under mutations related to Nakayama permutation.

Planar QPs provide a rich class of selfinjective QPs.

Abstract

We study quivers with potential (QPs) whose Jacobian algebras are finite dimensional selfinjective. They are an analogue of the `good QPs' studied by Bocklandt whose Jacobian algebras are 3-Calabi-Yau. We show that 2-representation-finite algebras are truncated Jacobian algebras of selfinjective QPs, which are factor algebras of Jacobian algebras by certain sets of arrows called cuts. We show that selfinjectivity of QPs is preserved under successive mutation with respect to orbits of the Nakayama permutation. We give a sufficient condition for all truncated Jacobian algebras of a fixed QP to be derived equivalent. We introduce planar QPs which provide us with a rich source of selfinjective QPs.