Relative D-cluster Tilting Subcategories In D-Calabi-Yau Triangulated Categories

Cluster algebras were introduced by Fomin and Zelevinsky.In recent years,the categorification of Fomin–Zelevinsky's cluster algebras is one of the research hotspots of the representation theory of algebras.Cluster tilting subcategories and maximal rigid subcategories play a pivotal role in the categorification of cluster algebras.By definitions,cluster tilting subcategories are maximal rigid subcategories,and it is generally incorrect conversely.However,in a triangulated category with cluster tilting subcategories and a Serre functor,it was proved that any maximal rigid subcategory is cluster tilting subcategory.Similarly,maximal relative rigid subcategories are equivalent to relative cluster tilting subcategories.This thesis introduces the notion of relative d-cluster tilting subcategories and maximal relative d-rigid subcategories in d-Calabi-Yau triangulated categories.It is proved that relative d-cluster tilting subcategories coincide with covariantly finite maximal relative d-rigid subcategories.