Algebraic structures and analysis in tensor categories and quantum groupoids

一类重要的张量范畴是有限维Hopf代数的表示范畴。我们在这里研究了它在张量范畴框架中的意义，这可以归结为张量范畴中的Frobenius代数的概念。然后用张量范畴中的Frobenius代数双模来表述量子对称上的对偶。主要结果发表在菲尔兹研究所公报上。2我们在考夫曼的报告的基础上研究了坦波利-利布类别，并澄清了所有已知的结构结果是如何从平面弦的基本操作中得出的。作为上述几何分析的应用，我们尝试在Temperley-Lieb类上确定纤维函子。结果表明，纤维函子是由非退化双线性形式决定的，直至相似等价，这与已知的旧定理有关。如果我们把自己限制在酉纤维函子的情况下，我们可以用核矩阵的谱数据重写这些条件，从而恢复了已知的关于关联紧量子群表示的理论。

1An important class of tensor categories arises as representation categories of finite-dimensional Hopf algebras. We here have studied its significance in the framework of tensor categories, which is boiled down to the notion of Frobenius algebra in tensor categories.A duality on quantum symmetry is then formulated in terms of Frobenius algebra bimodules in a tensor categories. The main result is published in Fields Institute Commun.2We have investigated Temperley-Lieb categories based on Kauffman's presentation and clarified how all the known structural results follow from elementary manipulations of planar strings.3As applications of the above mentioned geometrical analysis, we have tried to determine fiber functors on Temperley-Lieb categories.The result is that fiber functors are determined by non-degenerate bilinear forms up to the similarity equivalence, which turns out to be related with the known old theorems. If we restrict ourselves to the case of unitary fiber functors, we can rewrite the conditions in terms of spectral data of kernel matrices, which recovers the known theory on representations of the associaited compact quatum groups.