COLLABORATIVE RESEARCH: CLUSTER STRUCTURES ON POISSON-LIE GROUPS AND COMPLETE INTEGRABILITY

合作研究：泊松李群的簇结构和完全可积性

• 批准号: 1362801
• 负责人: Michael Gekhtman
• 金额: $ 20.97万
• 依托单位: University of Notre Dame
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2014
• 资助国家: 美国
• 起止时间: 2014-06-01 至 2017-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1362801&HistoricalAwards=false
• 关键词: COLLABORATIVE; RESEARCH; CLUSTER; STRUCTURES; POISSON

This project lies in Algebra and focuses on cluster algebras and Poisson-Lie groups. Since the invention of cluster algebras by Fomin and Zelevinsky in 2001, the mathematical community witnessed an explosion of interest in the subject due to the deep connections that were revealed between cluster algebras and a variety of branches of mathematics and theoretical physics ranging from quiver representations and algebraic geometry to string theory and statistical physics. The PIs will build upon their previous collaborations to continue a systematic study of multiple cluster structures in coordinate rings of Poisson-Lie groups and a number of other varieties of importance in algebraic geometry, representation theory and mathematical physics and study an interaction between corresponding cluster algebras. 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 community outreach, to expose high school students to mathematical research. The PIs will continue their work on applications of Poisson Geometry to the theory of cluster algebras. The main goals of the project include construction and study of (i) exotic cluster structures on simple Lie groups compatible with Poisson-Lie brackets described by the Belavin-Drinfeld classification; (ii) generalized cluster structures on the Drinfeld double and the Poisson-Lie dual of a simple Poisson-Lie group; (iii) combinatorics and inverse problems for higher genus nets dual to quivers arising in cluster structures above; (iv) discrete integrable systems arising as sequences of cluster transformations and elementary transformations of higher genus networks; (v) continuous limits for directed networks with applications to moduli spaces of flat connections.

该项目在于代数，专注于群集代数和泊松群组。自2001年Fomin和Zelevinsky发明集群代数以来，数学社区目睹了对主题的兴趣，这是由于群集代数之间的深厚联系和数学分支之间的深厚联系，以及各种分支。弦理论和统计物理学的代数几何形状。 PI将基于他们以前的合作，以继续对泊松组的坐标环的多个集群结构进行系统研究，以及代数几何学，代表性理论和数学物理学中的许多其他重要性，并研究相应的群集代数代数之间的相互作用。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群体，具有多种教育背景的研究生，并通过社区外展使高中生接触数学研究。 PI将继续对泊松几何形状应用于集群代数理论的应用。该项目的主要目标包括（i）与Belavin-Drinfeld分类所描述的Poisson-Lie括号兼容的简单谎言组上的（i）奇异的集群结构的构建和研究； （ii）Drinfeld Double上的广义集群结构和一个简单的泊松型组的泊松式二重奏； （iii）在上面的群集结构中引起的箭射的高属网络的组合和反问题； （iv）作为集群变换的序列和较高属网络的基本变换而产生的离散集成系统； （v）针对有向网络的连续限制，其应用程序可用于平坦连接的模量空间。