Quantum groups, integrable systems and dualities

这个项目是几个数学领域的交叉点：表示理论、经典和量子可积系统、数学物理、组合学和枚举代数几何。表示理论是研究具有额外重要结构的向量空间的对称性，例如三维欧几里得空间（或更一般地说，无限维空间）。这种对称性通常出现在由代数对象（如群、李代数或代数）编码的族中。以这种方式产生的一类重要代数是所谓的亲和量子群。自20世纪80年代以来，这些代数一直激发着数学和物理之间的积极研究和互动。本项目的主要目标是解决亲射量子群的仿射性质所固有的重要问题。该项目将增强我们对其内部代数结构的理解，并与上述领域建立新的联系。此外，该项目将通过对从高中到研究生院各级学生的培训和指导，对教育产生影响。更详细地说，该项目将开发量子环群研究的新方法，并将其应用于几何表示理论、可积系统、数学物理和量子簇代数。在最近成果的基础上，PI将在五个相关领域进行研究，每个领域都有具体目标。这些研究领域由对偶的一般概念和“洗牌代数方法”的使用统一起来。大致方案如下：1.；发展量子仿射代数和环面代数的关键结构；2. 继续研究量子化库仑分支；3. 继续研究可积自旋链；4. 发展有限量子群及其积分形式的新方法，允许任意统一根，以及模理论，从而推广和统一了deccini - kac - procesi的经典工作；和5。给出了弦理论中a (n)型颤振规范理论的严格数学公式和BPS/CFT对应证明，推广了a(1)-颤振的已知结果。该奖项反映了美国国家科学基金会的法定使命，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 at the intersection of several fields of mathematics: representation theory, classical and quantum integrable systems, mathematical physics, combinatorics, and enumerative algebraic geometry. Representation theory is the study of symmetries of a vector space such as a three-dimensional Euclidean space (or more generally, an infinite dimensional space) endowed with additional important structures. Such symmetries often arise in families encoded by algebraic objects like groups, Lie algebras, or algebras. One important class of algebras that arise in this way is the class of so-called affinized quantum groups. These algebras have been inspiring active research and interactions between mathematics and physics since the 1980s. The main goals of this project are to resolve important questions intrinsic to the affine nature of affinized quantum groups. The project will enhance our understanding of their internal algebraic structures and establish novel connections to the above fields. In addition, the project will have an educational impact through the training and mentoring of students at various levels from high school to graduate school. In more detail, the project will develop new methods in the study of quantum loop groups with applications to geometric representation theory, integrable systems, mathematical physics, and quantum cluster algebras. Building on recent results, the PI will pursue research in five related areas, with specific goals in each. These research areas are unified by the general notion of duality and the use of the "shuffle algebra approach." The rough plan is as follows: 1. Develop key structures of quantum affine and toroidal algebras; 2. Continue the study of quantized Coulomb branches; 3. Continue work on integrable spin chains; 4. Develop a new approach to finite quantum groups and their integral forms that allows for arbitrary roots of unity, as well as a modular theory, thus generalizing and unifying the classical work of DeConcini-Kac-Procesi; and 5. Provide a rigorous mathematical formulation and proof of the BPS/CFT correspondence for A(n)-type quiver gauge theories in string theory, generalizing the known results for the A(1)-quiver.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.