Finite-dimensional Hopf algebras and their representations

有限维 Hopf 代数及其表示

• 批准号: 1301860
• 负责人: Susan Montgomery
• 金额: $ 17.98万
• 依托单位: University of Southern California
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2013
• 资助国家: 美国
• 起止时间: 2013-06-01 至 2017-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1301860&HistoricalAwards=false
• 关键词: Finite; dimensional; Hopf; algebras; their

This proposal will study the structure and representations of certain Hopf algebras, especially those Hopf algebras constructed from finite groups. A major topic concerns the Frobenius-Schur indicators of the representations. These indicators are invariants of the tensor category of representations of the Hopf algebra, and can be extended to more general objects. They should give information not only on the representations of a Hopf algebra, but they also serve as a tool in proving more general facts about Hopf algebras themselves. The methods for many of the problems involve the theory of finite groups.Hopf algebras are special algebraic objects which arose in topology and algebraic groups in the 1940's and 1950's. However since then they have appeared in other parts of mathematics, such as geometry (knot theory), and in several parts of mathematical physics (conformal field theory and statistical mechanics). In many cases Hopf algebras can be used to give invariants of these structures which help to understand and classify them. Thus they are of general interest.

该建议将研究某些HOPF代数的结构和表示，尤其是由有限群体构建的HOPF代数。 一个主要主题涉及该代表的Frobenius-Schur指标。这些指标是Hopf代数的表示形式的张量类别的不变性，并且可以扩展到更通用的对象。他们不仅应该提供有关HOPF代数的表示信息的信息，而且还应该作为证明有关Hopf代数本身的更一般事实的工具。许多问题的方法涉及有限群体的理论。HOPF代数是特殊的代数对象，在1940年代和1950年代在拓扑和代数组中出现。然而，从那以后，它们出现在数学的其他部分，例如几何学（结理），以及数学物理学的几个部分（共形场理论和统计力学）。在许多情况下，Hopf代数可用于提供这些结构的不变，这些结构有助于理解和对它们进行分类。因此，它们具有普遍的兴趣。