Hopf algebras, Frobenius-Schur indicators and modular categories

Hopf 代数、Frobenius-Schur 指标和模类别

• 批准号: 1001566
• 负责人: Siu-Hung Ng
• 金额: $ 13.54万
• 依托单位: Iowa State University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2010
• 资助国家: 美国
• 起止时间: 2010-07-01 至 2014-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1001566&HistoricalAwards=false
• 关键词: Hopf; algebras; Frobenius; Schur; indicators

The principal investigator proposes to study Hopf algebras, modular tensor categories, and some of their invariants called Frobenius-Schur indicators. In particular, the following topics will be studied. (1) Classification of Hopf algebras of small dimension: It is fundamentally important to understand some basic examples in any mathematical theory. The PI will continue to work on the classification of Hopf algebras whose dimension admits a simple factorization.(2) Gauge invariants of finite-dimensional Hopf algebras: It is generally difficult to decide whether two given finite-dimensional Hopf algebras have monoidally inequivalent representation categories. The PI expects to discover some invariants under Drinfeld twists, which are generalizations of Frobenius-Schur indicators to arbitrary finite-dimensional Hopf algebras. (3) Arithmetic properties of Frobenius-Schur indicators: The arithmetic properties of Frobenius-Schur indicators have led to the quantum Cauchy theorem for integral fusion categories and a congruence subgroup theorem for modular tensor categories. The PI will continue to explore these properties and their applications to fusion categories.The appeal of symmetry has been guidance for understanding the nature of science. Some symmetry can be described in terms of algebraic structures. For instance, the symmetry of platonic solids can be described by some finite groups of spatial rotations. The symmetry of some physical systems can be described by Hopf algebras, and tensor categories, which are closely related to other areas of mathematics such as representation theory, topological invariants, and conformal field theory. The proposed research studies some fundamental questions in the field, with an eye on applications in other areas.

主要研究者建议研究霍普夫代数，模张量范畴，以及它们的一些不变量称为Frobenius-Schur指标。具体而言，将研究以下专题。(1)小维Hopf代数的分类：理解任何数学理论中的一些基本例子是非常重要的。PI将继续致力于其维数允许简单因子分解的Hopf代数的分类。(2)有限维霍普夫代数的规范不变量：通常很难决定两个给定的有限维霍普夫代数是否具有么半群不等价的表示范畴。PI期望发现Drinfeld扭曲下的一些不变量，这些不变量是Frobenius-Schur指标到任意有限维Hopf代数的推广。(3)Frobenius-Schur指标的算术性质：Frobenius-Schur指标的算术性质导致了积分融合范畴的量子柯西定理和模张量范畴的同余子群定理。PI将继续探索这些性质及其在聚变范畴中的应用。对称性的吸引力一直是理解科学本质的指导。一些对称性可以用代数结构来描述。例如，柏拉图立体的对称性可以用空间旋转的有限群来描述。一些物理系统的对称性可以用霍普夫代数和张量范畴来描述，它们与数学的其他领域如表示论、拓扑不变量和共形场论密切相关。拟议的研究研究领域的一些基本问题，着眼于在其他领域的应用。