Gr(?)bner-Shirshov Basis Of Irreducible Representation Of Quantum Group Of Type A_n

The reduction is an important problem in algebra and the Grobner-Shirshov basis theory provides a solution to the reduction problem for various kinds of algebras. But until now, there are only a few results about the Grobner-Shirshov basis of the irreducible representation of quantum groups. In this paper, first, we give a Grobner-Shirshov bases of finite dimensional irreducible representation Vq(λ) of Uq(An), the Drinfeld-Jimbo quantum group of type An, by using the double free module method and the known Grobner-Shirshov bases of Uq(An). Then, by specializing a suitable version of Uq(An) at q=1, we get a Grobner-Shirshov bases of the universal enveloping algebra U(An) of the simple Lie algebra of type An and the finite dimensional irreducible U(An)-module V(λ).At last, compute the Gelfand-kirillov dimension of quantum group of type An.