Set Of Generators Of Quantum Coordinate Algebra And Lift Of A Module

Let A = Z[v]_?, where v is an indeterminate and (?)is an ideal in Z[v] generated by v-1 and a fixed odd prime p.A' =Q(v) is fractional field of A, U' associated to Cartan matrix(a_ij) is a quantum algebra over A' .U is an A subquantum algebra of U' generated by E_i^N ,F_i^N ,K_i,K_i^-1 (i=1,…,n,N≥0).U' is an A'-Hopf algebra ,U is also A-Hopf algebra with related structure. In reference [1] the author introduced the quantum coordinatealgebra A[U] ,where A[U] is a non-commutative,non-cocommutativeA-Hopf algebra. In this paper , in the case of quantum algebra of rank 1,we give a basis of D(λ_m)(m≥0) which is made of weight vectors. We also prove that coefficient space c(V) is a U×U sub-module of A[U]for an arbitraryintegrable U module V .Then we discuss the set and the property of generators of coordinate algebra A [U] ,and prove some formulas with respect to the generator of A module A[U] X₁₁^d₁1X₁₂^d₁2X₂₁^d₂1X₂₂^d₂2 which act on E,F,K^±1 , and weight of generators of A algebra A[U] that has two structures of U moduleγ,δ. Finally, in the case of basic ring is fieldΓ, we prove the necessary and sufficient condition for the U_Γ^# module that can be lifted ,making use of natural representation D_Γ(λ₁) of U_Γ.