Algebras of finite representation type arising from maximal rigid objects

TL;DR

This paper classifies all endomorphism algebras of maximal rigid objects in certain 2-Calabi-Yau categories of finite type, revealing most are 2-Calabi-Yau-tilted and connecting to prior classifications.

Contribution

It provides a complete classification of these algebras and explains their structure through subcategory analysis, extending understanding of rigid objects in 2-Calabi-Yau categories.

Findings

Most algebras are 2-Calabi-Yau-tilted

All but one algebra are classified in earlier work

Subcategory analysis explains algebra structures

Abstract

We give a complete classification of all algebras appearing as endomorphism algebras of maximal rigid objects in standard 2-Calabi-Yau categories of finite type. Such categories are equivalent to certain orbit categories of derived categories of Dynkin algebras. It turns out that with one exception, all the algebras that occur are $2$-Calabi-Yau-tilted, and therefore appear in an earlier classification by Bertani-{\O}kland and Oppermann. We explain this phenomenon by investigating the subcategories generated by rigid objects in standard 2-Calabi-Yau categories of finite type.