超群表示理论

发布时间：2018-07-04 23:18

本文选题：超群 + 超空间　；参考：《青岛大学》2017年硕士论文

【摘要】：基于对群表示的理解,本文主要对群表示论的一些性质与结论在超群上进行对应的推广与研究。第一章主要介绍了超空间,超代数,超表示,李超代数等基本概念,这是整篇文章的基础。第二章主要对超群矩阵表示进行了相应的探究。首先,定义了超群G在超空间V上的一个超线性表示,其次,探讨了群的矩阵表示与超群的矩阵表示之间的诱导关系。第三章主要讨论了与超群代数联系紧密的超模的性质定理。类似于群表示理论的方法,在超空间V上构造出一个C[G]-超模结构,并定义超空间V上的内积,运算,讨论该内积关于G作用下不变时所具有的性质,同时对马施克定理及舒尔引理进行了推广。第四章主要研究了超群表示的特征标理论。给出超特征标的定义,并赋予其内积,运算,最终证明了超特征标关于此内积运算是正交关系,从而说明了超特征标的意义。
[Abstract]:Based on the understanding of group representation, some properties and conclusions of group representation theory are generalized and studied on hypergroup. The first chapter mainly introduces the basic concepts of hyperspace, superalgebra, superrepresentation, Li Chao algebra and so on, which is the basis of the whole paper. The second chapter mainly discusses the representation of hypergroup matrix. Firstly, a superlinear representation of the hypergroup G on the hyperspace V is defined. Secondly, the induced relationship between the matrix representation of a group and the matrix representation of a hypergroup is discussed. In chapter 3, we discuss the property theorem of supermodules which are closely related to hypergroup algebras. Similar to the group representation theory, we construct a C [G] -supermodule structure on the hyperspace V, define the inner product and operation on the hyperspace V, and discuss the properties of the inner product when it is invariant under the action of G. At the same time, the Marshick theorem and Schuler Lemma are generalized. In chapter 4, the characteristic theory of hypergroup representation is studied. In this paper, the definition of super feature scale is given, and its inner product and operation are given. Finally, it is proved that the super feature scale is orthogonal to the operation of this inner product, and the meaning of super feature scale is explained.
【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

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