Rigid Modules And Shurian Modules Over The Algebras Arising From Cluster Tubes

Let T be a tube,which is a hereditary abelian category.Let D be the bounded derived category of T with suspension functor[1]and Auslander-Reiten translation τ.The cluster tube C is defined to be the orbit category of D under the action of the cyclic group generated by the autoequivalence F=τ-1[1],this means that the objects of C are the same as those of D and that for two objects X and Y,the morphism space from X to Y in C is As everyone knows,C is a 2-Calabi-Yau trangulated category.It has been shown that the set of maximal rigid objects in C has a cluster structure.Let T be a maximal rigid object of C,and A=EndC(T)be the endomorphism algebra of T.The functor HomC(T,-):C→mod A induces an equivalence from a quotient category of some subcategory of C to mod A.In this thesis,we use the equivalence to identify the rigid and Schurian modules of mod A,and we prove that every indecomposable A-module which is rigid,but not τA-rigid,is Schurian.