The Ideal Intersection And Sum Of Quantum Group U_q (sl₂)

The research on the quantized enveloping algebra of finite-dimensional semisimple Lie algebra is an important research content in the field of algebra research in recent years.Furthermore,the research of quantized enveloping algebra U=Uq(sl2_of finite-dimensional simple Lie algebra sl2 is the basic content of study general quantized enveloping algebra theory,and plays an irreplaceable role in the quantum group and Drinfeld-Jimbo quantized enveloping algebra theory:it is not only a hint of general theory,but also provides the necessary results and tools in the general case theory.More importantly,it also offers a theoretical framework for the comprehensive study of general theory.In this paper,we mainly use the generators of every ideal of quantum group U = Ug(sl2)to give the descriptions of the generators of intersection and sum of the ideal of U.As an application,we give a sufficient and necessary condition for determining the direct sum of several simple modules tensor products.This paper is divided into four chapters.The first chapter is introduction,it briefly introduces the development history of Hopf algebra and quantum groups and some recent researches on quantum groups at home and abroad,and briefly introduces the research contents and main results of this paper.In the second chapter,we first introduce some basic concepts of U,such as the highest(lowest)weight vectors,the highest(lowest)weight module,the Verma module,the locally finite subalgebra F(U),etc.Also we give the most basic description of the submodule structure of F(U),and describe the generators of the ideals and prime ideals of U.In the third chapter,we first introduce some properties satisfied by the ideals of U.On this basis,we provide the descriptions of the generators of the intersections of a finite number of finite dimensional maximal prime ideals of U.Then,we give the generators of the sum of any two non-zero ideals in U.In the fourth chapter,firstly we give the direct sum decomposition of a finite number of finite dimensional simple modules tensor products and their annihilation ideals,so as to recommend the sufficient and necessary condition for a determining simple module which is a direct summand in a finite number of simple modules tensor products.Finally,we prove that U is a faithful action on a given quantum plane.