Endomorphism algebras for a class of negative Calabi-Yau categories

TL;DR

This paper characterizes and explicitly describes the endomorphism algebras of maximal m-rigid objects in a class of negative Calabi-Yau categories derived from type A_n, relating them to higher cluster-tilted and tiling algebras.

Contribution

It provides a full description of these endomorphism algebras in terms of quivers and relations, and introduces the new class of tiling algebras.

Findings

Connected endomorphism algebras characterized for maximal m-rigid objects.

Explicit descriptions of these algebras via quivers and relations.

Introduction of tiling algebras as a new class of algebras.

Abstract

We consider an orbit category of the bounded derived category of a path algebra of type A_n which can be viewed as a -(m+1)-cluster category, for m >= 1. In particular, we give a characterisation of those maximal m-rigid objects whose endomorphism algebras are connected, and then use it to explicitly study these algebras. Specifically, we give a full description of them in terms of quivers and relations, and relate them with (higher) cluster-tilted algebras of type A. As a by-product, we introduce a larger class of algebras, called 'tiling algebras'.