Brauer Graph Algebras And Associated Graded Algebras

A famous trichotomy result due to Drozd states that every finite dimensional algebra over algebraically closed field is either of finite,tame,or wild representation type.The representation type is a basic question in algebra representation theory.In this thesis,we will use the graded degrees of vertices in Brauer graph to give a criterion for when the graded algebras of Brauer graph algebras associated with the radical filtration(in other words,the graded Brauer graph algebras)are of finite representation type and when they are domestic.Besides,we describe the relationship between the Auslander-Reiten quivers of Brauer graph algebras and the Auslander-Reiten quivers of the graded Brauer graph algebras.In chapter 3,we study the finite representation type of the graded Brauer graph algebras.We note that the graded Brauer graph algebras are special biserial algebras and we get related string algebras such that they have the same representation type.We study the strings and bands over the related string algebras.According to some combined concepts:graded degree,unbalanced edge and -condition,we get a necessary and sufficient condition for the graded Brauer graph algebra to be of finite representation type.In other words,the graded Brauer graph algebra is of finite representation type if and only if the Brauer graph is a Brauer tree whose edge is not an unbalanced edge or is a Brauer tree satisfying ★-condition with respect to any unbalanced edge.In chapter 4,we give a characterization of the domestic graded Brauer graph al-gebras.According to the characterization of the domestic Brauer graph algebras,we only need to consider three kinds of Brauer graphs.The first kind of Brauer graph is Brauer tree.The second kind is the Brauer graph whose underlying graph is a tree and for exactly two vertices,the multiplicity is 2 and the multiplicity is 1 for other vertices.The last kind is the Brauer graph whose underlying graph is a graph with a unique cycle and m≡1.We study the domestic type case by case.If Brauer graph is a Brauer tree with an exceptional vertex of multiplicity m0,moreover,we define signs κ0 and κ1.Then the graded Brauer tree algebra is domestic if and only if κ0(m0-1)+κ1=1.For the other two kinds,the graded Brauer graph algebra is domestic if and only if there is no unbalanced edge in the Brauer graph or the Brauer graph satisfies ★-condition with respect to any unbalanced edge.In chapter 5,when the Brauer graph satisfies ★-condition with respect to any unbalanced edge,we describe the relationship between the Auslander-Reiten quiver of Brauer graph algebra and the Auslander-Reiten quiver of the graded Brauer graph algebra.In particular,when the Brauer graph satisfies ★-condition with respect to an unbalanced edge,we get a diamond associated with the unbalanced edge.