Simple Finite-dimensional U-modules And U_q-modules

Hopf algebras have become one of the most active studying fields in math-ematics. The notion of a Hopf algebra was developed by the study on cohomol-ogy of topological groups of German algebraic topologist H. Hopf in the early 1940s which equipped simultaneously with an algebra structure and a coalge-bra struture compatibly. In this paper, we major in simple finite-dimensional modules of two kinds of special Hopf algebras, which refers to simple finite-dimensional modules of enveloping algebra U(sl(2)) and quantum enveloping algebra Ug(sl(2)) of the Lie algebra sl(2).In the Introduction, We mainly introduce the historical background and researching status, and we describe the researching ideas and concrete work of this dissertation.In chapter 1, we give an introduction to the definitions and the main theorem which we needed in this dissertation.In chapter 2, we mainly prove two results:the dual module V(n)* of the simple finite-dimensional U-module is isomorphic to V(n), the dual module Vε,n* of the simple finite-dimensional Uq-odule is isomorphic to Vε,n. With respect to those two proofs, we both give a non-constructive proof firstly, then a constructive proof.In chapter 3, we mainly prove that for any two nonnegative integers m and n, we have Vn (?) Vm(?)Vm (?) Vn thanks to the quantum Clebsch-Gordan formula. In the last, for V2 (?) V1= Vi (?) V2, we find a concrete isomorphism of modules.