The Ideal Of U_q?sl₂? And The Center Of Its Subalgebra U_q^V

Let F denote a field and fix a nonzero q ? F such that q2 ?1.Let x,y±1,z denote the equitable generators for the F-algebra Uq(sl2)with the relationsyy-1=y-1y=1,qxy-q1yx/q-q-1 =1,qyx-q-1zy/q-q-1 =1,qzx-q-1xz/q-q-1 =1.In this paper,we consider a special ideal of Uq(Sl2),direct sum of Uq(sl2)as a vector space,and the subalgebra of Uq(sl2)denoted by x,y-1,z,the main results as follows.(1)Observe that for ? ? {1,-1},there exists an F-algebra homomorphism Uq(sl2)?F that sends x??,y??.For these two homomorphisms,let J denote the intersection of their kernels.Note that J is a ideal of Uq(sl2).Define the elements vx,vy,vz of Uq(sl2)by vx = q(1-yz),vy = q(1-zx),vz = q(1-xy).In this paper,we first show that J is generated by vx,vy,and vz.Then,we show Uq(sl2)= J + F1 +Fy(direct sum of vector spaces).(2)Let Uq? be the subalgebra of Uq(sl2)generated by x,y-1 and z.In this paper,we show that the center of Uq? is F1,provided that q is not a root of' unity.