The cluster category of a canonical algebra

TL;DR

This paper explores the structure of the cluster category associated with a canonical algebra, linking it to coherent sheaves over weighted projective lines, and analyzes its automorphisms and tilting graph connectivity.

Contribution

It characterizes the automorphism group of the cluster category and establishes the cluster structure formed by tilting objects, connecting it to the sheaf category's tilting graph.

Findings

Automorphism group of the cluster category determined.

Cluster-tilting objects form a cluster structure.

Tilting graph is connected for tame representation types.

Abstract

We study the cluster category of a canonical algebra A in terms of the hereditary category of coherent sheaves over the corresponding weighted projective line X. As an application we determine the automorphism group of the cluster category and show that the cluster-tilting objects form a cluster structure in the sense of Buan-Iyama-Reiten-Scott. The tilting graph of the sheaf category always coincides with the tilting or exchange graph of the cluster category. We show that this graph is connected if the Euler characteristic of X is non-negative, or equivalently, if A is of tame (domestic or tubular) representation type.