Torsion pairs in finite $2$-Calabi-Yau triangulated categories with maximal rigid objects

Abstract

We give a complete classification of (co)torsion pairs in finite $2$-Calabi-Yau triangulated categories with maximal rigid objects which are not cluster tilting. These finite $2$-Calabi-Yau triangulated categories are divided into two main classes: one denoted by ${\mathcal{A}}_{n,t}$ called of type $A$, and the other denoted by $D_{n,t}$ called of type $D$. By using the geometric model of cluster categories of type $A,$ or type $D$, we give a geometric description of (co)torsion pairs in ${\mathcal{A}}_{n,t}$ or $D_{n,t}$ respectively, via defining the periodic Ptolemy diagrams. This allows to count the number of (co)torsion pairs in these categories. Finally, we determine the hearts of (co)torsion pairs in all finite $2$-Calabi-Yau triangulated categories with maximal rigid objects which are not cluster tilting via quivers and relations.

25pages

References in corpus (2)

• On triangulated orbit categories
• A characterization of torsion theories in the cluster category of Dynkin type A_{\infty}

Cited by in corpus (1)

• Cotorsion pairs in cluster categories of type $A_{\infty }^{\infty }$