The Polynomial Modules Over Quantum Group U_q（sl₃）

Quantum groups were proposed in the 1980s.Quantum groups are a general term for a series of algebraic structures.And quantum groups are a special kind of Hopf algebras,which can be seen as the quantization of the universal enveloping algebra of Lie algebras.In the study of polynomial modules of quantum groups,in particular,the study of polynomial modules of quantum groups Uq(sl2)plays a crucial role in this paper.Let g be a finite dimensional complex simple Lie algebra with Cartan subalgebra b.Then C[b]has a g-module structure if and only if g is of type A or of type C,which is called the polynomial module of rank one.In the quantum version,the rank one polynomial module over Uq(sl2)has been classified.Starting from the results of the quantum group Uq(sl2),this paper studies the polynomial module of the quantum group Uq(sl3),and uses the exchange relation of the quantum group and the properties of the polynomial module to give nine possibilities for the rank one polynomial module of the quantum group Uq(sl3),and then proves that the quantum group Uq(sl3)has no polynomial module of rank one.