Geometric Characterization Of (Co)torsion Pairs In 2-Calabi-Yau Triangulated Categories

The main object in this thesis is to consider two kinds of 2-Calabi-Yau triangulated categories:(1)Finite 2-Calabi-Yau triangulated categories with non-zero maximal rigid objects which are not cluster tilting;(2)Cluster categories of type A_?~?.The first kind mainly contains two types of categories:one is type A,denoted by An,t,where n ? 1,t>1 when t = 1,An,1 contain cluster tilting objects,which we also consider in this paper;Another one is type D,denoted by Dn,t,where n,t ? 1.By using the geometric model of cluster categories of type A,or type D[1,2],and by the classification of torsion pairs in the cluster categories of type A,or type D[3,4],we give a geometric description of torsion pairs in An,t or Dn t respectively,via defining the periodic Ptolemy diagrams.This allows to count the number of torsion pairs in these categories.Besides these,we determine the hearts of torsion pairs in all finite 2-Calabi-Yau triangulated categories with maximal rigid objects which are not cluster tilting via quivers and relations.We show that the endomorphism algebra of the maximal rigid objects in An,t have quivers which do not have 2-cycles,then the collection of maximal rigid objects in An,t has a cluster structure in the sense of Buan-Marsh-Vatne[5].Moreover,We define the cluster complex of An,t,denoted by?(An,t).We show that there is an isomorphism from?(An,t)to the cluster complex of root system of type Bn(defined by Fomin-Zelevinsky).In particular,the maximal rigid objects are isomorphic to clusters.We have the similar results for Dn,t.This yields a result proved recently by Buan-Palu-Reiten[6]:Let RAn,t,resp.RDn,t,be the full subcategory of An,t,resp.Dn,t,generated by the rigid objects.Then RAn,t,(?)and RDn,t(?)RAn,1 as additive categories,for all t ? 1.For the cluster categories of type A_?~?,we define the Ptolemy diagrams of type A_?~?in an infinite strip with marked points,via the geometric description of the cluster cat-egories of type A_?~?[7]and show that there is a bijection between(co)torsion pairs in a cluster category of type A_?~?and Ptolemy diagrams of type A_?~?.As applications,we give a classification of t-structures and hearts of t-structures of the cluster category of type A_?~?.