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.

该项目属于代数领域，重点关注簇代数和泊松李群。自从 Fomin 和 Zelevinsky 于 2001 年发明簇代数以来，由于簇代数与数学和理论物理学的各个分支（从颤动表示和代数几何到弦理论和统计物理学）之间揭示出的深刻联系，数学界见证了对该学科的兴趣激增。 PI 将在之前的合作基础上继续系统研究泊松李群坐标环中的多簇结构以及代数几何、表示论和数学物理中的许多其他重要类型，并研究相应簇代数之间的相互作用。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，吸引来自弱势群体和具有不同教育背景的研究生，并通过社区外展，让高中生接触数学研究。 PI 将继续研究泊松几何在簇代数理论中的应用。该项目的主要目标包括构建和研究（i）与 Belavin-Drinfeld 分类描述的 Poisson-Lie 括号兼容的简单李群上的奇异簇结构； (ii) 简单泊松-李群的德林菲尔德对偶和泊松-李对偶的广义簇结构； (iii) 对于上述簇结构中出现的与颤动对偶的更高属网的组合和逆问题； (iv) 作为更高属网络的簇变换和初等变换序列而产生的离散可积系统； (v) 应用于平面连接模空间的有向网络的连续极限。