Whittaker Modules For U_q?sl₃?

Quantum group is a special kind of Hopf algebra,which can be regarded as q-quantized Lie algebra.Quantum envelope algebra Uq?sl2?plays a very important role in the theory of quantum groups and quantum envelope algebra,which not only plays a guiding role in the study of general theory,but also provides general results.This article starts from the Whittaker module on the quantum envelope algebra Uq?sl2?,defines and studies Whittaker modules on quantum envelope algebra U_q?sl₃?,then we will study the structure of the Whittaker module on the quantum envelope algebra U_q?sl₃?.We focuse on the research of its submodules and reducibility,which is of great significance for us to further understand and perfect the representation theory of Lie algebra,as well as various quantum algebra representation theories.By studying the Whittaker module on the quantum group U_q?sl₃?,we will constructed the Whittaker module of the quantum group U_q?sl₃?under degeneracy.With studying the irreducibility of this type of module and its submodule structure are studied,we will get a large class of irreducible modules,which is innovative for quantum group theory.