Hopf Algebras and Their Actions on Rings

Hopf 代数及其在环上的作用

• 批准号: 0401399
• 负责人: Susan Montgomery
• 金额: $ 10.5万
• 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: 2004
• 资助国家: 美国
• 起止时间: 2004-09-01 至 2007-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0401399&HistoricalAwards=false
• 关键词: Hopf; Algebras; Their; Actions; Rings

Abstract for DMS-0401399(1) The PI will study the prime ideals in a Hopf Galois extension; suchextensions include crossed products of Hopf algebras, as well as certainextensions of quantum groups obtained from quotients. She will also considerrepresentations of finite-dimensional semisimple Hopf algebras, withparticular interest in the Frobenius-Schur indicator of an irreduciblerepresentation. Progress on such problems might be of help in theclassification program. The PI will also study the possible actions of theTaft Hopf algebras on quantum polynomial rings. Finally the PI will studywhat happens to a Hopf algebra and an algebra on which it acts when the Hopfalgebra is twisted by a cocycle.(2) Hopf algebras are algebraic objects which have arisen in other subjects,both in mathmatics itself (such as in knot theory and geometry) and inmathematical physics (such in statistical mechanics and conformal fieldtheory). If one could classify certain Hopf algebras, it would help indescribing these other structures. Moreover Hopf algebras provide aunifying framework for groups, Lie algebras, and their recent quantumanalogs. Thus studying Hopf algebras themselves as well as their actionson other structures is an important problem.

DMS-0401399摘要（1）PI将研究Hopf Galois扩张中的素理想;这样的扩张包括Hopf代数的交叉积，以及从量子群中获得的某些扩张。她还将讨论有限维半单霍普夫代数的表示，特别是对不可约表示的Frobenius-Schur指标感兴趣。在这些问题上的进展可能对分类计划有所帮助。PI还将研究量子多项式环上的Taft Hopf代数的可能作用。最后，PI将研究当Hopfalgebra被一个上循环扭曲时，一个Hopf代数和一个它所作用的代数会发生什么。(2)霍普夫代数是在其他学科中出现的代数对象，包括数学本身（如纽结理论和几何学）和数学物理（如统计力学和共形场论）。如果能对某些Hopf代数进行分类，将有助于描述其他结构。此外，霍普夫代数提供了一个统一的框架群，李代数，和他们最近的quantumanalogs。因此研究Hopf代数本身以及它们在其它结构上的作用是一个重要的问题。