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

这个项目是关于代数的，主要研究簇代数和泊松-李群。自2001年Fomin和Zlevinsky发明簇代数以来，由于簇代数与数学和理论物理的各种分支之间的深刻联系，从箭图表示法、代数几何到弦论和统计物理，数学界见证了对该学科的兴趣的爆炸性增长。PI将在他们以前的合作的基础上，继续系统地研究泊松-李群的坐标环中的多个团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变化，并研究相应的团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，并通过社区外联，让高中生接触数学研究。PI将继续他们的工作，将泊松几何应用于簇代数理论。该项目的主要目标包括：(I)构造和研究(I)由Belavin-Drinfeld分类描述的与Poisson-Lie括号相容的单李群上的奇异簇结构；(Ii)单Poisson-Lie群的Drinfeld对偶和Poisson-Lie对偶上的广义簇结构；(Iii)高亏格网对偶到上述簇结构中出现的箭图的组合和反问题；(Iv)作为簇变换和高亏格网络的初等变换序列出现的离散可积系统；(V)有向网络的连续极限及其在平坦连接的模空间中的应用。

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.