STUDY ON FULL MATRIX ALGEBRAS WITH STRUCTURE SYSTEMS

对于具有结构系统的全矩阵代数，我们得到了以下研究结果。对于任意给定的结构系统，我们定义了它的(0，1)-极限全矩阵代数。设A是具有结构体系的n×n全矩阵代数。在n=3时，我们证明了A同构于它的(0，1)-极限，并得到了五个同构类的列表。对于每个n≧4，我们构造了一族互不同构的Frobenius全矩阵代数，它们具有相同的(0，1)-极限，它们由一个基域的元素来参数化。将结构系统集看作一个代数簇，证明了全矩阵代数的每个同构类对应于结构系统簇的一个G-轨道，其中G是作用在簇上的某个群。关于离散赋值环上的平铺序，Fujita(J.Algebra，2002)的一篇论文提出了一个问题，即研究，我们找到了这个问题的反例。我们注意到，这个问题是引入具有结构系统的全矩阵代数的动机之一，而反例在以前是没有预料到的。我们的例子还表明，基域的特征在具有结构系统的全矩阵代数的研究中起着重要作用。

Concerning full matrix algebras with structure systems, we obtained the following research results.1. For an arbitrarily given structure system, we defined its (0, 1)-limit full matrix algebra. Let A be an n × n full matrix algebra with a structure system. In the case of n=3, we proved that A is isomorphic to its (0, 1)-limit and we obtained the list of five isomorphism classes. For each n≧4, we constructed a family of mutually non-isomorphic Frobenius full matrix algebras having the same (0, 1)-limit, which are parameterized by elements of a base field. Considering the set of structure systems as an algebraic variety, we showed that each isomorphism class of full matrix algebras corresponds to a G-orbit of the variety of structure systems, where G is a certain group acting on the variety.2. Concerning tiled orders over a discrete valuation ring, a question was posed in a paper of Fujita (J. Algebra, 2002), that is, research, we found a counterexample to this question. We note that the question was one of the motivations to introduce full matrix algebras with structure systems, and that the counterexample had not been expected before. Our example also shows that the characteristic of a base filed plays an important role in the study of full matrix algebras with structure systems.