Collaborative Research: Generalized Cluster Structures on Poisson Varieties and Applications

该项目属于代数领域，正在开发中，旨在应用于数学和理论物理。一个特别的重点是泊松李群和集群代数的理论。前者长期以来一直是一个自然的框架，在其中可以研究经典和量子力学的重要精确可解模型。后者由Fomin和Zelevinsky在2001年发现，已经被证明与广泛的数学学科有许多令人兴奋的联系，包括组合学，表示论，代数和泊松几何，以及镜像对称和统计和高能物理。该PI将建立在他们以前的合作，继续系统地研究多个集群结构的坐标环的一些品种的重要性，在代数几何，表示论和数学物理和研究相应的集群代数之间的相互作用。这项研究将与本科生和研究生课程和研究项目的发展相联系。计划开展协同活动，促进机构间和部门间的合作，吸引来自代表性不足的群体和具有不同教育背景的研究生，并通过社区外联，使高中生接触数学研究。具体而言，本项目的主要目标包括：1）构造和研究Poisson-Lie群和Poisson-homogeneous簇上的广义团簇结构，包括带Belavin-Drinfeld括号的Poisson-Lie群，简单Poisson-Lie群的Drinfeld双和Poisson-Lie括号，以及3dN = 4超对称规范理论的K理论库仑分支;和2）Poisson簇上的广义簇结构在经典和非交换离散可积系统中的应用，这些系统作为簇变换的序列出现，量子群在单位根上的表示，该奖项反映了NSF的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

This project lies in an area of algebra that is being developed with a view toward applications in Mathematical and Theoretical Physics. A particular focus is on the theory of Poisson-Lie groups and cluster algebras. The former has long served as a natural framework in which important exactly solvable models of classical and quantum mechanics can be studied. The latter, discovered by Fomin and Zelevinsky in 2001, have since been shown to have numerous exciting connections with a wide range of mathematical subjects, including combinatorics, representation theory, algebraic and Poisson geometry, as well as mirror symmetry and statistical and high energy physics. The PIs will build upon their previous collaborations to continue 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. This research will be linked to the development of undergraduate and graduate courses and research projects. Synergistic activities are planned 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 work on applications of Poisson geometry to the theory of cluster algebras. In more detail, the main goals of the project include: 1) construction and study of generalized cluster structures on Poisson-Lie groups and Poisson homogeneous varieties including Poisson-Lie groups equipped with Belavin-Drinfeld brackets, Drinfeld doubles and Poisson-Lie duals of simple Poisson-Lie groups,and K-theoretic Coulomb branches of 3d N = 4 SUSY gauge theories; and 2) applications of generalized cluster structures on Poisson varieties to classical and non-commutative discrete, integrable systems that arise as sequences of cluster transformations, representations of quantum groups at roots of unity, and higher Teichmuller theory.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.