The cluster category of a tubular algebra

Dirk Kussin

This is a report on recent joint work with M. Barot and H. Lenzing. We study the cluster category C = C(A) of a canonical algebra A over an algebraically closed field, or equivalently the cluster category of the hereditary category H = coh(X) of coherent sheaves over a weighted projective line X. By a result of B. Keller C is a triangulated 2-Calabi-Yau category. Moreover, C admits a cluster structure in the sense of Buan-Iyama-Reiten-Scott. The category C can be identified with the category H plus certain extra morphisms. It follows that the tilting (exchange) graphs of C and H coincide.

Theorem 1.   Let A be tubular. The group Aut(C) of exact autoequivalences of C = C(A) admits in a natural way the group PSL₂(Z) as a homomorphic image.

Moreover, the corresponding kernel is a naturally described finite group. As an application of this result and of a rank additivity formula by T. Hübner we show the following.

Theorem 2.   Let A be a canonical algebra of domestic or of tubular type. Then the tilting graph of C(A) is connected.

The corresponding result for the wild canonical algebras is still open.