Research Of The Tilting Module Of The Quiver With Underlying Graph The Dynkin Graphs Of Type D₄

In mathematics, specifically representation theory, tilting theory describes a way to relate the module categories of two algebras using so-called tilting modules and associated tilting functors.In detail,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 need to construct an A-module T,called a tilting module,we set B=End TA,the the categories modA and modB are reasonably close to each other.First, we introduce the background development and future of the tilting theory. Then we briefly review some preliminaries, which consists of algebra,module,quiver,Auslander-Reiten quiver,tilting module and cluster algebra.In chapter3, we mainly discuss the methods to construct tilting module.First, we introduce the method of the definition of tilting module,but the actual operation is very difficult. Then we propose a new method,we use the cluster algebra,and construct the cluster variables and clusters.According to the bijection between cluster variables and isomorphism classes of indecomposable modules,we get the number of the isomorphis-m classes of indecomposable modules; and by using the according relationship between clusters and tilting modules, we then get a number and the specific form of tilting mod-ules.Then we calculate the cluster variables and clusters of the Dynkin graphs of type D4, and according to the corresponding relations,we calculate the number and specific forms of tilting modules of the Dynkin graphs of type D4.