Research On The Representation Of Some Special Algebras

Representation theory of algebras has been a new branch of algebras whichrose at the beginning of the1970’s. Its basic content is to study the categoryof modules over an artin algebra. The representation of quivers and almost splitsequences are the main tools in studying the representation theorem of algebras.Now, in the mathematical research, the subjects begin to be intersected and uni-fied. In this progress, the representation theory of algebras is more energetic andbecomes an important tool in many fields of mathematics. So, it has been amajor aspect in the international research on algebra. Today, the study of rep-resentations of associative algebras can be interpreted to be the classification ofindecomposable modules and the research on the homomorphisms. In view ofthe above mentioned, two important methods were introduced. One is the quivermethod by P. Gabriel, and another is Auslander-Reiten theory by M. Auslanderand I. Reiten etc.The concept of an almost split sequence was presented by M. Auslander andI. Reiten in the study the category of finitely generated modules over an artinalgebra. It laid the theoretical foundation on the representation theory of algebras.In fact, the almost split sequence is a special kind of exact sequences and it isconstructed by the right almost split morphism and left almost split morphism.The importance of the almost split sequence is reflected in the AR-quiver, whichis constructed by the almost split sequences. For an artin algebra, the AR-quivercan represent its category of modules. Hence one can see that the AR-quiveris an important topic in the representation theory of algebras. The irreducible morphism is a main tool when we construct the AR-quiver of an artin algebra.The arrows between two vertices are determined by the irreducible morphisms inan AR-quiver. Therefore, we will study the right almost split morphisms, leftalmost split morphisms and irreducible morphisms, and so, determine the almostsplit sequences and AR-quivers. In this paper, three kinds of algebras are themain research objects.In chapter2, the formal triangular matrix algebra is studied. The right al-most split morphisms, left almost split morphisms and irreducible morphisms aredetermined in this chapter. On one hand, we construct the above morphisms forthe indecomposable projective modules and injective modules. On the other hand,the above morphisms are constructed from the corresponding morphisms of therelated algebras. At the same time, the D-split sequences are established to studythe derive equivalence. At the end of this chapter, the almost split sequences inmodT2(T) are established from the definitions, the right almost split morphisms,left almost split morphisms and irreducible morphisms. Finally, we study theAR-quiver of the formal triangular matrix algebra.In chapter3, the trivial extension of a hereditary algebra is a main topic toresearch. Based on the conclusions of Tachikawa, the right almost split morphismsand left almost split morphisms can be constructed from the corresponding mor-phisms in the category of finitely generated modules over the hereditary algebra.At the same time, the irreducible morphisms between1st kind and2nd kind arealso established. At last, we study the almost split sequences.In chapter4, the Morita context is discussed. The the right almost split morphisms, left almost split morphisms and irreducible morphisms in the categoryof finitely generated modules over the Morita context are constructed, accordingto the corresponding morphisms of the related algebras. From the definition, westudy the minimal projective representation of the modules and compute theirAR-translations. Then, the almost split sequences can be established.In chapter5, a special Morita context is researched. We prove that it isa Gorenstein algebra, then from the properties of the Gorenstein algebra, theGorenstein-pojective modules are constructed. It can be seen that the class ofGorenstein-projective modules that constructed in this chapter contains the classof projective modules and the Gorenstein-pojective modules over the formal tri-angular matrix algebra.