On triangulated orbit categories

TL;DR

This paper proves that orbit categories of derived categories of hereditary categories under certain autoequivalences are naturally triangulated, providing new examples with Calabi-Yau properties relevant to tilting theory and cluster algebras.

Contribution

It establishes that these orbit categories are canonically triangulated, answering longstanding questions and linking to various areas like tilting theory, cluster algebras, and Calabi-Yau categories.

Findings

Orbit categories are canonically triangulated.

Provides examples with Calabi-Yau properties.

Connects orbit categories to representation theory and singularities.

Abstract

We show that the category of orbits of the bounded derived category of a hereditary category under a well-behaved autoequivalence is canonically triangulated. This answers a question by A. Buan, R. Marsh and I. Reiten which appeared in their study with M. Reineke and G. Todorov of the link between tilting theory and cluster algebras (closely related to work by Caldero-Chapoton-Schiffler) and a question by H. Asashiba about orbit categories. We observe that the resulting triangulated orbit categories provide many easy examples of triangulated categories with the Calabi-Yau property. These include the category of projective modules over a preprojective algebra of generalized Dynkin type in the sense of Happel-Preiser-Ringel, whose triangulated structure goes back to Auslander-Reiten's work on the representation-theoretic approach to rational singularities.