[Paper Review] The cluster category of a canonical algebra

Michael Barot, Dirk Kussin+1 more|arXiv (Cornell University)|Jan 29, 2008

Algebraic structures and combinatorial models18 references4 citations

TL;DR

This paper establishes a categorical equivalence between the cluster category of a canonical algebra and a graded extension of the hereditary category of coherent sheaves on a weighted projective line. It proves that the tilting graph of the cluster category is connected when the Euler characteristic is non-negative, and fully describes the automorphism group of the cluster category in terms of derived autoequivalences modulo the fundamental autoequivalence F.

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.

Motivation & Objective

• To establish a structural equivalence between the cluster category of a canonical algebra and the hereditary category of coherent sheaves on a weighted projective line.
• To determine the automorphism group of the cluster category in terms of derived category autoequivalences.
• To prove the connectedness of the tilting (or exchange) graph of the cluster category under non-negative Euler characteristic.
• To show that cluster-tilting objects in the cluster category form a cluster structure in the sense of Buan-Iyama-Reiten-Scott.
• To analyze the relationship between the tilting graph of the sheaf category and that of the cluster category, showing they coincide.

Proposed method

• Define the cluster category as the orbit category of the bounded derived category of the hereditary category of coherent sheaves under the autoequivalence F = τ⁻ ∘ [1].
• Construct a graded category 𝔻H with morphisms of degree 0 and 1, using Hom and Ext¹ spaces, and prove it is equivalent to the cluster category.
• Use the lifting property of exact autoequivalences from the cluster category to the derived category of the sheaf category.
• Apply results on cluster tubes and mutation sequences to analyze tilting objects and their connectivity.
• Employ automorphisms of the derived category to act on tilting objects and prove invariance of the connected component containing the canonical tilting object.
• Use Euler characteristic conditions (χ ≥ 0) to ensure the existence of mutation sequences connecting any tilting object to the canonical configuration.

Experimental results

Research questions

• RQ1 How is the cluster category of a canonical algebra related to the hereditary category of coherent sheaves on a weighted projective line?
• RQ2 Under what conditions is the tilting graph of the cluster category connected?
• RQ3 What is the structure of the automorphism group of the cluster category?
• RQ4 How do cluster-tilting objects in the cluster category relate to cluster structures in the sense of Buan-Iyama-Reiten-Scott?
• RQ5 What is the relationship between the tilting graph of the sheaf category and that of the cluster category?

Key findings

• The cluster category of a canonical algebra is equivalent to a graded category constructed from the hereditary category of coherent sheaves on a weighted projective line.
• The tilting graph of the cluster category is connected if and only if the Euler characteristic of the weighted projective line is non-negative.
• The automorphism group of the cluster category is canonically isomorphic to the group of exact autoequivalences of the derived category modulo the cyclic subgroup generated by F.
• In the non-tubular case, the automorphism group of the cluster category is isomorphic to the automorphism group of the sheaf category.
• In the tubular case, the coset space Aut(𝓒)/Aut(ℋ) is naturally in bijection with ℚ ∪ {∞}.
• The tilting graph of the sheaf category coincides with the tilting graph of the cluster category, and both are connected when χ ≥ 0.