本文選題：超群 + 自由超群��； 參考：《青島大學(xué)》2017年碩士論文

【摘要】：本文通過對代數(shù)學(xué)中群的理解與思考,模仿超向量空間與李超代數(shù)等諸多超數(shù)學(xué)概念的定義與處理方法,通過類比提出了超群的概念。本文對超群的研究還是初步的,主要討論了一些基本問題。例如,在第一章中討論了超群的可換性,超可換性相關(guān)的一些結(jié)論與性質(zhì),給出了超群的幾個比較基本的且重要的事實。文章還首次引入了自由超群的概念,并證明了自由超群的存在性。解決了比較常見的幾個群對象是否為超群以及是否具備超可換性的問題,并且還證明了超群的直積是超群,并做了某種推廣。然后借助于結(jié)合代數(shù)、張量代數(shù)、李代數(shù)及李超代數(shù)的基礎(chǔ)知識,對結(jié)合超代數(shù)、張量超代數(shù)等概念進行了進一步理解,討論了張量超代數(shù)的一些性質(zhì)。最后,通過對群的同態(tài)和表示的研究與類比,提出了超群的表示,超群的模的概念,并考慮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.
【學(xué)位授予單位】：青島大學(xué)
【學(xué)位級別】：碩士
【學(xué)位授予年份】：2017
【分類號】：O152.5

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本文編號：2076676