Algebraic structures and analysis in tensor categories and quantum groupoids

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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.