Quantum symmetry in tensor categories

张量范畴中的量子对称性

• 批准号: 12640201
• 负责人: YAMAGAMI Shigeru
• 金额: $ 1.73万
• 依托单位: IBARAKI UNIVERSITY
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 2000
• 资助国家: 日本
• 起止时间: 2000 至 2002
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-12640201/
• 关键词: tensor categories; orbifold; Hopf algebras; Frobenius algebra; bimodule; duality; 五角形方程式; 自由積; スペクトル流; 平面代数; 分岐則代数

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.

1.给出了半单张量范畴的多边形表示。然后将结果用于描述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中的对称构造。对于阿贝尔群的情形，我们进一步建立了第二轨道对偶，并建立了包含著名的AD-对偶作为特例的第二轨道对偶。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，并导出了循环张量积的各种公式。作为应用，证明了子因子论中的组合结构等价于互易。本文将张量范畴的对偶结构推广到由有限维Hopf代数的表示所支配的对称性，从而保持对偶性的有效性。通过在张量范畴中分裂Frobenius代数的概念，进一步推广了orbilold的对偶原理。在这里，orbillold结构被解释为带有相关范畴Frobenius代数的作用的双模扩张。

项目成果

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Shigeru Yamagami: "Tannaka duals in semisimple tensor categories"J. Algebra. 253. 350-391 (2002)

Shigeru Yamagami：“半单张量范畴中的 Tannaka 对偶”J.