Quantum symmetry in tensor categories

1.一个多边形表示制定半单张量范畴。然后利用所得结果描述了半单张量范畴上的对偶结构.给出了有限群的张量范畴的对称性，并将其应用于张量范畴的轨道构造。对于交换群的情形，我们进一步构造了第二轨道折叠，并建立了轨道折叠的对偶，其中包含了著名的AD-对偶.作为张量范畴刚性的一种变化，我们提出了Frobenius互易性的概念，并导出了循环张量积的各种公式。作为应用，证明了子因子理论中的组合结构与互反性是等价的.将张量范畴的轨道结构推广到有限维Hopf代数表示所支配的对称性，使得对偶性仍然成立.利用张量范畴中Frobenius代数的分裂概念，进一步推广了轨道折叠的对偶原理。轨道结构在这里被解释为取双模扩张与相关的范畴Frobenius代数的作用。

1. A polygonal presentation is formulated for semisimple tensor categories. The result is then used to describe duality structures on semisimple tensor categories.2. Symmetries of tensor categories are formulated for finite groups, which is applied to perform orbifold constructions in tensor categories. For the case of abelian groups, we further formulate the second orbifolds and have established the duality for orbifolds, which contains the famous AD-duality as a special case.3. As a variation of rigidity in tensor categories, we have formulated the notion of Frobenius reciprocity and derived various formulas on cyclic tensor products. As an application, the combinatorial structure in subfactor theory is proved to be equivalent to the reciprocity.4. The orbifold construction for tensor categories is extended to the symmetry governed by the representation of finite-dimensional Hopf algebras so that the duality remains valid.5. The duality principle for orbifolds is further extended in terms of the notion of splitting Frobenius algebras in tensor categories. The orbifold construction is here interpreted as taking bimodule extensions with actions of relevant categorical Frobenius algebras.