Ocneanu数量对称的代数与二维共形系统的分类
L'algebre des symetries quantiques d'Ocneanu et la classification des systemes conformes a 2D
摘要：
一个二维共形系统的配分函数-模不变函数或广义配分函数，来自缺陷线的引入-被表示为一组具有特殊性的系数，以形成某些代数的nimreps。这些系数定义了一类新的Hopf代数的各种结构映射，称为弱Hopf代数，并且可以编码在一组图中。第一章的目的是介绍关于这一主题的实际知识。第二章介绍了在二维共形系统研究中起关键作用的弱Hopf代数及其结构，特别是量子对称的Ocneanu代数。我们详细分析了这些结构的A3图，与仿射su（2）共形系统。在第三章中，我们给出了量子对称性的Ocneanu代数的一个实现，它被构造为一个图代数（仿射（2）模型的ADE图）的张量平方的商。这种实现允许获得用于确定与共形模型相关联的配分函数的非常简单的算法。我们的构造可以自然地推广到仿射su（n）情形，其中n > 2，其中很少有结果已知。第四章详细讨论了所有与su（2）-型相关的情形以及su（3）-型的三个特例。
英文摘要：
The partition functions of a 2D conformal system - the modular invariant one or the generalized ones, coming from the introduction of defect lines - are expressed in terms of a set of coefficients that have the particularity to form nimreps of certain algebras. These coefficients define the various structure maps of a new class of Hopf Algebras, called Weak Hopf Algebras, and can be encoded in a set of graphs. The aim of chapter 1 is a presentation of the actual knowledge on this subject. In chapter 2 the Weak Hopf Algebra together with its structures is introduced, in particular the Ocneanu algebra of quantum symmetries, that play a key role in the study of 2d conformal systems. We analyze in details these structures for the A3 diagram, associated to the affine su(2) conformal system. In chapter 3, we present a realization of the Ocneanu algebra of quantum symmetries, constructed as a quotient of the tensor square of a graph algebra (ADE graph for the affine (2) model). This realization allows obtaining a very simple algorithm for the determination of the partition functions associated with the conformal model. Our construction can be naturally generalized to the affine su(n) cases, with n > 2, where few results were known. All the cases related of su(2)-type as well as three particular examples of su(3)-type are explicitly treated in chapter 4.
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