Hopf algebras and their applications to quantum groups

Hopf 代数及其在量子群中的应用

• 批准号: 18540008
• 负责人: TAKEUCHI Mitsuhiro
• 金额: $ 2.66万
• 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: 2006
• 资助国家: 日本
• 起止时间: 2006 至 2009
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-18540008/
• 关键词: ホップ代数; 量子群; ガロア理論; FRT構成; bialgebroid, Picard-Vessiot理論; 組紐カテゴリー; bialgebroid; 環論; 森田理論; 余加群代数; 結晶基

Hopf algebras and quantum algebra theory are applied in the study of quasi-triangular Hopf algebras, braided categories and knot invariants. In this proposal, we have studied the following topics as well as reviewing the above studies from a new viewpoint.(1) We have studied H Morita theory, i.e., the theory of Morita equivalences between various Hcomodule algebras with a fixed Hopf algebra H from the viewpoint of quantization of Galois theory, and published a joint paper with Caenepeel, Crivei and Marcus in the Journal of Algebra.(2) We have studied Picard-Vessiot theory, i.e., Galois theory using algebraic groups (or commutative Hopf algebras) instead of usual groups and its generalization towards Galois theory using quantum groups. As its preliminaries we have studied Hopf algebraic approach to the Picard-Vessiot thery and published a joint paper with Katsutoshi Amano and Akira Masuoka in the Handbook of Algebra.(3) The principal investigator (Takeuchi) introduced the concept of bialgebroids in 1977. This concept is recently studied very actively by Boem, Brzezinski and so on. We have studied the FRT (Faddeev, Reshetikhin, Takhtajan) construction which plays an mportant role in the quantum group theory in the frame work of bialgebroids and published a joint paper with Youichi Shibukawa, Hokkaido University in the Journal of Algebra.

Hopf代数和量子代数理论应用于拟三角Hopf代数、编织范畴和结不变量的研究。在本提案中，我们研究了以下主题，并从新的角度回顾了上述研究。（1）我们研究了H Morita理论，即具有固定Hopf代数H的各种Hco模代数之间的Morita等价理论提出了伽罗瓦理论的量子化观点，并与Caenepeel、Crivei和Marcus在Journal of Algebra上发表了联合论文。（2）我们研究了Picard-Vessiot理论，即使用代数群（或交换霍普夫代数）而不是通常群的伽罗瓦理论及其对使用量子群的伽罗瓦理论的推广。作为其初步研究，我们研究了 Picard-Vessiot 理论的 Hopf 代数方法，并在代数手册中发表了与 Katsutoshi Amano 和 Akira Masuoka 的联合论文。(3) 首席研究员 (Takeuchi) 在 1977 年引入了双代数胚的概念。 Boem、Brzezinski 等人最近对这一概念进行了非常积极的研究。我们研究了在双代数胚框架下的量子群论中发挥重要作用的FRT（Faddeev，Reshetikhin，Takhtajan）构造，并与北海道大学的Youichi Shibukawa在《代数杂志》上发表了联合论文。

项目成果

コサイクル変形としての量子包絡環

作为共循环变形的量子包络环

• 发表时间: 2008
• 作者: Naito;Satoshi;Sagaki;Daisuke;増岡彰
• 通讯作者: 増岡彰

A lost mathematician, Takeo Nakasawa

迷失的数学家中泽武夫

• 发表时间: 2009
• 作者: H.Nishimura;S.Kuroda
• 通讯作者: S.Kuroda

A lost mathematician

迷失的数学家

• 发表时间: 2009
• 作者: H. Nishimura;S. Kuroda
• 通讯作者: S. Kuroda

Hopf algebraic approach to Picard-Vessiot theory

Picard-Vessiot 理论的 Hopf 代数方法

• 发表时间: 2009
• 期刊: Handbook of Algebra Vol. 6
• 作者: K. Amano;A. Masuoka;M. Takeuchi
• 通讯作者: M. Takeuchi

Synthetic vector analysis III. From vector analysis to differential forms

合成载体分析 III．

• 发表时间: 2009
• 期刊: Far East J.Math.Sci.(FJMS)
• 作者: K. Amano;A. Masuoka;M. Takeuchi;H. Nishimura
• 通讯作者: H. Nishimura