Semisimple Hopf algebras: Their representations and actions

这个项目的主要部分是关于半单Hopf代数表示的某些不变量。这些不变量，Frobenius-Schur指示符及其更高的类似物，不仅应该给出关于Hopf代数的表示的信息，而且应该给出关于它可能的结构的信息，因此可能有助于分类程序。它们还应该作为工具，通过寻找表示论证明来证明关于Hopf代数的更一般的事实。该项目还研究了Hopf代数在非交换代数上的作用，特别是关于Jacobson根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，有助于做Hopf代数的‘不变理论’。最后，我们将讨论一些关于Hopf代数的扭曲及其在代数上的作用的问题，以及更一般的关于通过扭曲保持哪些代数性质的问题。Hopf代数不仅因为它们本身的原因而感兴趣，而且因为它们在其他数学结构中作为不变量出现，例如在几何中理解纽结，以及在数学物理的部分中，例如保形场论。因此，对Hopf代数本身的更好的理解最终可能在这些领域中有用。该项目还将让一些女学生参与南加州大学的女性科学与工程(WISE)计划。

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.