本文選題：擬Hopf代數(shù) + 量子群��； 參考：《山東大學(xué)》2016年博士論文

【摘要】：本文主要研究了有限維的點(diǎn)化Majid代數(shù)的分類(lèi)理論和結(jié)構(gòu)理論,以及有限群上扭Yetter-Drinfeld范疇中具有有限根系的對(duì)角型Nichols代數(shù)的分類(lèi)理論。我們給出了有限群的交換3階上循環(huán)消解的一般性方法,再利用張量范疇的規(guī)范變換,從而把有限群上的扭Yetter-Dringeld范疇中的對(duì)角型Nichols代數(shù)的分類(lèi)問(wèn)題,轉(zhuǎn)化為有限群上通常的Yetter-Drinfeld范疇中對(duì)角型Nichols代數(shù)的分類(lèi)問(wèn)題。進(jìn)而結(jié)合Heckenberger關(guān)于算術(shù)根系的分類(lèi),我們給出了有限群上的扭Yetter-Dringeld范疇中具有有限根系的對(duì)角型Nichols代數(shù)的分類(lèi)。特別地,我們得到了這類(lèi)范疇中所有的有限維對(duì)角型Nichols代數(shù)的分類(lèi)。然后,我們證明了所有的有限維對(duì)角型點(diǎn)化Majid代數(shù)都是由群樣元和協(xié)本原元生成的,從而部份肯定回答了廣義Andruskiewitsch-Schneider猜想。最后,利用我們?cè)趶V義Andruskiewitsch-Schneider猜想方面的證明結(jié)果,以及我們對(duì)有限群上的扭Yetter-Drinfeld范疇中具有有限根系的對(duì)角型Nichols代數(shù)的分類(lèi),我們給出了所有有限維連通的對(duì)角型分次點(diǎn)化Majid代數(shù)的分類(lèi)。本文共分為五章。第一章,我們主要介紹擬量子群的歷史來(lái)源和發(fā)展?fàn)顩r。我們著重介紹了該領(lǐng)域當(dāng)前的研究進(jìn)展和研究方法,以及本文所取得的主要結(jié)果。第二章,我們?cè)敿?xì)地介紹了擬量子群,張量范疇,算術(shù)根系,Weyl群胚和Nichols代數(shù)等本文需要用到的概念,以及一些基本的結(jié)論。我們近期所取得的一些關(guān)于點(diǎn)化Majid代數(shù)的結(jié)果,比如Majid玻色子化的具體公式等,也放在這一章節(jié)進(jìn)行介紹。第三章,我們主要研究扭Yetter-Drinfeld范疇KGKGyDΦ中的對(duì)角型Nichol代數(shù),對(duì)其中具有有限根系的對(duì)角型Nichols代數(shù)進(jìn)行分類(lèi)。Yetter-Drinfeld范疇KGKGyD中的結(jié)合子是由G的3-上循環(huán)西來(lái)決定的。首先我們證明了如果KGKGyD中中的一個(gè)對(duì)角型Nichols代數(shù)的支撐子群是G,則G是交換群,中是G的一個(gè)交換3階上循環(huán)。這相當(dāng)于說(shuō)任何一個(gè)對(duì)角型Nichols代數(shù)B(V)都可以實(shí)現(xiàn)在這樣一個(gè)Yetter-Drinfeld范疇KGKGyDΦ中,其中G是交換群,Φ是G的一個(gè)交換3階上循環(huán)。接下來(lái),我們對(duì)交換群的交換3階上循環(huán)進(jìn)行了細(xì)致的研究,給出了交換3階上循環(huán)的消解方法,成功地把KGKGyDΦ中的對(duì)角型Nichols代數(shù)和某個(gè)更大的交換群G對(duì)應(yīng)的通常的、Yetter-Drinfeld范疇KGKGgyD中的對(duì)角型Nichols代數(shù)聯(lián)系起來(lái),進(jìn)而得到KGKGyDΦ中具有限根系的對(duì)角型Nichols代數(shù)的分類(lèi)。特別的,考慮具有有限根系的對(duì)角型Nichols代數(shù)的每一個(gè)正根對(duì)應(yīng)的根向量的冪零指數(shù),我們得到了KGKGyDΦ中所有的有限維對(duì)角型Nichols代數(shù)的分類(lèi)。第四章,我們給出了有限維連通的余根分次對(duì)角型點(diǎn)化Ma.jid代數(shù)的分類(lèi)。我們稱(chēng)一個(gè)Majid代數(shù)為連通的,當(dāng)且僅當(dāng)其Gabriel箭圖是連通的。一般的有限維余根分次點(diǎn)化Majid代數(shù)的分類(lèi),總是可以約化成有限維連通的余根分次點(diǎn)化Majid代數(shù)的分類(lèi)。要給出有限維點(diǎn)化Majid代數(shù)的分類(lèi),一個(gè)必須要回答的問(wèn)題就是猜想1.2。作為本文的主要結(jié)果之一,我們部份肯定地回答了這個(gè)猜想,即我們證明了任何一個(gè)有限維對(duì)角型點(diǎn)化Majid代數(shù)都是由群樣元和協(xié)本原元生成的。從而我們可以把有限維連通的余根分次對(duì)角型點(diǎn)化Majid代數(shù)的分類(lèi)問(wèn)題轉(zhuǎn)化為有限維對(duì)角型Nichols代數(shù)的分類(lèi)問(wèn)題,再結(jié)合上一章的結(jié)果,我們得到了本章關(guān)于點(diǎn)化Majid代數(shù)的分類(lèi)結(jié)果。這一章,我們還給出了一些有限維連通的分次點(diǎn)化Majid代數(shù)的結(jié)構(gòu)定理。第五章,我們對(duì)Carta n型和標(biāo)準(zhǔn)型的點(diǎn)化Majid代數(shù)做了細(xì)致的研究。我們證明了從任何有限Cartan矩陣出發(fā),都存在無(wú)限多個(gè)有限維的Cartan型點(diǎn)化Majid代數(shù),其相應(yīng)的Nichols代數(shù)的根系就是該Cartan矩陣對(duì)應(yīng)的復(fù)半單李代數(shù)的根系。與此同時(shí),我們也提供了一套具體的從有限Cartan矩陣出發(fā),構(gòu)造有限維點(diǎn)化Majid代數(shù)的方法。我們還對(duì)標(biāo)準(zhǔn)型點(diǎn)化Majid代數(shù)進(jìn)行了研究,對(duì)其類(lèi)別和結(jié)構(gòu)進(jìn)行了更為細(xì)致的刻畫(huà)。最后我們提供了大量的秩為2的具有有限PBW生成元的點(diǎn)化Majid代數(shù)的例子,列出了所有秩為2的有限維標(biāo)準(zhǔn)型分次點(diǎn)化Majid代數(shù)。
[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.
【學(xué)位授予單位】：山東大學(xué)
【學(xué)位級(jí)別】：博士
【學(xué)位授予年份】：2016
【分類(lèi)號(hào)】：O152.5

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本文編號(hào)：1932183