Représentations des algèbres et algèbres amassées - Representations of algebras and cluster algebras

Representations des algèbres et algèbres amassées - 代数和簇代数的表示

基本信息

批准号：
41227-2011
负责人：
Assem, Ibrahim
金额：
$ 1.89万
依托单位：
Université de Sherbrooke
依托单位国家：
加拿大
项目类别：
Discovery Grants Program - Individual
财政年份：
2015
资助国家：
加拿大
起止时间：
2015-01-01 至 2016-12-31
项目状态：
已结题

来源：
https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=568841
关键词：
Repr sentations des alg bres

项目摘要

REPRESENTATIONS OF ALGEBRAS AND CLUSTER ALGEBRAS This project has two facets, closely linked together: (a) The representation theory of finite dimensional algebras. The modern form of the theory goes back to the 1930s, when E. Noether showed that representations may be understood as modules. Thus we study the category of modules over an algebra and the morphisms between them. This theory has undergone a very fast development since the 1970s, thanks to the works of Auslander, Gabriel, Reiten, Ringel and others. We are mostly interested in 3 subdomains of the theory: 1. The study of algebras defined by homological properties. 2. Tilting theory. 3. Hochschild cohomology in relation with simple connectedness. (b) Cluster algebras. Since their introduction 10 years ago by Fomin and Zelevinsky, their study has taken more and more importance and has been related to several areas of mathematics, notably geometry, combinatorics and mathematical physics. Among others, their categorification by Buan, Reiten, Amiot and others has revealed links with the representation theory of algebras, and notably with tilting theory. We are mostly interested in 3 subdomains of the theory: 1. The links with representation theory, in particular with tilting theory. 2. The effective computation of cluster variables. 3. The study of algebras defined by triangulations of surfaces.

代数和集群代数的表示该项目有两个方面，紧密相连：（a）有限维代数的表示理论。理论的现代形式可以追溯到 1930年代，当Noether表明表示形式可以理解为模块时。因此，我们研究了代数的模块类别及其之间的形态。这个理论经历了非常快的自1970年代以来的发展，感谢Auslander，Gabriel，Reiten，Ringel等的作品。我们是对该理论的三个子域感兴趣： 1。由同源性质定义的代数研究。 2。倾斜理论。 3。与简单连接有关的Hochschild共同体。（b）群集代数。自从10年前引入Fomin和Zelevinsky以来，他们的研究就进行了越来越重要，与数学的几个领域有关，尤其是几何学，组合和数学物理学。除其他外，它们由Buan，Reiten，Amiot和其他人则揭示了与代数的表示理论的联系，尤其是倾斜理论。我们是对该理论的三个子域感兴趣： 1。与表示理论的联系，特别是与倾斜理论的联系。 2。群集变量的有效计算。 3。由表面三角剖分定义的代数研究。