2002 Fiscal Year Final Research Report Summary
Quantum symmetry in tensor categories
| Project/Area Number |
12640201
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Global analysis
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| Research Institution | IBARAKI UNIVERSITY |
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Principal Investigator |
YAMAGAMI Shigeru Ibaraki Univ., college of Science, Professor, 理学部, 教授 (90175654)
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| Co-Investigator(Kenkyū-buntansha) |
OHTSUKA Fumiko Ibaraki Univ., college of Science, Associate Prof., 理学部, 助教授 (90194208)
FUJIWARA Takanori Ibaraki Univ., college of Science, Professor, 理学部, 教授 (50183596)
HIAI Fumio Tohoku Univ., Information Science, Professor, 情報科学研究科, 教授 (30092571)
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| Project Period (FY) |
2000 – 2002
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| Keywords | tensor categories / orbifold / Hopf algebras / Frobenius algebra / bimodule / duality |
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| Research Abstract |
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.
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Research Products
(12 results)
