Quantum symmetry in tensor categories

1. 给出了半简单张量范畴的多边形表示。该结果被用于描述半简单张量范畴上的对偶结构。给出了有限群张量范畴的对称性，并将其应用于张量范畴的轨道构造。对于阿贝尔群，我们进一步构造了第二轨道，并建立了轨道的对偶性，其中包含了著名的ad对偶性。作为张量范畴中刚性的一种变化，我们提出了Frobenius互易的概念，并推导了关于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是等价的。将张量范畴的轨道构造推广到有限维Hopf代数表示所支配的对称，使对偶性保持有效。根据张量范畴中分裂Frobenius代数的概念，进一步扩展了轨道的对偶原理。轨道结构在这里被解释为在相关的范畴Frobenius代数的作用下采取双模扩展。

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.