Cluster Algebras in Representation Theory and Symplectic Geometry

表示论和辛几何中的簇代数

• 批准号: 1702489
• 负责人: Harold Williams
• 金额: $ 10.7万
• 依托单位: University of Texas at Austin
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2017
• 资助国家: 美国
• 起止时间: 2017-08-01 至 2017-12-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1702489&HistoricalAwards=false
• 关键词: Cluster; Algebras; Representation; Theory; Symplectic

Representation theory is the study and classification of different kinds of symmetry, specifically in the context of linear algebra. Historically it has played an important role in mathematics because analyzing a given mathematical object, for example a system of equations, is often dramatically simplified when the object is symmetric in a suitable abstract sense. Symplectic geometry, on the other hand, grew out of Hamiltonian mechanics and today occupies a prominent place in mathematics due to the wide range of contexts, for example in topology and algebraic geometry, where the underlying geometric structures of Hamiltonian mechanics appear. This research project aims to advance understanding of both subjects, as well as unearthing deep new connections between them, by leveraging new ideas from algebraic combinatorics, in particular the recently developed theory of cluster algebras. These connections are in turn all informed by recent advances in mathematical physics, specifically string theory and supersymmetric gauge theory.On the side of representation theory, the main objects of study in this project include the affine Grassmannian and its relatives, in particular their derived categories of equivariant coherent sheaves. On the side of symplectic geometry, the main objects are categories of Lagrangian branes or microlocal sheaves in noncompact symplectic 4- and 6-manifolds. In the former context, cluster structures appear at the level of the equivariant K-rings of the geometric objects involved, and describe certain combinatorial structures inherited by these rings. In the latter, cluster structures appear on moduli spaces of Lagrangian branes in 4 dimensions, or through Hall algebra-type constructions based on Fukaya categories in 6 dimensions. Through its combinatorial nature the language of cluster algebras provides a means of isolating tractable aspects of otherwise very rich and complicated mathematical objects, as well as uncovering new relations between these objects.

表示理论是不同类型的对称性的研究和分类，特别是在线性代数的背景下。从历史上看，它在数学中起着重要的作用，因为分析给定的数学对象（例如方程式系统）通常在适当的抽象意义上对象对称时通常会大大简化。另一方面，象征性的几何形状源于哈密顿力学，今天由于各种环境而在数学中占据着重要地位，例如在拓扑和代数几何学中，汉密尔顿力学的基本几何结构出现了。该研究项目旨在通过利用代数组合学的新思想，特别是最近开发的集群代数理论来提高对这两个学科的理解，并发掘他们之间的深刻新联系。这些联系反过来又源于数学物理学的最新进展，尤其是弦理论和超对称仪表理论。在代表理论的一面，该项目的研究主要对象包括仿生的司司曼尼亚及其亲戚，尤其是其衍生的等质激素冰淇淋类别。在符号几何形状的侧面，主要对象是非紧密符号4和6个manifolds中的拉格朗日麸皮或微斜纹滑轮的类别。在前面的情况下，群集结构出现在所涉及的几何对象的模棱两可的k环的层面上，并描述了这些环继承的某些组合结构。在后者中，群集结构出现在拉格朗日麸皮的模量空间上，或者通过基于6个维度的福卡亚类别的大厅代数型结构出现。通过其组合性质，集群代数的语言提供了一种隔离原本非常丰富和复杂的数学对象的可拖动方面的方法，并揭示了这些对象之间的新关系。