有限维单U-模和U_q-模

发布时间：2018-05-25 22:10

本文选题：Lie代数 + Hopf代数　；参考：《山东大学》2015年硕士论文

【摘要】：Hopf代数是如今数学中最活跃的研究领域之一,它是德国代数拓扑学家H. Hopf于20世纪40年代初在研究拓扑群中的上同调问题时构造出的一种兼容代数结构和余代数结构的代数系统.本文我们主要研究了两种特殊的Hopf代数上的有限维单模,即Lie代数sl(2)的包络代数U(sl(2))和量子包络代数Uq(sl(2))上的有限维单模.在引言中,我们主要介绍了本文的历史背景和研究现状,并且阐述了本文的研究思路和具体工作.第一章是基础知识,我们主要介绍了本文将会用到的一些代数结构的基本概念和结论.在第二章中,我们主要证明了两个结论：即有限维单U-模V(n)同构于它的对偶模V(n)*,有限维单Uq-模Vε,n同构于它的对偶模Vε,n*.对于这两个结论,我们都是先给出了一个存在性的证明,然后又给出了一个构造性的证明.在第三章中,我们主要借助量子Clebsch-Gordan公式证明了对于任意的非负整数m,n都有Vn(?)Vm≌Vm(?)Vn.最后对于V2(?)V1≌V1(?)V2,我们找出了一个具体的模同构映射.属性不符
[Abstract]:Hopf algebra is one of the most active research fields in mathematics today. It is an algebraic system of compatible algebraic structure and coalgebraic structure constructed by German algebraic topology scientist H. Hopf in the early 1940s when studying cohomology problems in topological groups. In this paper, we mainly study the finite-dimensional simple modules on two special Hopf algebras, that is, the enveloping algebra U _ (n ~ (2) of the Lie algebra Sl _ (2) and the quantum envelope algebra U _ Q _ (I _ (2) ~ (2). In the introduction, we mainly introduce the historical background and research status of this paper, and elaborate the research ideas and specific work of this paper. The first chapter is the basic knowledge, we mainly introduce some basic concepts and conclusions of algebraic structure that will be used in this paper. In the second chapter, we mainly prove two conclusions: that is, the finite dimensional simple U-module is isomorphic to its dual module, and the finite-dimensional simple Uq-module V 蔚 -n is isomorphic to its dual module V 蔚 n. For these two conclusions, we give a proof of existence and then a proof of constructivity. In the third chapter, we mainly prove that for any nonnegative integer m ~ n, we have Vn(?)Vm = V _ m ~ n ~ n by means of quantum Clebsch-Gordan formula. Finally, we find a specific module isomorphism map for V2(?)V1 = V1 / V2. Attribute discrepancy
【学位授予单位】：山东大学
【学位级别】：硕士
【学位授予年份】：2015
【分类号】：O153.3

【相似文献】