Gr(?)bner-Shirshov Pair Of Irreducible Modules Over The Quantum Group Of Type B₂

In this thesis,we accomplish three tasks.Firstly,using the Gr(?)bner-Shirshov basis of quantum group Uq(B2)and the Chibrikov's double free module method,we construct a Gr(?)bner-Shirshov pair of finite dimensional irreducible Uq(B2)-module Vq(?).Secondly,we define a suitable equivalent version of Uq(B2),and specialize it at q=1,we get a Gr(?)bner-Shirshov basis of U(B2),and this basis is consistent with the one given by Klein and Bokut.Finally,we specialize Uq(B2)at q=1 again,and get a Gr(?)bner-Shirshov pair of finite dimensional irreducible U(B2)-module V(?),and this basis is much smaller then the one given by Lee and our method is much simpler.