Rosso-Yamane Theorem on PBW basis of $U_q(A_N)$

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

Let $U_q^+(A_N)$ (resp. $U_q^-(A_N)$) be the $(+)$-part (resp. $(-)$-part) of the Drinfeld-Jimbo quantum group of type $A_N$ over a field $K$. With respect to 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 an appropriate monomial ordering $\prec$ on ${\cal B}$, that $U_q^+(A_N)$ (resp. $U_q^-(A_N)$) is a solvable polynomial algebra. Consequently, further structural properties of $U_q^+(A_N)$ (resp. $U_q^-(A_N)$) and their modules may be established and realized in a constructive-computational way.