Representation and category theoretic aspects of logarithmic conformal field theories

物理学中对称结构的表示，在大量的例子中，是完全可约的。对数共形场理论是一类量子场理论，在统计力学和固体物理中有着广泛的应用；在这些理论中出现了不可分解但不是不可约的表征。一些相关的表示范畴在数学上被很好地理解，例如通过量子化的全称包络代数。一方面，我们建议使新的示例类显式可访问。另一方面，我们建议使用表征论和范畴论的工具来构建一般和类的物理相关量。我们有三个详细的目标论文项目：1。作为物理相关子候选的映射类群的作用的不变量的构造。在具体的例子中计算这些不变量，例如Nichols代数的表示范畴或群代数的“bad”特征。用分类量描述物理上相关的量，如边界条件或缺陷类型。

Representations of symmetry structures in physics are, in large classes of examples, completely reducible. Logarithmic conformal _eld theories are a class of quantum _eld theories with numerous applications in statistical mechanics and solid state physics; in these theories indecomposable but not irreducible representations arise.Some relevant representation categories are mathematically well-understood, e.g. via quantized universal envelopping algebras. We propose on the one hand side to make new classes of examples explicitly accessible. On the other hand, we propose to construct in general and in classes of examples physically relevant quantities using tools from representation theory and category theory.We have three detailed goals for dissertation projects:1. Construction of invariants of actions of mapping class groups which are candidates for physical correlators.2. Calculations of these invariants in concrete examples, e.g. representation categories of Nichols algebras or group algebras in \bad" characteristic.3. Description of physically relevant quantities like boundary conditions or defect types in terms of categorical quantities.