The N-APR Tilting Theory Of Dual ?-slice Algebra

The higher algebraic representation theory is an important research field of representation theory of algebra at present,and there are important applications in non-commutative algebraic geometry.Iyama introduces and develops the higher n-AR theory,and studies the n-APR tilts of n-representation finite algebras of n-translation quivers.And then,Guo Jin Yun introduces n-translation algebras of n-translation quivers,and generalizes some concepts in translation quivers for studying the high-dimensional representation theory.In this thesis,we study two aspects of higher representation theory:Firstly,we study the n-APR tilts and n-APR cotilts related to the acyclic n-translation algebras,and the relationships between these n-APR tilts and n-APR cotilts and ?-mutations;Secondly,we study a class of stable 2-translation algebras given by three-dimensional McKay quivers and discuss their relations.In this paper,starting from the conclusion that there is a Koszul complex for in the category of the finite generated projective modules of the quadratic dual of an n-translation algebra.According to the method of constructing the tilt modules related to the D-split sequences discussed by Hu W.and Xi C.C.,we construct the n-APR tilt modules and the n-APR cotilt modules at a vertex related to the Koszul complex on the quadratic dual algebra of the truncation of an acyclic n-translation algebra.And we generalize the ?mutations,and prove that such n-APR tilts and n-APR cotilts can be relazied by corresponding ?-mutations.As an application,we futher prove that the n-APR tilts and n-APR cotilts of a dual ?-slice algebra can be realized by?-mutations.In additions,we also discuss the stable 2-translation algebras related to a class of three-dimensional McKay quivers.We use the properties of the Loewy matrix of the stable n-translation algebra to give its relations.And then,we construct a dual ?-slice algebra,and obtain all its 2-APR tilts by ?-mutations.