On Derived Unique Gentle Algebras

Gentle algebras play an important role in the research of the representation theory of algebra and many other fields.The derived category of'gentle algebras have been studied extensively in recent years,including the classification of'gentle algebras up to derived equivalence.In particular,Avella-Alaminos and Geiss provided a new combinatorial derived invariant f'or gentle algebra,which is a excellent invariant to judge the derived equivalences for gentle algebras with at most one cycle.In this thesis,we mainly classify the derived unique gentle algebras with at most one cycle.A k-algebra A is derived unique if for each k-algebra B which satisfies Db(A)(?)Db(B)is necessarily Morita equivalent to A.We further study the derived classifications of gentle algebras at most one cycle base on the Avella-Alaminos-Geiss invariant.We first define three elemental transformations of gentle algebra,and recall the Avella-Alaminos-Geiss invariant ? of gentle algebras.Moreover,we provide a complete classification theorem up to derived equivalences for gentle algebras with at most one cycle,which unifies the known classification results.Finally,we establish that the derived unique gentle algebras with at most one cycle are those directed cycle algebras with m arrows such that the number of relations are exactly m or m-1.