Finite-dimensional Hopf algebras and their representations

本文将研究一类Hopf代数的结构和表示，特别是那些由有限群构造的Hopf代数。一个主要的话题涉及表征的Frobenius-Schur指标。这些指标是Hopf代数表示的张量范畴的不变量，并且可以扩展到更一般的对象。它们不仅应该提供关于Hopf代数表示的信息，而且还可以作为证明关于Hopf代数本身的更一般事实的工具。许多问题的解决方法都涉及到有限群理论。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。然而，从那时起，它们出现在数学的其他部分，如几何（结理论）和数学物理的几个部分（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，它们具有普遍的意义。

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.