Classification of Hopf algebras and quantum groups by tensor equivalences

通过张量等价对 Hopf 代数和量子群进行分类

• 批准号: 12640008
• 负责人: MASUOKA Akira
• 金额: $ 1.15万
• 依托单位: University of Tsukuba
• 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 2001
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-12640008/
• 关键词: Hopf algebra; Quantum group; Tensor equivalence; Cocfuasi-bialgebra; Crossed product; Hopf cohomology; Hochschild cohomology; 組紐構造; ガロア対象

Our new results are divided roughly into two. First, we have classified some range of coquasi-bialgebras, which generalize Hopf algebras, up to tensor equivalences of their comodule categories ; this was done by constructing related cohomology groups, in a very concrete way with use of braid diagrams. Second, we have obtained a new result on crossed products, and thereby proved in a unified way three famous theorems for affine group schemes ; this was based on the idea that the Hopf cohomology, which is non-abelian and multiplicative, can be approximated, in some sense, by the HochschiId cohomology, which is abelian and additive.The results were annonced at the following conferences.The 46-th Symposium on Algebra, Osaka Univ., July 30-August 2, 2001.Non-commutative Geometry and Quantum Groups, Banach Center (Warsaw), September 17-29, 2001.

我们的新结果大致分为两部分。首先，我们对一些 coquasi-bialgebras 进行了分类，这些 coquasi-bialgebras 概括了 Hopf 代数，直到其余模类别的张量等价；这是通过使用辫子图以非常具体的方式构建相关的上同调群来完成的。其次，我们在交叉积上得到了新的结果，从而统一证明了仿射群方案的三个著名定理；这是基于这样的想法，即非阿贝尔和乘法的 Hopf 上同调在某种意义上可以通过阿贝尔和加法的 HochschiId 上同调来近似。结果在以下会议上公布。 46- th Symposium on Algebra，大阪大学，2001 年 7 月 30 日至 8 月 2 日。Non-commutative Geometry and Quantum团体，巴纳赫中心（华沙），2001 年 9 月 17-29 日。

项目成果

Jun MORITA and Eugene PLOTKIN: "Prescribed Gauss decompositions for Kac-Moody aroups over fields"Rend. Sem. Mat. Univ. Padova. 106. 153-163 (2001)

Jun MORITA 和 Eugene PLOTKIN：“Kac-Moody 域上的规定高斯分解”Rend。

Mitsuhiro Takeuchi: "Survey of braided Hopf algebras"Contemporary Mathematics. 267. 301-323 (2000)

竹内光宏：《辫状Hopf代数综述》当代数学。

Kenichiro TANABE: "A new proof of the Assmus-Mattson theorem for non-binary codes"Design, Codes and Cyptography. 22. 149-155 (2001)

Kenichiro TANABE：“非二进制代码的 Assmus-Mattson 定理的新证明”设计、代码和密码学。

森田 純: "Kac-Moody群講義"上智大学数学教室. 116 (2001)

森田淳：“Kac-Moody小组讲座”上智大学数学系116（2001）。

R.Moody & Jun Morita: "Positivity for K_1 and K_2"Journal of Algebra. 229. 1-24 (2000)

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