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，我们已经分类了一些范围的余拟双代数，它推广了Hopf代数，直到它们的余同调范畴的张量等价;这是通过构造相关的上同调群来完成的，以一种非常具体的方式使用辫子图。第二，得到了关于交叉积的一个新结果，从而统一证明了仿射群概型的三个著名定理;这是基于这样的思想，即非交换的和乘法的Hopf上同调在某种意义上可以近似为交换的和加法的HochemberId上同调。该结果在以下会议上公布：第46届代数研讨会，大坂大学，2001年7月30日至8月2日。非交换几何和量子群，巴拿赫中心（华沙），2001年9月17日至29日。

项目成果

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。

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。

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

穆迪

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）。