Algebraic structures and analysis in tensor categories and quantum groupoids

[1]一类重要的张量范畴作为有限维Hopf代数的表示范畴而产生。本文在张量范畴的框架下研究了它的意义，将其归结为张量范畴中的Frobenius代数的概念，并利用张量范畴中的Frobenius代数双模建立了量子对称性的对偶.主要结果发表在Fields Institute Commun.2上。我们研究了基于Kauffman提出的Temperley-Lieb范畴，并阐明了所有已知的结构结果是如何从平面弦的初等操作中得出的。我们试图确定Temperley-Lieb范畴上的纤维函子，结果是纤维函子由非-退化的双线性型的相似等价，这原来是与已知的旧定理。如果我们仅限于酉纤维函子的情形，我们可以用核矩阵的谱数据来改写条件，这就恢复了关于相伴紧四元群表示的已知理论.

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.