Several Cases About Tilting Theory

Tilting theory is one of the main tools in the representation theory of algebras. When the representation theory of an algebra A is difficult to study directly, it may be convenient to replace A with another simpler algebra B and to reduce the problem on A to a problem on B. We then construct an A-module T^, called a tilting-module.By the method of previous research, given a quiver of the Dynkin graphs A2, A2, we have calculated all the tilting modules and quivers of the tilting-module endomorphism algebra. The main structure of the thesis as follows:In chapter1, we first introduce our research background, including the history and research situation of tilting theory. Then we give some elementary concepts about quiver, path algebra, tilting theory and Morita theory, cluster algebra which will be used in this thesis.In chapter2and chapter3, we calculate tilting modules of the Dynkin graphs A2, As by two methods. One is the defining method, the other is the cluster algebra, we use a initial cluster variable to make mutations, then calculate the corresponding tilting modules according to the correspondence between clusters and tilting modules.In chapter4, we give the description about tilting modules of the Dynkin graphs An, we give some properties about the indecomposable A-modules in a same tilting module.In chapter5, we calculate quivers corresponding to the tilting-module endomorphism algebra of the Dynkin graph A2, A3and get the following conclusions:(1) the representation of is Morita equivalent.(2) the representation of and is Morita equivalent.