Hopf algebras and quantum groups

对称性的研究是数学和物理科学的核心，对称性是用群体的数学语言来表示的。更微妙的对称形式需要更复杂的数学概念。其中之一是量子对称性的概念，例如，不是简单的ab等于ba的关系，我们有ab是qba，换句话说，ab和ba等于q的因子。这个概念在物理世界中是自然存在的。 该项目将继续研究Hopf代数和量子群的代数理论。Hopf代数是由H.Hopf在20世纪40年代的S工作中命名的，40年前由Milnor和Moore、Sweedler、Kaplansky和Abe等几位环论学者研究。但是，在过去的二十年里，自从V.G.德林费尔在国际数学家大会上发表演讲，描述L.D.法迪耶夫及其物理学派的工作时提到量子群以来，人们对量子群以及霍普夫代数代数理论的一般领域的认识和兴趣都有了很大的提高。因此，一个正在进行的问题是描述给定类型的所有Hopf代数--点的、余的、半单的等等--以便理解这些Hopf代数自然出现的与物理学和其他数学分支(如纽结理论)的联系。

The study of symmetry is central to the mathematical and physical sciences and symmetry is expressed mathematically in the language of groups. More subtle forms of symmetry require more sophisticated mathematical notions. One of these is the notion of quantum symmetry, so that for example, rather than having a simple relation of the form ab equals ba, we have that ab is qba, in other words ab and ba are equal up to a factor of q. This notion occurs naturally in the physical world. This project will continue research in the algebraic theory of Hopf algebras and quantum groups. Hopf algebras, named from work of H. Hopf in the 1940's, were studied by a few ring theorists such as Milnor and Moore, Sweedler, Kaplansky and Abe forty years ago. But, in the last twenty years, since V.G.Drinfel'd spoke of quantum groups in his address to the International Congress of Mathematicians, describing work of L.D.Faddeev and his school of physics, there has been greatly increased awareness of and interest in the study of quantum groups and in the general area of the algebraic theory of Hopf algebras. An ongoing problem then, is to describe all Hopf algebras of a given type-- pointed, copointed, semisimple, etc.-- in order to understand the links with physics and with other branches of mathematics such as knot theory, in which these Hopf algebras appear naturally.