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)研究了由重复代数盖上伽罗瓦覆盖的代数的稳定范畴，证明了自注入代数具有由拟倾斜代数的重复代数盖上伽罗瓦覆盖的性质的不变性。(2)研究了非frobenius自注入（=拟frobenius）代数，得到了这些代数的一个性质，并给出了一个具有任意大维数的非frobenius自注入代数的例子。通过这个例子，我们知道中山在1939年给出的例子是最小维度的。(3)代数具有预内射分量是一个开放问题。研究了一类单点可拓代数问题。证明了单点扩展的预内射分量的存在性取决于定义该扩展的模的和所属的ar分量。

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.