Collaborative Research: Cluster Algebras Approach to Poisson-Lie Groups and Higher Genus Directed Networks

协作研究：泊松李群和更高属有向网络的簇代数方法

• 批准号: 1101462
• 负责人: Michael Gekhtman
• 金额: $ 14.5万
• 依托单位: University of Notre Dame
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2011
• 资助国家: 美国
• 起止时间: 2011-06-01 至 2014-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1101462&HistoricalAwards=false
• 关键词: Collaborative; Research; Cluster; Algebras; Approach

The proposed research builds upon PIs' previous collaboration on applications of Poisson geometry to cluster algebras. In the current project we will undertake a systematic study of multiple cluster structures in coordinate rings of a number of varieties of importance in algebraic geometry, representation theory and mathematical physics and study an interaction between corresponding cluster algebras. Important examples include simple Lie groups, homogeneous spaces, configuration spaces of points, and are related to discrete and continuous integrable systems. The problems to be considered include cluster structures on simple Lie groups compatible with Poisson-Lie structures associated with the Belavin-Drinfeld classification, inverse problems for directed nets on surfaces of higher genus, continuous limits for directed networks with applications to moduli spaces of flat connections and growth rate classification of cluster algebras.The rapid development of the cluster algebra theory in recent years revealed relations between cluster algebras and a variety of areas including, among others, commutative and non-commutative algebraic geometry, quiver representations and Teichmuller theory. The proposed research is linked to the development of undergraduate and graduate courses and research projects. Synergistic activities are planned with the goal to promote inter-institutional and inter-departmental cooperation, to attract graduate students from underrepresented groups and with diverse educational backgrounds, and, through a community outreach, at the early exposure of high school students to mathematical research.

拟议的研究建立在 PI 先前在泊松几何在聚类代数应用方面的合作基础上。在当前的项目中，我们将对代数几何、表示论和数学物理中多种重要的坐标环中的多重簇结构进行系统研究，并研究相应簇代数之间的相互作用。重要的例子包括简单的李群、齐次空间、点的配置空间，并且与离散和连续可积系统相关。需要考虑的问题包括与 Belavin-Drinfeld 分类相关的 Poisson-Lie 结构兼容的简单李群上的簇结构、更高属表面上的有向网的反演问题、有向网络的连续极限及其在平连接模空间中的应用以及簇代数的增长率分类。 近年来簇代数理论的快速发展揭示了簇代数与簇代数之间的关系。 各种领域，其中包括交换和非交换代数几何、箭袋表示和 Teichmuller 理论。拟议的研究与本科生和研究生课程及研究项目的开发相关。协同活动的目的是促进机构间和部门间的合作，吸引来自代表性不足群体和具有不同教育背景的研究生，并通过社区外展，让高中生尽早接触数学研究。