Semisimple Hopf algebras: Their representations and actions

这个项目的主要部分是关于半简单Hopf代数表示的某些不变量。这些不变量，即Frobenius-Schur指示器和它们的高级类似物，不仅应该给出Hopf代数的表示，而且应该给出其可能的结构的信息，因此可能有助于分类程序。它们还可以作为一种工具，通过寻找表征理论证明来证明关于Hopf代数的更一般的事实。本课题还研究了Hopf代数对非交换代数的作用，特别是在这种作用下Jacobson根或素根何时是稳定的；这些问题与素数理想和半直积的根有关，有助于对Hopf代数进行“不变理论”。最后，我们将讨论Hopf代数的扭曲及其对代数的作用，以及更一般的关于扭曲保留了哪些代数性质的问题。Hopf代数之所以引起人们的兴趣，不仅是因为它们本身，还因为它们在其他数学结构中作为不变量出现，比如理解几何中的结，以及在数学物理的某些领域，比如共形场论。因此，更好地理解Hopf代数本身可能最终会在这些领域中有所帮助。该项目还将涉及一些参与南加州大学女性科学与工程（WiSE）项目的女学生。

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.