  We give the Heisenberg realization for the quantum algebra $U_q(sl_n)$, which
is written by the $q$-difference operator on the flag manifold. We construct it
from the action of $U_q(sl_n)$ on the $q$-symmetric algebra $A_q(Mat_n)$ by the
Borel-Weil like approach. Our realization is applicable to the construction of
the free field realization for the $U_q(\widehat{sl_n})$ [AOS].
