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In the paper, we further realize the higher rank quantized universalenveloping algebra $U_q(sl_{n+1})$ as certain quantum differential operators in$\mathcal W_q(2n)$ defined over the quantum divided power algebra$\mathcal{A}_q(n)$ of rank $n$. We give the quantum differential operatorsrealization for both the simple root vectors and the non-simple root vectors of$U_q(sl_{n+1})$. The nice behavior of the quantum root vectors formulas underthe action of the Lusztig symmetries once again indicates that our realizationmodel is naturally matched.

Matching realization of $U_q(sl_{n+1})$ of higher rank in the quantum Weyl algebra $\mathcal W_q(2n)$