Representations and actions of finite dimensional Hopf algebras

这个项目的主要部分涉及表示的半单Hopf代数，特别是Frobenius-Schur指标的表示。指标的值不仅应该给出关于Hopf代数的表示范畴的信息，而且还应该给出关于Hopf代数本身的结构的信息。一个相关的有用的不变量是一个Hopf代数的对极的迹。该项目还关注的行动的霍普夫代数上的非交换代数，特别是当雅各布森或素根是稳定的，在这样的行动。一个基本的问题是关系的频谱的原始理想，其稳定的模拟，并原始理想的一个霍普夫半直积。 这里的进展将有助于做“不变理论”的霍普夫代数。Hopf代数是20世纪40年代和50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，如几何学（纽结理论）和数学物理学（共形场论）。经常Hopf代数给出这些结构的不变量。因此，对Hopf代数本身的更深入理解最终可能在这些领域有用。该项目将涉及几名女学生，作为南加州大学妇女参与科学和工程方案的一部分。

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.