Cluster structures from maximal rigid objects in 2-CY categories

Dagfinn F. Vatne

For a Hom-finite 2-CY triangulated category C, we generalise the notion of cluster structures from [BIRS] to include situations where the quivers of the clusters may have loops.

An object in C is said to be maximal rigid if Ext¹(T,T) = 0 and whenever Ext¹(T,T) (T Å X,T Å X) = 0, we have that X lies in add T. We show that the set of maximal rigid objects in C forms a generalised cluster structure. This is a generalisation of a result in [BIRS].

As an example, we will show that the set of maximal rigid objects in the cluster category of a tube (none of which are cluster-tilted) forms a generalised cluster structure of type B.

This is joint work with Aslak Bakke Buan and Robert Marsh.

References
[BIRS]  Buan A B, Iyama O, Reiten I, Scott J, Cluster structures for 2-Calabi-Yau categories and unipotent groups, preprint v. 3 arXiv:math/0701557 (2007)