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

• 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

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.

代数和簇代数的表示 该项目有两个紧密相连的方面： (a) 有限维代数的表示论。该理论的现代形式可以追溯到 20 世纪 30 年代，E. Noether 证明表征可以被理解为模块。因此，我们研究代数上的模范畴以及它们之间的态射。这个理论经历了非常快的发展 自 20 世纪 70 年代以来的发展，得益于 Auslander、Gabriel、Reiten、Ringel 等人的工作。我们是 主要对该理论的 3 个子领域感兴趣： 1. 由同调性质定义的代数的研究。 2.倾斜理论。 3. Hochschild 上同调与简单连通性的关系。 (b) 簇代数。自从 10 年前 Fomin 和 Zelevinsky 提出这一观点以来，他们的研究取得了长足的进步。 越来越重要，并且与数学的几个领域相关，特别是几何， 组合数学和数学物理。其中，按 Buan、Reiten、Amiot 和 其他人揭示了与代数表示论的联系，特别是与倾斜理论的联系。我们是 主要对该理论的 3 个子领域感兴趣： 1. 与表征理论的联系，特别是与倾斜理论的联系。 2.聚类变量的有效计算。 3. 研究由曲面三角剖分定义的代数。