Quantum symmetry in tensor categories

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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.