Investigation into algebraic structure of the category of representations of finite demensional algebras

该项目的目的是研究有限维代数表示范畴的代数结构。(1)利用重复代数研究了具有Galois覆盖的代数的稳定范畴，证明了拟倾斜代数的重复代数对自射代数具有Galois覆盖的性质的不变性. (2)研究了非Frobenius自内射（=拟Frobenius）代数，得到了这类代数的一个特征，并给出了任意大维数的非Frobenius自内射代数的一个例子.通过这个例子，我们知道Nakayama在1939年给出的例子是具有最小维数的例子。(3)当代数有预内射分支时，这是一个公开问题。研究了单点扩张代数的问题。我们澄清了一点扩张的预内射分支的存在性依赖于定义扩张的模的和项所属的AR-分支。

The aim of the project is to study algebraic structures of the category of representations of finite dimensional algebras.(1) We studied the stable categories of the algebras with Galois coveings by repetitive algebras, and proved the invariance of the property that a self-injecitve algebra has a Galois covering by the repetitive algebra of a quasi-tilted algebra.(2) We studied non-Frobenius self-injecitve (=quasi-Frobenius) algebras, so that we found a characterization of those algebras and showed an example of non-Frobenius self-injective algebras with arbitrarily large dimension. By this example, we know that the example given by Nakayama in 1939 is the one with the smallest dimension.(3) It is an open problem when an algebra has a preinjective component. We studied the problem for one-point extension algebras. We clarified that the existence of preinjective components of a one-point extension depends on AR-components where summands of the module defining the extension belong.