The Polynomial Modules over Quantum Group U-q(sl(3))

Let g be a finite dimensional complex simple Lie algebra with Cartan subalgebra h. Then C[h] has a g-module structure if and only if g is of type A or of type C; this is called the polynomial module of rank one. In the quantum version, the rank one polynomial modules over Uq(sl2) have been classified. In this paper, we prove that the quantum group Uq(sl3) has no rank one polynomial module.