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.

拟议的研究基于PIS先前在Poisson几何形状到集群代数的应用的合作。在当前项目中，我们将对代数几何，表示理论和数学物理学中多种重要性的坐标环中的多个群集结构进行系统研究，并研究相应群集代数之间的相互作用。重要的例子包括简单的谎言组，均匀的空间，点的配置空间，并且与离散和连续的集成系统有关。要考虑的问题包括与与Belavin-Drinfeld分类相关的Poisson-Lie结构兼容的群集结构包括交换性和非共同代数几何形状，颤抖的代表和Teichmuller理论在内的各种领域。拟议的研究与本科和研究生课程和研究项目的发展有关。计划进行协同活动，目的是促进机构间和部门间合作，吸引来自代表性不足的团体和具有多种教育背景的研究生，并通过社区宣传在高中生的早期暴露于数学研究中。