Singularity categories of some 2-CY-tilted algebras

TL;DR

This paper introduces simple polygon-tree algebras, a class of 2-CY-tilted Jacobian algebras, and analyzes their singularity categories through mutations and extensions, providing classification results.

Contribution

It defines simple polygon-tree algebras as a generalization of cluster-tilted algebras of type D and explores their singularity categories using mutation techniques.

Findings

Connected selfinjective Nakayama algebra's stable category is equivalent to the singularity category of a simple polygon-tree algebra.

Classification of these algebras up to representation type.

Mutation and extension methods preserve singularity equivalences.

Abstract

We define a class of finite-dimensional Jacobian algebras, which are called (simple) polygon-tree algebras, as a generalization of cluster-tilted algebras of type $\D$. They are $2$-CY-tilted algebras. Using a suitable process of mutations of quivers with potentials (which are also BB-mutations) inducing derived equivalences, and one-pointed (co)extensions which preserve singularity equivalences, we find a connected selfinjective Nakayama algebra whose stable category is equivalent to the singularity category of a simple polygon-tree algebra. Furthermore, we also give a classification of algebras of this kind up to representation type.