本文關(guān)鍵詞：Gr-范疇中的Azumaya代數(shù)　出處：《山東大學(xué)》2016年博士論文　論文類型：學(xué)位論文

  更多相關(guān)文章： (辮子)線性Gr-范疇 Azumaya代數(shù) 廣義Clifford代數(shù) 扭群代數(shù) 規(guī)范變換 八元數(shù)代數(shù) 弱Hopf代數(shù)

【摘要】：本博士論文的主要研究對象是Gr-范疇中的Azumaya代數(shù)。作為結(jié)合代數(shù)的自然推廣,范疇中代數(shù)理論的研究是近年來研究的一個(gè)熱點(diǎn),許多專家和學(xué)者在此領(lǐng)域做了大量的工作,并取得了很多的進(jìn)展。具體到本文,我們首先研究了一類Z。-分次Azumaya代數(shù)——廣義Clifford代數(shù)作為Gr-范疇中的代數(shù)所具有的性質(zhì)。然后,我們給出了辮子Gr-范疇中Azumaya代數(shù)的結(jié)構(gòu)定理,介紹了如何借助計(jì)算機(jī)編程將八元數(shù)代數(shù)等扭群代數(shù)看作某些恰當(dāng)辮子Gr-范疇中的Azumaya代數(shù),并給出了一類簡單Gr-范疇中Azumaya代數(shù)的分類。本文由四章組成,主要內(nèi)容如下：在第一章中,我們回顧了Gr-范疇和Azumaya代數(shù)等相關(guān)內(nèi)容的歷史起源和發(fā)展現(xiàn)狀。然后,我們介紹了論文的主要結(jié)果和結(jié)構(gòu)。在第二章中,我們介紹了monoidal范疇中的Azumaya代數(shù),張量范疇,Gr-范疇等基本定義和相關(guān)結(jié)論。特別地,我們重點(diǎn)介紹了Gr-范疇中的扭群代數(shù)理論、規(guī)范變換這些在以后的章節(jié)中要用到的工具。在第三章中,我們討論了廣義Clifford代數(shù)作為適當(dāng)?shù)膶ΨQ線性Gr-范疇中的代數(shù)所具有的性質(zhì)。通過將Clifford代數(shù)作為群Z2n的扭群代數(shù),Albuquerque和Majid利用新的方法對Clifford代數(shù)的性質(zhì)進(jìn)行了新的研究[6]。利用上述結(jié)果,Bulacu觀察到Clifford代數(shù)實(shí)際上是某些對稱線性Gr-范疇中的弱Hopf代數(shù)[18]。在本章中,我們將廣義Clifford代數(shù)作為群Znm的扭群代數(shù),利用這一新的觀點(diǎn),推導(dǎo)出了廣義Clifford代數(shù)的周期性,并得到了構(gòu)造廣義Clifford代數(shù)的一種新方法——廣義Clifford進(jìn)程,將Albuquerque,Majid和Bulacu的結(jié)果推廣到更一般的情形。特別地,利用對稱線性Gr-范疇中的規(guī)范變換,我們得到了廣義Clifford代數(shù)的分解定理和它在Gr-范疇中的弱Hopf代數(shù)結(jié)構(gòu),推廣并簡化了Bulacu等人的工作。在第四章中,我們研究了一般辮子線性Gr-范疇中的Azumaya代數(shù)。首先,我們證明了辮子線性Gr-范疇中的Azumaya代數(shù)就是該范疇中的中心單代數(shù),推廣了群分次代數(shù)的結(jié)果。其次,利用規(guī)范變換,我們發(fā)現(xiàn)可以將八元數(shù)代數(shù)看做適當(dāng)范疇中的結(jié)合代數(shù),或更準(zhǔn)確地說,將八元數(shù)代數(shù)用扭群代數(shù)來刻畫,則八元數(shù)代數(shù)可看作某些恰當(dāng)辮子Gr-范疇中的Azumaya代數(shù),并可將這種方法推廣到一般的扭群代數(shù)上。最后,我們得到了一類簡單的辮子線性Gr-范疇(VecZ2Φ,R)中Azumaya代數(shù)的具體結(jié)構(gòu)定理和分類。
[Abstract]:The main object of this thesis is the Azumaya algebra in the Gr- category. As a natural generalization of associative algebra, the research of algebraic theory in category is a hot topic in recent years. Many experts and scholars have done a lot of work in this field, and have made many progress. In this article, we first study a class of Z. - graded Azumaya algebra - generalized Clifford algebra as a property of algebra in the category of Gr-. Then, we give the structure theorem of Azumaya algebras in braided Gr- category, describes how to use computer programming will be eight yuan number of twisted group algebra Azumaya algebra algebra as some appropriate braided Gr- category, and gives a simple classification of Azumaya Gr- in the category of algebras. This article is composed of four chapters. The main contents are as follows: in the first chapter, we review the historical origin and development status of Gr- category and Azumaya algebra. Then, we introduce the main results and structure of the paper. In the second chapter, we introduce the basic definitions and related conclusions of Azumaya algebra, tensor category, Gr- category in the monoidal category. In particular, we focus on the theory of the torsional group algebra in the Gr- category and the tools to be used in the later chapters. In the third chapter, we discuss the properties of generalized Clifford algebra as an algebra in a proper symmetric linear Gr- category. By using Clifford algebra as a torsion group algebra of group Z2n, Albuquerque and Majid have made a new study of the properties of Clifford algebra by new methods, [6]. Using the above results, Bulacu observed that the Clifford algebra is actually a weak Hopf algebra [18] in some symmetric linear Gr- categories. In this chapter, we generalized Clifford algebras as group Znm twisted group algebras, using this new perspective, deduces the periodicity of generalized Clifford algebras, and obtained a new method to construct the generalized Clifford algebras of generalized Clifford process, Albuquerque, Majid and Bulacu are generalized to the more general the case. In particular, by using the gauge transformation in the symmetric linear Gr- category, we get the decomposition theorem of the generalized Clifford algebra and its weak Hopf algebra structure in Gr- category, and generalize and simplify the work of Bulacu et al. In the fourth chapter, we study the Azumaya algebra in the general braid linear Gr- category. First, we prove that the Azumaya algebra in the braid linear Gr- category is the central single algebra in the category, which generalizes the results of the group sub algebra. Secondly, using the canonical transformation, we find that the number of eight yuan as appropriate in the category of algebraic algebra, or more precisely, the number will be eight yuan with twisted group algebra to describe the algebra, algebraic number is eight yuan can be regarded as some Azumaya algebra in the appropriate braided Gr- category, and this method is extended to the general twisted group algebras. Finally, we obtain the specific structure theorems and classifications of a simple class of braid linear Gr- category (VecZ2 (R)) Azumaya algebra.
【學(xué)位授予單位】：山東大學(xué)
【學(xué)位級別】：博士
【學(xué)位授予年份】：2016
【分類號】：O154.1

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