关于有限拟量子群的分类

发布时间：2018-05-25 05:03

本文选题：拟Hopf代数 + 量子群　；参考：《山东大学》2016年博士论文

【摘要】：本文主要研究了有限维的点化Majid代数的分类理论和结构理论,以及有限群上扭Yetter-Drinfeld范畴中具有有限根系的对角型Nichols代数的分类理论。我们给出了有限群的交换3阶上循环消解的一般性方法,再利用张量范畴的规范变换,从而把有限群上的扭Yetter-Dringeld范畴中的对角型Nichols代数的分类问题,转化为有限群上通常的Yetter-Drinfeld范畴中对角型Nichols代数的分类问题。进而结合Heckenberger关于算术根系的分类,我们给出了有限群上的扭Yetter-Dringeld范畴中具有有限根系的对角型Nichols代数的分类。特别地,我们得到了这类范畴中所有的有限维对角型Nichols代数的分类。然后,我们证明了所有的有限维对角型点化Majid代数都是由群样元和协本原元生成的,从而部份肯定回答了广义Andruskiewitsch-Schneider猜想。最后,利用我们在广义Andruskiewitsch-Schneider猜想方面的证明结果,以及我们对有限群上的扭Yetter-Drinfeld范畴中具有有限根系的对角型Nichols代数的分类,我们给出了所有有限维连通的对角型分次点化Majid代数的分类。本文共分为五章。第一章,我们主要介绍拟量子群的历史来源和发展状况。我们着重介绍了该领域当前的研究进展和研究方法,以及本文所取得的主要结果。第二章,我们详细地介绍了拟量子群,张量范畴,算术根系,Weyl群胚和Nichols代数等本文需要用到的概念,以及一些基本的结论。我们近期所取得的一些关于点化Majid代数的结果,比如Majid玻色子化的具体公式等,也放在这一章节进行介绍。第三章,我们主要研究扭Yetter-Drinfeld范畴KGKGyDΦ中的对角型Nichol代数,对其中具有有限根系的对角型Nichols代数进行分类。Yetter-Drinfeld范畴KGKGyD中的结合子是由G的3-上循环西来决定的。首先我们证明了如果KGKGyD中中的一个对角型Nichols代数的支撑子群是G,则G是交换群,中是G的一个交换3阶上循环。这相当于说任何一个对角型Nichols代数B(V)都可以实现在这样一个Yetter-Drinfeld范畴KGKGyDΦ中,其中G是交换群,Φ是G的一个交换3阶上循环。接下来,我们对交换群的交换3阶上循环进行了细致的研究,给出了交换3阶上循环的消解方法,成功地把KGKGyDΦ中的对角型Nichols代数和某个更大的交换群G对应的通常的、Yetter-Drinfeld范畴KGKGgyD中的对角型Nichols代数联系起来,进而得到KGKGyDΦ中具有限根系的对角型Nichols代数的分类。特别的,考虑具有有限根系的对角型Nichols代数的每一个正根对应的根向量的幂零指数,我们得到了KGKGyDΦ中所有的有限维对角型Nichols代数的分类。第四章,我们给出了有限维连通的余根分次对角型点化Ma.jid代数的分类。我们称一个Majid代数为连通的,当且仅当其Gabriel箭图是连通的。一般的有限维余根分次点化Majid代数的分类,总是可以约化成有限维连通的余根分次点化Majid代数的分类。要给出有限维点化Majid代数的分类,一个必须要回答的问题就是猜想1.2。作为本文的主要结果之一,我们部份肯定地回答了这个猜想,即我们证明了任何一个有限维对角型点化Majid代数都是由群样元和协本原元生成的。从而我们可以把有限维连通的余根分次对角型点化Majid代数的分类问题转化为有限维对角型Nichols代数的分类问题,再结合上一章的结果,我们得到了本章关于点化Majid代数的分类结果。这一章,我们还给出了一些有限维连通的分次点化Majid代数的结构定理。第五章,我们对Carta n型和标准型的点化Majid代数做了细致的研究。我们证明了从任何有限Cartan矩阵出发,都存在无限多个有限维的Cartan型点化Majid代数,其相应的Nichols代数的根系就是该Cartan矩阵对应的复半单李代数的根系。与此同时,我们也提供了一套具体的从有限Cartan矩阵出发,构造有限维点化Majid代数的方法。我们还对标准型点化Majid代数进行了研究,对其类别和结构进行了更为细致的刻画。最后我们提供了大量的秩为2的具有有限PBW生成元的点化Majid代数的例子,列出了所有秩为2的有限维标准型分次点化Majid代数。
[Abstract]:In this paper, we mainly study the classification theory and structure theory of the finite dimensional Majid algebra, and the classification theory of diagonal Nichols algebras with finite roots in a finite group of twisted Yetter-Drinfeld categories. We give the general square method for the exchange of the 3 order cyclic digestion of the finite groups, and then use the normal transformation of the tensor category, so that the normal transformation of the tensor category is used. The classification problem of diagonal Nichols algebra in the torsional Yetter-Dringeld category on a finite group is transformed into a classification problem of diagonal Nichols algebra in the normal Yetter-Drinfeld category on a finite group. Then, combined with the classification of the arithmetic roots of Heckenberger, we give a finite group of twisted Yetter-Dringeld categories with finite elements. The classification of diagonal Nichols algebras of the root system. In particular, we get the classification of all the finite dimensional diagonal Nichols algebras in this category. Then, we prove that all the finite dimensional diagonal Majid algebras are generated by the group and co primitive elements, and the part certainly answers the generalized Andruskiewitsch-Schneider Conjecture. Finally, using our proof of the generalized Andruskiewitsch-Schneider conjecture, and the classification of diagonal Nichols algebras with finite roots in the torsional Yetter-Drinfeld category on a finite group, we give the classification of all finite dimensional connected diagonalised Majid algebras. This paper is divided into five Chapter 1. We mainly introduce the historical origin and development of quasi quantum groups. We focus on the current research progress and research methods in this field, and the main results obtained in this paper. In the second chapter, we introduce the quasi quantum group, tensor category, arithmetic root, Weyl group embryo and Nichols algebra in detail. The concept, and some basic conclusions, we have recently obtained some results on Majid algebra, such as the specific formula for Majid bosonalization, and so on. In the third chapter, we mainly study the diagonal Nichol algebra in the twisted Yetter-Drinfeld category KGKGyD, which has a finite root system. The combination of angular Nichols algebras in the category of.Yetter-Drinfeld category KGKGyD is determined by the circular west of G's 3-. First we prove that if the support subgroup of a diagonal Nichols algebra in KGKGyD is G, then G is a commutative group and a commutative 3 order on G, which is equivalent to any of the diagonal Nichols. The algebraic B (V) can be implemented in such a Yetter-Drinfeld category KGKGyD diameter, where G is an exchange group and a commutative 3 order loop of G. Next, we have studied the commutative 3 order upper loop of the exchange group carefully, and give the elimination square method of exchanging the 3 order upper loop, and successfully put the diagonal Nichols algebra in the KGKGyD diameter. A larger exchange group G corresponds to the diagonal Nichols algebra in the Yetter-Drinfeld category KGKGgyD, and then the classification of diagonal Nichols algebras with finite roots in KGKGyD is obtained. In particular, the nilpotent exponents of the root vectors corresponding to each positive root of a diagonal Nichols Algebra with a finite root are considered. We get the classification of all the finite dimensional diagonal Nichols algebras in KGKGyD. Fourth, we give the classification of the finite dimensional connected complementary root fractional diagonal Ma.jid algebra. We call a Majid algebra connected, if and only if its Gabriel arrows are connected. A kind of finite dimensional residual root fractional Majid algebra. Classification can always be reduced to a classification of Majid algebras of a finite dimensional connectedness. To give a classification of Majid algebras of finite dimension points, a problem to be answered is to conjecture that 1.2. is one of the main results of this article, and we partly answer the guess that we have proved any finite dimensional diagonal. All type of Majid algebras are generated by group like and co primitive elements, so we can convert the classification problem of the finite dimensional connected diagonal Majid algebra into the classification problem of the finite dimensional diagonal Nichols algebra, and then combine the results of the last chapter, and we get the classification results of the Majid algebra in this chapter. In this chapter, we also give some structural theorems for finite dimensional connected fractional Majid algebras. Fifth, we make a careful study of Carta N and standard type Majid algebras. We prove that from any finite Cartan matrix, there are infinitely many Cartan type Majid algebras with limited dimensions, and their corresponding Nich The root of OLS algebra is the root of the complex semisimple Lie algebra corresponding to the Cartan matrix. At the same time, we also provide a set of concrete methods to construct the finite dimension Majid algebra from the finite Cartan matrix. We also study the standard type Majid algebra, and describe its category and structure more carefully. In the end, we provide an example of a large number of ordered Majid algebras with a finite PBW generating element with rank 2, and lists all the finite dimensional standard type graded Majid algebras with rank of 2.
【学位授予单位】：山东大学
【学位级别】：博士
【学位授予年份】：2016
【分类号】：O152.5

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