Survey on matched pairs of groups-an elementary approach to the ESS-LYZ theory
Survey on matched pairs of groups-an elementary approach to the ESS-LYZ theory
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配对群体的调查——ESS-LYZ理论的基本方法
DOI：
10.4064/bc61-0-19
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发表时间：
2003
期刊：
影响因子：
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通讯作者：
M. Takeuchi
中科院分区：
文献类型：
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作者：
M. Takeuchi
本文综述了群的匹配对理论，并特别讨论了它在双砸积Hopf代数上的辫子群和拟三角结构中的应用。导言。1981年，在W·M·Singer关于∗⊗代数扩张的工作[Si]的启发下，当S是有限群[T]时，我引入了匹配对群的概念，并定义了域k上的双砸积Hopf代数(KS)Hopf代数kB。后来，S.Majid[Mj]将这一工作扩展并应用于量子群的研究。进一步，I.Hofstetter[H]和A.Masuoka[MS1]，[MS2]研究了匹配Hopf代数对的上同调理论。在这项研究中，Masuoka发现，配对群的概念基本上是由G.I.Kac[K]在1968年得到的，并在那里研究了配对群的上同调理论。最近，两组代数学家ESS(Etingof，Schedler，Soloviev)和Lyz(Lu，Yan，朱)在研究Yang-Baxter方程的集论解时，在群的匹配对的研究方面取得了一些有趣的进展。特别是，群ESS[ESS]在有限辫子集的分类方面得到了一些深刻的结果。在这项调查中，我对ESS-Lyz理论的引言部分给出了一个基本的和图解的方法。这将使读者能够更容易地欣赏他们的更深层次的结果(这里不讨论)。另一方面，后一群LYZ得到了双碰撞积Hopf代数H(S，B)=(KS)∗⊗kB[LYZ2]上拟三角结构的一些有趣的构造。我还根据Masuoka的观察[MS3]，对这个结构给出了一个自然的直截了当的解释。在§1中，我们回顾了群的匹配对(S，B)和匹配积B/S的概念，并用2000年数学学科分类：16W35，18D10的图表解释了匹配对的条件。这篇论文是最终版本，不会在其他地方发表。
The theory of matched pairs of groups is surveyed with special interest in applications to braided groups and quasitriangular structures on the bi-smash product Hopf algebra. Introduction. In 1981, I introduced the notion of a matched pair of groups (S,B) and defined the bi-smash product Hopf algebra (kS)∗ ⊗ kB over a field k, when S is a finite group [T], motivated by W. M. Singer’s work [Si] on Hopf algebra extensions. Later, S. Majid [Mj] extended and applied this work in the study of quantum groups. Further, the cohomology theory of matched pairs of Hopf algebras was studied by I. Hofstetter [H] and A. Masuoka [Ms1], [Ms2]. During this research, Masuoka found out that the notion of a matched pair of groups had been obtained essentially by G. I. Kac [K] in 1968 and that the cohomology theory of matched pairs of groups had been studied there. Recently, some interesting progress in the study of matched pairs of groups has been made by two groups of algebraists, ESS (Etingof, Schedler, Soloviev) and LYZ (Lu, Yan, Zhu), during their study of set-theoretical solutions of the Yang-Baxter equation. Especially, the group ESS [ESS] has obtained some deep results on classification of finite braided sets. In this survey, I give an elementary and diagrammatic approach to the introductory part of the ESS-LYZ theory. This will enable the reader to appreciate their deeper results (which are not discussed here) more easily. On the other hand, the latter group LYZ has obtained some interesting construction of quasitriangular structures on the bi-smash product Hopf algebra H(S,B) = (kS)∗ ⊗ kB [LYZ2]. I also give a natural categorical explanation of this construction, based on Masuoka’s observation [Ms3]. In §1, we review the notion of a matched pair of groups (S,B) and matched product B ./ S, and explain the matched pair condition by means of diagrams of the form 2000 Mathematics Subject Classification: 16W35, 18D10. The paper is in final form and no version of it will be published elsewhere.