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

这个项目的方向是代数，主要研究聚类代数和泊松-李群。自从2001年Fomin和Zelevinsky发明了簇代数以来，数学界见证了对这一主题的兴趣爆发，因为揭示了簇代数与数学和理论物理的各种分支之间的深刻联系，从颤振表示和代数几何到弦理论和统计物理。pi将在他们之前的合作基础上继续系统地研究泊松-李群的坐标环中的多个簇结构，以及代数几何、表示理论和数学物理中许多其他重要的变化，并研究相应簇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。计划开展协同活动，目的是促进机构间和部门间的合作，吸引来自代表性不足和教育背景不同的群体的研究生，并通过社区外展使高中生接触数学研究。他们将继续研究泊松几何在聚类代数理论中的应用。该项目的主要目标包括构建和研究(i)与Belavin-Drinfeld分类描述的泊松-李括号相容的简单李群上的奇异簇结构；（ii）单泊松-李群的Drinfeld对偶和泊松-李对偶上的广义团簇结构；(3)上述簇结构中偶到颤振的高属网的组合学和逆问题；（iv）由高属网络的簇变换和初等变换序列产生的离散可积系统；(v)有向网络的连续极限及其在平面连接模空间上的应用。

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.