Endomorphism algebras for a class of negative Calabi-Yau categories

We consider an orbit category of the bounded derived category of a path algebra of type An 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.