三角矩阵代数的表示论

发布时间：2018-01-22 01:24

本文关键词： 三角矩阵代数 Gorenstein投射模反射函子局部自由模根系 APR-倾斜模丛代数丛变量　出处：《浙江大学》2016年博士论文　论文类型：学位论文

【摘要】：本文主要做了三方面的工作.在第一部分,我们介绍了一类三角矩阵代数,其中的每个代数叫做一个正规三角矩阵代数.我们刻画了这类代数的Gorenstein投射模.并且给出了正规三角矩阵代数的模成为强Gorenstein投射模的一个充分条件.正规三角矩阵代数的重要性在于它包含了所谓的代数上的路代数和广义路代数.因此,我们刻画了代数上的路代数和广义路代数上的Gorenstein投射模和强Gorenstein投射模作为在正规三角矩阵代数上的主要结果的应用.然后,我们给了一个例子来说明如何通过在广义路代数上的定理来求一个给定代数的所有不可分解Gorenstein投射模.在第二部分,我们通过对Frobenius-型三角矩阵代数做反射函子的方法建立了一个有限根系的新的表示论的实现.然后,我给出了这类代数中类似于APR-倾斜模的概念.其中的主要结论包含一些已知的结果,比如代数闭域上的路代数和由可对称化的Cartan矩阵构造的带关系的路代数.同时,它也包含一些其他的重要的代数类,如Frobenius代数上的路代数和顶点上放Frobenius代数的广义路代数.在第三部分,我们通过广义路代数来研究可斜对称化的丛代数.首先利用广义路代数与一个路代数的同构,给出了它们做了丛代数的变异之后还能保持同构.接着利用斜对称情形下非初始丛变量与路代数的不可分解刚性模的一一对应,建立起可斜对称化情形下非初始丛变量与广义路代数的不可分解局部自由刚性模之间的对应.并证明了在有限型的情形下,这是一个一一对应.
[Abstract]:In the first part, we introduce a class of triangular matrix algebra. Each of these algebras is called a normal triangular matrix algebra. We characterize the Gorenstein projective modules of these algebras and give the modules of the normal triangular matrix algebras to be strong Gorenste. The importance of the normal triangular matrix algebra is that it contains the path algebra and the generalized path algebra on the algebra. We characterize the applications of path algebras on algebras and Gorenstein projective modules and strong Gorenstein projective modules on generalized path algebras as main results on normal triangular matrix algebras. After. We give an example of how to obtain all indecomposable Gorenstein projective modules of a given algebra by the theorem on a generalized path algebra. In the second part. We establish a new representation theory of finite roots by making a reflection functor for Frobenius-type triangular matrix algebra. Then. I have given a concept similar to APR-tilting modules in such algebras, in which the main conclusions contain some known results. For example, the path algebra on an algebraic closed field and the path algebra with relation constructed from a symmetric Cartan matrix. At the same time, it also contains some other important algebraic classes. For example, the path algebra on the Frobenius algebra and the generalized path algebra of the Frobenius algebra on the vertex. We study skew symmetric bundle algebras by means of generalized path algebras. Firstly, we use the isomorphism of generalized path algebras and a path algebra. It is shown that they can keep isomorphism after making the variation of bundle algebras, and then use the one-to-one correspondence between the non-initial bundle variables and the indecomposable rigid modules of path algebras in the case of skew symmetry. The correspondence between the noninitial bundle variables and the indecomposable local free rigid modules of generalized path algebras in the case of skew symmetrization is established, and it is proved that in the case of finite type, this is a one-to-one correspondence.
【学位授予单位】：浙江大学
【学位级别】：博士
【学位授予年份】：2016
【分类号】：O153.3

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