Twisted Stable Calabi-Yau Property Of Selfinjective Algebras Of Finite Representation Type

In this paper,we focus on the twisted stable Calabi-Yau property of selfinjective algebras of finite representation type.A selfinjective algebra A is called a stable Calabi-Yau algebra,if the stable category of A is a weakly Calabi-Yau category.The notion of twisted stable Calabi-Yau algebras is a generalization of this definition.In Chapter 3,we obtain that selfinjective algebras of finite representation type are twisted stable CalabiYau algebras.Especially,we discuss the twisted stable Calabi-Yau property of trivial extensions of path algebras of Dynkin type and Nakayama selfinjective algebras.To discribe the twisted Calabi-Yau property of objects in triangulated categories,in Chapter 4,we introduce the notion of twisted Calabi-Yau objects.This definition is a generalization of Calabi-Yau objects.We first discuss the relation between twisted Calabi-Yau objects and Auslander-Reiten triangles.Then we study twisted Calabi-Yau objects in the stable categories of selfinjective algebras of finite representation type.Such objects are called twisted Calabi-Yau modules.We prove that Nakayama selfinjective algebras always have indecomposable twisted Calabi-Yau modules.For other selfinjective algebras of finite representation type,we give sufficient conditions for having indecomposable twisted Calabi-Yau modules.