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

Huimin Chang Department of Mathematics, Tsinghua University Bin Zhu Department of Mathematics, Tsinghua University

Representation Theory mathscidoc:1609.30001

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 $\A_{n,t}$ 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 $\A_{n,t}$ 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.
Finite $2$-Calabi-Yau triangulated category; Periodic Ptolemy diagram; Torsion pair, Heart of torsion pair.
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@inproceedings{huimintorsion,
  title={Torsion pairs in finite $2$-Calabi-Yau triangulated categories with maximal rigid objects},
  author={Huimin Chang, and Bin Zhu},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20160920085013098357024},
}
Huimin Chang, and Bin Zhu. Torsion pairs in finite $2$-Calabi-Yau triangulated categories with maximal rigid objects. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20160920085013098357024.
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