Generalized Higher Koszul Algebras And Some Theses On Koszul Algebras

In this thesis,the Koszul theory is studied by tools of algebra representation theory.It is organized as follows:1.Firstly,we generalized the definitions of higher Koszul modules and higher Koszul algebras,and introduce generalized Koszul modules and genralzied Koszul algebras.Using this theory,we.prove that if A is a.d-Koszul algebra and M is a generalized d-Koszul module,then E^ev(M) is a KOszul E^ev(A)-module.Then we point out that the odd Ext of a d-Koszul module is a Koszul module and the Poincare series of some nice d-Koszul modules are rational.We also study the one point extension of generalized higher Koszul algebras and give a.description of the module on the one point extension algebra.2.Secondly,graded Morita theory are studied in chapter 3.We.point out that in general graded equivalence does not necessarily preserve the Koszulity of algebras.It is proved that if a graded equivalence preserves the pure modules then it preserve the Koszulity.3.Finally,we consider some specific examples of Koszul algebras and higher Koszul algebras including symmetric algebras,exterior algebras and Yang-Mills algebras.We,prove that the symmetric algebras are all Calabi-Yau algebras and classify the Yang-Mills algebras.