Semisimple Hopf algebras: Their representations and actions

半简单 Hopf 代数：它们的表示和作用

• 批准号: 0701291
• 负责人: Susan Montgomery
• 金额: $ 21.13万
• 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: 2007
• 资助国家: 美国
• 起止时间: 2007-07-01 至 2011-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0701291&HistoricalAwards=false
• 关键词: Semisimple; Hopf; algebras; Their; representations

The main part of this project concerns certain invariants of the representations of semisimple Hopf algebras. These invariants, the Frobenius-Schur indicators and their higher analogs, should give information not only about the representations of a Hopf algebra, but also about its possible structure and so may help in the classification program. They should also serve as a tool in proving more general facts about Hopf algebras, by looking for representation-thoeretic proofs. The project also concerns actions of Hopf algebras on non-commutative algebras, especially as to when the Jacobson or prime radicals are stable under such actions; these questions are related to prime ideals and the radical of semidirect products, and would help in doing ``invariant theory'' for Hopf algebras. Finally some questions will be considered about twistings of Hopf algebras and of their actions on algebras, as well as more general questions about which algebra properties are preserved by twisting. Hopf algebras are of interest not only for their own sake, but because they appear as invariants in other mathematical structures, such as understanding knots in geometry, and in parts of mathematical physics, such as conformal field theory. Thus a greater understanding of the Hopf algebras themselves might eventually be useful in these areas. This project will also involve some women students involved with the Women in Science and Engineering (WiSE) program at USC.

这个项目的主要部分涉及某些不变量的表示半单Hopf代数。这些不变量，Frobenius-Schur指标和它们的高级类似物，不仅应该提供有关Hopf代数表示的信息，而且还应该提供有关其可能结构的信息，因此可能有助于分类程序。他们也应该作为一种工具，在证明更一般的事实，霍普夫代数，通过寻找代表thoeretic证明。该项目还关注了Hopf代数在非交换代数上的作用，特别是当Jacobson或素根在这种作用下是稳定的;这些问题与素理想和半直积的根有关，并将有助于为Hopf代数做“不变理论”。最后将考虑一些问题的扭曲的霍普夫代数和他们的行动对代数，以及更一般的问题，其中代数性质的保留扭曲。霍普夫代数之所以令人感兴趣，不仅是因为它们本身，而且因为它们在其他数学结构中作为不变量出现，例如理解几何中的结，以及数学物理的部分，例如共形场论。因此，对Hopf代数本身有更深入的了解最终可能在这些领域中有用。该项目还将涉及南加州大学妇女参与科学和工程方案的一些女学生。