A new quality theorem for tensor category and the equivarence of topological quantum field theories

张量范畴的新质量定理与拓扑量子场论的等价性

• 批准号: 10640019
• 负责人: HAYASHI Takahiro
• 金额: $ 2.05万
• 依托单位: Graduate School of Mathematics, Nagoya University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 1998
• 资助国家: 日本
• 起止时间: 1998 至 2001
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-10640019/
• 关键词: quantum group; tensor category; Tannaka duality; classical invariant theory; inverse matrix; Littlewood-Richardson 則; Littlewood-Richardson則; 6j-symbol; 位相的場の量子論

For each finite split semisimple tensor category C, the canonical Tannaka duality gives a quantum group (face algebra) whose comodule category is equivalent to C. The duality gives a unified understanding of tensor categories arising from mathematics and physics, and also, it gives a new picture of the representation theory of the ordinary groups. By applying the duality and other techniques in the quantum group theory, we obtained the following results.1. We showed that the quantum ej-symbo, is a sum of the partition functions of the ABF mode, of finite size. Also we gave a simple summation formula for the ordinary 6j-symbols.2. By using the duality (or rather the canonical fiber functor), we constructed a bases for invariants and semiinvariants of binary quadratics and binary cubics.3. For each tensor products of two irreducible representations of the general linear group, we gave their explicit irreducible decomposition at the module level.4. We classified the braiding and th'e ribbon structure on each quantum classica, groups and the tensor category of type A of level L.5. For each linearly reductive matrix group G, we give an "economical" inverse matrix formula for each elements of G.

对于每个有限分裂半单张量范畴 C，规范的 Tannaka 对偶性给出了一个量子群（面代数），其余模范畴相当于 C。对偶性给出了对数学和物理产生的张量范畴的统一理解，并且还给出了普通群体表征论的新图景。应用量子群论中的对偶性等技术，我们得到了以下结果： 1．我们证明了量子 ej-symbo 是有限大小的 ABF 模式的配分函数之和。我们还给出了普通6j-符号的简单求和公式。2．利用对偶性（或者更确切地说是规范纤维函子），我们构造了二元二次和二元三次不变量和半不变量的基。 3．对于一般线性群的两个不可约表示的每个张量积，我们在模块级别给出了它们的显式不可约分解。 4．我们对每个量子经典、群和 L.5 级 A 型张量范畴上的编织和带状结构进行了分类。对于每个线性约简矩阵组 G，我们为 G 的每个元素给出一个“经济”的逆矩阵公式。

项目成果

T.Hayashi: "Coribbon Hopf (face) algebras generated by lattice models"Journal of Algebra. 233. 614-641 (2000)

T.Hayashi：“由晶格模型生成的 Coribbon Hopf（面）代数”代数杂志。

Takahiro Hayashi: "Galois quantum groups of II_1-subfactors"Tohoku Mathematical Journal . 51. 365-389 (1999)

Takahiro Hayashi：“II_1-子因子的伽罗瓦量子群”东北数学杂志。

Takahiro Hayashi: Communications in Mathematical Physics.

Takahiro Hayashi：数学物理通讯。

T. Hayashi: "Coribbon Hopf (face) algebras generated by lattice models"Journal of Algebra. 233. 614-641 (2000)

T. Hayashi：“由晶格模型生成的 Coribbon Hopf（面）代数”代数杂志。

T. Hayashi: "Galois quantum groups of II-subfactors"Tohoku Mathematical Journal. 51. 365-389 (1999)

T. Hayashi：“II 子因子的伽罗瓦量子群”东北数学杂志。