Gr-范畴中的Azumaya代数

发布时间：2017-12-26 18:24

本文关键词：Gr-范畴中的Azumaya代数　出处：《山东大学》2016年博士论文　论文类型：学位论文

【摘要】：本博士论文的主要研究对象是Gr-范畴中的Azumaya代数。作为结合代数的自然推广,范畴中代数理论的研究是近年来研究的一个热点,许多专家和学者在此领域做了大量的工作,并取得了很多的进展。具体到本文,我们首先研究了一类Z。-分次Azumaya代数——广义Clifford代数作为Gr-范畴中的代数所具有的性质。然后,我们给出了辫子Gr-范畴中Azumaya代数的结构定理,介绍了如何借助计算机编程将八元数代数等扭群代数看作某些恰当辫子Gr-范畴中的Azumaya代数,并给出了一类简单Gr-范畴中Azumaya代数的分类。本文由四章组成,主要内容如下：在第一章中,我们回顾了Gr-范畴和Azumaya代数等相关内容的历史起源和发展现状。然后,我们介绍了论文的主要结果和结构。在第二章中,我们介绍了monoidal范畴中的Azumaya代数,张量范畴,Gr-范畴等基本定义和相关结论。特别地,我们重点介绍了Gr-范畴中的扭群代数理论、规范变换这些在以后的章节中要用到的工具。在第三章中,我们讨论了广义Clifford代数作为适当的对称线性Gr-范畴中的代数所具有的性质。通过将Clifford代数作为群Z2n的扭群代数,Albuquerque和Majid利用新的方法对Clifford代数的性质进行了新的研究[6]。利用上述结果,Bulacu观察到Clifford代数实际上是某些对称线性Gr-范畴中的弱Hopf代数[18]。在本章中,我们将广义Clifford代数作为群Znm的扭群代数,利用这一新的观点,推导出了广义Clifford代数的周期性,并得到了构造广义Clifford代数的一种新方法——广义Clifford进程,将Albuquerque,Majid和Bulacu的结果推广到更一般的情形。特别地,利用对称线性Gr-范畴中的规范变换,我们得到了广义Clifford代数的分解定理和它在Gr-范畴中的弱Hopf代数结构,推广并简化了Bulacu等人的工作。在第四章中,我们研究了一般辫子线性Gr-范畴中的Azumaya代数。首先,我们证明了辫子线性Gr-范畴中的Azumaya代数就是该范畴中的中心单代数,推广了群分次代数的结果。其次,利用规范变换,我们发现可以将八元数代数看做适当范畴中的结合代数,或更准确地说,将八元数代数用扭群代数来刻画,则八元数代数可看作某些恰当辫子Gr-范畴中的Azumaya代数,并可将这种方法推广到一般的扭群代数上。最后,我们得到了一类简单的辫子线性Gr-范畴(VecZ2Φ,R)中Azumaya代数的具体结构定理和分类。
[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.
【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】：O154.1

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