U < Sub > Q < / Sub > (sl < Sub > 2 < / Sub >) In C < Sub > Q < / Sub > [x < Sup > + 1 < / Sup >, Y] On The Model Of Algebraic Structure

Quantum group theory is one of the important branches and important research content in algebra, which develops in the mid of the eighties. It has been widely discussed and developed during the last twenty years. The aim of this thesis is to describe the module algebra of the quantum universal enveloping algebra Uq (sl2) on the quantum polynomial algebra Cq[x±1,y], and the invariant subalgebras of these module algebras when q is not a root of unity. The quantum universal enveloping algebr Uq(sl2) is an unital associative algebra determined by its generators k, k-1, e, f and the relations are: the quantum polynomial algebra Cq[x±1,y] is determined by its generators x,x-1,y and the relations are:yx=qxy, xx’1=x-1x=1.Specifically, the first part of this thesis will intuoduce the background of the quantum universal enveloping algebra Uq(sl2) and some relative conclusions about the module algebra of the quantum universal enveloping algebra Uq(sl2) on the quantum polynomial algebra. Then we introduce the object of this test.In the second part, we list some important results on the quantum universal enveloping algebra Uq (sl2) and the quantum polynomial algebra Cq[x±1,y].· The standard Hopf algebra structure on the quantum universal enveloping algebra Uq(sl2).(lemma2.1.)· Some important equations are obtained among the generators of the quantum polynomial algebra Cq[x±1,y].(lemma2.2)· The automorphisms of the quantum polynomial algebra Cq [x±1,y](lemma2.3), Namely,any automorphism of Cq[x±1,y]has the following form: φ(x)=αx,φ(y)=βxny,α,β∈C*,n∈Z.In the third part,we discuss the module algebra of the quantum universal enveloping algebra Uq(sl2) on the quantum polynomial algebra Cq[x±1,y]when k is described as k·x=αx,k·y=βxny,α,β∈C*,n∈Z,n≠0,and the main conclusions are:Theorem3.1Assume that Cq[X±1,y] is a module algebra over Uq(sl2)and k is described as: k.x=αx,k·y=βxny,α,β∈C*,n∈Z,n≠0. then the module algebra is determined by parameters α,β,n,a and the formulas are: e·x=axm+1,e·y=cxm+ny+dxmy, f·x=bx-m+1,f·y=hx-my+gx-m-ny, where a,b∈C*,c,d,h,g∈C,and the relations of these coefficients are:In addition,the module algebras determined by two groups of parameters (α,β,n,a) and (α’,β’,n’,a’)are isomorphism when the parameters satisfy the following formulass: α=α’,n=n’.In the fourth part,we will discussS six types of the module algebra of the quantum envelopm-ent algebra Uq(sl2)on the quantum polynomial algebra Cq[x±1,y]when k is described as k·x=αx,k·y=βy,α,β∈C*.In the fifth part,we will discuss the invariant subalgebras of these module algebras that discussed in the third and fourth part in detail.