Classification results for $n$-hereditary monomial algebras

TL;DR

This paper classifies $n$-hereditary monomial algebras across different contexts, identifying specific classes such as $n$-representation-finite Nakayama algebras and quadratic monomial algebras, with detailed results for $n=2$ and $n extgreater 2$.

Contribution

It provides a comprehensive classification of $n$-hereditary monomial algebras, including new results for quadratic monomial algebras and the uniqueness of certain $n$-representation-finite algebras.

Findings

Classified $n$-hereditary truncated path algebras as $n$-representation-finite Nakayama algebras.

Identified only two $n$-hereditary quadratic monomial algebras for $n=2$ under specific conditions.

For $n extgreater 2$, the only $n$-representation-finite algebras are Nakayama with quadratic relations.

Abstract

We classify $n$-hereditary monomial algebras in three natural contexts: First, we give a classification of the $n$-hereditary truncated path algebras. We show that they are exactly the $n$-representation-finite Nakayama algebras classified by Vaso. Next, we classify partially the $n$-hereditary quadratic monomial algebras. In the case $n=2$, we prove that there are only two examples, provided that the preprojective algebra is a planar quiver with potential. The first one is a Nakayama algebra and the second one is obtained by mutating $\mathbb A_3\otimes_k \mathbb A_3$, where $\mathbb A_3$ is the Dynkin quiver of type $A$ with bipartite orientation. In the case $n\geq 3$, we show that the only $n$-representation finite algebras are the $n$-representation-finite Nakayama algebras with quadratic relations.