Representations and actions of finite dimensional Hopf algebras

这个项目的主要部分是关于半简单Hopf代数的表示，特别是其表示的Frobenius-Schur指标。这些指标的值不仅要给出Hopf代数的表示范畴的信息，还要给出Hopf代数本身的结构信息。一个相关的有用的不变量是Hopf代数对正的迹。研究了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。一个基本问题是原始理想的谱与其稳定类似物的关系，以及与Hopf半直积的原始理想的关系。这方面的进展将有助于研究Hopf代数的“不变理论”。Hopf代数是20世纪四五十年代在拓扑学和代数群中出现的一种特殊的代数对象。最近，它们出现在数学的其他部分，如几何（结理论）和数学物理（共形场论）。Hopf代数通常给出这些结构的不变量。因此，更好地理解Hopf代数本身可能最终在这些领域有用。该项目将涉及几名女学生，作为南加州大学科学与工程女性（WiSE）项目的一部分。

The main part of this project concerns representations of a semisimple Hopf algebra, particularly the Frobenius-Schur indicators of its representations. The values of the indicators should give information not only about the representation category of a Hopf algebra, but also about the structure of the Hopf algebra itself. A related useful invariant is the trace of the antipode of a Hopf algebra. The project also concerns actions of Hopf algebras on non-commutative algebras, especially when the Jacobson or prime radical is stable under such an action. A basic problem is the relationship of the spectrum of primitive ideals to its stable analog, and to the primitive ideals of a Hopf semidirect product. Progress here would help in doing "invariant theory" for Hopf algebras. Hopf algebras are special algebraic objects which arose in topology and algebraic groups in the 1940's and 1950's. More recently they have appeared in other parts of mathematics, such as geometry (knot theory), and in mathematical physics (conformal field theory). Frequently Hopf algebras give invariants of these structures. Thus a greater understanding of Hopf algebras themselves may eventually be useful in these areas. This project will involve several women students as part of the Women in Science and Engineering (WiSE) program at USC.