Hom-configurations in triangulated categories generated by spherical objects

TL;DR

This paper classifies Hom- and Riedtmann configurations in categories generated by spherical objects, linking them to noncrossing partitions and providing a new geometric model for higher orbit categories.

Contribution

It introduces a classification of configurations in categories generated by spherical objects and connects these to combinatorial structures, extending understanding of higher cluster and orbit categories.

Findings

Classified higher Hom- and Riedtmann configurations in these categories.

Linked configurations with noncrossing partitions for w = -1.

Developed a new geometric model for higher orbit categories.

Abstract

Hom- and Riedtmann configurations were studied in the context of stable module categories of selfinjective algebras and a certain orbit category C of the bounded derived category of a Dynkin quiver, which is highly reminiscent of the cluster category. The category C is (-1)-Calabi-Yau. Holm and Jorgensen introduced a family of triangulated categories generated by $w$-spherical objects. When $w \geq 2$, these may be regarded as higher cluster categories of type A infinity. When $w \leq -1$, they are higher analogues of the orbit category C. In this paper, we classify the (higher) Hom- and Riedtmann configurations for these categories, and link them with noncrossing partitions in the case $w = -1$. Along the way, we obtain a new geometric model for the higher versions of the orbit category C.