广义路代数及其在表示论中的意义  

Generalized path algebra and its significance in representation theory

作　　者：李方[1] 叶昌 Fang Li;Chang Ye

机构地区：[1]浙江大学数学科学学院,杭州310027 [2]湖州师范学院数学系,湖州313000

出　　处：《中国科学：数学》2018年第11期1595-1614,共20页Scientia Sinica：Mathematica

基　　金：国家自然科学基金(批准号:11671350和11571173)资助项目

摘　　要：刘绍学和Coelho于2000年引进了广义路代数的概念,以期对代数的结构和表示进行更直接的刻画.在随后的十几年,广义路代数的方法获得了发展和应用.本文是对这方面的一个总结和展示,包括对广义路代数及其相关概念(如三角矩阵代数等)的结构和表示的介绍、用广义路(余)代数提出的对非基本(非点) Hopf代数的刻画的一个设想、广义路代数的变异以及一类广义路代数决定的丛代数的加法范畴化.In 2000, Shaoxue Liu and Coelho introduced the concept of generalized path algebras in order to make a more direct characterization of the structure and representations of algebras. Over the following decade,the method for generalized path algebras received some developments and applications. This article is a summary and demonstration in this respect. Firstly, we give an introduction to structures and representations of generalized path algebras and some related notions, such as triangular matrix algebras, etc. We show the characterization of generalized path coalgebras with Hopf structures and then suggest a programme for classification of non-basic（resp. non-pointed） Hopf algebras by generalized path（co-）algebras. We also define the mutation of generalized path algebras and then investigate the additive categorification of cluster algebras determined by a class of generalized path algebras.

关 键 词：广义路代数 Gabriel定理 模块 HOPF代数 丛代数 

分 类 号：O152.6[理学—数学]