Hopf Algebras and Their Actions on Rings

对于DMS-0401399(1)， PI将研究Hopf伽罗瓦扩展中的素数理想；这种扩展包括Hopf代数的交叉积，以及由商得到的量子群的某些扩展。她还将考虑有限维半简单Hopf代数的表示，对不可约表示的Frobenius-Schur指示特别感兴趣。这些问题的进展可能对分类程序有所帮助。PI还将研究taft Hopf代数在量子多项式环上的可能作用。最后，PI将研究Hopf代数和Hopf代数被一个环扭转时所作用的代数会发生什么。Hopf代数是在其他学科中出现的代数对象，包括数学本身（如结理论和几何）和数学物理（如统计力学和共形场论）。如果能对Hopf代数进行分类，将有助于描述这些其他结构。此外，霍普夫代数为群、李代数及其最近的量子类似性提供了统一框架。因此，研究Hopf代数本身以及它们对其他结构的作用是一个重要的问题。

Abstract for DMS-0401399(1) The PI will study the prime ideals in a Hopf Galois extension; suchextensions include crossed products of Hopf algebras, as well as certainextensions of quantum groups obtained from quotients. She will also considerrepresentations of finite-dimensional semisimple Hopf algebras, withparticular interest in the Frobenius-Schur indicator of an irreduciblerepresentation. Progress on such problems might be of help in theclassification program. The PI will also study the possible actions of theTaft Hopf algebras on quantum polynomial rings. Finally the PI will studywhat happens to a Hopf algebra and an algebra on which it acts when the Hopfalgebra is twisted by a cocycle.(2) Hopf algebras are algebraic objects which have arisen in other subjects,both in mathmatics itself (such as in knot theory and geometry) and inmathematical physics (such in statistical mechanics and conformal fieldtheory). If one could classify certain Hopf algebras, it would help indescribing these other structures. Moreover Hopf algebras provide aunifying framework for groups, Lie algebras, and their recent quantumanalogs. Thus studying Hopf algebras themselves as well as their actionson other structures is an important problem.