On The McKay Quiver And Selfinjective Algebras

In 1980, McKay introduces the concept of McKay quiver. He also observesthat for a finite subgroup G of SL(2, C), the McKay quiver is one of the extendedDynkin graphs (?)_n, (?)_n, (?)₆, (?)₇, (?)₈, and classical McKay correspondence asserts thatthere exists one to one correspondence between irreducible representations of finitesubgroup G of SL(2, C) and the cohomology of the minimal resolution of Kleinsingularities C²/G.In this paper we will consider the McKay quiver in higher dimensional case,what do they look like? We get some relationship between McKay quiet and KoszulArtin-Schelter regular algebras and Koszul selfinjective algebras via skew group al-gebras. Then we find some description of McKay quier by the property of Loewymatrix. In fact, we will determine the McKay quiver in higher dimensional caseby the properties of Loewy matrix, and we also determine finite complexity Koszulselfinjective algebras at the same time.In chapter 1, we introduce some preliminary results, recall the defnition ofMcKay quiver and describe its' properties in term of graph. We get the relation-ship between McKay quier and Koszul Artin-Schelter regular algebras and Koszulselfinjective algebras via skew group algebras obtained by the finite subgroup G ofGL(m, C). For selfinjective albebras, we can definite its' Loewy matrix. So we canestablish the connection betwcen McKay quiver, selfinjective albebras and Loewymatrix, and derive some properties of Loewy matrix. The eigenvalue of Loewy ma-trix has a good property, and we prove that it is exactly the root of cyclotomicpolynomials.In chapter 2, we recall the properties of cyclotomic polynomials, and provean easily result obout Euler function: for n＞2 and n≠6,φ(n)≥n^1/2. Weinvestigate closely, relationship between characteristic polynomial of Loewy. Matrix and cyclotomic polynomials: f(x)=multiply from∑d_i=degf(x)λ_di(x).In chapter 3, we definite 3-eigenvalue, and determine the quiver correspondingto 3-eigenvalue whose norm is 1. Then to determine the McKay quiver of subgroup ofSL(3, C) who has 3 nonisomorphic irreducible representations. Finally, we describesome Koszul selfinjective algebras whose four power of radical is zero.