Hopf Algebra Structure And Representations Over The Subalgebra U_A Of The Two-parameter Quantum Group U_r,s(sl_n)

G.Benkart and S.Witherspoon introduced and investigated the two-parameter quantum group Ur,s(sln) in their paper [1] at 2004. It is a associative algebra defined over the alge-braically closed field k,We call it U in short. In this paper, we mainly make an Promotion of Lusztig Z[v,v-1]-form for U denoted UA,it is a subalgebra of U,where A is Z[r±1,s±1].In this paper, we investigate the basic properties of UA and show its Hopf algebra struc-ture,and prove that there also exists the triangular decomposition of UA as a vector space, i.e. UA(?)UA-(?)UA0(?)UA+.LetΛ=Σi=1nZεi is weigh lattice of special linear lie algebra sln,Λ+ is a subset of A which is called dominated weigh. Under the base of the triangular decomposition of UA,λ∈Λ+, we can define an finite dimensional simple highest weigh UA-module with the form LA(λ)= UA-x,where x is a primitive element of L(A),λ∈Λ+. We prove that finite di-mensional highest weigh U- modules are integrable, and we show that any UA-submodule of an integrable U- module is also integrable. In addition, we show that F(M)={m∈(?)λ∈ΛMλ|ej(p)m=0= fj(p)m.(?)p>>0, j= 1,…,n-1} is an integrable UA-submodule of M.The important work of this paper is with respect to LA (λ). Firstly, we show that it is an integrable UA-module. Secondly, we prove that it is a free A- lattice. Finally,let A= k0[r±1,s±1] where k0 is subfield of k and Q(r, s) (?)k0,Furthermore, we construct a k0- module (?)= k0(?)A LA(λ), and gain the important result about L, i.e. we can view it as a U(sln)-module.So We can look a UA- module LA(λ) as the quantum deformation of a U(sln)-module (?).