超群及其基本性质

发布时间：2018-06-28 04:11

本文选题：超群 + 自由超群　；参考：《青岛大学》2017年硕士论文

【摘要】：本文通过对代数学中群的理解与思考,模仿超向量空间与李超代数等诸多超数学概念的定义与处理方法,通过类比提出了超群的概念。本文对超群的研究还是初步的,主要讨论了一些基本问题。例如,在第一章中讨论了超群的可换性,超可换性相关的一些结论与性质,给出了超群的几个比较基本的且重要的事实。文章还首次引入了自由超群的概念,并证明了自由超群的存在性。解决了比较常见的几个群对象是否为超群以及是否具备超可换性的问题,并且还证明了超群的直积是超群,并做了某种推广。然后借助于结合代数、张量代数、李代数及李超代数的基础知识,对结合超代数、张量超代数等概念进行了进一步理解,讨论了张量超代数的一些性质。最后,通过对群的同态和表示的研究与类比,提出了超群的表示,超群的模的概念,并考虑了在子超群上的限制表示。
[Abstract]:Through understanding and thinking of groups in algebra, this paper imitates the definition and processing methods of many hypermathematics concepts, such as hypervector space and Li Chao algebra, and puts forward the concept of hypergroup by analogy. In this paper, the study of hypergroup is preliminary, and some basic problems are discussed. For example, in the first chapter, we discuss some conclusions and properties related to the commutativity and supercommutativity of hypergroups, and give some basic and important facts about hypergroups. The concept of free hypergroup is introduced for the first time, and the existence of free hypergroup is proved. The problems of whether some group objects are hypergroup and whether they have supercommutability are solved, and it is proved that the direct product of hypergroup is hypergroup, and some generalization is given. Based on the basic knowledge of associative algebra, Zhang Liang algebra, lie algebra and Li Chao algebra, the concepts of associative superalgebra and tensor superalgebra are further understood, and some properties of tensor superalgebra are discussed. Finally, through the study and analogy of the homomorphism and representation of a group, the concepts of the representation of a hypergroup and the module of a hypergroup are proposed, and the restricted representation on a subhypergroup is considered.
【学位授予单位】：青岛大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O152.5

【参考文献】