Finite-dimensional Hopf algebras and their representations

本计划将研究某些Hopf代数的结构和表示，特别是那些由有限群构造的Hopf代数。 一个主要的主题是关于Frobenius-Schur指标的表示。这些指标是不变量的张量范畴表示的霍普夫代数，并可以扩展到更一般的对象。他们应该提供的信息不仅代表了一个霍普夫代数，但他们也作为一种工具，在证明更一般的事实本身的霍普夫代数。许多问题的解决方法都涉及到有限群理论，而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。然而，从那时起，他们已经出现在其他部分的数学，如几何（纽结理论），并在几个部分的数学物理（共形场论和统计力学）。在许多情况下，Hopf代数可以用来给出这些结构的不变量，这有助于理解和分类它们。因此，它们具有普遍意义。

This proposal will study the structure and representations of certain Hopf algebras, especially those Hopf algebras constructed from finite groups. A major topic concerns the Frobenius-Schur indicators of the representations. These indicators are invariants of the tensor category of representations of the Hopf algebra, and can be extended to more general objects. They should give information not only on the representations of a Hopf algebra, but they also serve as a tool in proving more general facts about Hopf algebras themselves. The methods for many of the problems involve the theory of finite groups.Hopf algebras are special algebraic objects which arose in topology and algebraic groups in the 1940's and 1950's. However since then they have appeared in other parts of mathematics, such as geometry (knot theory), and in several parts of mathematical physics (conformal field theory and statistical mechanics). In many cases Hopf algebras can be used to give invariants of these structures which help to understand and classify them. Thus they are of general interest.