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Let $U_q^+(A_N)$ (resp. $U_q^-(A_N)$) be the $(+)$-part (resp. $(-)$-part) ofthe Drinfeld-Jimbo quantum group of type $A_N$ over a field $K$. With respectto Jimbo relations and the PBW $K$-basis ${\cal B}$ of $U_q^+(A_N)$ (resp.$U_q^-(A_N)$) established by Yamane, it is shown, by constructing anappropriate monomial ordering $\prec$ on ${\cal B}$, that $U_q^+(A_N)$ (resp.$U_q^-(A_N)$) is a solvable polynomial algebra. Consequently, furtherstructural properties of $U_q^+(A_N)$ (resp. $U_q^-(A_N)$) and their modulesmay be established and realized in a constructive-computational way.

On The Algebras $U_q^{\pm}(A_N)$: From A Constructive-Computational Viewpoint