本文从经典 McKay 对应出发，主要研究 McKay 箭图及其与 Dynkin 图的关系，并由此推出一类斜群代数与矩阵代数以及它们的有限生成模范畴的性质。我们考虑任意一个有限子群G及它的一个F-线性表示，其中F为特征不整除 G的阶的代数闭域。一方面，我们对满足一定条件的表示给出了其 McKay 图的一个描述。首先，我们发现对自反轭忠实表示，McKay 矩阵是一个不可分解的余秩为 1 的半正定实对称矩阵。通过抽象出这类 McKay 矩阵的性质来作为关联矩阵的性质，我们定义了广义 McKay 图，然后由广义 McKay 图的分类得到：特征为实数的二维忠实表示对应的 McKay 图为仿射ADE型 Dynkin 图或JK型图。特别地，我们验证了3-二面体群D_3的二维不可约表示对应的 Mckay 图为K型图。作为这个定理的一个应用，我们建立了经典 McKay 对应。我们的这一方法避免了R. Steinberg分析仿射 Coxeter 元素的特征值这一复杂过程。另一方面，我们对任意表示得到了其可分 McKay 箭图的一个描述。首先，对一个 Jacobson 根平方为零的有限维不可分解遗传F-代数A且具有 G 的F-线性作用，我们证明了AG的 Gabriel 箭图的底图的连通分支的型可以由A的的 Gabriel 箭图的底图型来确定。再借助于形式幂级数环的一个商环R，我们证明了可分McKay箭图同构于RG的矩阵代数的Gabriel 箭图。然后通过证明R的矩阵代数为满足上述性质的代数并具体给出R 的矩阵代数的Gabriel箭图，我们就得到了可分McKay箭图的结构。特别地，在 Auslander 和 Reiten 推广结果的基础上，我们补充了一维情形下可分McKay箭图的底图为有限ADE 型 Dynkin 图的不交并。更进一步，我们证明了一维情形下可分McKay箭图的底图是 n 个有限A_2型 Dynkin 图的不交并（n为 G中共轭类的个数）。基于此结果，对一类斜群代数与有限秩自由矩阵代数，我们证明了它们的有限生成模范畴是等价的，同时它们F-代数同构当且仅当G是交换群。最后，作为对可分McKay箭图描述结果的应用，我们描述了McKay箭图，与前面对 McKay 图的描述相比这里不要求F为复数域。

In this thesis, we study the structure of McKay quivers and the relationship between McKay quivers and Dynkin diagrams coming from the classical McKay correspondence. Based on this, we deduce some properties of some skew group algebras, matrix algebras and their finitely generated modular categories.We consider a finite group G and its F-linear representation, where F is a algebraic closed field of characteristic not dividing the order of G. On the one hand, we describe the McKay diagram of a representation satisfying some conditions. First, we find that the McKay matrix of a self-contragredient faithful representation is an 1-corank indecomposable positive semi-definite real symmetric matrix. By abstracting the properties of these McKay matrixes as the properties of incidence matrixes, we define the generalized McKay diagrams. Second, by the classification of generalized McKay diagrams, we obtain: the McKay diagram of a 2-dimensional faithful representation with real character is the Dynkin diagram of affine type ADE or the generalized McKay diagram of type JK. In particular, we verify that the McKay diagram of the 2-dimensional indecomposable representation of D_3 is the generalized McKay diagram of type K_3. As an application of this result, we prove the classical McKay correspondence. Our method avoids analysing the eigenvalues of affine Coxeter elements by R. Steinberg.On the other hand, we describe the separated McKay quiver for any representation. First, for a finite dimensional F-algebra A with Jacobson radical square zero, we prove the separated Gabriel quiver of A is isomorphic to the Gabriel quiver of the matrix algebra of A, and the skew algebra of the matrix algebra of A is isomorphic to the matrix algebra of the skew algebra of A. At the same time, if A is indecomposable hereditary and G can F-linear act on it, we prove the type of connected components of the underlying diagram of the Gabriel quiver of AG can be determined by the type of connected components of the underlying diagram of the Gabriel quiver of A. Second, with the help of formal power series rings, we prove the separated McKay quiver is isomorphic to the Gabriel quiver of the matrix algebra of RG. Third, by describing the Gabriel quiver of the matrix algebra of RG and proving the matrix algebra of RG is the algebra satisfying the above properties, we describe the underlying diagram of the separated McKay quiver. Particularly, based on the result of Auslander and Reiten, we add the result of 1-dimension: any connected component of the underlying diagram of the separated McKay quiver is the Dynkin diagram of finite type ADE. Furthermore, we prove the underlying diagram of the separated McKay quiver is the union of n Dynkin diagrams of type A_2 (n denotes the number of conjugate classes in G). Based on this result, for some skew group algebras and free matrix algebras of finitely rank, we prove their finitely generated modular categories are equivalent, and they are F-algebraic isomorphic if and only if G is Abelian. At last, as an application of the description of underlying diagrams of separated McKay quivers, we describe McKay diagrams without the assumption F=C comparing with the above description of McKay diagrams.